Subgroups — #11755316

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	C₃×#4149	C₂	1	1	yes	maximal, characteristic, complemented
2	#92018	C₂	1	1	yes	maximal, characteristic, complemented
3	#91839	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#949	C₂	1	1	yes	maximal, characteristic, complemented
5	#91514	C₂	1	2	yes	maximal, complemented
6	#91502	C₂	1	2	yes	maximal, complemented
7	#91489	C₂	1	2	yes	maximal, complemented
8	D₅×#213	C₂	1	2	yes	maximal, complemented
9	#91312	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#6652	C₂	1	1	yes	maximal, characteristic, complemented
11	#91284	C₂	1	1	yes	maximal, characteristic, complemented
12	#66108	—	3	1	no	maximal, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
14	#6817	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
16	C₃×#713	C₂²	1	2	yes	complemented
17	C₃×D₄×D₅	—	2	2	no	—
18	C₃×D₄×D₅	C₂²	1	2	yes	complemented
19	C₃×#709	—	2	1	no	—
20	C₃×#709	C₂²	1	1	yes	characteristic
21	C₂×C₃×D₂₀	—	2	1	no	—
22	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
23	D₃×C₂²⋊D₅	—	2	2	no	—
24	#6777	—	2	1	no	—
25	D₅×D₁₂	C₂²	1	2	yes	complemented
26	#6772	C₂²	1	2	yes	complemented
27	#6768	—	2	1	no	—
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	C₅×#215	C₂²	1	2	yes	complemented
34	C₅×#213	C₂²	1	2	yes	complemented
35	C₅×#208	C₂²	1	1	yes	characteristic, complemented
36	#6657	C₂²	1	2	yes	complemented
37	#6653	C₂²	1	2	yes	complemented
38	#6652	C₂²	1	2	yes	complemented
39	#6652	C₂²	1	2	yes	complemented
40	#6649	C₂²	1	2	yes	complemented
41	D₅×C₃⋊C₈	C₂²	1	2	yes	complemented
42	#5835	—	5	1	no	maximal, Hall
43	#4149	D₃	1	1	yes	characteristic, complemented
44	#4144	—	3	1	no	—
45	C₅×AGL(1,ℤ/8)	—	3	1	no	—
46	#4096	—	3	1	no	—
47	C₂×#686	—	3	1	no	—
48	#4062	—	3	2	no	—
49	#4060	—	3	2	no	—
50	#4057	—	3	2	no	—
51	D₅×D₈	—	3	2	no	—
52	#4054	—	3	1	no	—
53	#4053	—	3	1	no	—
54	C₂²×C₅×D₃	—	2	1	no	—
55	C₂²×C₃×D₅	—	2	1	no	—
56	C₂²×C₃×D₅	—	2	2	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×D₁₂	—	2	1	no	—
68	C₅×D₁₂	C₂³	1	2	yes	—
69	C₃×C₂²⋊D₅	—	2	2	no	—
70	C₃×C₂²⋊D₅	D₄	1	2	yes	—
71	C₃×D₂₀	—	2	2	no	—
72	C₃×D₂₀	—	2	1	no	—
73	C₃×D₂₀	D₄	1	1	yes	characteristic
74	C₃×D₂₀	C₂³	1	2	yes	—
75	C₃×C₄×D₅	—	2	2	no	—
76	C₃×C₄×D₅	C₂³	1	2	yes	—
77	C₃×Dic₁₀	D₄	1	1	yes	characteristic
78	#1203	—	2	2	no	—
79	#1201	—	2	2	no	—
80	D₃×Dic₅	—	2	2	no	—
81	C₃⋊#168	C₂³	1	1	yes	characteristic
82	C₃⋊#168	C₂³	1	1	yes	characteristic
83	C₅×C₃⋊C₈	C₂³	1	2	yes	—
84	C₃×H(5,𝔽₂)	—	5	1	no	—
85	#1009	—	5	1	no	—
86	#950	—	5	1	no	—
87	#950	—	5	2	no	—
88	#949	—	5	2	no	—
89	#949	D₅	1	1	yes	characteristic, complemented
