Subgroups — C₃×C₅×#3294 #14276328

Order 1920.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#480	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#466	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#447	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₅×#102	C₂	1	2	yes	maximal, complemented
12	C₅×#3294	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	—	2	1	no	—
14	C₂²×C₃×C₅×D₄	—	2	1	no	—
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
18	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	complemented
19	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₅×#127	C₂²	1	1	yes	characteristic
21	C₃×C₅×#123	—	2	2	no	—
22	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
23	C₃×C₅×#121	—	2	1	no	—
24	C₃×C₅×#121	C₂²	1	2	yes	—
25	C₃×C₅×#121	C₂²	1	2	yes	complemented
26	C₃×C₅×#120	—	2	2	no	—
27	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
28	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
31	C₃×C₅×#107	C₂²	1	2	yes	complemented
32	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃×C₅×#102	C₂²	1	2	yes	—
35	C₃×C₅×#102	C₂²	1	1	yes	characteristic
36	C₃×C₅×#102	C₂²	1	2	yes	complemented
37	C₃×C₅×#102	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₅×#102	C₂²	1	2	yes	complemented
39	C₃×C₅×#102	C₂²	1	2	yes	complemented
40	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃×#3294	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#547	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×D₄²	C₂×C₃	1	1	yes	characteristic, complemented
45	C₅×#486	C₂×C₃	1	1	yes	characteristic, complemented
46	C₅×#480	C₂×C₃	1	2	yes	complemented
47	C₅×#474	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#466	C₂×C₃	1	2	yes	complemented
49	C₅×#459	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#447	C₂×C₃	1	2	yes	complemented
51	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic, complemented
52	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₅×#102	C₂×C₃	1	2	yes	complemented
54	C₂⁴×C₃×C₅	—	2	2	no	—
55	C₃×C₅×#41	—	4	4	no	—
56	C₂×C₃×C₅×Q₈	—	2	1	no	—
57	C₂×C₃×C₅×D₄	—	2	4	no	—
58	C₂×C₃×C₅×D₄	—	2	2	no	—
59	C₂×C₃×C₅×D₄	—	2	2	no	—
60	C₂×C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₅×D₄	—	2	1	no	—
62	C₂×C₃×C₅×D₄	—	2	1	no	—
63	C₂×C₃×C₅×D₄	—	4	2	no	—
64	C₂×C₃×C₅×D₄	—	4	4	no	—
65	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
66	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
68	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	—	2	2	no	—
70	C₂²×C₃×C₄×C₅	—	2	1	no	—
71	C₂²×C₃×C₄×C₅	—	2	1	no	—
72	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
73	C₃×C₅×D₈	—	2	2	no	—
74	C₃×C₅×D₈	—	4	2	no	—
75	C₂×C₃×C₅×C₈	—	4	2	no	—
76	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
78	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₃×C₅×C₄⋊C₄	—	2	2	no	—
80	C₃×C₅×C₄⋊C₄	—	2	1	no	—
81	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
82	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
84	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
90	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
91	C₃×#547	C₂×C₅	1	1	yes	characteristic, complemented
92	C₃×D₄²	C₂×C₅	1	1	yes	characteristic, complemented
93	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
94	C₃×#480	C₂×C₅	1	2	yes	complemented
