Subgroups — C₃×C₅×#2626 #14275660

Order 1920.

1336 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#399	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#396	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#393	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#385	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#379	C₂	1	2	yes	maximal
9	C₅×#2626	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊Q₈	—	2	4	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
19	C₂×C₃×C₅×C₂²⋊C₄	—	2	4	no	—
20	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃×C₅×#95	C₂²	1	2	yes	complemented
27	C₃×C₅×#95	C₂²	1	4	yes	complemented
28	C₃×#2626	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₅×#399	C₂×C₃	1	2	yes	complemented
32	C₅×#396	C₂×C₃	1	4	yes	complemented
33	C₅×#393	C₂×C₃	1	2	yes	complemented
34	C₅×#385	C₂×C₃	1	2	yes	complemented
35	C₅×#380	C₂×C₃	1	1	yes	characteristic, complemented
36	C₅×#379	C₂×C₃	1	2	yes	—
37	C₂⁴×C₃×C₅	—	2	2	no	—
38	C₂⁴×C₃×C₅	—	2	2	no	—
39	C₂⁴×C₃×C₅	—	4	2	no	—
40	C₂⁴×C₃×C₅	—	4	2	no	—
41	C₂⁴×C₃×C₅	Q₈	1	4	yes	complemented
42	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
43	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×Q₈	—	2	2	no	—
45	C₂×C₃×C₅×Q₈	—	4	2	no	—
46	C₂²×C₃×C₄×C₅	—	2	2	no	—
47	C₂²×C₃×C₄×C₅	—	4	2	no	—
48	C₂²×C₃×C₄×C₅	—	4	2	no	—
49	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
50	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
52	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₃×C₅×C₄⋊C₄	—	2	2	no	—
57	C₃×C₅×C₄⋊C₄	—	2	4	no	—
58	C₃×C₅×C₄⋊C₄	—	4	2	no	—
59	C₃×C₅×C₄⋊C₄	—	4	4	no	—
60	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
63	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
65	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
66	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
68	C₃×#399	C₂×C₅	1	2	yes	complemented
69	C₃×#396	C₂×C₅	1	4	yes	complemented
70	C₃×#393	C₂×C₅	1	2	yes	complemented
71	C₃×#385	C₂×C₅	1	2	yes	complemented
72	C₃×#380	C₂×C₅	1	1	yes	characteristic, complemented
73	C₃×#379	C₂×C₅	1	2	yes	—
74	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
75	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
76	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
77	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
78	C₅×C₂²⋊Q₈	—	2	4	no	—
79	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
80	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
81	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
83	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
84	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
89	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
90	C₅×#95	C₂²×C₃	1	2	yes	complemented
91	C₅×#95	C₂²×C₃	1	4	yes	complemented
92	#2626	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
93	C₂³×C₃×C₅	—	2	2	no	—
94	C₂³×C₃×C₅	—	2	2	no	—
95	C₂³×C₃×C₅	—	2	8	no	—
96	C₂³×C₃×C₅	—	2	2	no	—
97	C₂³×C₃×C₅	—	2	2	no	—
98	C₂³×C₃×C₅	—	2	4	no	—
99	C₂³×C₃×C₅	—	2	4	no	—
100	C₂³×C₃×C₅	—	4	8	no	—
101	C₂³×C₃×C₅	—	4	4	no	—
102	C₂³×C₃×C₅	—	4	4	no	—
103	C₂³×C₃×C₅	—	4	8	no	—
104	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
105	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
106	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
107	C₂³×C₃×C₅	#41	1	2	yes	—
108	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
109	C₃×C₅×Q₈	—	4	4	no	—
110	C₂×C₃×C₄×C₅	—	2	2	no	—
111	C₂×C₃×C₄×C₅	—	2	2	no	—
112	C₂×C₃×C₄×C₅	—	2	2	no	—
113	C₂×C₃×C₄×C₅	—	2	2	no	—
114	C₂×C₃×C₄×C₅	—	2	2	no	—
115	C₂×C₃×C₄×C₅	—	2	4	no	—
