Subgroups — #11752729

Order 960.

676 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6777	C₂	1	1	yes	maximal, characteristic
2	#91761	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×#685	C₂	1	2	yes	maximal, complemented
6	D₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
7	#91441	C₂	1	2	yes	maximal, complemented
8	#91440	C₂	1	2	yes	maximal, complemented
9	#91435	C₂	1	1	yes	maximal, characteristic, complemented
10	#91434	C₂	1	1	yes	maximal, characteristic, complemented
11	#91347	C₂	1	1	yes	maximal, characteristic, complemented
12	#65879	—	3	1	no	maximal, Hall
13	C₂²×D₃×D₅	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	complemented
17	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	complemented
19	C₂×#1201	C₂²	1	2	yes	—
20	C₂×D₃×Dic₅	C₂²	1	2	yes	complemented
21	C₂×D₅×Dic₃	C₂²	1	2	yes	complemented
22	#6777	C₄	1	8	yes	complemented
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
24	#6714	C₂²	1	1	yes	characteristic
25	#6714	C₂²	1	1	yes	characteristic
26	#6712	C₂²	1	1	yes	characteristic, complemented
27	#6712	C₂²	1	1	yes	characteristic, complemented
28	C₄×Dic₁₅	C₂²	1	1	yes	characteristic, complemented
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
30	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
31	C₃×#685	C₂²	1	2	yes	complemented
32	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
33	#6667	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	complemented
35	#6665	C₂²	1	2	yes	complemented
36	Dic₃×Dic₅	C₂²	1	2	yes	complemented
37	#5545	—	5	1	no	maximal, Hall
38	C₂×D₄×D₅	—	3	1	no	—
39	C₄×C₅×D₄	—	3	1	no	—
40	D₄×Dic₅	—	3	1	no	—
41	C₄×C₂²⋊D₅	—	3	2	no	—
42	C₂×#685	—	3	2	no	—
43	#4039	—	3	1	no	—
44	#4038	—	3	1	no	—
45	#4027	—	3	2	no	—
46	D₅×C₂²⋊C₄	—	3	2	no	—
47	C₄×D₂₀	D₃	1	1	yes	characteristic, complemented
48	#4018	—	3	1	no	—
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₃	C₂³	1	2	yes	—
51	C₂²×C₃×D₅	C₂³	1	2	yes	—
52	C₂×D₃×D₅	—	4	2	no	—
53	C₂×D₃×D₅	—	4	2	no	—
54	C₂×D₃×D₅	C₂×C₄	1	8	yes	complemented
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
61	C₄×D₁₅	C₂×C₄	1	4	yes	complemented
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₅×D₁₂	C₂×C₄	1	4	yes	complemented
64	C₄×C₅×D₃	—	4	2	no	—
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₃×D₂₀	C₂×C₄	1	4	yes	complemented
67	C₃×C₄×D₅	—	4	2	no	—
68	#1201	C₂×C₄	1	8	yes	complemented
69	D₃×Dic₅	—	4	2	no	—
70	D₅×Dic₃	—	4	2	no	—
71	C₂×D₃×D₄	—	5	1	no	—
72	C₃×C₄×D₄	—	5	1	no	—
73	D₄×Dic₃	—	5	1	no	—
74	C₄×#67	—	5	2	no	—
75	C₂×(C₂×D₃)⋊C₄	—	5	2	no	—
76	#893	—	5	1	no	—
77	#892	—	5	1	no	—
78	#67⋊C₄	—	5	2	no	—
79	D₃×C₂²⋊C₄	—	5	2	no	—
80	C₄×D₁₂	D₅	1	1	yes	characteristic, complemented
81	#872	—	5	1	no	—
82	C₂³×D₅	—	3	2	no	—
83	C₂×C₅×D₄	—	3	1	no	—
84	C₂²×C₄×C₅	—	3	2	no	—