95	C₃×#474	C₂×C₅	1	1	yes	characteristic, complemented
96	C₃×#466	C₂×C₅	1	2	yes	complemented
97	C₃×#459	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#447	C₂×C₅	1	2	yes	complemented
99	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic, complemented
101	C₂×C₃×#102	C₂×C₅	1	2	yes	complemented
102	C₂×C₅×#41	—	2	1	no	—
103	C₂²×C₅×D₄	—	2	1	no	—
104	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
105	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
107	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	complemented
108	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
109	C₅×#127	C₂²×C₃	1	1	yes	characteristic
110	C₅×#123	—	2	2	no	—
111	C₅×C₂²⋊Q₈	—	2	1	no	—
112	C₅×#121	—	2	1	no	—
113	C₅×#121	C₂²×C₃	1	2	yes	—
114	C₅×#121	C₂²×C₃	1	2	yes	complemented
115	C₅×#120	—	2	2	no	—
116	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
117	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
118	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
120	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
121	C₅×#107	C₂²×C₃	1	2	yes	complemented
122	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₅×#102	C₂²×C₃	1	1	yes	characteristic
124	C₅×#102	C₂²×C₃	1	2	yes	—
125	C₅×#102	C₂²×C₃	1	2	yes	complemented
126	C₅×#102	C₂²×C₃	1	2	yes	complemented
127	C₅×#102	C₂²×C₃	1	1	yes	characteristic, complemented
128	C₅×#102	C₂²×C₃	1	2	yes	complemented
129	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
130	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
131	#3294	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₃×C₅	—	2	1	no	—
133	C₂³×C₃×C₅	—	2	1	no	—
134	C₂³×C₃×C₅	—	2	2	no	—
135	C₂³×C₃×C₅	—	4	4	no	—
136	C₂³×C₃×C₅	—	4	4	no	—
137	C₂³×C₃×C₅	—	8	2	no	—
138	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
139	C₃×C₅×Q₈	—	4	2	no	—
140	C₃×C₅×D₄	—	2	1	no	—
141	C₃×C₅×D₄	—	2	2	no	—
142	C₃×C₅×D₄	—	2	1	no	—
143	C₃×C₅×D₄	—	2	1	no	—
144	C₃×C₅×D₄	—	2	2	no	—
145	C₃×C₅×D₄	—	4	4	no	—
146	C₃×C₅×D₄	—	4	4	no	—
147	C₃×C₅×D₄	—	4	4	no	—
148	C₃×C₅×D₄	—	4	2	no	—
149	C₃×C₅×D₄	—	8	1	no	—
150	C₃×C₅×D₄	D₈	1	4	yes	complemented
151	C₃×C₅×D₄	#41	1	2	yes	complemented
152	C₂×C₃×C₄×C₅	—	2	2	no	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	2	1	no	—
155	C₂×C₃×C₄×C₅	—	2	1	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	—	4	4	no	—
158	C₂×C₃×C₄×C₅	—	4	2	no	—
159	C₂×C₃×C₄×C₅	—	4	2	no	—
160	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
162	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
164	C₃×C₅×C₈	—	2	1	no	—
165	C₃×C₅×C₈	—	2	1	no	—
166	C₃×C₅×C₈	—	2	2	no	—
167	C₂×C₃×#41	—	2	1	no	—
168	C₂²×C₃×D₄	—	2	1	no	—
169	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
170	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
172	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	complemented
173	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
174	C₃×#127	C₂²×C₅	1	1	yes	characteristic
175	C₃×#123	—	2	2	no	—
176	C₃×C₂²⋊Q₈	—	2	1	no	—
177	C₃×#121	—	2	1	no	—
178	C₃×#121	C₂²×C₅	1	2	yes	—
179	C₃×#121	C₂²×C₅	1	2	yes	complemented
180	C₃×#120	—	2	2	no	—
181	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
182	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
183	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
184	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
