Subgroups — #13077753

Order 1408.

356 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623059	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622740	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622740	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622724	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622613	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622453	C₂	1	1	yes	maximal, characteristic, complemented
9	#10622452	C₂	1	1	yes	maximal, characteristic, complemented
10	#10622449	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622448	C₂	1	1	yes	maximal, characteristic, complemented
12	#10622447	C₂	1	1	yes	maximal, characteristic, complemented
13	D₁₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#10622444	C₂	1	1	yes	maximal, characteristic, complemented
15	#10622441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	C₁₁×#117	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
21	C₂×#4313	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₁	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₁	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₁	C₂²	1	1	yes	characteristic
25	#66951	C₄	1	1	yes	characteristic
26	#66949	C₄	1	1	yes	characteristic
27	#66947	C₄	1	1	yes	characteristic
28	#66942	C₄	1	1	yes	characteristic
29	#66938	C₄	1	1	yes	characteristic
30	#66937	C₄	1	1	yes	characteristic
31	#66936	C₄	1	1	yes	characteristic
32	#66932	C₄	1	1	yes	characteristic
33	C₄×D₄₄	C₂²	1	1	yes	characteristic
34	#66925	C₂²	1	1	yes	characteristic
35	C₄²×D₁₁	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₂	C₂²	1	1	yes	characteristic
37	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
42	C₂²⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
43	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
47	#66879	C₂²	1	1	yes	characteristic
48	#66879	C₂²	1	1	yes	characteristic
49	#66879	C₂²	1	1	yes	characteristic
50	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₁	C₂²	1	1	yes	characteristic
55	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
57	#66867	C₂²	1	1	yes	characteristic
58	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄₄	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
68	C₁₁×M₁₆	—	4	1	no	—
69	C₂×C₈×C₁₁	—	4	1	no	—
70	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₁₁×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₁₁×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₄²×C₁₁	C₂³	1	1	yes	characteristic
79	C₄²×C₁₁	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
90	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
93	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
94	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
95	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
100	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
104	#4313	—	4	1	no	—
105	C₈×D₁₁	—	4	1	no	—
106	C₈×D₁₁	—	4	1	no	—
107	C₈×D₁₁	—	4	1	no	—
108	#2992	—	11	1	no	maximal, Sylow
109	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	—	2	1	no	—
113	C₂×C₄×C₁₁	—	2	1	no	—
