Subgroups — C₂×C₅×#5534 #14481322

Order 1920.

2264 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#930	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5534	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅×#898	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#896	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#885	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#883	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#542	—	3	1	no	maximal, Hall
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×#67	C₂²	1	2	yes	—
16	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
20	C₃×C₅×#117	C₂²	1	8	yes	complemented
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₂×C₅×#217	C₂²	1	2	yes	—
28	C₅×#930	C₂²	1	8	yes	—
29	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
31	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
32	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
34	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
35	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
36	C₅×#924	C₂²	1	8	yes	complemented
37	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
38	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
39	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
40	C₅×#898	C₂²	1	16	yes	complemented
41	C₅×#896	C₂²	1	16	yes	complemented
42	C₅×#885	C₂²	1	16	yes	complemented
43	C₅×#883	C₂²	1	16	yes	complemented
44	C₅×#875	C₂²	1	8	yes	complemented
45	C₄×C₅×D₁₂	C₂²	1	8	yes	complemented
46	C₅×#869	C₂²	1	8	yes	complemented
47	C₄×C₅×Dic₆	C₂²	1	8	yes	complemented
48	C₂×#5534	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#542	—	3	16	no	—
50	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
51	C₂×C₅×#126	—	3	2	no	—
52	C₂×C₅×#124	—	3	2	no	—
53	C₂×C₅×#124	—	3	1	no	—
54	C₂×C₅×#123	—	3	2	no	—
55	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
56	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
57	C₂×C₅×#121	—	3	1	no	—
58	C₂×C₄×C₅×Q₈	—	3	1	no	—
59	C₂×C₄×C₅×D₄	—	3	1	no	—
60	C₂×C₅×#117	D₃	1	1	yes	characteristic, complemented
61	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
62	C₂³×C₅×D₃	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	—	2	4	no	—
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂×C₅×#67	—	4	4	no	—
72	C₂×C₅×#67	C₂³	1	8	yes	—
73	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
74	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
75	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
76	C₂×C₅×D₁₂	—	2	4	no	—
77	C₂×C₅×D₁₂	C₂³	1	4	yes	—
78	C₂×C₄×C₅×D₃	—	4	4	no	—
79	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
80	C₂×C₅×Dic₆	—	2	4	no	—
81	C₂×C₅×Dic₆	—	4	4	no	—
82	C₂×C₅×Dic₆	C₂³	1	4	yes	—
83	C₂×C₅×Dic₆	C₂³	1	8	yes	—
84	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
87	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
88	C₃×C₄²×C₅	C₂³	1	4	yes	—
89	C₃×C₄²×C₅	C₂³	1	4	yes	—
90	C₅×#217	C₂³	1	8	yes	—
91	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
92	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
93	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
94	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
95	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
96	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
97	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
98	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
99	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
100	C₄×C₅×Dic₃	C₂³	1	8	yes	—
101	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂×#930	C₂×C₅	1	1	yes	characteristic
103	C₂×#924	C₂×C₅	1	1	yes	characteristic, complemented
104	#5534	C₂×C₅	1	16	yes	direct product factor
105	C₂×#898	C₂×C₅	1	2	yes	complemented