85	C₂×C₂²⋊D₅	—	3	2	no	—
86	C₂²×Dic₅	—	3	2	no	—
87	D₄×D₅	—	3	8	no	—
88	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
89	C₂×C₄×D₅	—	3	1	no	—
90	C₂×C₄×D₅	—	3	2	no	—
91	C₂×C₄×D₅	C₂×D₃	1	2	yes	complemented
92	C₅×C₄⋊C₄	—	3	1	no	—
93	C₅×C₂²⋊C₄	—	3	2	no	—
94	C₄²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
95	C₂²⋊Dic₅	—	3	2	no	—
96	#685	—	3	2	no	—
97	#685	—	3	1	no	—
98	#685	—	3	1	no	—
99	#685	—	3	2	no	—
100	#685	C₂×D₃	1	2	yes	complemented
101	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
102	#683	—	3	1	no	—
103	#683	—	3	1	no	—
104	C₄×Dic₅	—	3	1	no	—
105	C₄×Dic₅	—	3	2	no	—
106	#518	—	15	1	no	Sylow
107	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
108	C₂×D₁₅	—	2	1	no	—
109	C₂×D₁₅	—	2	1	no	—
110	C₂×D₁₅	C₂²×C₄	1	2	yes	—
111	C₂×C₅×D₃	—	2	2	no	—
112	C₂×C₅×D₃	—	2	2	no	—
113	C₂×C₅×D₃	C₂²×C₄	1	4	yes	—
114	C₂×C₃×D₅	—	2	2	no	—
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	C₂²×C₄	1	4	yes	—
117	D₃×D₅	—	4	8	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
121	Dic₁₅	—	2	1	no	—
122	Dic₁₅	—	2	1	no	—
123	Dic₁₅	C₂²×C₄	1	2	yes	—
124	C₃×Dic₅	—	2	2	no	—
125	C₅×Dic₃	—	2	2	no	—
126	C₂³×D₃	—	5	2	no	—
127	C₂×C₃×D₄	—	5	1	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂×#67	—	5	2	no	—
130	C₂²×Dic₃	—	5	2	no	—
131	D₃×D₄	—	5	8	no	—
132	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic
133	C₂×C₄×D₃	—	5	1	no	—
134	C₂×C₄×D₃	—	5	2	no	—
135	C₂×C₄×D₃	C₂×D₅	1	2	yes	complemented
136	C₃×C₄⋊C₄	—	5	1	no	—
137	C₃×C₂²⋊C₄	—	5	2	no	—
138	C₃×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
139	#217	—	5	2	no	—
140	(C₂×D₃)⋊C₄	—	5	2	no	—
141	(C₂×D₃)⋊C₄	—	5	1	no	—
142	(C₂×D₃)⋊C₄	—	5	1	no	—
143	(C₂×D₃)⋊C₄	—	5	2	no	—
144	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	complemented
145	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
146	Dic₃⋊C₄	—	5	1	no	—
147	Dic₃⋊C₄	—	5	1	no	—
148	C₄×Dic₃	—	5	1	no	—
149	C₄×Dic₃	—	5	2	no	—
150	C₂³×C₅	—	3	2	no	—
151	C₂²×D₅	—	3	1	no	—
152	C₂²×D₅	—	3	8	no	—
153	C₂²×D₅	—	12	2	no	—
154	C₂²×D₅	—	12	2	no	—
155	C₂²×D₅	C₂²×D₃	1	2	yes	—
156	C₅×D₄	—	3	4	no	—
157	C₂×C₄×C₅	—	3	2	no	—
158	C₂×C₄×C₅	—	12	2	no	—
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂²⋊D₅	—	3	8	no	—
163	C₂×Dic₅	—	3	1	no	—
164	C₂×Dic₅	—	3	1	no	—
165	C₂×Dic₅	—	3	1	no	—
166	C₂×Dic₅	—	12	2	no	—
167	C₂×Dic₅	C₂²×D₃	1	2	yes	—
168	D₂₀	C₄×D₃	1	4	yes	complemented
169	C₄×D₅	—	3	4	no	—
170	C₄×D₅	—	4	2	no	—
171	C₄×D₅	—	12	2	no	—
172	C₂²×D₄	—	15	1	no	—
173	C₄×D₄	—	3	1	no	—
174	C₄×D₄	—	5	1	no	—
175	C₄×D₄	—	15	2	no	—
176	C₄×D₄	—	15	2	no	—
177	C₄×D₄	—	15	1	no	—
178	C₄×D₄	—	15	1	no	—
179	#117	—	15	1	no	—
180	#117	—	15	1	no	—
181	C₂×C₂²⋊C₄	—	15	2	no	—
182	C₂×C₂²⋊C₄	—	15	2	no	—
183	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
186	D₁₅	—	2	2	no	—
