Subgroups — #11756529

Order 960.

516 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#92101	C₂	1	1	yes	maximal, characteristic, complemented
2	#92100	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	#91840	C₂	1	1	yes	maximal, characteristic, complemented
5	#91838	C₂	1	1	yes	maximal, characteristic, complemented
6	#91807	C₂	1	2	yes	maximal, complemented
7	#91805	C₂	1	2	yes	maximal, complemented
8	SD₁₆×D₁₅	C₂	1	2	yes	maximal, complemented
9	#91800	C₂	1	2	yes	maximal, complemented
10	C₂×#6706	C₂	1	1	yes	maximal, characteristic, complemented
11	#91790	C₂	1	1	yes	maximal, characteristic, complemented
12	#66104	—	3	1	no	maximal, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
15	#6821	C₂²	1	2	yes	—
16	Q₈×D₁₅	C₂²	1	2	yes	complemented
17	#6819	—	2	2	no	—
18	#6819	C₂²	1	2	yes	complemented
19	D₄×D₁₅	—	2	2	no	—
20	D₄×D₁₅	C₂²	1	2	yes	—
21	#6817	—	2	1	no	—
22	#6817	—	2	1	no	—
23	#6817	C₂²	1	1	yes	characteristic
24	C₂×D₆₀	—	2	1	no	—
25	C₂×D₆₀	C₂²	1	1	yes	characteristic
26	C₂×Dic₃₀	—	2	1	no	—
27	C₂×Dic₃₀	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
33	#6718	C₂²	1	2	yes	complemented
34	#6717	C₂²	1	2	yes	complemented
35	#6716	C₂²	1	2	yes	complemented
36	#6715	C₂²	1	2	yes	complemented
37	#6710	C₂²	1	1	yes	characteristic, complemented
38	#6706	C₂²	1	2	yes	complemented
39	#6706	C₂²	1	2	yes	complemented
40	#6705	C₂²	1	2	yes	complemented
41	C₈×D₁₅	C₂²	1	2	yes	complemented
42	#5770	—	5	1	no	maximal, Hall
43	#4150	—	3	1	no	—
44	#4149	—	3	1	no	—
45	C₅×#135	D₃	1	1	yes	characteristic, complemented
46	#4097	—	3	1	no	—
47	#4095	—	3	1	no	—
48	#4064	—	3	2	no	—
49	#4062	—	3	2	no	—
50	D₅×SD₁₆	—	3	2	no	—
51	#4057	—	3	2	no	—
52	C₂×#677	—	3	1	no	—
53	#4047	—	3	1	no	—
54	C₂²×D₁₅	—	2	1	no	—
55	C₂²×D₁₅	—	2	2	no	—
56	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
57	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
58	C₃×C₅×D₄	—	2	1	no	—
59	C₃×C₅×D₄	—	2	1	no	—
60	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
61	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	#1220	—	2	2	no	—
66	#1220	—	2	2	no	—
67	#1220	D₄	1	2	yes	—
68	C₂×Dic₁₅	—	2	1	no	—
69	C₂×Dic₁₅	—	2	2	no	—
70	D₆₀	—	2	2	no	—
71	D₆₀	—	2	1	no	—
72	D₆₀	D₄	1	1	yes	characteristic
73	D₆₀	C₂³	1	2	yes	—
74	C₄×D₁₅	—	2	2	no	—
75	C₄×D₁₅	—	2	2	no	—
76	C₄×D₁₅	C₂³	1	2	yes	—
77	Dic₃₀	—	2	1	no	—
78	Dic₃₀	—	2	2	no	—
79	Dic₃₀	D₄	1	1	yes	characteristic
80	Dic₃₀	C₂³	1	2	yes	—
81	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
83	C₃⋊#168	C₂³	1	2	yes	—
84	#1010	—	5	1	no	—
85	#1009	—	5	1	no	—
86	C₃×#135	D₅	1	1	yes	characteristic, complemented
87	#951	—	5	1	no	—
88	#949	—	5	1	no	—
89	#918	—	5	2	no	—
90	#916	—	5	2	no	—
91	D₃×SD₁₆	—	5	2	no	—
92	#911	—	5	2	no	—
93	C₂×#204	—	5	1	no	—
94	#901	—	5	1	no	—
95	C₅×#41	C₂×D₃	1	1	yes	characteristic
96	C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
97	#713	—	3	2	no	—
98	Q₈×D₅	—	3	2	no	—
99	#711	—	3	2	no	—
100	#711	—	6	2	no	—
101	D₄×D₅	—	3	2	no	—