106	C₂×#896	C₂×C₅	1	2	yes	complemented
107	C₂×#885	C₂×C₅	1	2	yes	complemented
108	C₂×#883	C₂×C₅	1	2	yes	complemented
109	C₂×#875	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
112	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
113	C₂²×C₅×Q₈	—	3	1	no	—
114	C₂²×C₅×Q₈	—	3	2	no	—
115	C₂²×C₅×D₄	—	3	2	no	—
116	C₂²×C₅×D₄	—	3	1	no	—
117	C₂³×C₄×C₅	—	3	2	no	—
118	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
119	C₅×C₄⋊Q₈	—	3	8	no	—
120	C₅×#126	—	3	16	no	—
121	C₅×#124	—	3	16	no	—
122	C₅×#124	—	3	8	no	—
123	C₅×#123	—	3	16	no	—
124	C₅×C₂²⋊Q₈	—	3	8	no	—
125	C₅×C₂²⋊Q₈	—	3	16	no	—
126	C₅×#121	—	3	8	no	—
127	C₄×C₅×Q₈	—	3	8	no	—
128	C₄×C₅×D₄	—	3	8	no	—
129	C₅×#117	C₂×D₃	1	8	yes	complemented
130	C₂×C₅×C₄⋊C₄	—	3	2	no	—
131	C₂×C₅×C₄⋊C₄	—	3	2	no	—
132	C₂×C₅×C₄⋊C₄	—	3	1	no	—
133	C₂×C₅×C₄⋊C₄	—	3	1	no	—
134	C₂×C₅×C₄⋊C₄	—	3	2	no	—
135	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
136	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
137	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
138	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
139	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
140	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
141	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
142	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
143	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
144	C₂×C₄²×C₅	—	3	2	no	—
145	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
146	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
147	C₂×#542	—	3	1	no	Sylow
148	C₂³×C₃×C₅	—	2	2	no	—
149	C₂³×C₃×C₅	—	2	2	no	—
150	C₂³×C₃×C₅	—	2	2	no	—
151	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
152	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
153	C₂²×C₅×D₃	—	2	4	no	—
154	C₂²×C₅×D₃	—	2	4	no	—
155	C₂²×C₅×D₃	—	2	4	no	—
156	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
157	C₂×C₃×C₄×C₅	—	2	2	no	—
158	C₂×C₃×C₄×C₅	—	2	2	no	—
159	C₂×C₃×C₄×C₅	—	2	2	no	—
160	C₂×C₃×C₄×C₅	—	2	2	no	—
161	C₂×C₃×C₄×C₅	—	2	2	no	—
162	C₂×C₃×C₄×C₅	—	2	8	no	—
163	C₂×C₃×C₄×C₅	#41	1	4	yes	—
164	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
165	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
167	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
168	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
169	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
170	C₅×#67	—	4	8	no	—
171	C₂×C₅×Dic₃	—	2	4	no	—
172	C₂×C₅×Dic₃	—	2	4	no	—
173	C₂×C₅×Dic₃	—	2	4	no	—
174	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
175	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
176	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
177	C₅×D₁₂	—	2	8	no	—
178	C₄×C₅×D₃	—	4	8	no	—
179	C₅×Dic₆	—	2	8	no	—
180	C₅×Dic₆	—	4	8	no	—
181	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
182	C₂²×#67	C₂²×C₅	1	2	yes	—
183	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
184	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
185	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
186	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
187	C₃×#117	C₂²×C₅	1	8	yes	complemented
188	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
189	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
190	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
191	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
192	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
193	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
194	C₂×#217	C₂²×C₅	1	2	yes	—
195	#930	C₂²×C₅	1	8	yes	—
196	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
197	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
198	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