116	C₂×C₃×C₄×C₅	—	2	4	no	—
117	C₂×C₃×C₄×C₅	—	2	4	no	—
118	C₂×C₃×C₄×C₅	—	2	4	no	—
119	C₂×C₃×C₄×C₅	—	2	4	no	—
120	C₂×C₃×C₄×C₅	—	2	4	no	—
121	C₂×C₃×C₄×C₅	—	4	4	no	—
122	C₂×C₃×C₄×C₅	—	4	4	no	—
123	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
125	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
126	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
127	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
128	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
129	C₃×C₂²⋊Q₈	—	2	4	no	—
130	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
131	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
132	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
133	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
134	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
135	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
136	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
137	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
138	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
139	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
140	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
141	C₃×#95	C₂²×C₅	1	2	yes	complemented
142	C₃×#95	C₂²×C₅	1	4	yes	complemented
143	C₂⁴×C₅	—	2	2	no	—
144	C₂⁴×C₅	—	2	2	no	—
145	C₂⁴×C₅	—	4	2	no	—
146	C₂⁴×C₅	—	4	2	no	—
147	C₂⁴×C₅	C₃×Q₈	1	4	yes	complemented
148	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
149	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₅×Q₈	—	2	2	no	—
151	C₂×C₅×Q₈	—	4	2	no	—
152	C₂²×C₄×C₅	—	2	2	no	—
153	C₂²×C₄×C₅	—	4	2	no	—
154	C₂²×C₄×C₅	—	4	2	no	—
155	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
156	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
157	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
158	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
159	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
160	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
162	C₅×C₄⋊C₄	—	2	2	no	—
163	C₅×C₄⋊C₄	—	2	4	no	—
164	C₅×C₄⋊C₄	—	4	2	no	—
165	C₅×C₄⋊C₄	—	4	4	no	—
166	C₅×C₂²⋊C₄	—	2	4	no	—
167	C₅×C₂²⋊C₄	—	2	4	no	—
168	C₅×C₂²⋊C₄	—	2	4	no	—
169	C₅×C₂²⋊C₄	—	2	4	no	—
170	C₅×C₂²⋊C₄	—	4	4	no	—
171	C₅×C₂²⋊C₄	—	4	4	no	—
172	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
173	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
174	#399	C₂×C₃×C₅	1	2	yes	complemented
175	#396	C₂×C₃×C₅	1	4	yes	complemented
176	#393	C₂×C₃×C₅	1	2	yes	complemented
177	#385	C₂×C₃×C₅	1	2	yes	complemented
178	#380	C₂×C₃×C₅	1	1	yes	characteristic, complemented
179	#379	C₂×C₃×C₅	1	2	yes	—
180	C₂²×C₃×C₅	—	2	2	no	—
181	C₂²×C₃×C₅	—	2	2	no	—
182	C₂²×C₃×C₅	—	2	4	no	—
183	C₂²×C₃×C₅	—	2	4	no	—
184	C₂²×C₃×C₅	—	2	4	no	—
185	C₂²×C₃×C₅	—	2	8	no	—
186	C₂²×C₃×C₅	—	4	16	no	—
187	C₂²×C₃×C₅	—	4	2	no	—
188	C₂²×C₃×C₅	—	4	2	no	—
189	C₂²×C₃×C₅	—	4	4	no	—
190	C₂²×C₃×C₅	C₂²⋊Q₈	1	4	yes	—
191	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
192	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
193	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
194	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
195	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
196	C₃×C₄×C₅	—	2	2	no	—
197	C₃×C₄×C₅	—	2	2	no	—
198	C₃×C₄×C₅	—	4	2	no	—
199	C₃×C₄×C₅	—	4	2	no	—
200	C₃×C₄×C₅	—	4	2	no	—
201	C₃×C₄×C₅	—	4	4	no	—
202	C₂⁴×C₃	—	2	2	no	—
203	C₂⁴×C₃	—	2	2	no	—
204	C₂⁴×C₃	—	4	2	no	—
205	C₂⁴×C₃	—	4	2	no	—
206	C₂⁴×C₃	C₅×Q₈	1	4	yes	complemented
207	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
208	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
209	C₂×C₃×Q₈	—	2	2	no	—