185	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
186	C₃×#107	C₂²×C₅	1	2	yes	complemented
187	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
188	C₃×#102	C₂²×C₅	1	1	yes	characteristic
189	C₃×#102	C₂²×C₅	1	2	yes	—
190	C₃×#102	C₂²×C₅	1	2	yes	complemented
191	C₃×#102	C₂²×C₅	1	2	yes	complemented
192	C₃×#102	C₂²×C₅	1	1	yes	characteristic, complemented
193	C₃×#102	C₂²×C₅	1	2	yes	complemented
194	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
195	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
196	C₂⁴×C₅	—	2	2	no	—
197	C₅×#41	—	4	4	no	—
198	C₂×C₅×Q₈	—	2	1	no	—
199	C₂×C₅×D₄	—	2	4	no	—
200	C₂×C₅×D₄	—	2	2	no	—
201	C₂×C₅×D₄	—	2	2	no	—
202	C₂×C₅×D₄	—	2	2	no	—
203	C₂×C₅×D₄	—	2	1	no	—
204	C₂×C₅×D₄	—	2	1	no	—
205	C₂×C₅×D₄	—	4	2	no	—
206	C₂×C₅×D₄	—	4	4	no	—
207	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
208	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
209	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
210	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
211	C₂²×C₄×C₅	—	2	2	no	—
212	C₂²×C₄×C₅	—	2	1	no	—
213	C₂²×C₄×C₅	—	2	1	no	—
214	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
215	C₅×D₈	—	2	2	no	—
216	C₅×D₈	—	4	2	no	—
217	C₂×C₅×C₈	—	4	2	no	—
218	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
219	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
220	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	—	2	2	no	—
222	C₅×C₄⋊C₄	—	2	1	no	—
223	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
224	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
225	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
226	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
227	C₅×C₂²⋊C₄	—	2	1	no	—
228	C₅×C₂²⋊C₄	—	2	1	no	—
229	C₅×C₂²⋊C₄	—	2	2	no	—
230	C₅×C₂²⋊C₄	—	2	2	no	—
231	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
232	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
233	#547	C₂×C₃×C₅	1	1	yes	characteristic, complemented
234	D₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
235	#486	C₂×C₃×C₅	1	1	yes	characteristic, complemented
236	#480	C₂×C₃×C₅	1	2	yes	complemented
237	#474	C₂×C₃×C₅	1	1	yes	characteristic, complemented
238	#466	C₂×C₃×C₅	1	2	yes	complemented
239	#459	C₂×C₃×C₅	1	1	yes	characteristic, complemented
240	#447	C₂×C₃×C₅	1	2	yes	complemented
241	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
242	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
243	C₂×#102	C₂×C₃×C₅	1	2	yes	complemented
244	C₂²×C₃×C₅	—	2	2	no	—
245	C₂²×C₃×C₅	—	2	2	no	—
246	C₂²×C₃×C₅	—	2	2	no	—
247	C₂²×C₃×C₅	—	4	1	no	—
248	C₂²×C₃×C₅	—	4	1	no	—
249	C₂²×C₃×C₅	—	4	2	no	—
250	C₂²×C₃×C₅	—	4	1	no	—
251	C₂²×C₃×C₅	—	4	1	no	—
252	C₂²×C₃×C₅	—	4	2	no	—
253	C₂²×C₃×C₅	—	4	2	no	—
254	C₂²×C₃×C₅	—	4	1	no	—
255	C₂²×C₃×C₅	—	8	8	no	—
256	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
257	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
258	C₃×C₄×C₅	—	2	2	no	—
259	C₃×C₄×C₅	—	2	1	no	—
260	C₃×C₄×C₅	—	4	1	no	—
261	C₃×C₄×C₅	—	4	1	no	—
262	C₃×C₄×C₅	—	4	2	no	—
263	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
264	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
265	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂⁴×C₃	—	2	2	no	—
267	C₃×#41	—	4	4	no	—
268	C₂×C₃×Q₈	—	2	1	no	—
269	C₂×C₃×D₄	—	2	4	no	—