199	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
200	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
201	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
202	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
203	#924	C₂²×C₅	1	8	yes	complemented
204	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
205	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
206	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
207	#898	C₂²×C₅	1	16	yes	complemented
208	#896	C₂²×C₅	1	16	yes	complemented
209	#885	C₂²×C₅	1	16	yes	complemented
210	#883	C₂²×C₅	1	16	yes	complemented
211	#875	C₂²×C₅	1	8	yes	complemented
212	C₄×D₁₂	C₂²×C₅	1	8	yes	complemented
213	#869	C₂²×C₅	1	8	yes	complemented
214	C₄×Dic₆	C₂²×C₅	1	8	yes	complemented
215	C₂⁴×C₅	—	3	2	no	—
216	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
217	C₂×C₅×Q₈	—	3	4	no	—
218	C₂×C₅×Q₈	—	3	8	no	—
219	C₂×C₅×Q₈	—	6	4	no	—
220	C₂×C₅×Q₈	—	12	4	no	—
221	C₂×C₅×D₄	—	3	8	no	—
222	C₂×C₅×D₄	—	3	4	no	—
223	C₂×C₅×D₄	—	6	4	no	—
224	C₂×C₅×D₄	—	12	4	no	—
225	C₂²×C₄×C₅	—	2	4	no	—
226	C₂²×C₄×C₅	—	3	2	no	—
227	C₂²×C₄×C₅	—	3	2	no	—
228	C₂²×C₄×C₅	—	3	2	no	—
229	C₂²×C₄×C₅	—	3	8	no	—
230	C₂²×C₄×C₅	—	12	4	no	—
231	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
232	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
233	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
234	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
235	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
236	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
237	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
238	C₅×C₄⋊C₄	—	3	4	no	—
239	C₅×C₄⋊C₄	—	3	4	no	—
240	C₅×C₄⋊C₄	—	3	8	no	—
241	C₅×C₄⋊C₄	—	3	8	no	—
242	C₅×C₄⋊C₄	—	3	8	no	—
243	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
244	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
245	C₅×C₂²⋊C₄	—	3	8	no	—
246	C₅×C₂²⋊C₄	—	3	8	no	—
247	C₅×C₂²⋊C₄	—	3	8	no	—
248	C₅×C₂²⋊C₄	—	3	8	no	—
249	C₅×C₂²⋊C₄	—	3	8	no	—
250	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
251	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
252	C₄²×C₅	—	3	8	no	—
253	C₄²×C₅	C₂²×D₃	1	4	yes	—
254	C₄²×C₅	C₂²×D₃	1	4	yes	—
255	#542	—	3	16	no	—
256	C₂×C₄⋊Q₈	—	3	1	no	—
257	C₂×#126	—	3	2	no	—
258	C₂×#124	—	3	2	no	—
259	C₂×#124	—	3	1	no	—
260	C₂×#123	—	3	2	no	—
261	C₂×C₂²⋊Q₈	—	3	1	no	—
262	C₂×C₂²⋊Q₈	—	3	2	no	—
263	C₂×#121	—	3	1	no	—
264	C₂×C₄×Q₈	—	3	1	no	—
265	C₂×C₄×D₄	—	3	1	no	—
266	C₂×#117	C₅×D₃	1	1	yes	characteristic, complemented
267	C₂²×C₃×C₅	—	2	2	no	—
268	C₂²×C₃×C₅	—	2	2	no	—
269	C₂²×C₃×C₅	—	2	2	no	—
270	C₂²×C₃×C₅	—	4	4	no	—
271	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
272	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
273	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
274	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
275	C₂×C₅×D₃	—	2	4	no	—
276	C₂×C₅×D₃	—	2	4	no	—
277	C₂×C₅×D₃	—	2	4	no	—
278	C₂×C₅×D₃	—	4	8	no	—
279	C₃×C₄×C₅	—	2	2	no	—
280	C₃×C₄×C₅	—	2	2	no	—
281	C₃×C₄×C₅	—	2	2	no	—
282	C₃×C₄×C₅	—	2	2	no	—
283	C₃×C₄×C₅	—	2	2	no	—
284	C₃×C₄×C₅	C₂×#41	1	4	yes	—
285	C₅×Dic₃	—	2	4	no	—
286	C₅×Dic₃	—	2	4	no	—
287	C₅×Dic₃	—	2	4	no	—
288	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
289	C₂³×D₃	C₂³×C₅	1	2	yes	—
290	C₂²×C₃×C₄	—	2	4	no	—
291	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
292	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
293	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
294	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
295	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
296	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
297	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
298	C₂×#67	—	4	4	no	—
299	C₂×#67	C₂³×C₅	1	8	yes	—
