Subgroups — #11755323

Order 960.

536 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1009	C₂	1	1	yes	maximal, characteristic, complemented
2	#92017	C₂	1	1	yes	maximal, characteristic, complemented
3	#91839	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4095	C₂	1	1	yes	maximal, characteristic, complemented
5	#91517	C₂	1	2	yes	maximal, complemented
6	#91504	C₂	1	2	yes	maximal, complemented
7	#91494	C₂	1	2	yes	maximal, complemented
8	D₃×#686	C₂	1	2	yes	maximal, complemented
9	#91304	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#6652	C₂	1	1	yes	maximal, characteristic, complemented
11	#91288	C₂	1	1	yes	maximal, characteristic, complemented
12	#66169	—	3	1	no	maximal, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
14	#6817	C₂²	1	1	yes	characteristic, complemented
15	C₅×#239	C₂²	1	2	yes	complemented
16	C₅×D₃×D₄	—	2	2	no	—
17	C₅×D₃×D₄	C₂²	1	2	yes	complemented
18	C₅×#235	—	2	1	no	—
19	C₅×#235	C₂²	1	1	yes	characteristic
20	C₂×C₅×D₁₂	—	2	1	no	—
21	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
23	D₅×#67	—	2	2	no	—
24	#6777	—	2	1	no	—
25	D₃×D₂₀	C₂²	1	2	yes	complemented
26	#6766	—	2	1	no	—
27	#6765	C₂²	1	2	yes	complemented
28	#6718	C₂²	1	1	yes	characteristic, complemented
29	#6717	C₂²	1	1	yes	characteristic, complemented
30	#6716	C₂²	1	1	yes	characteristic, complemented
31	#6715	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic, complemented
33	C₃×#688	C₂²	1	2	yes	complemented
34	C₃×#686	C₂²	1	2	yes	complemented
35	C₃×#681	C₂²	1	1	yes	characteristic, complemented
36	#6659	C₂²	1	2	yes	complemented
37	#6654	C₂²	1	2	yes	complemented
38	#6652	C₂²	1	2	yes	complemented
39	#6652	C₂²	1	2	yes	complemented
40	#6650	C₂²	1	2	yes	complemented
41	D₃×#168	C₂²	1	2	yes	complemented
42	#5774	—	5	1	no	maximal, Hall
43	C₅×H(5,𝔽₂)	—	3	1	no	—
44	#4149	—	3	1	no	—
45	#4096	—	3	1	no	—
46	#4096	—	3	2	no	—
47	#4095	—	3	2	no	—
48	#4095	D₃	1	1	yes	characteristic, complemented
49	#4094	—	3	1	no	—
50	#4078	—	3	2	no	—
51	#4078	—	3	1	no	—
52	C₂×#686	—	3	1	no	—
53	C₂×#686	—	3	2	no	—
54	C₂²×C₅×D₃	—	2	1	no	—
55	C₂²×C₅×D₃	—	2	2	no	—
56	C₂²×C₃×D₅	—	2	1	no	—
57	C₂×D₃×D₅	—	2	2	no	—
58	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
59	C₃×C₅×D₄	—	2	1	no	—
60	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	#1220	—	2	1	no	—
64	D₆₀	C₂³	1	1	yes	characteristic
65	C₄×D₁₅	—	2	1	no	—
66	Dic₃₀	C₂³	1	1	yes	characteristic
67	C₅×#67	—	2	2	no	—
68	C₅×#67	D₄	1	2	yes	—
69	C₅×D₁₂	—	2	2	no	—
70	C₅×D₁₂	—	2	1	no	—
71	C₅×D₁₂	D₄	1	1	yes	characteristic
72	C₅×D₁₂	C₂³	1	2	yes	—
73	C₄×C₅×D₃	—	2	2	no	—
74	C₄×C₅×D₃	C₂³	1	2	yes	—
75	C₅×Dic₆	D₄	1	1	yes	characteristic
76	C₃×D₂₀	—	2	1	no	—
77	C₃×D₂₀	C₂³	1	2	yes	—
78	#1202	—	2	2	no	—
79	#1201	—	2	2	no	—
80	D₅×Dic₃	—	2	2	no	—
81	C₃⋊#168	C₂³	1	1	yes	characteristic
82	C₃⋊#168	C₂³	1	1	yes	characteristic
83	C₃×#168	C₂³	1	2	yes	—
84	#1009	D₅	1	1	yes	characteristic, complemented
85	#1004	—	5	1	no	—
86	C₃×AGL(1,ℤ/8)	—	5	1	no	—
87	#950	—	5	1	no	—
88	C₂×#213	—	5	1	no	—
89	#916	—	5	2	no	—