210	C₂×C₃×Q₈	—	4	2	no	—
211	C₂²×C₃×C₄	—	2	2	no	—
212	C₂²×C₃×C₄	—	4	2	no	—
213	C₂²×C₃×C₄	—	4	2	no	—
214	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
215	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
216	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
217	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
218	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
219	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
220	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
221	C₃×C₄⋊C₄	—	2	2	no	—
222	C₃×C₄⋊C₄	—	2	4	no	—
223	C₃×C₄⋊C₄	—	4	2	no	—
224	C₃×C₄⋊C₄	—	4	4	no	—
225	C₃×C₂²⋊C₄	—	2	4	no	—
226	C₃×C₂²⋊C₄	—	2	4	no	—
227	C₃×C₂²⋊C₄	—	2	4	no	—
228	C₃×C₂²⋊C₄	—	2	4	no	—
229	C₃×C₂²⋊C₄	—	4	4	no	—
230	C₃×C₂²⋊C₄	—	4	4	no	—
231	C₂³×C₅	—	2	2	no	—
232	C₂³×C₅	—	2	2	no	—
233	C₂³×C₅	—	2	8	no	—
234	C₂³×C₅	—	2	2	no	—
235	C₂³×C₅	—	2	2	no	—
236	C₂³×C₅	—	2	4	no	—
237	C₂³×C₅	—	2	4	no	—
238	C₂³×C₅	—	4	8	no	—
239	C₂³×C₅	—	4	4	no	—
240	C₂³×C₅	—	4	4	no	—
241	C₂³×C₅	—	4	8	no	—
242	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
243	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
244	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
245	C₂³×C₅	C₃×#41	1	2	yes	—
246	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
247	C₅×Q₈	—	4	4	no	—
248	C₂×C₄×C₅	—	2	2	no	—
249	C₂×C₄×C₅	—	2	2	no	—
250	C₂×C₄×C₅	—	2	2	no	—
251	C₂×C₄×C₅	—	2	2	no	—
252	C₂×C₄×C₅	—	2	2	no	—
253	C₂×C₄×C₅	—	2	4	no	—
254	C₂×C₄×C₅	—	2	4	no	—
255	C₂×C₄×C₅	—	2	4	no	—
256	C₂×C₄×C₅	—	2	4	no	—
257	C₂×C₄×C₅	—	2	4	no	—
258	C₂×C₄×C₅	—	2	4	no	—
259	C₂×C₄×C₅	—	4	4	no	—
260	C₂×C₄×C₅	—	4	4	no	—
261	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
262	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
263	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
264	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
265	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
266	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
267	C₂²⋊Q₈	—	2	4	no	—
268	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	complemented
269	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
270	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	complemented
271	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	complemented
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
274	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
275	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
277	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
278	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
279	#95	C₂²×C₃×C₅	1	2	yes	complemented
280	#95	C₂²×C₃×C₅	1	4	yes	complemented
281	C₂×C₃×C₅	—	2	4	no	—
282	C₂×C₃×C₅	—	4	4	no	—
283	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₃×C₅	#534	1	1	yes	characteristic
285	C₂×C₃×C₅	#535	1	1	yes	characteristic
286	C₂×C₃×C₅	#538	1	2	yes	—
287	C₂×C₃×C₅	#543	1	2	yes	—
288	C₂³×C₃	—	2	2	no	—
289	C₂³×C₃	—	2	2	no	—
290	C₂³×C₃	—	2	8	no	—
291	C₂³×C₃	—	2	2	no	—
292	C₂³×C₃	—	2	2	no	—
293	C₂³×C₃	—	2	4	no	—
294	C₂³×C₃	—	2	4	no	—
295	C₂³×C₃	—	4	8	no	—
296	C₂³×C₃	—	4	4	no	—
297	C₂³×C₃	—	4	4	no	—
298	C₂³×C₃	—	4	8	no	—
299	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
300	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
301	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
302	C₂³×C₃	C₅×#41	1	2	yes	—
303	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
304	C₃×Q₈	—	4	4	no	—
305	C₂×C₃×C₄	—	2	2	no	—
306	C₂×C₃×C₄	—	2	2	no	—
307	C₂×C₃×C₄	—	2	2	no	—
308	C₂×C₃×C₄	—	2	2	no	—
309	C₂×C₃×C₄	—	2	2	no	—
310	C₂×C₃×C₄	—	2	4	no	—
311	C₂×C₃×C₄	—	2	4	no	—