90	#948	—	5	1	no	—
91	#932	—	5	2	no	—
92	#932	—	5	1	no	—
93	C₂×#213	—	5	1	no	—
94	C₂×#213	—	5	2	no	—
95	C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
96	C₂×C₅×D₄	—	3	1	no	—
97	C₂×C₂²⋊D₅	—	6	2	no	—
98	#713	C₂×D₃	1	2	yes	complemented
99	#711	—	3	2	no	—
100	#711	—	6	1	no	—
101	D₄×D₅	—	2	2	no	—
102	D₄×D₅	—	3	2	no	—
103	D₄×D₅	—	6	1	no	—
104	D₄×D₅	C₂×D₃	1	2	yes	complemented
105	#709	—	2	1	no	—
106	#709	—	3	1	no	—
107	#709	C₂×D₃	1	1	yes	characteristic
108	C₂×D₂₀	—	2	1	no	—
109	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic, complemented
110	C₅×SD₁₆	—	3	2	no	—
111	C₅×D₈	—	3	2	no	—
112	C₅×M₁₆	—	3	1	no	—
113	#689	—	3	1	no	—
114	#688	—	3	1	no	—
115	#687	—	3	1	no	—
116	#686	—	3	1	no	—
117	#686	—	3	2	no	—
118	#686	—	3	2	no	—
119	C₂×#168	—	3	1	no	—
120	D₄₀	—	3	2	no	—
121	#677	—	3	2	no	—
122	#676	—	3	2	no	—
123	C₈×D₅	—	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	1	no	—
130	C₂×C₃×D₅	—	2	2	no	—
131	C₂×C₃×D₅	—	8	2	no	—
132	C₂×C₃×D₅	—	8	1	no	—
133	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
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	C₃×Dic₅	C₂×D₄	1	2	yes	—
140	C₃×#41	—	5	1	no	—
141	C₃×#41	—	5	2	no	—
142	C₃×#41	—	10	1	no	—
143	C₃×#41	C₂×D₅	1	1	yes	characteristic, complemented
144	C₂×C₃×D₄	—	5	2	no	—
145	C₂×C₃×D₄	—	5	1	no	—
146	C₂×C₃×D₄	—	10	1	no	—
147	C₂×C₃×D₄	—	10	2	no	—
148	#239	—	10	1	no	—
149	D₃×D₄	—	10	1	no	—
150	D₃×D₄	—	10	2	no	—
151	#235	—	5	1	no	—
152	#235	—	5	2	no	—
153	C₂×D₁₂	—	5	2	no	—
154	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
155	#216	—	5	1	no	—
156	#215	—	5	1	no	—
157	#215	C₂×D₅	1	2	yes	complemented
158	#214	—	5	1	no	—
159	#214	—	5	2	no	—
160	#213	—	5	1	no	—
161	#213	—	5	2	no	—
162	#213	—	5	2	no	—
163	#213	—	5	2	no	—
164	#213	C₂×D₅	1	2	yes	complemented
165	#208	—	5	2	no	—
166	#208	C₂×D₅	1	1	yes	characteristic, complemented
167	C₂×C₃⋊C₈	—	5	1	no	—
168	C₂×C₃⋊C₈	—	5	2	no	—
169	C₂³×C₅	—	6	1	no	—
170	C₂²×D₅	—	2	1	no	—
171	C₂²×D₅	—	2	2	no	—
172	C₂²×D₅	—	6	2	no	—
173	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
174	C₅×D₄	—	2	1	no	—
175	C₅×D₄	—	3	2	no	—
176	C₅×D₄	—	6	1	no	—
177	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
180	C₂²⋊D₅	—	2	2	no	—
181	C₂²⋊D₅	—	6	1	no	—
182	C₂²⋊D₅	—	6	2	no	—
183	C₂²⋊D₅	—	6	2	no	—
184	C₂²⋊D₅	#67	1	2	yes	—
185	C₂×Dic₅	—	6	2	no	—
186	D₂₀	—	2	2	no	—
187	D₂₀	—	2	1	no	—
188	D₂₀	—	3	1	no	—
189	D₂₀	#67	1	1	yes	characteristic
190	D₂₀	C₂²×D₃	1	2	yes	—
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	6	1	no	—
193	C₄×D₅	C₂²×D₃	1	2	yes	—
194	Dic₁₀	—	3	1	no	—
195	Dic₁₀	#67	1	1	yes	characteristic