270	C₂×C₃×D₄	—	2	2	no	—
271	C₂×C₃×D₄	—	2	2	no	—
272	C₂×C₃×D₄	—	2	2	no	—
273	C₂×C₃×D₄	—	2	1	no	—
274	C₂×C₃×D₄	—	2	1	no	—
275	C₂×C₃×D₄	—	4	2	no	—
276	C₂×C₃×D₄	—	4	4	no	—
277	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
278	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
279	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
280	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
281	C₂²×C₃×C₄	—	2	2	no	—
282	C₂²×C₃×C₄	—	2	1	no	—
283	C₂²×C₃×C₄	—	2	1	no	—
284	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
285	C₃×D₈	—	2	2	no	—
286	C₃×D₈	—	4	2	no	—
287	C₂×C₃×C₈	—	4	2	no	—
288	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
289	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
290	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
291	C₃×C₄⋊C₄	—	2	2	no	—
292	C₃×C₄⋊C₄	—	2	1	no	—
293	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
294	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
295	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
296	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
297	C₃×C₂²⋊C₄	—	2	1	no	—
298	C₃×C₂²⋊C₄	—	2	1	no	—
299	C₃×C₂²⋊C₄	—	2	2	no	—
300	C₃×C₂²⋊C₄	—	2	2	no	—
301	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
302	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
303	C₂³×C₅	—	2	1	no	—
304	C₂³×C₅	—	2	1	no	—
305	C₂³×C₅	—	2	2	no	—
306	C₂³×C₅	—	4	4	no	—
307	C₂³×C₅	—	4	4	no	—
308	C₂³×C₅	—	8	2	no	—
309	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
310	C₅×Q₈	—	4	2	no	—
311	C₅×D₄	—	2	1	no	—
312	C₅×D₄	—	2	2	no	—
313	C₅×D₄	—	2	1	no	—
314	C₅×D₄	—	2	1	no	—
315	C₅×D₄	—	2	2	no	—
316	C₅×D₄	—	4	4	no	—
317	C₅×D₄	—	4	4	no	—
318	C₅×D₄	—	4	4	no	—
319	C₅×D₄	—	4	2	no	—
320	C₅×D₄	—	8	1	no	—
321	C₅×D₄	C₃×D₈	1	4	yes	complemented
322	C₅×D₄	C₃×#41	1	2	yes	complemented
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	1	no	—
326	C₂×C₄×C₅	—	2	1	no	—
327	C₂×C₄×C₅	—	2	2	no	—
328	C₂×C₄×C₅	—	4	4	no	—
329	C₂×C₄×C₅	—	4	2	no	—
330	C₂×C₄×C₅	—	4	2	no	—
331	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
332	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
333	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
334	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
335	C₅×C₈	—	2	1	no	—
336	C₅×C₈	—	2	1	no	—
337	C₅×C₈	—	2	2	no	—
338	C₂×#41	—	2	1	no	—
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
341	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
342	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
343	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	complemented
344	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
345	#127	C₂²×C₃×C₅	1	1	yes	characteristic
346	#123	—	2	2	no	—
347	C₂²⋊Q₈	—	2	1	no	—
348	#121	—	2	1	no	—
349	#121	C₂²×C₃×C₅	1	2	yes	—
350	#121	C₂²×C₃×C₅	1	2	yes	complemented
351	#120	—	2	2	no	—
352	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
353	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
354	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
355	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	complemented
356	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
357	#107	C₂²×C₃×C₅	1	2	yes	complemented
358	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
359	#102	C₂²×C₃×C₅	1	1	yes	characteristic
360	#102	C₂²×C₃×C₅	1	2	yes	—