312	C₂×C₃×C₄	—	2	4	no	—
313	C₂×C₃×C₄	—	2	4	no	—
314	C₂×C₃×C₄	—	2	4	no	—
315	C₂×C₃×C₄	—	2	4	no	—
316	C₂×C₃×C₄	—	4	4	no	—
317	C₂×C₃×C₄	—	4	4	no	—
318	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
319	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
320	C₂²×C₅	—	2	2	no	—
321	C₂²×C₅	—	2	2	no	—
322	C₂²×C₅	—	2	4	no	—
323	C₂²×C₅	—	2	4	no	—
324	C₂²×C₅	—	2	4	no	—
325	C₂²×C₅	—	2	8	no	—
326	C₂²×C₅	—	4	16	no	—
327	C₂²×C₅	—	4	2	no	—
328	C₂²×C₅	—	4	2	no	—
329	C₂²×C₅	—	4	4	no	—
330	C₂²×C₅	C₃×C₂²⋊Q₈	1	4	yes	—
331	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
332	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
333	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
334	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
335	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
336	C₄×C₅	—	2	2	no	—
337	C₄×C₅	—	2	2	no	—
338	C₄×C₅	—	4	2	no	—
339	C₄×C₅	—	4	2	no	—
340	C₄×C₅	—	4	2	no	—
341	C₄×C₅	—	4	4	no	—
342	C₂⁴	—	2	2	no	—
343	C₂⁴	—	2	2	no	—
344	C₂⁴	—	4	2	no	—
345	C₂⁴	—	4	2	no	—
346	C₂⁴	C₃×C₅×Q₈	1	4	yes	complemented
347	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
348	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
355	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
356	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
357	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
358	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
359	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
360	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	—	4	4	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	2	4	no	—
367	C₂²⋊C₄	—	2	4	no	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	4	4	no	—
370	C₂²⋊C₄	—	4	4	no	—
371	C₃×C₅	#2626	1	1	yes	characteristic, direct product factor, Hall
372	C₂²×C₃	—	2	2	no	—
373	C₂²×C₃	—	2	2	no	—
374	C₂²×C₃	—	2	4	no	—
375	C₂²×C₃	—	2	4	no	—
376	C₂²×C₃	—	2	4	no	—
377	C₂²×C₃	—	2	8	no	—
378	C₂²×C₃	—	4	16	no	—
379	C₂²×C₃	—	4	2	no	—
380	C₂²×C₃	—	4	2	no	—
381	C₂²×C₃	—	4	4	no	—
382	C₂²×C₃	C₅×C₂²⋊Q₈	1	4	yes	—
383	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
384	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
385	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
386	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
387	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
388	C₃×C₄	—	2	2	no	—
389	C₃×C₄	—	2	2	no	—
390	C₃×C₄	—	4	2	no	—
391	C₃×C₄	—	4	2	no	—
392	C₃×C₄	—	4	2	no	—
393	C₃×C₄	—	4	4	no	—
394	C₂×C₅	—	2	4	no	—
395	C₂×C₅	—	4	4	no	—
396	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
397	C₂×C₅	C₃×#534	1	1	yes	characteristic
398	C₂×C₅	C₃×#535	1	1	yes	characteristic
399	C₂×C₅	C₃×#538	1	2	yes	—
400	C₂×C₅	C₃×#543	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	8	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	4	no	—
407	C₂³	—	2	4	no	—
408	C₂³	—	4	8	no	—
409	C₂³	—	4	4	no	—
410	C₂³	—	4	4	no	—
411	C₂³	—	4	8	no	—
412	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
413	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
414	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
415	C₂³	C₃×C₅×#41	1	2	yes	—
416	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
417	Q₈	—	4	4	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
432	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
433	C₂×C₃	—	2	4	no	—
434	C₂×C₃	—	4	4	no	—
435	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
436	C₂×C₃	C₅×#534	1	1	yes	characteristic
437	C₂×C₃	C₅×#535	1	1	yes	characteristic
438	C₂×C₃	C₅×#538	1	2	yes	—
439	C₂×C₃	C₅×#543	1	2	yes	—