196	C₅×C₈	—	3	2	no	—
197	#168	—	3	1	no	—
198	#168	—	3	1	no	—
199	H(5,𝔽₂)	—	5	1	no	—
200	H(5,𝔽₂)	—	15	1	no	—
201	AGL(1,ℤ/8)	—	3	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	—	2	2	no	—
215	C₃×D₅	—	4	2	no	—
216	C₅×D₃	—	4	2	no	—
217	C₂³×C₃	—	10	2	no	—
218	C₂³×C₃	—	10	1	no	—
219	C₂²×D₃	—	2	1	no	—
220	C₂²×D₃	—	10	2	no	—
221	C₃×Q₈	—	5	1	no	—
222	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
223	C₃×D₄	—	2	1	no	—
224	C₃×D₄	—	5	2	no	—
225	C₃×D₄	—	5	1	no	—
226	C₃×D₄	—	5	2	no	—
227	C₃×D₄	—	10	2	no	—
228	C₃×D₄	—	10	2	no	—
229	C₃×D₄	—	10	1	no	—
230	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	5	2	no	—
233	C₂×C₃×C₄	—	10	2	no	—
234	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
235	#67	—	10	2	no	—
236	#67	—	10	1	no	—
237	D₁₂	—	2	1	no	—
238	D₁₂	—	5	1	no	—
239	D₁₂	—	10	2	no	—
240	D₁₂	C₂²×D₅	1	2	yes	—
241	C₄×D₃	—	10	2	no	—
242	C₄×D₃	—	10	1	no	—
243	Dic₆	—	5	1	no	—
244	C₃⋊C₈	—	5	1	no	—
245	C₃⋊C₈	—	5	1	no	—
246	C₃⋊C₈	C₂²×D₅	1	2	yes	—
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	6	2	no	—
249	C₂²×C₅	—	24	1	no	—
250	C₂²×C₅	C₂×#67	1	1	yes	characteristic
251	C₂×D₅	—	2	2	no	—
252	C₂×D₅	—	6	1	no	—
253	C₂×D₅	—	8	2	no	—
254	C₂×D₅	—	8	1	no	—
255	C₂×D₅	—	24	2	no	—
256	C₂×D₅	C₂×#67	1	2	yes	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	C₂×#67	1	1	yes	characteristic
259	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
260	Dic₅	—	6	1	no	—
261	Dic₅	C₂×#67	1	2	yes	—
262	#41	—	5	1	no	—
263	#41	—	5	2	no	—
264	#41	—	10	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	1	no	—
270	C₂×D₄	—	5	2	no	—
271	C₂×D₄	—	5	1	no	—
272	C₂×D₄	—	10	1	no	—
273	C₂×D₄	—	10	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	—	3	2	no	—
279	SD₁₆	—	15	2	no	—
280	SD₁₆	—	15	1	no	—
281	SD₁₆	—	15	1	no	—
282	D₈	—	3	2	no	—
283	D₈	—	15	2	no	—
284	D₈	—	15	2	no	—
285	D₈	—	15	2	no	—
286	D₈	—	15	1	no	—
287	M₁₆	—	3	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	—	5	2	no	—
294	C₂²×C₃	—	10	2	no	—
295	C₂²×C₃	—	40	1	no	—
296	C₂²×C₃	—	40	2	no	—
297	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
298	C₂×D₃	—	2	2	no	—
299	C₂×D₃	—	8	1	no	—
300	C₂×D₃	—	10	1	no	—
301	C₂×D₃	—	40	2	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	5	2	no	—
304	C₃×C₄	D₄×D₅	1	1	yes	characteristic
305	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
306	Dic₃	—	10	1	no	—
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	4	1	no	—
309	C₂×C₅	—	12	2	no	—
310	C₂×C₅	C₂²×#67	1	1	yes	characteristic
311	D₅	—	2	2	no	—
312	D₅	—	4	2	no	—
313	D₅	—	12	1	no	—
314	C₂³	—	6	1	no	—
315	C₂³	—	10	2	no	—
316	C₂³	—	10	1	no	—