361	#102	C₂²×C₃×C₅	1	2	yes	complemented
362	#102	C₂²×C₃×C₅	1	2	yes	complemented
363	#102	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
364	#102	C₂²×C₃×C₅	1	2	yes	complemented
365	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
366	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
367	C₂×C₃×C₅	—	2	4	no	—
368	C₂×C₃×C₅	—	4	2	no	—
369	C₂×C₃×C₅	—	8	1	no	—
370	C₂×C₃×C₅	—	8	2	no	—
371	C₂×C₃×C₅	#546	1	1	yes	characteristic
372	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
373	C₂×C₃×C₅	#577	1	1	yes	characteristic
374	C₂³×C₃	—	2	1	no	—
375	C₂³×C₃	—	2	1	no	—
376	C₂³×C₃	—	2	2	no	—
377	C₂³×C₃	—	4	4	no	—
378	C₂³×C₃	—	4	4	no	—
379	C₂³×C₃	—	8	2	no	—
380	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
381	C₃×Q₈	—	4	2	no	—
382	C₃×D₄	—	2	1	no	—
383	C₃×D₄	—	2	2	no	—
384	C₃×D₄	—	2	1	no	—
385	C₃×D₄	—	2	1	no	—
386	C₃×D₄	—	2	2	no	—
387	C₃×D₄	—	4	4	no	—
388	C₃×D₄	—	4	4	no	—
389	C₃×D₄	—	4	4	no	—
390	C₃×D₄	—	4	2	no	—
391	C₃×D₄	—	8	1	no	—
392	C₃×D₄	C₅×D₈	1	4	yes	complemented
393	C₃×D₄	C₅×#41	1	2	yes	complemented
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	2	no	—
396	C₂×C₃×C₄	—	2	1	no	—
397	C₂×C₃×C₄	—	2	1	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	4	4	no	—
400	C₂×C₃×C₄	—	4	2	no	—
401	C₂×C₃×C₄	—	4	2	no	—
402	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
403	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
404	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
405	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
406	C₃×C₈	—	2	1	no	—
407	C₃×C₈	—	2	1	no	—
408	C₃×C₈	—	2	2	no	—
409	C₂²×C₅	—	2	2	no	—
410	C₂²×C₅	—	2	2	no	—
411	C₂²×C₅	—	2	2	no	—
412	C₂²×C₅	—	4	1	no	—
413	C₂²×C₅	—	4	1	no	—
414	C₂²×C₅	—	4	1	no	—
415	C₂²×C₅	—	4	2	no	—
416	C₂²×C₅	—	4	2	no	—
417	C₂²×C₅	—	4	1	no	—
418	C₂²×C₅	—	4	2	no	—
419	C₂²×C₅	—	4	1	no	—
420	C₂²×C₅	—	8	8	no	—
421	C₂²×C₅	C₂×C₃×D₈	1	4	yes	—
422	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
423	C₄×C₅	—	2	1	no	—
424	C₄×C₅	—	2	2	no	—
425	C₄×C₅	—	4	1	no	—
426	C₄×C₅	—	4	1	no	—
427	C₄×C₅	—	4	2	no	—
428	C₄×C₅	C₂×C₃×D₈	1	2	yes	—
429	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
430	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
431	C₂⁴	—	2	2	no	—
432	#41	—	4	4	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	4	no	—
442	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
443	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
444	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
445	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
450	D₈	—	2	2	no	—
451	D₈	—	4	2	no	—
452	C₂×C₈	—	4	2	no	—
453	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
454	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
455	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
459	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
460	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
461	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
467	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
468	C₃×C₅	#3294	1	1	yes	characteristic, direct product factor, Hall
469	C₂²×C₃	—	2	2	no	—
470	C₂²×C₃	—	2	2	no	—
471	C₂²×C₃	—	2	2	no	—
472	C₂²×C₃	—	4	1	no	—
473	C₂²×C₃	—	4	1	no	—
474	C₂²×C₃	—	4	2	no	—
475	C₂²×C₃	—	4	1	no	—
476	C₂²×C₃	—	4	1	no	—
477	C₂²×C₃	—	4	2	no	—
478	C₂²×C₃	—	4	2	no	—