440	C₅	C₃×#2626	1	1	yes	characteristic, direct product factor, Sylow
441	C₂²	—	2	2	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	8	no	—
447	C₂²	—	4	16	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	4	no	—
451	C₂²	C₃×C₅×C₂²⋊Q₈	1	4	yes	—
452	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
453	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
454	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
455	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
456	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
457	C₄	—	2	2	no	—
458	C₄	—	2	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	4	no	—
463	C₃	C₅×#2626	1	1	yes	characteristic, direct product factor, Sylow
464	C₂	—	2	4	no	—
465	C₂	—	4	4	no	—
466	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
467	C₂	C₃×C₅×#534	1	1	yes	characteristic
468	C₂	C₃×C₅×#535	1	1	yes	characteristic
469	C₂	C₃×C₅×#538	1	2	yes	—
470	C₂	C₃×C₅×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₂²⋊Q₈	C₂	semidirect
3. C₃×C₅×#399	C₂	semidirect
4. C₃×C₅×#396	C₂	semidirect
5. C₃×C₅×#393	C₂	semidirect
6. C₃×C₅×#385	C₂	semidirect
7. C₃×C₅×#380	C₂	semidirect
9. C₅×#2626	C₃	direct (N × H)
11. C₂²×C₃×C₅×Q₈	C₂²	semidirect
15. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
16. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
18. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
26. C₃×C₅×#95	C₂²	semidirect
27. C₃×C₅×#95	C₂²	semidirect
28. C₃×#2626	C₅	direct (N × H)
29. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₅×#399	C₂×C₃	semidirect
32. C₅×#396	C₂×C₃	semidirect
33. C₅×#393	C₂×C₃	semidirect
34. C₅×#385	C₂×C₃	semidirect
35. C₅×#380	C₂×C₃	semidirect
41. C₂⁴×C₃×C₅	Q₈	semidirect
66. C₂×C₃×C₂²⋊Q₈	C₂×C₅	semidirect
68. C₃×#399	C₂×C₅	semidirect
69. C₃×#396	C₂×C₅	semidirect
70. C₃×#393	C₂×C₅	semidirect
71. C₃×#385	C₂×C₅	semidirect
72. C₃×#380	C₂×C₅	semidirect
75. C₂²×C₅×Q₈	C₂²×C₃	semidirect
79. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
80. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
81. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
82. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
90. C₅×#95	C₂²×C₃	semidirect
91. C₅×#95	C₂²×C₃	semidirect
92. #2626	C₃×C₅	direct (N × H)
126. C₂²×C₃×Q₈	C₂²×C₅	semidirect
130. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
131. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
132. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
133. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
141. C₃×#95	C₂²×C₅	semidirect
142. C₃×#95	C₂²×C₅	semidirect
147. C₂⁴×C₅	C₃×Q₈	semidirect
172. C₂×C₂²⋊Q₈	C₂×C₃×C₅	semidirect
174. #399	C₂×C₃×C₅	semidirect
175. #396	C₂×C₃×C₅	semidirect
176. #393	C₂×C₃×C₅	semidirect
177. #385	C₂×C₃×C₅	semidirect
178. #380	C₂×C₃×C₅	semidirect
206. C₂⁴×C₃	C₅×Q₈	semidirect
264. C₂²×Q₈	C₂²×C₃×C₅	semidirect
268. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
269. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
270. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
271. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
279. #95	C₂²×C₃×C₅	semidirect
280. #95	C₂²×C₃×C₅	semidirect
346. C₂⁴	C₃×C₅×Q₈	semidirect
371. C₃×C₅	#2626	direct (N × H)
440. C₅	C₃×#2626	direct (N × H)
463. C₃	C₅×#2626	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₂²⋊Q₈	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#399	2
C₃×C₅×#396	2
C₃×C₅×#393	2
C₃×C₅×#385	2
C₃×C₅×#380	2
C₃×C₅×#379	2
C₅×#2626	3
C₃×#2626	5

Subgroups — C3×C5×#2626 #14275660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2626 #14275660