Subgroups — #11755274

Order 960.

668 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	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6777	C₂	1	1	yes	maximal, characteristic, complemented
3	#91835	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#947	C₂	1	1	yes	maximal, characteristic, complemented
5	#91474	C₂	1	2	yes	maximal, complemented
6	D₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
7	#91449	C₂	1	1	yes	maximal, characteristic, complemented
8	#91447	C₂	1	2	yes	maximal, complemented
9	C₄×#1202	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#91408	C₂	1	2	yes	maximal, complemented
12	#66010	—	3	1	no	maximal, Hall
13	C₂²×D₃×D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₃×#713	—	2	4	no	—
18	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×C₃×D₂₀	C₂²	1	2	yes	complemented
20	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₄×D₅	C₂²	1	2	yes	complemented
22	C₂×#1202	C₂²	1	1	yes	characteristic
23	C₂×#1201	C₂²	1	2	yes	—
24	C₂×#1201	C₂²	1	2	yes	complemented
25	C₂×D₅×Dic₃	C₂²	1	2	yes	complemented
26	#6777	—	2	4	no	—
27	#6714	C₂²	1	2	yes	—
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic, complemented
29	#6712	C₂²	1	2	yes	complemented
30	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
31	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
32	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
33	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
34	#6667	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	complemented
36	#6665	C₂²	1	1	yes	characteristic, complemented
37	#6665	C₂²	1	2	yes	complemented
38	#5829	—	5	1	no	maximal, Hall
39	C₂×#713	D₃	1	1	yes	characteristic, complemented
40	C₂×D₄×D₅	—	3	1	no	—
41	C₅×#124	—	3	1	no	—
42	#4092	—	3	1	no	—
43	#4084	—	3	1	no	—
44	#4030	—	3	2	no	—
45	#4029	—	3	2	no	—
46	#4028	—	3	2	no	—
47	D₅×C₂²⋊C₄	—	3	2	no	—
48	C₄×D₂₀	—	3	1	no	—
49	C₄×D₂₀	—	3	1	no	—
50	C₂²×D₁₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	C₂³	1	2	yes	—
52	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
53	C₂²×C₃×D₅	C₂³	1	2	yes	—
54	C₂×D₃×D₅	—	2	4	no	—
55	C₂×D₃×D₅	—	2	4	no	—
56	C₂×D₃×D₅	—	4	2	no	—
57	C₃×C₅×Q₈	—	2	2	no	—
58	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	2	yes	—
61	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
62	C₄×D₁₅	—	2	2	no	—
63	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₅×D₁₂	—	2	2	no	—
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₃×D₂₀	—	2	4	no	—
68	C₃×D₂₀	D₄	1	4	yes	complemented
69	C₃×C₄×D₅	—	2	4	no	—
70	C₃×C₄×D₅	—	2	2	no	—
71	#1202	—	2	2	no	—
72	#1201	—	2	4	no	—
73	#1201	—	4	2	no	—
74	D₅×Dic₃	—	4	2	no	—
75	C₂×C₃×#41	—	5	1	no	—
76	C₂×D₃×D₄	—	5	1	no	—
77	#947	D₅	1	1	yes	characteristic, complemented
78	#946	—	5	1	no	—
79	#939	—	5	2	no	—
80	#937	—	5	2	no	—
81	D₄×Dic₃	—	5	1	no	—
82	#930	—	5	1	no	—
83	#929	—	5	2	no	—
84	C₄×#67	—	5	1	no	—
85	C₂×(C₂×D₃)⋊C₄	—	5	2	no	—
86	C₂³×D₅	—	3	2	no	—
87	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂×C₅×D₄	—	3	1	no	—