479	C₂²×C₃	—	4	1	no	—
480	C₂²×C₃	—	8	8	no	—
481	C₂²×C₃	C₂×C₅×D₈	1	4	yes	—
482	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
483	C₃×C₄	—	2	2	no	—
484	C₃×C₄	—	2	1	no	—
485	C₃×C₄	—	4	1	no	—
486	C₃×C₄	—	4	1	no	—
487	C₃×C₄	—	4	2	no	—
488	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
489	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
490	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
491	C₂×C₅	—	2	4	no	—
492	C₂×C₅	—	4	2	no	—
493	C₂×C₅	—	8	2	no	—
494	C₂×C₅	—	8	1	no	—
495	C₂×C₅	C₃×#546	1	1	yes	characteristic
496	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
497	C₂×C₅	C₃×#577	1	1	yes	characteristic
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	8	2	no	—
504	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
505	Q₈	—	4	2	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	2	no	—
508	D₄	—	2	1	no	—
509	D₄	—	2	1	no	—
510	D₄	—	2	2	no	—
511	D₄	—	4	4	no	—
512	D₄	—	4	4	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	2	no	—
515	D₄	—	8	1	no	—
516	D₄	C₃×C₅×D₈	1	4	yes	complemented
517	D₄	C₃×C₅×#41	1	2	yes	complemented
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
527	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
528	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
529	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	2	no	—
533	C₂×C₃	—	2	4	no	—
534	C₂×C₃	—	4	2	no	—
535	C₂×C₃	—	8	2	no	—
536	C₂×C₃	—	8	1	no	—
537	C₂×C₃	C₅×#546	1	1	yes	characteristic
538	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
539	C₂×C₃	C₅×#577	1	1	yes	characteristic
540	C₅	C₃×#3294	1	1	yes	characteristic, direct product factor, Sylow
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	8	8	no	—
553	C₂²	C₂×C₃×C₅×D₈	1	4	yes	—
554	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
555	C₄	—	2	2	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	2	no	—
560	C₄	C₂×C₃×C₅×D₈	1	2	yes	—
561	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
562	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
563	C₃	C₅×#3294	1	1	yes	characteristic, direct product factor, Sylow
564	C₂	—	2	4	no	—
565	C₂	—	4	2	no	—
566	C₂	—	8	2	no	—
567	C₂	—	8	1	no	—
568	C₂	C₃×C₅×#546	1	1	yes	characteristic
569	C₂	C₂²×C₃×C₅×D₈	1	1	yes	characteristic
570	C₂	C₃×C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄×C₅×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₅×#547	C₂	semidirect
2. C₃×C₅×D₄²	C₂	semidirect
3. C₃×C₅×#486	C₂	semidirect
4. C₃×C₅×#480	C₂	semidirect
5. C₃×C₅×#474	C₂	semidirect
6. C₃×C₅×#466	C₂	semidirect
7. C₃×C₅×#459	C₂	semidirect
8. C₃×C₅×#447	C₂	semidirect
9. C₃×C₄×C₅×D₈	C₂	semidirect
10. C₃×C₅×C₈×D₄	C₂	semidirect
11. C₂×C₃×C₅×#102	C₂	semidirect
12. C₅×#3294	C₃	direct (N × H)
17. C₂×C₃×C₅×D₈	C₂²	semidirect
18. C₂²×C₃×C₅×C₈	C₂²	semidirect
19. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
25. C₃×C₅×#121	C₂²	semidirect
29. C₃×C₄×C₅×D₄	C₂²	semidirect
30. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
31. C₃×C₅×#107	C₂²	semidirect
32. C₃×C₅×#107	C₂²	semidirect
33. C₃×C₅×C₄⋊C₈	C₂²	semidirect
36. C₃×C₅×#102	C₂²	semidirect
37. C₃×C₅×#102	C₂²	semidirect
38. C₃×C₅×#102	C₂²	semidirect
39. C₃×C₅×#102	C₂²	semidirect
40. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₅×C₈	C₂²	semidirect
42. C₃×#3294	C₅	direct (N × H)
43. C₅×#547	C₂×C₃	semidirect