102	D₄×D₅	—	6	2	no	—
103	#709	—	3	1	no	—
104	#709	—	6	1	no	—
105	#709	—	6	1	no	—
106	C₂×D₂₀	—	3	1	no	—
107	C₂×D₂₀	—	6	1	no	—
108	C₂×Dic₁₀	—	3	1	no	—
109	C₂×Dic₁₀	—	6	1	no	—
110	C₅×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
111	C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
112	C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
113	C₅×D₈	C₂×D₃	1	1	yes	characteristic, complemented
114	C₂×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
115	#689	—	3	2	no	—
116	#688	—	3	2	no	—
117	#687	—	3	2	no	—
118	#686	—	3	2	no	—
119	#681	—	3	1	no	—
120	#677	—	3	2	no	—
121	#677	—	3	2	no	—
122	#676	—	3	2	no	—
123	C₈×D₅	—	3	2	no	—
124	#577	—	15	1	no	Sylow
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
128	C₂×D₁₅	—	2	2	no	—
129	C₂×D₁₅	—	8	2	no	—
130	C₂×D₁₅	—	8	1	no	—
131	C₂×D₁₅	C₂×D₄	1	2	yes	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
136	Dic₁₅	—	2	2	no	—
137	Dic₁₅	C₂×D₄	1	2	yes	—
138	C₃×#41	C₂×D₅	1	1	yes	characteristic
139	C₃×#41	C₂×D₅	1	1	yes	characteristic, complemented
140	#239	—	5	2	no	—
141	D₃×Q₈	—	5	2	no	—
142	#237	—	5	2	no	—
143	#237	—	10	2	no	—
144	D₃×D₄	—	5	2	no	—
145	D₃×D₄	—	10	2	no	—
146	#235	—	5	1	no	—
147	#235	—	10	1	no	—
148	#235	—	10	1	no	—
149	C₂×D₁₂	—	5	1	no	—
150	C₂×D₁₂	—	10	1	no	—
151	C₂×Dic₆	—	5	1	no	—
152	C₂×Dic₆	—	10	1	no	—
153	C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
154	C₃×SD₁₆	C₂×D₅	1	1	yes	characteristic, complemented
155	C₃×SD₁₆	C₂×D₅	1	1	yes	characteristic, complemented
156	C₃×D₈	C₂×D₅	1	1	yes	characteristic, complemented
157	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic, complemented
158	#216	—	5	2	no	—
159	#215	—	5	2	no	—
160	#214	—	5	2	no	—
161	#213	—	5	2	no	—
162	#208	—	5	1	no	—
163	#204	—	5	2	no	—
164	#204	—	5	2	no	—
165	ΓL(1,𝔽₂₅)	—	5	2	no	—
166	D₃×C₈	—	5	2	no	—
167	C₂²×D₅	—	6	2	no	—
168	C₂²×D₅	—	6	1	no	—
169	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
170	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
171	C₅×D₄	—	2	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
174	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
178	C₂²⋊D₅	—	3	2	no	—
179	C₂²⋊D₅	—	6	2	no	—
180	C₂²⋊D₅	—	6	2	no	—
181	C₂×Dic₅	—	6	2	no	—
182	C₂×Dic₅	—	6	1	no	—
183	D₂₀	—	3	1	no	—
184	D₂₀	—	3	2	no	—
185	D₂₀	—	6	2	no	—
186	D₂₀	—	6	1	no	—
187	C₄×D₅	—	3	2	no	—
188	C₄×D₅	—	6	2	no	—
189	C₄×D₅	—	6	2	no	—
190	Dic₁₀	—	3	1	no	—
191	Dic₁₀	—	3	2	no	—
192	Dic₁₀	—	6	2	no	—
193	Dic₁₀	—	6	1	no	—
194	C₅×C₈	C₂²×D₃	1	1	yes	characteristic
195	C₅×C₈	C₂²×D₃	1	1	yes	characteristic
196	#168	—	3	2	no	—
197	2₋¹⁺⁴	—	15	1	no	—
198	H(5,𝔽₂)	—	15	1	no	—
199	#137	—	15	1	no	—
200	#137	—	15	2	no	—
201	AGL(1,ℤ/8)	—	15	1	no	—
202	AGL(1,ℤ/8)	—	15	2	no	—
203	#135	—	15	2	no	—
204	#135	D₁₅	1	1	yes	characteristic, complemented
205	C₂×SD₁₆	—	15	2	no	—
206	C₂×SD₁₆	—	15	1	no	—
207	#131	—	15	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	—	4	1	no	—
210	C₂×C₃×C₅	—	4	1	no	—
211	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