317	C₂³	—	30	2	no	—
318	Q₈	—	5	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	—	5	2	no	—
324	D₄	—	5	1	no	—
325	D₄	—	5	2	no	—
326	D₄	—	6	1	no	—
327	D₄	—	10	2	no	—
328	D₄	—	10	2	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	—	5	2	no	—
337	C₂×C₄	—	10	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	—	3	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	—	10	2	no	—
347	C₂×C₃	—	20	2	no	—
348	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
349	D₃	—	4	2	no	—
350	D₃	—	20	1	no	—
351	C₅	#5835	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	5	2	no	—
354	C₂²	—	6	2	no	—
355	C₂²	—	10	2	no	—
356	C₂²	—	24	1	no	—
357	C₂²	—	30	1	no	—
358	C₂²	—	40	1	no	—
359	C₂²	—	40	2	no	—
360	C₂²	—	120	2	no	—
361	C₂²	D₅×#67	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	5	2	no	—
364	C₄	—	30	1	no	—
365	C₄	D₅×#67	1	1	yes	characteristic
366	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
367	C₃	#66108	1	1	yes	characteristic, complemented, Sylow
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	10	2	no	—
371	C₂	—	12	2	no	—
372	C₂	—	20	2	no	—
373	C₂	—	60	1	no	—
374	C₂	C₂×D₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅×#67
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. C₃×#4149	C₂	semidirect
2. #92018	C₂	semidirect
3. #91839	C₂	semidirect
4. C₅×#949	C₂	semidirect
5. #91514	C₂	semidirect
6. #91502	C₂	semidirect
7. #91489	C₂	semidirect
8. D₅×#213	C₂	semidirect
9. #91312	C₂	semidirect
10. C₂×#6652	C₂	semidirect
11. #91284	C₂	semidirect
13. C₃×C₅×#41	C₂²	semidirect
14. #6817	C₂²	semidirect
15. C₂×C₅×D₁₂	C₂²	semidirect
16. C₃×#713	C₂²	semidirect
18. C₃×D₄×D₅	C₂²	semidirect
22. C₂×C₃×D₂₀	C₂²	semidirect
25. D₅×D₁₂	C₂²	semidirect
26. #6772	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. C₅×#215	C₂²	semidirect
34. C₅×#213	C₂²	semidirect
35. C₅×#208	C₂²	semidirect
36. #6657	C₂²	semidirect
37. #6653	C₂²	semidirect
38. #6652	C₂²	semidirect
39. #6652	C₂²	semidirect
40. #6649	C₂²	semidirect
41. D₅×C₃⋊C₈	C₂²	semidirect
43. #4149	D₃	semidirect
89. #949	D₅	semidirect
95. C₅×#41	C₂×D₃	semidirect
98. #713	C₂×D₃	semidirect
104. D₄×D₅	C₂×D₃	semidirect
109. C₂×D₂₀	C₂×D₃	semidirect
143. C₃×#41	C₂×D₅	semidirect
154. C₂×D₁₂	C₂×D₅	semidirect
157. #215	C₂×D₅	semidirect
164. #213	C₂×D₅	semidirect
166. #208	C₂×D₅	semidirect
268. #41	D₃×D₅	semidirect
291. C₃×C₅	#576	semidirect
351. C₅	#5835	semidirect
367. C₃	#66108	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4149	2
#92018	2
#91839	2
C₅×#949	2
#91514	2
#91502	2
#91489	2
D₅×#213	2
#91312	2
C₂×#6652	2
#91284	2
#66108	3
#5835	5