90	#914	—	5	2	no	—
91	#911	—	5	2	no	—
92	D₃×D₈	—	5	2	no	—
93	#908	—	5	1	no	—
94	#907	—	5	1	no	—
95	C₅×#41	—	3	1	no	—
96	C₅×#41	—	3	2	no	—
97	C₅×#41	—	6	1	no	—
98	C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
99	C₂×C₅×D₄	—	3	2	no	—
100	C₂×C₅×D₄	—	3	1	no	—
101	C₂×C₅×D₄	—	6	1	no	—
102	C₂×C₅×D₄	—	6	2	no	—
103	#713	—	6	1	no	—
104	D₄×D₅	—	6	1	no	—
105	D₄×D₅	—	6	2	no	—
106	#709	—	3	1	no	—
107	#709	—	3	2	no	—
108	C₂×D₂₀	—	3	2	no	—
109	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic, complemented
110	#689	—	3	1	no	—
111	#688	—	3	1	no	—
112	#688	C₂×D₃	1	2	yes	complemented
113	#687	—	3	1	no	—
114	#687	—	3	2	no	—
115	#686	—	3	1	no	—
116	#686	—	3	2	no	—
117	#686	—	3	2	no	—
118	#686	—	3	2	no	—
119	#686	C₂×D₃	1	2	yes	complemented
120	#681	—	3	2	no	—
121	#681	C₂×D₃	1	1	yes	characteristic, complemented
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	3	2	no	—
124	#576	—	15	1	no	Sylow
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×D₁₅	—	2	1	no	—
128	C₂×C₅×D₃	—	2	2	no	—
129	C₂×C₅×D₃	—	8	2	no	—
130	C₂×C₅×D₃	—	8	1	no	—
131	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
132	C₂×C₃×D₅	—	2	2	no	—
133	C₂×C₃×D₅	—	8	1	no	—
134	D₃×D₅	—	8	2	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
138	Dic₁₅	—	2	1	no	—
139	C₅×Dic₃	C₂×D₄	1	2	yes	—
140	C₃×#41	C₂×D₅	1	1	yes	characteristic, complemented
141	C₂×C₃×D₄	—	5	1	no	—
142	C₂×#67	—	10	2	no	—
143	#239	C₂×D₅	1	2	yes	complemented
144	#237	—	5	2	no	—
145	#237	—	10	1	no	—
146	D₃×D₄	—	2	2	no	—
147	D₃×D₄	—	5	2	no	—
148	D₃×D₄	—	10	1	no	—
149	D₃×D₄	C₂×D₅	1	2	yes	complemented
150	#235	—	2	1	no	—
151	#235	—	5	1	no	—
152	#235	C₂×D₅	1	1	yes	characteristic
153	C₂×D₁₂	—	2	1	no	—
154	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
155	C₃×SD₁₆	—	5	2	no	—
156	C₃×D₈	—	5	2	no	—
157	C₃×M₁₆	—	5	1	no	—
158	#216	—	5	1	no	—
159	#215	—	5	1	no	—
160	#214	—	5	1	no	—
161	#213	—	5	1	no	—
162	#213	—	5	2	no	—
163	#213	—	5	2	no	—
164	C₂×C₃⋊C₈	—	5	1	no	—
165	D₂₄	—	5	2	no	—
166	#204	—	5	2	no	—
167	ΓL(1,𝔽₂₅)	—	5	2	no	—
168	D₃×C₈	—	5	2	no	—
169	C₂³×C₅	—	6	2	no	—
170	C₂³×C₅	—	6	1	no	—
171	C₂²×D₅	—	2	1	no	—
172	C₂²×D₅	—	6	2	no	—
173	C₅×Q₈	—	3	1	no	—
174	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
175	C₅×D₄	—	2	1	no	—
176	C₅×D₄	—	3	2	no	—
177	C₅×D₄	—	3	1	no	—
178	C₅×D₄	—	3	2	no	—
179	C₅×D₄	—	6	2	no	—
180	C₅×D₄	—	6	2	no	—
181	C₅×D₄	—	6	1	no	—
182	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	—	2	1	no	—
184	C₂×C₄×C₅	—	3	2	no	—
185	C₂×C₄×C₅	—	6	2	no	—
186	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
187	C₂²⋊D₅	—	6	2	no	—
188	C₂²⋊D₅	—	6	1	no	—
189	D₂₀	—	2	1	no	—
190	D₂₀	—	3	1	no	—
191	D₂₀	—	6	2	no	—
192	D₂₀	C₂²×D₃	1	2	yes	—
193	C₄×D₅	—	6	2	no	—
194	C₄×D₅	—	6	1	no	—
195	Dic₁₀	—	3	1	no	—
196	#168	—	3	1	no	—
197	#168	—	3	1	no	—
198	#168	C₂²×D₃	1	2	yes	—
199	H(5,𝔽₂)	—	3	1	no	—
200	H(5,𝔽₂)	—	15	1	no	—
201	AGL(1,ℤ/8)	—	5	1	no	—
202	AGL(1,ℤ/8)	—	15	1	no	—
203	AGL(1,ℤ/8)	—	15	2	no	—
204	AGL(1,ℤ/8)	—	15	2	no	—
205	#135	—	15	2	no	—
206	#135	—	15	1	no	—
207	C₂×D₈	—	15	2	no	—