212	D₁₅	—	2	2	no	—
213	D₁₅	—	4	2	no	—
214	C₂²×D₃	—	10	2	no	—
215	C₂²×D₃	—	10	1	no	—
216	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
217	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	—	2	1	no	—
220	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
221	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
225	#67	—	5	2	no	—
226	#67	—	10	2	no	—
227	#67	—	10	2	no	—
228	C₂×Dic₃	—	10	2	no	—
229	C₂×Dic₃	—	10	1	no	—
230	D₁₂	—	5	1	no	—
231	D₁₂	—	5	2	no	—
232	D₁₂	—	10	2	no	—
233	D₁₂	—	10	1	no	—
234	C₄×D₃	—	5	2	no	—
235	C₄×D₃	—	10	2	no	—
236	C₄×D₃	—	10	2	no	—
237	Dic₆	—	5	1	no	—
238	Dic₆	—	5	2	no	—
239	Dic₆	—	10	2	no	—
240	Dic₆	—	10	1	no	—
241	C₃×C₈	C₂²×D₅	1	1	yes	characteristic
242	C₃×C₈	C₂²×D₅	1	1	yes	characteristic
243	C₃⋊C₈	—	5	2	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
247	C₂×D₅	—	3	2	no	—
248	C₂×D₅	—	6	2	no	—
249	C₂×D₅	—	24	2	no	—
250	C₂×D₅	—	24	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	D₃×D₄	1	1	yes	characteristic
254	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
255	Dic₅	—	3	2	no	—
256	Dic₅	—	6	2	no	—
257	#41	—	15	1	no	—
258	#41	—	15	2	no	—
259	#41	—	15	2	no	—
260	#41	—	30	1	no	—
261	#41	—	30	1	no	—
262	#41	—	30	2	no	—
263	#41	C₂×D₁₅	1	1	yes	characteristic
264	#41	C₂×D₁₅	1	1	yes	characteristic, complemented
265	C₂×Q₈	—	15	2	no	—
266	C₂×Q₈	—	15	1	no	—
267	C₂×Q₈	—	30	1	no	—
268	C₂×D₄	—	15	2	no	—
269	C₂×D₄	—	15	1	no	—
270	C₂×D₄	—	30	1	no	—
271	C₂×D₄	—	30	2	no	—
272	Q₁₆	—	15	2	no	—
273	Q₁₆	C₂×D₁₅	1	1	yes	characteristic, complemented
274	SD₁₆	—	15	2	no	—
275	SD₁₆	—	15	2	no	—
276	SD₁₆	—	15	2	no	—
277	SD₁₆	—	15	2	no	—
278	SD₁₆	C₂×D₁₅	1	1	yes	characteristic, complemented
279	SD₁₆	C₂×D₁₅	1	1	yes	characteristic, complemented
280	D₈	—	15	2	no	—
281	D₈	C₂×D₁₅	1	1	yes	characteristic, complemented
282	M₁₆	—	15	1	no	—
283	M₁₆	—	15	2	no	—
284	C₂×C₈	—	15	2	no	—
285	C₂×C₈	C₂×D₁₅	1	1	yes	characteristic, complemented
286	C₃×C₅	#577	1	1	yes	characteristic, complemented, Hall
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	2	1	no	—
289	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
290	C₂×D₃	—	5	2	no	—
291	C₂×D₃	—	10	2	no	—
292	C₂×D₃	—	40	2	no	—
293	C₂×D₃	—	40	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	D₄×D₅	1	1	yes	characteristic
297	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
298	Dic₃	—	5	2	no	—
299	Dic₃	—	10	2	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	—	4	1	no	—
302	C₂×C₅	—	4	1	no	—
303	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
304	D₅	—	6	2	no	—
305	D₅	—	12	2	no	—
306	C₂³	—	30	2	no	—
307	C₂³	—	30	1	no	—
308	Q₈	—	15	1	no	—
309	Q₈	—	15	2	no	—
310	Q₈	—	30	2	no	—
311	Q₈	—	30	1	no	—
312	Q₈	C₂²×D₁₅	1	1	yes	characteristic
313	Q₈	C₂²×D₁₅	1	1	yes	characteristic
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	15	2	no	—
317	D₄	—	15	1	no	—
318	D₄	—	15	2	no	—
319	D₄	—	30	2	no	—
320	D₄	—	30	2	no	—
321	D₄	—	30	2	no	—
322	D₄	—	30	1	no	—