89	C₂×C₂²⋊D₅	—	3	2	no	—
90	C₂×C₂²⋊D₅	—	3	2	no	—
91	#713	—	2	4	no	—
92	D₄×D₅	—	6	4	no	—
93	C₂×D₂₀	—	3	1	no	—
94	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
95	C₂×D₂₀	C₂×D₃	1	2	yes	complemented
96	C₂×C₄×D₅	—	3	1	no	—
97	C₂×C₄×D₅	—	3	2	no	—
98	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic, complemented
99	C₂×C₄×D₅	C₂×D₃	1	2	yes	complemented
100	C₅×C₂²⋊C₄	—	3	2	no	—
101	C₅×C₂²⋊C₄	—	3	2	no	—
102	C₄²×C₅	—	3	1	no	—
103	C₂²⋊Dic₅	—	3	2	no	—
104	#685	—	3	2	no	—
105	#685	—	3	1	no	—
106	#685	—	3	2	no	—
107	#685	—	3	1	no	—
108	C₄⋊Dic₅	—	3	1	no	—
109	C₄⋊Dic₅	—	3	1	no	—
110	#683	—	3	2	no	—
111	#546	—	15	1	no	Sylow
112	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×D₁₅	—	2	1	no	—
114	C₂×D₁₅	—	2	1	no	—
115	C₂×D₁₅	#41	1	2	yes	complemented
116	C₂×C₅×D₃	—	2	2	no	—
117	C₂×C₅×D₃	—	2	2	no	—
118	C₂×C₅×D₃	—	2	2	no	—
119	C₂×C₃×D₅	—	2	2	no	—
120	C₂×C₃×D₅	—	2	2	no	—
121	C₂×C₃×D₅	—	2	1	no	—
122	C₂×C₃×D₅	—	2	1	no	—
123	C₂×C₃×D₅	—	2	1	no	—
124	C₂×C₃×D₅	C₂×D₄	1	4	yes	—
125	D₃×D₅	—	4	8	no	—
126	C₃×C₄×C₅	—	2	2	no	—
127	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
128	Dic₁₅	—	2	1	no	—
129	Dic₁₅	—	2	2	no	—
130	C₃×Dic₅	—	2	1	no	—
131	C₅×Dic₃	—	2	1	no	—
132	C₅×Dic₃	—	2	1	no	—
133	C₂³×D₃	—	5	2	no	—
134	C₃×#41	—	10	4	no	—
135	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
136	C₂×C₃×D₄	—	5	1	no	—
137	C₂×C₃×D₄	—	5	2	no	—
138	C₂²×C₃×C₄	—	5	2	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂×#67	—	5	2	no	—
141	C₂×#67	—	5	2	no	—
142	C₂×#67	—	5	1	no	—
143	C₂²×Dic₃	—	5	2	no	—
144	D₃×D₄	—	10	4	no	—
145	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic
146	C₂×C₄×D₃	—	5	1	no	—
147	#217	—	5	1	no	—
148	#217	—	5	2	no	—
149	(C₂×D₃)⋊C₄	—	5	2	no	—
150	(C₂×D₃)⋊C₄	—	5	2	no	—
151	(C₂×D₃)⋊C₄	—	5	1	no	—
152	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	complemented
153	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	complemented
154	C₄⋊Dic₃	—	5	1	no	—
155	Dic₃⋊C₄	—	5	2	no	—
156	Dic₃⋊C₄	—	5	1	no	—
157	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
158	C₂³×C₅	—	3	2	no	—
159	C₂²×D₅	—	3	1	no	—
160	C₂²×D₅	—	6	4	no	—
161	C₂²×D₅	—	6	4	no	—
162	C₂²×D₅	—	12	2	no	—
163	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
164	C₂²×D₅	C₂²×D₃	1	2	yes	—
165	C₅×Q₈	—	2	2	no	—
166	C₅×D₄	—	6	2	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	1	no	—
169	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
170	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
171	C₂²⋊D₅	—	6	4	no	—
172	C₂²⋊D₅	—	12	2	no	—
173	C₂×Dic₅	—	3	1	no	—
174	C₂×Dic₅	—	3	2	no	—
175	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
176	D₂₀	—	2	4	no	—
177	D₂₀	—	6	2	no	—
178	D₂₀	#67	1	4	yes	complemented
179	C₄×D₅	—	2	4	no	—
180	C₄×D₅	—	2	2	no	—
181	C₄×D₅	—	6	2	no	—
182	C₄×D₅	—	12	2	no	—
183	C₂×#41	—	5	1	no	—
184	C₂²×D₄	—	15	1	no	—
185	#124	—	3	1	no	—
186	#123	—	15	2	no	—
187	C₂²⋊Q₈	—	15	1	no	—
188	#121	—	15	2	no	—
189	#121	—	15	1	no	—
190	#120	—	15	2	no	—
191	C₄×D₄	—	15	1	no	—
192	C₄×D₄	—	15	1	no	—