208	C₂×D₈	—	15	1	no	—
209	#131	—	15	1	no	—
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	—	4	1	no	—
212	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
213	D₁₅	—	4	1	no	—
214	C₃×D₅	—	4	2	no	—
215	C₅×D₃	—	2	2	no	—
216	C₅×D₃	—	4	2	no	—
217	C₂³×C₃	—	10	1	no	—
218	C₂²×D₃	—	2	1	no	—
219	C₂²×D₃	—	2	2	no	—
220	C₂²×D₃	—	10	2	no	—
221	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
222	C₃×D₄	—	2	1	no	—
223	C₃×D₄	—	5	2	no	—
224	C₃×D₄	—	10	1	no	—
225	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
228	#67	—	2	2	no	—
229	#67	—	10	1	no	—
230	#67	—	10	2	no	—
231	#67	—	10	2	no	—
232	#67	C₂²⋊D₅	1	2	yes	—
233	C₂×Dic₃	—	10	2	no	—
234	D₁₂	—	2	2	no	—
235	D₁₂	—	2	1	no	—
236	D₁₂	—	5	1	no	—
237	D₁₂	C₂²⋊D₅	1	1	yes	characteristic
238	D₁₂	C₂²×D₅	1	2	yes	—
239	C₄×D₃	—	2	2	no	—
240	C₄×D₃	—	10	1	no	—
241	C₄×D₃	C₂²×D₅	1	2	yes	—
242	Dic₆	—	5	1	no	—
243	Dic₆	C₂²⋊D₅	1	1	yes	characteristic
244	C₃×C₈	—	5	2	no	—
245	C₃⋊C₈	—	5	1	no	—
246	C₃⋊C₈	—	5	1	no	—
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	3	2	no	—
249	C₂²×C₅	—	6	2	no	—
250	C₂²×C₅	—	24	1	no	—
251	C₂²×C₅	—	24	2	no	—
252	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
253	C₂×D₅	—	2	2	no	—
254	C₂×D₅	—	6	1	no	—
255	C₂×D₅	—	8	1	no	—
256	C₂×D₅	—	24	2	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	3	2	no	—
259	C₄×C₅	D₃×D₄	1	1	yes	characteristic
260	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
261	Dic₅	—	6	1	no	—
262	#41	—	3	1	no	—
263	#41	—	3	2	no	—
264	#41	—	6	1	no	—
265	#41	—	15	2	no	—
266	#41	—	15	1	no	—
267	#41	—	30	1	no	—
268	#41	D₃×D₅	1	1	yes	characteristic, complemented
269	C₂×D₄	—	3	2	no	—
270	C₂×D₄	—	3	1	no	—
271	C₂×D₄	—	5	1	no	—
272	C₂×D₄	—	6	1	no	—
273	C₂×D₄	—	6	2	no	—
274	C₂×D₄	—	15	2	no	—
275	C₂×D₄	—	30	1	no	—
276	C₂×D₄	—	30	2	no	—
277	Q₁₆	—	15	1	no	—
278	SD₁₆	—	5	2	no	—
279	SD₁₆	—	15	2	no	—
280	SD₁₆	—	15	1	no	—
281	SD₁₆	—	15	1	no	—
282	D₈	—	5	2	no	—
283	D₈	—	15	2	no	—
284	D₈	—	15	2	no	—
285	D₈	—	15	2	no	—
286	D₈	—	15	1	no	—
287	M₁₆	—	5	1	no	—
288	M₁₆	—	15	2	no	—
289	C₂×C₈	—	15	2	no	—
290	C₂×C₈	—	15	1	no	—
291	C₃×C₅	#576	1	1	yes	characteristic, complemented, Hall
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	10	2	no	—
294	C₂²×C₃	—	40	1	no	—
295	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
296	C₂×D₃	—	2	2	no	—
297	C₂×D₃	—	8	2	no	—
298	C₂×D₃	—	8	1	no	—
299	C₂×D₃	—	10	1	no	—
300	C₂×D₃	—	40	2	no	—
301	C₂×D₃	C₂×C₂²⋊D₅	1	2	yes	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
304	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
305	Dic₃	—	10	1	no	—
306	Dic₃	C₂×C₂²⋊D₅	1	2	yes	—
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	4	1	no	—
309	C₂×C₅	—	6	2	no	—
310	C₂×C₅	—	12	2	no	—
311	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
312	D₅	—	4	2	no	—
313	D₅	—	12	1	no	—
314	C₂³	—	6	2	no	—
315	C₂³	—	6	1	no	—
316	C₂³	—	10	1	no	—
317	C₂³	—	30	2	no	—
318	Q₈	—	3	1	no	—
319	Q₈	—	15	1	no	—
320	Q₈	C₂×D₃×D₅	1	1	yes	characteristic
321	D₄	—	2	1	no	—
322	D₄	—	3	2	no	—
323	D₄	—	3	1	no	—
324	D₄	—	3	2	no	—