323	D₄	C₂²×D₁₅	1	1	yes	characteristic
324	D₄	C₂²×D₁₅	1	1	yes	characteristic
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	15	2	no	—
328	C₂×C₄	—	30	2	no	—
329	C₂×C₄	—	30	2	no	—
330	C₂×C₄	—	30	2	no	—
331	C₂×C₄	—	30	1	no	—
332	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
333	C₈	—	15	2	no	—
334	C₈	C₂²×D₁₅	1	1	yes	characteristic
335	C₈	C₂²×D₁₅	1	1	yes	characteristic
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	4	1	no	—
339	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
340	D₃	—	10	2	no	—
341	D₃	—	20	2	no	—
342	C₅	#5770	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	15	2	no	—
346	C₂²	—	30	2	no	—
347	C₂²	—	120	1	no	—
348	C₂²	—	120	2	no	—
349	C₂²	D₄×D₁₅	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	15	2	no	—
353	C₄	—	30	2	no	—
354	C₄	D₄×D₁₅	1	1	yes	characteristic
355	C₄	C₂³×D₁₅	1	1	yes	characteristic
356	C₃	#66104	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	4	1	no	—
360	C₂	—	30	2	no	—
361	C₂	—	60	2	no	—
362	C₂	C₂×D₄×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₅
commutator, cosocle	C₃×C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₁₅
Fitting	C₃×C₅×#135	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	#135	D₁₅
hypocenter	C₃×C₅	#577
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #92101	C₂	semidirect
2. #92100	C₂	semidirect
3. C₃×C₅×#135	C₂	semidirect
4. #91840	C₂	semidirect
5. #91838	C₂	semidirect
6. #91807	C₂	semidirect
7. #91805	C₂	semidirect
8. SD₁₆×D₁₅	C₂	semidirect
9. #91800	C₂	semidirect
10. C₂×#6706	C₂	semidirect
11. #91790	C₂	semidirect
14. C₃×C₅×#41	C₂²	semidirect
16. Q₈×D₁₅	C₂²	semidirect
18. #6819	C₂²	semidirect
27. C₂×Dic₃₀	C₂²	semidirect
28. C₃×C₅×Q₁₆	C₂²	semidirect
29. C₃×C₅×SD₁₆	C₂²	semidirect
30. C₃×C₅×SD₁₆	C₂²	semidirect
31. C₃×C₅×D₈	C₂²	semidirect
32. C₂×C₃×C₅×C₈	C₂²	semidirect
33. #6718	C₂²	semidirect
34. #6717	C₂²	semidirect
35. #6716	C₂²	semidirect
36. #6715	C₂²	semidirect
37. #6710	C₂²	semidirect
38. #6706	C₂²	semidirect
39. #6706	C₂²	semidirect
40. #6705	C₂²	semidirect
41. C₈×D₁₅	C₂²	semidirect
45. C₅×#135	D₃	semidirect
86. C₃×#135	D₅	semidirect
96. C₅×#41	C₂×D₃	semidirect
110. C₅×Q₁₆	C₂×D₃	semidirect
111. C₅×SD₁₆	C₂×D₃	semidirect
112. C₅×SD₁₆	C₂×D₃	semidirect
113. C₅×D₈	C₂×D₃	semidirect
114. C₂×C₅×C₈	C₂×D₃	semidirect
139. C₃×#41	C₂×D₅	semidirect
153. C₃×Q₁₆	C₂×D₅	semidirect
154. C₃×SD₁₆	C₂×D₅	semidirect
155. C₃×SD₁₆	C₂×D₅	semidirect
156. C₃×D₈	C₂×D₅	semidirect
157. C₂×C₃×C₈	C₂×D₅	semidirect
204. #135	D₁₅	semidirect
264. #41	C₂×D₁₅	semidirect
273. Q₁₆	C₂×D₁₅	semidirect
278. SD₁₆	C₂×D₁₅	semidirect
279. SD₁₆	C₂×D₁₅	semidirect
281. D₈	C₂×D₁₅	semidirect
285. C₂×C₈	C₂×D₁₅	semidirect
286. C₃×C₅	#577	semidirect
342. C₅	#5770	semidirect
356. C₃	#66104	semidirect

Maximal subgroups

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

subgroup	index
#92101	2
#92100	2
C₃×C₅×#135	2
#91840	2
#91838	2
#91807	2
#91805	2
SD₁₆×D₁₅	2
#91800	2
C₂×#6706	2
#91790	2
#66104	3
#5770	5