300	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
301	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
302	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
303	C₂×D₁₂	—	2	4	no	—
304	C₂×D₁₂	C₂³×C₅	1	4	yes	—
305	C₂×C₄×D₃	—	4	4	no	—
306	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
307	C₂×Dic₆	—	2	4	no	—
308	C₂×Dic₆	—	4	4	no	—
309	C₂×Dic₆	C₂³×C₅	1	4	yes	—
310	C₂×Dic₆	C₂³×C₅	1	8	yes	—
311	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
312	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
313	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
314	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
315	C₃×C₄²	C₂³×C₅	1	4	yes	—
316	C₃×C₄²	C₂³×C₅	1	4	yes	—
317	#217	C₂³×C₅	1	8	yes	—
318	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
319	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
320	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
321	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
322	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
323	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
324	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
325	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
326	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
327	C₄×Dic₃	C₂³×C₅	1	8	yes	—
328	C₂³×C₅	—	2	2	no	—
329	C₂³×C₅	—	2	2	no	—
330	C₂³×C₅	—	2	2	no	—
331	C₂³×C₅	—	3	4	no	—
332	C₂³×C₅	—	6	4	no	—
333	C₂³×C₅	—	6	4	no	—
334	C₂³×C₅	—	6	4	no	—
335	C₂³×C₅	C₂³×D₃	1	2	yes	—
336	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
337	C₅×Q₈	—	6	8	no	—
338	C₅×Q₈	—	12	8	no	—
339	C₅×D₄	—	6	8	no	—
340	C₅×D₄	—	12	8	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	2	no	—
343	C₂×C₄×C₅	—	2	2	no	—
344	C₂×C₄×C₅	—	2	2	no	—
345	C₂×C₄×C₅	—	2	2	no	—
346	C₂×C₄×C₅	—	2	8	no	—
347	C₂×C₄×C₅	—	3	4	no	—
348	C₂×C₄×C₅	—	3	4	no	—
349	C₂×C₄×C₅	—	3	4	no	—
350	C₂×C₄×C₅	—	6	4	no	—
351	C₂×C₄×C₅	—	6	4	no	—
352	C₂×C₄×C₅	—	6	4	no	—
353	C₂×C₄×C₅	—	12	8	no	—
354	C₂×C₄×C₅	#235	1	4	yes	—
355	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
356	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
357	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
358	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
359	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
360	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
361	C₂²×Q₈	—	3	1	no	—
362	C₂²×Q₈	—	3	2	no	—
363	C₂²×D₄	—	3	2	no	—
364	C₂²×D₄	—	3	1	no	—
365	C₂³×C₄	—	3	2	no	—
366	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
367	C₄⋊Q₈	—	3	8	no	—
368	#126	—	3	16	no	—
369	#124	—	3	16	no	—
370	#124	—	3	8	no	—
371	#123	—	3	16	no	—
372	C₂²⋊Q₈	—	3	8	no	—
373	C₂²⋊Q₈	—	3	16	no	—
374	#121	—	3	8	no	—
375	C₄×Q₈	—	3	8	no	—
376	C₄×D₄	—	3	8	no	—
377	#117	C₂×C₅×D₃	1	8	yes	complemented
378	C₂×C₄⋊C₄	—	3	1	no	—
379	C₂×C₄⋊C₄	—	3	1	no	—
380	C₂×C₄⋊C₄	—	3	2	no	—
381	C₂×C₄⋊C₄	—	3	2	no	—
382	C₂×C₄⋊C₄	—	3	2	no	—
383	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	—	3	2	no	—
386	C₂×C₂²⋊C₄	—	3	2	no	—
387	C₂×C₂²⋊C₄	—	3	2	no	—
388	C₂×C₂²⋊C₄	—	3	2	no	—
389	C₂×C₂²⋊C₄	—	3	2	no	—
390	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
391	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
392	C₂×C₄²	—	3	2	no	—
393	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
394	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
395	C₂×C₃×C₅	—	4	2	no	—
396	C₂×C₃×C₅	#542	1	4	yes	complemented
397	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
398	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
399	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
400	C₅×D₃	—	4	4	no	—
401	C₂³×C₃	—	2	2	no	—
402	C₂³×C₃	—	2	2	no	—
403	C₂³×C₃	—	2	2	no	—
404	C₂³×C₃	C₂⁴×C₅	1	2	yes	—
405	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