193	C₂×C₂²⋊C₄	—	15	2	no	—
194	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
196	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
197	D₁₅	—	2	2	no	—
198	C₃×D₅	—	2	4	no	—
199	C₃×D₅	—	4	1	no	—
200	C₅×D₃	—	4	2	no	—
201	C₂³×C₃	—	5	2	no	—
202	C₂³×C₃	—	5	1	no	—
203	C₂²×D₃	—	5	1	no	—
204	C₂²×D₃	—	10	4	no	—
205	C₂²×D₃	—	10	4	no	—
206	C₂²×D₃	—	20	2	no	—
207	C₂²×D₃	C₂²×D₅	1	2	yes	—
208	C₃×Q₈	—	2	2	no	—
209	C₃×D₄	—	5	4	no	—
210	C₃×D₄	—	10	4	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	—	10	2	no	—
214	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
215	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
216	#67	—	10	2	no	—
217	#67	—	10	4	no	—
218	#67	—	20	2	no	—
219	C₂×Dic₃	—	5	1	no	—
220	C₂×Dic₃	—	5	2	no	—
221	C₂×Dic₃	—	20	2	no	—
222	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
223	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
224	D₁₂	—	2	2	no	—
225	C₄×D₃	—	10	2	no	—
226	C₂²×C₅	—	6	2	no	—
227	C₂²×C₅	—	6	2	no	—
228	C₂²×C₅	—	6	2	no	—
229	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
230	C₂×D₅	—	2	2	no	—
231	C₂×D₅	—	2	2	no	—
232	C₂×D₅	—	2	1	no	—
233	C₂×D₅	—	2	1	no	—
234	C₂×D₅	—	2	1	no	—
235	C₂×D₅	—	3	2	no	—
236	C₂×D₅	—	6	1	no	—
237	C₂×D₅	—	6	1	no	—
238	C₂×D₅	—	12	8	no	—
239	C₂×D₅	C₂×#67	1	4	yes	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	6	1	no	—
242	C₄×C₅	—	6	1	no	—
243	C₄×C₅	C₂×#67	1	2	yes	—
244	Dic₅	—	2	1	no	—
245	Dic₅	—	6	2	no	—
246	Dic₅	—	6	1	no	—
247	C₂⁴	—	15	2	no	—
248	#41	—	10	4	no	—
249	C₂×Q₈	D₃×D₅	1	1	yes	characteristic, complemented
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	5	1	no	—
252	C₂×D₄	—	5	2	no	—
253	C₂×D₄	—	15	1	no	—
254	C₂×D₄	—	15	2	no	—
255	C₂×D₄	—	15	2	no	—
256	C₂×D₄	—	30	4	no	—
257	C₂²×C₄	—	5	2	no	—
258	C₂²×C₄	—	5	1	no	—
259	C₂²×C₄	—	15	2	no	—
260	C₂²×C₄	—	15	1	no	—
261	C₄⋊C₄	—	15	1	no	—
262	C₄⋊C₄	—	15	1	no	—
263	C₄⋊C₄	—	15	2	no	—
264	C₂²⋊C₄	—	3	2	no	—
265	C₂²⋊C₄	—	3	2	no	—
266	C₂²⋊C₄	—	15	1	no	—
267	C₂²⋊C₄	—	15	2	no	—
268	C₂²⋊C₄	—	15	1	no	—
269	C₂²⋊C₄	—	15	2	no	—
270	C₂²⋊C₄	—	15	2	no	—
271	C₄²	—	3	1	no	—
272	C₃×C₅	#546	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	5	4	no	—
274	C₂²×C₃	—	10	2	no	—
275	C₂²×C₃	—	10	2	no	—
276	C₂²×C₃	—	10	1	no	—
277	C₂²×C₃	—	10	1	no	—
278	C₂²×C₃	—	10	1	no	—
279	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
280	C₂×D₃	—	2	2	no	—
281	C₂×D₃	—	2	2	no	—
282	C₂×D₃	—	2	2	no	—
283	C₂×D₃	—	5	2	no	—
284	C₂×D₃	—	10	1	no	—
285	C₂×D₃	—	10	1	no	—
286	C₂×D₃	—	20	8	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	10	1	no	—
289	C₃×C₄	D₄×D₅	1	2	yes	—
290	Dic₃	—	2	1	no	—
291	Dic₃	—	2	1	no	—
292	Dic₃	—	10	1	no	—
293	Dic₃	—	10	2	no	—
294	C₂×C₅	—	12	2	no	—
295	C₂×C₅	D₃×#41	1	1	yes	characteristic
296	C₂×C₅	#1009	1	1	yes	characteristic
297	C₂×C₅	C₂²×#67	1	1	yes	characteristic
298	D₅	—	2	4	no	—
299	D₅	—	4	1	no	—
300	D₅	—	6	2	no	—
301	C₂³	—	3	2	no	—
302	C₂³	—	5	2	no	—
303	C₂³	—	5	1	no	—
304	C₂³	—	15	1	no	—
305	C₂³	—	30	4	no	—
306	C₂³	—	30	4	no	—
307	C₂³	—	60	2	no	—
308	Q₈	—	2	2	no	—