325	D₄	—	5	2	no	—
326	D₄	—	6	2	no	—
327	D₄	—	6	2	no	—
328	D₄	—	6	1	no	—
329	D₄	—	10	1	no	—
330	D₄	—	15	1	no	—
331	D₄	—	30	2	no	—
332	D₄	—	30	2	no	—
333	D₄	—	30	1	no	—
334	D₄	C₂×D₃×D₅	1	1	yes	characteristic
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	3	2	no	—
337	C₂×C₄	—	6	2	no	—
338	C₂×C₄	—	30	2	no	—
339	C₂×C₄	—	30	1	no	—
340	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
341	C₈	—	5	2	no	—
342	C₈	—	15	1	no	—
343	C₈	—	15	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	4	1	no	—
346	C₂×C₃	—	20	2	no	—
347	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
348	D₃	—	2	2	no	—
349	D₃	—	4	2	no	—
350	D₃	—	20	1	no	—
351	C₅	#5774	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	3	2	no	—
354	C₂²	—	6	2	no	—
355	C₂²	—	10	2	no	—
356	C₂²	—	24	1	no	—
357	C₂²	—	24	2	no	—
358	C₂²	—	30	1	no	—
359	C₂²	—	40	1	no	—
360	C₂²	—	120	2	no	—
361	C₂²	D₃×C₂²⋊D₅	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	3	2	no	—
364	C₄	—	30	1	no	—
365	C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
366	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
367	C₃	#66169	1	1	yes	characteristic, complemented, Sylow
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	6	2	no	—
371	C₂	—	12	2	no	—
372	C₂	—	20	2	no	—
373	C₂	—	60	1	no	—
374	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×C₂²⋊D₅
commutator, cosocle	C₃×C₄×C₅	C₂⁴
Frattini	C₄	C₂²×D₃×D₅
Fitting	C₃×C₅×#41	C₂²
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	#41	D₃×D₅
hypocenter	C₃×C₅	#576
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₅×#1009	C₂	semidirect
2. #92017	C₂	semidirect
3. #91839	C₂	semidirect
4. C₃×#4095	C₂	semidirect
5. #91517	C₂	semidirect
6. #91504	C₂	semidirect
7. #91494	C₂	semidirect
8. D₃×#686	C₂	semidirect
9. #91304	C₂	semidirect
10. C₂×#6652	C₂	semidirect
11. #91288	C₂	semidirect
13. C₃×C₅×#41	C₂²	semidirect
14. #6817	C₂²	semidirect
15. C₅×#239	C₂²	semidirect
17. C₅×D₃×D₄	C₂²	semidirect
21. C₂×C₅×D₁₂	C₂²	semidirect
22. C₂×C₃×D₂₀	C₂²	semidirect
25. D₃×D₂₀	C₂²	semidirect
27. #6765	C₂²	semidirect
28. #6718	C₂²	semidirect
29. #6717	C₂²	semidirect
30. #6716	C₂²	semidirect
31. #6715	C₂²	semidirect
32. C₂×C₃⋊#168	C₂²	semidirect
33. C₃×#688	C₂²	semidirect
34. C₃×#686	C₂²	semidirect
35. C₃×#681	C₂²	semidirect
36. #6659	C₂²	semidirect
37. #6654	C₂²	semidirect
38. #6652	C₂²	semidirect
39. #6652	C₂²	semidirect
40. #6650	C₂²	semidirect
41. D₃×#168	C₂²	semidirect
48. #4095	D₃	semidirect
84. #1009	D₅	semidirect
98. C₅×#41	C₂×D₃	semidirect
109. C₂×D₂₀	C₂×D₃	semidirect
112. #688	C₂×D₃	semidirect
119. #686	C₂×D₃	semidirect
121. #681	C₂×D₃	semidirect
140. C₃×#41	C₂×D₅	semidirect
143. #239	C₂×D₅	semidirect
149. D₃×D₄	C₂×D₅	semidirect
154. C₂×D₁₂	C₂×D₅	semidirect
268. #41	D₃×D₅	semidirect
291. C₃×C₅	#576	semidirect
351. C₅	#5774	semidirect
367. C₃	#66169	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1009	2
#92017	2
#91839	2
C₃×#4095	2
#91517	2
#91504	2
#91494	2
D₃×#686	2
#91304	2
C₂×#6652	2
#91288	2
#66169	3
#5774	5