406	C₂²×D₃	—	2	4	no	—
407	C₂²×D₃	—	2	4	no	—
408	C₂²×D₃	—	2	4	no	—
409	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
410	C₂×C₃×C₄	—	2	2	no	—
411	C₂×C₃×C₄	—	2	2	no	—
412	C₂×C₃×C₄	—	2	2	no	—
413	C₂×C₃×C₄	—	2	2	no	—
414	C₂×C₃×C₄	—	2	2	no	—
415	C₂×C₃×C₄	—	2	8	no	—
416	C₂×C₃×C₄	C₅×#41	1	4	yes	—
417	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
418	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
419	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
420	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
421	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
422	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
423	#67	—	4	8	no	—
424	C₂×Dic₃	—	2	4	no	—
425	C₂×Dic₃	—	2	4	no	—
426	C₂×Dic₃	—	2	4	no	—
427	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
428	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
429	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
430	D₁₂	—	2	8	no	—
431	C₄×D₃	—	4	8	no	—
432	Dic₆	—	2	8	no	—
433	Dic₆	—	4	8	no	—
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	2	2	no	—
437	C₂²×C₅	—	4	4	no	—
438	C₂²×C₅	—	6	4	no	—
439	C₂²×C₅	—	6	4	no	—
440	C₂²×C₅	—	6	4	no	—
441	C₂²×C₅	—	12	8	no	—
442	C₂²×C₅	C₂×#235	1	2	yes	—
443	C₂²×C₅	#1009	1	2	yes	—
444	C₂²×C₅	#1010	1	2	yes	—
445	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	2	no	—
448	C₄×C₅	—	2	2	no	—
449	C₄×C₅	—	2	2	no	—
450	C₄×C₅	—	2	2	no	—
451	C₄×C₅	—	6	4	no	—
452	C₄×C₅	—	6	4	no	—
453	C₄×C₅	—	6	4	no	—
454	C₄×C₅	C₂×#235	1	4	yes	—
455	C₂⁴	—	3	2	no	—
456	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
457	C₂×Q₈	—	3	4	no	—
458	C₂×Q₈	—	3	8	no	—
459	C₂×Q₈	—	6	4	no	—
460	C₂×Q₈	—	12	4	no	—
461	C₂×D₄	—	3	8	no	—
462	C₂×D₄	—	3	4	no	—
463	C₂×D₄	—	6	4	no	—
464	C₂×D₄	—	12	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	3	2	no	—
467	C₂²×C₄	—	3	2	no	—
468	C₂²×C₄	—	3	2	no	—
469	C₂²×C₄	—	3	8	no	—
470	C₂²×C₄	—	12	4	no	—
471	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
473	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
475	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
477	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
478	C₄⋊C₄	—	3	4	no	—
479	C₄⋊C₄	—	3	4	no	—
480	C₄⋊C₄	—	3	8	no	—
481	C₄⋊C₄	—	3	8	no	—
482	C₄⋊C₄	—	3	8	no	—
483	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
484	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
485	C₂²⋊C₄	—	3	8	no	—
486	C₂²⋊C₄	—	3	8	no	—
487	C₂²⋊C₄	—	3	8	no	—
488	C₂²⋊C₄	—	3	8	no	—
489	C₂²⋊C₄	—	3	8	no	—
490	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
491	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
492	C₄²	—	3	8	no	—
493	C₄²	C₂²×C₅×D₃	1	4	yes	—
494	C₄²	C₂²×C₅×D₃	1	4	yes	—
495	C₃×C₅	C₂×#542	1	1	yes	characteristic, complemented, Hall
496	C₂²×C₃	—	2	2	no	—
497	C₂²×C₃	—	2	2	no	—
498	C₂²×C₃	—	2	2	no	—
499	C₂²×C₃	—	4	4	no	—
500	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
501	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
502	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
503	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
504	C₂×D₃	—	2	4	no	—
505	C₂×D₃	—	2	4	no	—
506	C₂×D₃	—	2	4	no	—
507	C₂×D₃	—	4	8	no	—
508	C₃×C₄	—	2	2	no	—
509	C₃×C₄	—	2	2	no	—
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	2	2	no	—
512	C₃×C₄	—	2	2	no	—
513	C₃×C₄	C₂×C₅×#41	1	4	yes	—
514	Dic₃	—	2	4	no	—
515	Dic₃	—	2	4	no	—
516	Dic₃	—	2	4	no	—
517	C₂×C₅	—	4	2	no	—
518	C₂×C₅	—	12	4	no	—
519	C₂×C₅	#5534	1	4	yes	direct product factor
520	C₂×C₅	C₂²×#235	1	1	yes	characteristic
521	C₂×C₅	C₂×#1009	1	1	yes	characteristic
522	C₂×C₅	C₂×#1010	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	3	4	no	—
527	C₂³	—	6	4	no	—
528	C₂³	—	6	4	no	—
529	C₂³	—	6	4	no	—
530	C₂³	C₂³×C₅×D₃	1	2	yes	—