187	C₃×D₅	—	2	4	no	—
188	C₅×D₃	—	2	4	no	—
189	C₂³×C₃	—	5	2	no	—
190	C₂²×D₃	—	5	1	no	—
191	C₂²×D₃	—	5	8	no	—
192	C₂²×D₃	—	20	2	no	—
193	C₂²×D₃	—	20	2	no	—
194	C₂²×D₃	C₂²×D₅	1	2	yes	—
195	C₃×D₄	—	5	4	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	—	20	2	no	—
198	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
201	#67	—	5	8	no	—
202	C₂×Dic₃	—	5	1	no	—
203	C₂×Dic₃	—	5	1	no	—
204	C₂×Dic₃	—	5	1	no	—
205	C₂×Dic₃	—	20	2	no	—
206	C₂×Dic₃	C₂²×D₅	1	2	yes	—
207	D₁₂	C₄×D₅	1	4	yes	complemented
208	C₄×D₃	—	4	2	no	—
209	C₄×D₃	—	5	4	no	—
210	C₄×D₃	—	20	2	no	—
211	C₂²×C₅	—	3	4	no	—
212	C₂²×C₅	—	6	2	no	—
213	C₂²×C₅	—	6	2	no	—
214	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
215	C₂×D₅	—	2	2	no	—
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	3	2	no	—
218	C₂×D₅	—	6	1	no	—
219	C₂×D₅	—	6	1	no	—
220	C₂×D₅	—	12	8	no	—
221	C₂×D₅	C₂×C₄×D₃	1	4	yes	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	6	2	no	—
225	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
226	Dic₅	—	2	2	no	—
227	Dic₅	—	3	2	no	—
228	Dic₅	—	6	1	no	—
229	Dic₅	—	6	1	no	—
230	C₂⁴	—	15	2	no	—
231	C₂×D₄	—	3	1	no	—
232	C₂×D₄	—	5	1	no	—
233	C₂×D₄	—	15	8	no	—
234	C₂×D₄	—	15	2	no	—
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	15	2	no	—
238	C₂²×C₄	—	15	2	no	—
239	C₂²×C₄	—	15	1	no	—
240	C₄⋊C₄	—	3	1	no	—
241	C₄⋊C₄	—	5	1	no	—
242	C₄⋊C₄	—	15	1	no	—
243	C₄⋊C₄	—	15	1	no	—
244	C₂²⋊C₄	—	3	2	no	—
245	C₂²⋊C₄	—	5	2	no	—
246	C₂²⋊C₄	—	15	2	no	—
247	C₂²⋊C₄	—	15	2	no	—
248	C₂²⋊C₄	—	15	2	no	—
249	C₂²⋊C₄	—	15	1	no	—
250	C₂²⋊C₄	—	15	1	no	—
251	C₄²	—	15	2	no	—
252	C₄²	—	15	1	no	—
253	C₄²	D₃×D₅	1	1	yes	characteristic, complemented
254	C₃×C₅	#518	1	1	yes	characteristic, complemented, Hall
255	C₂²×C₃	—	5	4	no	—
256	C₂²×C₃	—	10	2	no	—
257	C₂²×C₃	—	10	2	no	—
258	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
259	C₂×D₃	—	2	2	no	—
260	C₂×D₃	—	2	2	no	—
261	C₂×D₃	—	5	2	no	—
262	C₂×D₃	—	10	1	no	—
263	C₂×D₃	—	10	1	no	—
264	C₂×D₃	—	20	8	no	—
265	C₂×D₃	C₂×C₄×D₅	1	4	yes	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	10	2	no	—
269	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
270	Dic₃	—	2	2	no	—
271	Dic₃	—	5	2	no	—
272	Dic₃	—	10	1	no	—
273	Dic₃	—	10	1	no	—
274	C₂×C₅	—	6	4	no	—
275	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
276	C₂×C₅	#1004	1	1	yes	characteristic
277	C₂×C₅	#1009	1	1	yes	characteristic
278	D₅	—	2	4	no	—
279	D₅	—	6	2	no	—
280	C₂³	—	3	2	no	—
281	C₂³	—	5	2	no	—
282	C₂³	—	15	8	no	—
283	C₂³	—	15	1	no	—
284	C₂³	—	60	2	no	—
285	C₂³	—	60	2	no	—
286	D₄	—	3	4	no	—
287	D₄	—	5	4	no	—
288	D₄	—	15	8	no	—
289	C₂×C₄	—	3	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	12	2	no	—
292	C₂×C₄	—	15	1	no	—
293	C₂×C₄	—	15	4	no	—
294	C₂×C₄	—	15	1	no	—
295	C₂×C₄	—	15	1	no	—