44. C₅×D₄²	C₂×C₃	semidirect
45. C₅×#486	C₂×C₃	semidirect
46. C₅×#480	C₂×C₃	semidirect
47. C₅×#474	C₂×C₃	semidirect
48. C₅×#466	C₂×C₃	semidirect
49. C₅×#459	C₂×C₃	semidirect
50. C₅×#447	C₂×C₃	semidirect
51. C₄×C₅×D₈	C₂×C₃	semidirect
52. C₅×C₈×D₄	C₂×C₃	semidirect
53. C₂×C₅×#102	C₂×C₃	semidirect
91. C₃×#547	C₂×C₅	semidirect
92. C₃×D₄²	C₂×C₅	semidirect
93. C₃×#486	C₂×C₅	semidirect
94. C₃×#480	C₂×C₅	semidirect
95. C₃×#474	C₂×C₅	semidirect
96. C₃×#466	C₂×C₅	semidirect
97. C₃×#459	C₂×C₅	semidirect
98. C₃×#447	C₂×C₅	semidirect
99. C₃×C₄×D₈	C₂×C₅	semidirect
100. C₃×C₈×D₄	C₂×C₅	semidirect
101. C₂×C₃×#102	C₂×C₅	semidirect
106. C₂×C₅×D₈	C₂²×C₃	semidirect
107. C₂²×C₅×C₈	C₂²×C₃	semidirect
108. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
114. C₅×#121	C₂²×C₃	semidirect
118. C₄×C₅×D₄	C₂²×C₃	semidirect
119. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
120. C₅×#107	C₂²×C₃	semidirect
121. C₅×#107	C₂²×C₃	semidirect
122. C₅×C₄⋊C₈	C₂²×C₃	semidirect
125. C₅×#102	C₂²×C₃	semidirect
126. C₅×#102	C₂²×C₃	semidirect
127. C₅×#102	C₂²×C₃	semidirect
128. C₅×#102	C₂²×C₃	semidirect
129. C₅×C₂²⋊C₈	C₂²×C₃	semidirect
130. C₄×C₅×C₈	C₂²×C₃	semidirect
131. #3294	C₃×C₅	direct (N × H)
150. C₃×C₅×D₄	D₈	semidirect
151. C₃×C₅×D₄	#41	semidirect
171. C₂×C₃×D₈	C₂²×C₅	semidirect
172. C₂²×C₃×C₈	C₂²×C₅	semidirect
173. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
179. C₃×#121	C₂²×C₅	semidirect
183. C₃×C₄×D₄	C₂²×C₅	semidirect
184. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
185. C₃×#107	C₂²×C₅	semidirect
186. C₃×#107	C₂²×C₅	semidirect
187. C₃×C₄⋊C₈	C₂²×C₅	semidirect
190. C₃×#102	C₂²×C₅	semidirect
191. C₃×#102	C₂²×C₅	semidirect
192. C₃×#102	C₂²×C₅	semidirect
193. C₃×#102	C₂²×C₅	semidirect
194. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
195. C₃×C₄×C₈	C₂²×C₅	semidirect
233. #547	C₂×C₃×C₅	semidirect
234. D₄²	C₂×C₃×C₅	semidirect
235. #486	C₂×C₃×C₅	semidirect
236. #480	C₂×C₃×C₅	semidirect
237. #474	C₂×C₃×C₅	semidirect
238. #466	C₂×C₃×C₅	semidirect
239. #459	C₂×C₃×C₅	semidirect
240. #447	C₂×C₃×C₅	semidirect
241. C₄×D₈	C₂×C₃×C₅	semidirect
242. C₈×D₄	C₂×C₃×C₅	semidirect
243. C₂×#102	C₂×C₃×C₅	semidirect
321. C₅×D₄	C₃×D₈	semidirect
322. C₅×D₄	C₃×#41	semidirect
342. C₂×D₈	C₂²×C₃×C₅	semidirect
343. C₂²×C₈	C₂²×C₃×C₅	semidirect
344. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
350. #121	C₂²×C₃×C₅	semidirect
354. C₄×D₄	C₂²×C₃×C₅	semidirect
355. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
356. #107	C₂²×C₃×C₅	semidirect
357. #107	C₂²×C₃×C₅	semidirect
358. C₄⋊C₈	C₂²×C₃×C₅	semidirect
361. #102	C₂²×C₃×C₅	semidirect
362. #102	C₂²×C₃×C₅	semidirect
363. #102	C₂²×C₃×C₅	semidirect
364. #102	C₂²×C₃×C₅	semidirect
365. C₂²⋊C₈	C₂²×C₃×C₅	semidirect
366. C₄×C₈	C₂²×C₃×C₅	semidirect
392. C₃×D₄	C₅×D₈	semidirect
393. C₃×D₄	C₅×#41	semidirect
468. C₃×C₅	#3294	direct (N × H)
516. D₄	C₃×C₅×D₈	semidirect
517. D₄	C₃×C₅×#41	semidirect
540. C₅	C₃×#3294	direct (N × H)
563. C₃	C₅×#3294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#547	2
C₃×C₅×D₄²	2
C₃×C₅×#486	2
C₃×C₅×#480	2
C₃×C₅×#474	2
C₃×C₅×#466	2
C₃×C₅×#459	2
C₃×C₅×#447	2
C₃×C₄×C₅×D₈	2
C₃×C₅×C₈×D₄	2
C₂×C₃×C₅×#102	2
C₅×#3294	3
C₃×#3294	5

Subgroups — C3×C5×#3294 #14276328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3294 #14276328