114	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₁	—	2	2	no	—
121	C₂²⋊D₁₁	—	4	1	no	—
122	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
128	D₄₄	—	2	2	no	—
129	C₄×D₁₁	—	2	2	no	—
130	C₄×D₁₁	—	2	1	no	—
131	C₄×D₁₁	—	2	1	no	—
132	C₄×D₁₁	—	2	1	no	—
133	C₄×D₁₁	—	2	1	no	—
134	C₄×D₁₁	—	4	1	no	—
135	Dic₂₂	—	4	1	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	—	2	1	no	—
138	C₈×C₁₁	—	2	1	no	—
139	C₈×C₁₁	—	2	1	no	—
140	C₁₁⋊C₈	—	2	1	no	—
141	C₁₁⋊C₈	—	2	1	no	—
142	C₁₁⋊C₈	—	2	1	no	—
143	C₁₁⋊C₈	—	2	1	no	—
144	#529	—	11	1	no	—
145	#446	—	11	1	no	—
146	#446	—	11	1	no	—
147	#435	—	11	1	no	—
148	#435	—	11	1	no	—
149	#435	—	11	1	no	—
150	#435	—	11	1	no	—
151	C₈×D₄	—	11	1	no	—
152	C₈×D₄	—	11	1	no	—
153	#433	—	11	1	no	—
154	#433	—	11	1	no	—
155	#432	—	11	1	no	—
156	#424	—	11	1	no	—
157	#424	D₁₁	1	1	yes	characteristic, complemented
158	C₂×C₄⋊C₈	—	11	1	no	—
159	C₂²×C₁₁	—	2	1	no	—
160	C₂²×C₁₁	—	2	1	no	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
163	C₂×D₁₁	—	2	1	no	—
164	C₂×D₁₁	—	2	1	no	—
165	C₂×D₁₁	—	2	1	no	—
166	C₂×D₁₁	—	2	1	no	—
167	C₂×D₁₁	—	2	1	no	—
168	C₂×D₁₁	#131	1	2	yes	—
169	C₄×C₁₁	—	2	1	no	—
170	C₄×C₁₁	—	2	1	no	—
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	—	2	1	no	—
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
176	Dic₁₁	—	2	1	no	—
177	Dic₁₁	—	2	1	no	—
178	Dic₁₁	—	2	1	no	—
179	Dic₁₁	—	2	1	no	—
180	Dic₁₁	—	2	1	no	—
181	Dic₁₁	—	2	1	no	—
182	C₂×M₁₆	—	11	1	no	—
183	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
184	C₂²×C₈	—	11	1	no	—
185	C₂²×C₈	—	11	1	no	—
186	C₂²×C₈	—	11	1	no	—
187	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
188	#126	—	11	1	no	—
189	#126	—	11	1	no	—
190	#125	—	11	1	no	—
191	#124	—	11	1	no	—
192	#123	—	11	1	no	—
193	#123	—	11	1	no	—
194	C₂²⋊Q₈	—	11	1	no	—
195	#121	—	11	1	no	—
196	C₄×Q₈	—	11	1	no	—
197	C₄×D₄	—	11	1	no	—
198	C₄×D₄	—	11	1	no	—
199	C₄×D₄	—	11	1	no	—
200	#117	—	11	1	no	—
201	#117	C₂×D₁₁	1	1	yes	characteristic
202	C₂×C₄²	—	11	1	no	—
203	C₄⋊C₈	—	11	1	no	—
204	C₄⋊C₈	—	11	1	no	—
205	C₄⋊C₈	—	11	1	no	—
206	C₄⋊C₈	—	11	1	no	—
207	C₄⋊C₈	—	11	1	no	—
208	C₄⋊C₈	—	11	1	no	—
209	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
211	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
212	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
213	C₂²⋊C₈	—	11	1	no	—
214	C₂²⋊C₈	—	11	1	no	—
215	C₂²⋊C₈	—	11	1	no	—
216	C₂²⋊C₈	—	11	1	no	—
217	C₂²⋊C₈	—	11	1	no	—
218	C₂²⋊C₈	—	11	1	no	—
219	#97	—	11	1	no	—
220	#97	—	11	1	no	—
221	#97	—	11	1	no	—
222	#97	—	11	1	no	—
223	C₄×C₈	—	11	1	no	—
224	C₄×C₈	—	11	1	no	—
225	C₂×C₁₁	—	4	1	no	—
226	C₂×C₁₁	#520	1	1	yes	characteristic
227	C₂×C₁₁	C₂×#131	1	1	yes	characteristic
228	C₂×C₁₁	#568	1	1	yes	characteristic
229	D₁₁	—	2	2	no	—
230	D₁₁	—	4	1	no	—
231	C₂×Q₈	—	11	1	no	—
232	C₂×D₄	—	11	1	no	—
233	C₂×D₄	—	11	1	no	—
234	C₂×D₄	—	11	1	no	—
235	C₂²×C₄	—	11	1	no	—
236	C₂²×C₄	—	11	1	no	—
237	C₂²×C₄	—	11	1	no	—
238	C₂²×C₄	—	11	1	no	—
239	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
240	M₁₆	—	4	1	no	—
241	M₁₆	—	44	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	11	1	no	—