309	D₄	—	5	4	no	—
310	D₄	—	6	2	no	—
311	D₄	—	10	4	no	—
312	D₄	—	30	2	no	—
313	D₄	—	30	4	no	—
314	D₄	—	60	2	no	—
315	C₂×C₄	—	3	1	no	—
316	C₂×C₄	—	3	1	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	10	4	no	—
319	C₂×C₄	—	10	2	no	—
320	C₂×C₄	—	15	2	no	—
321	C₂×C₄	—	15	1	no	—
322	C₂×C₄	—	30	2	no	—
323	C₂×C₄	—	60	2	no	—
324	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
325	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
326	C₂×C₃	—	10	4	no	—
327	C₂×C₃	—	20	1	no	—
328	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
329	C₂×C₃	D₅×#41	1	1	yes	characteristic
330	C₂×C₃	#4149	1	1	yes	characteristic
331	D₃	—	4	2	no	—
332	D₃	—	10	2	no	—
333	C₅	#5829	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	5	4	no	—
335	C₂²	—	6	2	no	—
336	C₂²	—	6	2	no	—
337	C₂²	—	6	2	no	—
338	C₂²	—	10	2	no	—
339	C₂²	—	10	2	no	—
340	C₂²	—	10	1	no	—
341	C₂²	—	10	1	no	—
342	C₂²	—	10	1	no	—
343	C₂²	—	15	2	no	—
344	C₂²	—	30	1	no	—
345	C₂²	—	30	1	no	—
346	C₂²	—	60	8	no	—
347	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	6	1	no	—
350	C₄	—	6	1	no	—
351	C₄	—	10	1	no	—
352	C₄	—	30	2	no	—
353	C₄	—	30	1	no	—
354	C₄	D₅×#67	1	2	yes	—
355	C₃	#66010	1	1	yes	characteristic, complemented, Sylow
356	C₂	—	10	4	no	—
357	C₂	—	12	2	no	—
358	C₂	—	20	1	no	—
359	C₂	—	30	2	no	—
360	C₂	#92034	1	1	yes	characteristic
361	C₂	#92035	1	1	yes	characteristic
362	C₂	C₂×D₅×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂×C₃×C₅×Q₈	C₂²
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	#546
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#713	C₂	semidirect
2. C₂×#6777	C₂	semidirect
3. #91835	C₂	semidirect
4. C₅×#947	C₂	semidirect
5. #91474	C₂	semidirect
6. D₅×(C₂×D₃)⋊C₄	C₂	semidirect
7. #91449	C₂	semidirect
8. #91447	C₂	semidirect
9. C₄×#1202	C₂	semidirect
10. Dic₃×D₂₀	C₂	semidirect
11. #91408	C₂	semidirect
14. C₂×C₃×C₅×Q₈	C₂²	semidirect
19. C₂×C₃×D₂₀	C₂²	semidirect
20. C₂×C₃×C₄×D₅	C₂²	semidirect
21. C₂×C₃×C₄×D₅	C₂²	semidirect
24. C₂×#1201	C₂²	semidirect
25. C₂×D₅×Dic₃	C₂²	semidirect
28. C₄⋊Dic₁₅	C₂²	semidirect
29. #6712	C₂²	semidirect
30. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
31. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
32. C₄×C₅×Dic₃	C₂²	semidirect
33. Dic₃⋊Dic₅	C₂²	semidirect
35. (C₂×D₃)⋊Dic₅	C₂²	semidirect
36. #6665	C₂²	semidirect
37. #6665	C₂²	semidirect
39. C₂×#713	D₃	semidirect
68. C₃×D₂₀	D₄	semidirect
77. #947	D₅	semidirect
87. C₂×C₅×Q₈	C₂×D₃	semidirect
95. C₂×D₂₀	C₂×D₃	semidirect
98. C₂×C₄×D₅	C₂×D₃	semidirect
99. C₂×C₄×D₅	C₂×D₃	semidirect
115. C₂×D₁₅	#41	semidirect
135. C₂×C₃×Q₈	C₂×D₅	semidirect
152. (C₂×D₃)⋊C₄	C₂×D₅	semidirect
153. (C₂×D₃)⋊C₄	C₂×D₅	semidirect
157. C₄×Dic₃	C₂×D₅	semidirect
178. D₂₀	#67	semidirect
249. C₂×Q₈	D₃×D₅	semidirect
272. C₃×C₅	#546	semidirect
333. C₅	#5829	semidirect
355. C₃	#66010	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#713	2
C₂×#6777	2
#91835	2
C₅×#947	2
#91474	2
D₅×(C₂×D₃)⋊C₄	2
#91449	2
#91447	2
C₄×#1202	2
Dic₃×D₂₀	2
#91408	2
#66010	3
#5829	5