296	C₂×C₄	—	20	2	no	—
297	C₂×C₄	—	60	2	no	—
298	C₂×C₄	—	60	2	no	—
299	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
302	C₂×C₃	—	10	4	no	—
303	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
304	C₂×C₃	#4144	1	1	yes	characteristic
305	C₂×C₃	#4149	1	1	yes	characteristic
306	D₃	—	2	4	no	—
307	D₃	—	10	2	no	—
308	C₅	#5545	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	3	4	no	—
310	C₂²	—	5	4	no	—
311	C₂²	—	6	2	no	—
312	C₂²	—	6	2	no	—
313	C₂²	—	10	2	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	15	2	no	—
316	C₂²	—	30	1	no	—
317	C₂²	—	30	1	no	—
318	C₂²	—	60	8	no	—
319	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	6	2	no	—
323	C₄	—	10	2	no	—
324	C₄	—	15	2	no	—
325	C₄	—	30	1	no	—
326	C₄	—	30	1	no	—
327	C₄	C₄×D₃×D₅	1	2	yes	—
328	C₃	#65879	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	6	4	no	—
330	C₂	—	10	4	no	—
331	C₂	—	30	2	no	—
332	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
333	C₂	#92017	1	1	yes	characteristic
334	C₂	#92018	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₄²×C₅	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×#6777	C₂
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#518
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #91761	C₂	semidirect
3. C₄×C₅×D₁₂	C₂	semidirect
4. C₃×C₄×D₂₀	C₂	semidirect
5. D₃×#685	C₂	semidirect
6. D₅×(C₂×D₃)⋊C₄	C₂	semidirect
7. #91441	C₂	semidirect
8. #91440	C₂	semidirect
9. #91435	C₂	semidirect
10. #91434	C₂	semidirect
11. #91347	C₂	semidirect
16. C₂×C₄×C₅×D₃	C₂²	semidirect
18. C₂×C₃×C₄×D₅	C₂²	semidirect
20. C₂×D₃×Dic₅	C₂²	semidirect
21. C₂×D₅×Dic₃	C₂²	semidirect
22. #6777	C₄	semidirect
23. C₃×C₄²×C₅	C₂²	semidirect
26. #6712	C₂²	semidirect
27. #6712	C₂²	semidirect
28. C₄×Dic₁₅	C₂²	semidirect
29. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
30. C₅×C₄⋊Dic₃	C₂²	semidirect
31. C₃×#685	C₂²	semidirect
32. C₃×C₄⋊Dic₅	C₂²	semidirect
34. (C₂×D₃)⋊Dic₅	C₂²	semidirect
35. #6665	C₂²	semidirect
36. Dic₃×Dic₅	C₂²	semidirect
47. C₄×D₂₀	D₃	semidirect
54. C₂×D₃×D₅	C₂×C₄	semidirect
61. C₄×D₁₅	C₂×C₄	semidirect
63. C₅×D₁₂	C₂×C₄	semidirect
66. C₃×D₂₀	C₂×C₄	semidirect
68. #1201	C₂×C₄	semidirect
80. C₄×D₁₂	D₅	semidirect
91. C₂×C₄×D₅	C₂×D₃	semidirect
94. C₄²×C₅	C₂×D₃	semidirect
100. #685	C₂×D₃	semidirect
101. C₄⋊Dic₅	C₂×D₃	semidirect
135. C₂×C₄×D₃	C₂×D₅	semidirect
138. C₃×C₄²	C₂×D₅	semidirect
144. (C₂×D₃)⋊C₄	C₂×D₅	semidirect
145. C₄⋊Dic₃	C₂×D₅	semidirect
168. D₂₀	C₄×D₃	semidirect
207. D₁₂	C₄×D₅	semidirect
253. C₄²	D₃×D₅	semidirect
254. C₃×C₅	#518	semidirect
308. C₅	#5545	semidirect
328. C₃	#65879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6777	2
#91761	2
C₄×C₅×D₁₂	2
C₃×C₄×D₂₀	2
D₃×#685	2
D₅×(C₂×D₃)⋊C₄	2
#91441	2
#91440	2
#91435	2
#91434	2
#91347	2
#65879	3
#5545	5