244	C₂×C₈	—	11	1	no	—
245	C₂×C₈	—	11	1	no	—
246	C₂×C₈	—	11	1	no	—
247	C₂×C₈	—	44	1	no	—
248	C₂×C₈	—	44	1	no	—
249	C₂×C₈	—	44	1	no	—
250	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
254	C₄⋊C₄	—	11	1	no	—
255	C₄⋊C₄	—	11	1	no	—
256	C₄⋊C₄	—	11	1	no	—
257	C₄⋊C₄	—	11	1	no	—
258	C₄⋊C₄	—	11	1	no	—
259	C₄⋊C₄	—	11	1	no	—
260	C₄⋊C₄	—	11	1	no	—
261	C₄⋊C₄	—	11	1	no	—
262	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	11	1	no	—
265	C₂²⋊C₄	—	11	1	no	—
266	C₂²⋊C₄	—	11	1	no	—
267	C₂²⋊C₄	—	11	1	no	—
268	C₂²⋊C₄	—	11	1	no	—
269	C₂²⋊C₄	—	11	1	no	—
270	C₂²⋊C₄	—	11	1	no	—
271	C₂²⋊C₄	—	11	1	no	—
272	C₂²⋊C₄	C₄×D₁₁	1	1	yes	characteristic
273	C₂²⋊C₄	C₄×D₁₁	1	1	yes	characteristic
274	C₄²	—	11	1	no	—
275	C₄²	—	11	1	no	—
276	C₄²	—	11	1	no	—
277	C₄²	—	11	1	no	—
278	C₄²	C₂²×D₁₁	1	1	yes	characteristic
279	C₄²	C₂²×D₁₁	1	1	yes	characteristic
280	C₁₁	#2992	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	11	1	no	—
282	C₂³	—	11	1	no	—
283	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
284	Q₈	—	44	1	no	—
285	D₄	—	22	2	no	—
286	D₄	—	22	2	no	—
287	D₄	—	44	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	11	1	no	—
291	C₂×C₄	—	11	1	no	—
292	C₂×C₄	—	11	1	no	—
293	C₂×C₄	—	11	1	no	—
294	C₂×C₄	—	11	1	no	—
295	C₂×C₄	—	11	1	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	1	no	—
298	C₂×C₄	—	22	1	no	—
299	C₂×C₄	—	22	1	no	—
300	C₂×C₄	—	22	1	no	—
301	C₂×C₄	—	44	1	no	—
302	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	22	1	no	—
313	C₈	—	22	1	no	—
314	C₈	—	22	1	no	—
315	C₈	—	22	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	11	2	no	—
320	C₂²	—	22	1	no	—
321	C₂²	—	22	1	no	—
322	C₂²	—	22	1	no	—
323	C₂²	—	22	1	no	—
324	C₂²	—	22	1	no	—
325	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	22	1	no	—
332	C₄	—	22	1	no	—
333	C₄	—	22	1	no	—
334	C₄	—	22	1	no	—
335	C₄	—	22	1	no	—
336	C₄	—	22	1	no	—
337	C₄	#67037	1	1	yes	characteristic
338	C₄	#67040	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	22	2	no	—
341	C₂	—	44	1	no	—
342	C₂	#10623035	1	1	yes	characteristic
343	C₂	D₁₁×#131	1	1	yes	characteristic
344	C₂	#10623281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#424	C₂
prime radical	#66937	C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#424	D₁₁
hypocenter	C₁₁	#2992
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. C₁₁×#424	C₂	semidirect
3. #10622740	C₂	semidirect
4. #10622740	C₂	semidirect
5. #10622724	C₂	semidirect
6. C₈×C₂²⋊D₁₁	C₂	semidirect
7. #10622613	C₂	semidirect
8. #10622453	C₂	semidirect
9. #10622452	C₂	semidirect
10. #10622449	C₂	semidirect
11. #10622448	C₂	semidirect
12. #10622447	C₂	semidirect
13. D₁₁×C₄⋊C₈	C₂	semidirect
14. #10622444	C₂	semidirect
15. #10622441	C₂	semidirect
157. #424	D₁₁	semidirect
280. C₁₁	#2992	semidirect

Maximal subgroups

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

subgroup	index
#10623059	2
C₁₁×#424	2
#10622740	2
#10622724	2
C₈×C₂²⋊D₁₁	2
#10622613	2
#10622453	2
#10622452	2
#10622449	2
#10622448	2
#10622447	2
D₁₁×C₄⋊C₈	2
#10622444	2
#10622441	2
#2992	11