531	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
532	Q₈	—	6	8	no	—
533	Q₈	—	12	8	no	—
534	D₄	—	6	8	no	—
535	D₄	—	12	8	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	3	4	no	—
543	C₂×C₄	—	3	4	no	—
544	C₂×C₄	—	3	4	no	—
545	C₂×C₄	—	6	4	no	—
546	C₂×C₄	—	6	4	no	—
547	C₂×C₄	—	6	4	no	—
548	C₂×C₄	—	12	8	no	—
549	C₂×C₄	C₅×#235	1	4	yes	—
550	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
551	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
552	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
553	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
554	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
555	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
556	C₂×C₃	—	4	2	no	—
557	C₂×C₃	C₅×#542	1	4	yes	complemented
558	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
559	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
560	C₂×C₃	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
561	D₃	—	4	4	no	—
562	C₅	C₂×#5534	1	1	yes	characteristic, direct product factor, Sylow
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	6	4	no	—
568	C₂²	—	6	4	no	—
569	C₂²	—	6	4	no	—
570	C₂²	—	12	8	no	—
571	C₂²	C₂×C₅×#235	1	2	yes	—
572	C₂²	C₅×#1009	1	2	yes	—
573	C₂²	C₅×#1010	1	2	yes	—
574	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	6	4	no	—
581	C₄	—	6	4	no	—
582	C₄	—	6	4	no	—
583	C₄	C₂×C₅×#235	1	4	yes	—
584	C₃	C₂×C₅×#542	1	1	yes	characteristic, complemented, Sylow
585	C₂	—	4	2	no	—
586	C₂	—	12	4	no	—
587	C₂	C₅×#5534	1	4	yes	direct product factor
588	C₂	C₂²×C₅×#235	1	1	yes	characteristic
589	C₂	C₂×C₅×#1009	1	1	yes	characteristic
590	C₂	C₂×C₅×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#930	C₂
hypercenter	C₂×C₅×#117	D₃
hypocenter	C₃	C₂×C₅×#542
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#117	C₂	semidirect
3. C₂×C₅×#924	C₂	semidirect
4. C₅×#5534	C₂	direct (N × H)
5. C₂×C₅×#898	C₂	semidirect
6. C₂×C₅×#896	C₂	semidirect
7. C₂×C₅×#885	C₂	semidirect
8. C₂×C₅×#883	C₂	semidirect
9. C₂×C₅×#875	C₂	semidirect
10. C₂×C₄×C₅×D₁₂	C₂	semidirect
11. C₂×C₅×#869	C₂	semidirect
12. C₂×C₄×C₅×Dic₆	C₂	semidirect
20. C₃×C₅×#117	C₂²	semidirect
36. C₅×#924	C₂²	semidirect
40. C₅×#898	C₂²	semidirect
41. C₅×#896	C₂²	semidirect
42. C₅×#885	C₂²	semidirect
43. C₅×#883	C₂²	semidirect
44. C₅×#875	C₂²	semidirect
45. C₄×C₅×D₁₂	C₂²	semidirect
46. C₅×#869	C₂²	semidirect
47. C₄×C₅×Dic₆	C₂²	semidirect
48. C₂×#5534	C₅	direct (N × H)
60. C₂×C₅×#117	D₃	semidirect
101. C₂×C₃×#117	C₂×C₅	semidirect
103. C₂×#924	C₂×C₅	semidirect
104. #5534	C₂×C₅	direct (N × H)
105. C₂×#898	C₂×C₅	semidirect
106. C₂×#896	C₂×C₅	semidirect
107. C₂×#885	C₂×C₅	semidirect
108. C₂×#883	C₂×C₅	semidirect
109. C₂×#875	C₂×C₅	semidirect
110. C₂×C₄×D₁₂	C₂×C₅	semidirect
111. C₂×#869	C₂×C₅	semidirect
112. C₂×C₄×Dic₆	C₂×C₅	semidirect
129. C₅×#117	C₂×D₃	semidirect
187. C₃×#117	C₂²×C₅	semidirect
203. #924	C₂²×C₅	semidirect
207. #898	C₂²×C₅	semidirect
208. #896	C₂²×C₅	semidirect
209. #885	C₂²×C₅	semidirect
210. #883	C₂²×C₅	semidirect
211. #875	C₂²×C₅	semidirect
212. C₄×D₁₂	C₂²×C₅	semidirect
213. #869	C₂²×C₅	semidirect
214. C₄×Dic₆	C₂²×C₅	semidirect
266. C₂×#117	C₅×D₃	semidirect
377. #117	C₂×C₅×D₃	semidirect
396. C₂×C₃×C₅	#542	semidirect
495. C₃×C₅	C₂×#542	semidirect
519. C₂×C₅	#5534	direct (N × H)
557. C₂×C₃	C₅×#542	semidirect
562. C₅	C₂×#5534	direct (N × H)
584. C₃	C₂×C₅×#542	semidirect
587. C₂	C₅×#5534	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×#117	2
C₂×C₅×#930	2
C₂×C₅×#924	2
C₅×#5534	2
C₂×C₅×#898	2
C₂×C₅×#896	2
C₂×C₅×#885	2
C₂×C₅×#883	2
C₂×C₅×#875	2
C₂×C₄×C₅×D₁₂	2
C₂×C₅×#869	2
C₂×C₄×C₅×Dic₆	2
C₂×C₅×#542	3
C₂×#5534	5

Subgroups — C2×C5×#5534 #14481322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5534 #14481322