Subgroups — #13077652

Order 1408.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622741	C₂	1	2	yes	maximal, complemented
5	#10622740	C₂	1	2	yes	maximal, complemented
6	#10622343	C₂	1	2	yes	maximal, complemented
7	#10622339	C₂	1	1	yes	maximal, characteristic, complemented
8	(C₂²⋊D₁₁)⋊C₈	C₂	1	2	yes	maximal, complemented
9	D₁₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#10622332	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₁	C₄	1	1	yes	characteristic
14	C₂²×C₄×D₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×M₁₆	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Dic₁₁	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₁	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₁	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₁	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₁	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₁₁)⋊C₄	C₄	1	1	yes	characteristic
26	C₂×Dic₁₁⋊C₄	C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
28	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	complemented
29	C₂×#4313	C₂²	1	2	yes	complemented
30	C₂×C₈×D₁₁	C₂²	1	2	yes	complemented
31	C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₂²⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
35	C₂²⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
36	C₂²⋊(C₁₁⋊C₈)	C₂²	1	2	yes	complemented
37	(C₂×D₁₁)⋊C₈	C₂²	1	2	yes	complemented
38	(C₂×D₁₁)⋊C₈	C₂²	1	2	yes	complemented
39	#66879	C₂²	1	2	yes	complemented
40	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
41	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
42	C₈×Dic₁₁	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×D₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	—	4	2	no	—
46	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₁	—	2	4	no	—
53	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₁	—	4	2	no	—
61	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
65	C₁₁×M₁₆	—	4	2	no	—
66	C₂×C₈×C₁₁	C₂³	1	2	yes	—
67	C₂×C₈×C₁₁	C₂³	1	2	yes	—
68	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
72	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
73	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
74	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
75	(C₂×D₁₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
84	C₂×C₁₁⋊C₈	—	4	2	no	—
85	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
86	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
87	#4313	—	4	2	no	—
88	C₈×D₁₁	—	4	2	no	—
89	#2974	—	11	1	no	maximal, Sylow
90	C₂³×C₁₁	—	2	2	no	—
91	C₂³×C₁₁	—	2	2	no	—
92	C₂³×C₁₁	—	4	1	no	—
93	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₁₁	—	2	2	no	—
97	C₂²×D₁₁	—	2	2	no	—
98	C₂²×D₁₁	—	4	1	no	—
99	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	—	2	2	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	4	1	no	—
105	C₂×C₄×C₁₁	—	4	1	no	—
106	C₂×C₄×C₁₁	—	4	1	no	—
107	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
111	C₂²⋊D₁₁	—	2	2	no	—
112	C₂²⋊D₁₁	—	2	2	no	—
113	C₂²⋊D₁₁	—	2	2	no	—
114	C₂²⋊D₁₁	M₁₆	1	4	yes	complemented
115	C₂×Dic₁₁	—	2	2	no	—
116	C₂×Dic₁₁	—	2	2	no	—
117	C₂×Dic₁₁	—	4	1	no	—
118	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×D₁₁	—	2	2	no	—
125	C₄×D₁₁	—	2	2	no	—
126	C₄×D₁₁	—	4	1	no	—
127	C₄×D₁₁	—	4	1	no	—
128	C₈×C₁₁	—	2	2	no	—
129	C₈×C₁₁	—	2	2	no	—
130	C₁₁⋊C₈	—	2	2	no	—
131	C₁₁⋊C₈	—	2	2	no	—
132	C₂×C₄×D₄	—	11	1	no	—
133	#436	—	11	2	no	—
134	#435	—	11	2	no	—
135	#435	—	11	2	no	—
136	C₈×D₄	—	11	2	no	—
137	#432	—	11	1	no	—
138	#431	—	11	1	no	—
139	#407	—	11	1	no	—
140	#407	D₁₁	1	1	yes	characteristic, complemented
141	C₂×C₂²⋊C₈	—	11	1	no	—
142	C₂×C₂²⋊C₈	—	11	1	no	—
143	C₂²×C₁₁	—	2	1	no	—
144	C₂²×C₁₁	—	2	1	no	—
145	C₂²×C₁₁	—	2	1	no	—
146	C₂²×C₁₁	—	2	1	no	—
147	C₂²×C₁₁	—	2	1	no	—
148	C₂²×C₁₁	—	2	1	no	—
149	C₂²×C₁₁	—	2	1	no	—
150	C₂²×C₁₁	—	4	4	no	—
151	C₂²×C₁₁	C₂×M₁₆	1	2	yes	—
152	C₂²×C₁₁	#131	1	2	yes	—
153	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
154	C₂×D₁₁	—	2	1	no	—
155	C₂×D₁₁	—	2	1	no	—
156	C₂×D₁₁	—	2	1	no	—
157	C₂×D₁₁	—	2	1	no	—
158	C₂×D₁₁	—	2	1	no	—
159	C₂×D₁₁	—	4	4	no	—
160	C₂×D₁₁	C₂×M₁₆	1	2	yes	—
161	C₄×C₁₁	—	2	1	no	—
162	C₄×C₁₁	—	2	1	no	—
163	C₄×C₁₁	—	2	1	no	—
164	C₄×C₁₁	H(5,𝔽₂)	1	2	yes	—
165	Dic₁₁	—	2	1	no	—
166	Dic₁₁	—	2	1	no	—
167	Dic₁₁	—	2	1	no	—
168	Dic₁₁	—	2	1	no	—
169	Dic₁₁	—	2	1	no	—
170	Dic₁₁	C₂×M₁₆	1	2	yes	—
171	C₂²×D₄	—	11	1	no	—
172	C₂³×C₄	—	11	1	no	—
173	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
174	C₂×M₁₆	—	11	2	no	—
175	C₂×M₁₆	C₂×D₁₁	1	2	yes	complemented
176	C₂²×C₈	—	11	2	no	—
177	C₂²×C₈	—	11	2	no	—
178	C₄×D₄	—	11	1	no	—
179	C₄×D₄	—	11	1	no	—
180	C₄×D₄	—	11	1	no	—
181	C₄×D₄	—	11	1	no	—
182	C₄×D₄	—	11	1	no	—
183	C₄×D₄	—	11	1	no	—
184	C₄×D₄	—	11	1	no	—
185	C₄×D₄	—	11	1	no	—
186	C₂×C₄⋊C₄	—	11	1	no	—
187	C₂×C₂²⋊C₄	—	11	1	no	—
188	C₂×C₂²⋊C₄	—	11	1	no	—
189	C₂×C₄²	—	11	1	no	—
190	C₄⋊C₈	—	11	2	no	—
191	C₄⋊C₈	—	11	2	no	—
192	C₂²⋊C₈	—	11	1	no	—
193	C₂²⋊C₈	—	11	1	no	—
194	C₂²⋊C₈	—	11	2	no	—
195	C₂²⋊C₈	—	11	2	no	—
196	C₂²⋊C₈	—	11	2	no	—
197	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
198	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
199	C₂²⋊C₈	C₂×D₁₁	1	2	yes	complemented
200	#97	—	11	2	no	—
201	C₄×C₈	—	11	2	no	—
202	C₂×C₁₁	—	2	2	no	—
203	C₂×C₁₁	—	2	2	no	—
204	C₂×C₁₁	—	4	1	no	—
205	C₂×C₁₁	#518	1	1	yes	characteristic
206	C₂×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
207	C₂×C₁₁	C₂×#131	1	1	yes	characteristic
208	D₁₁	—	2	2	no	—
209	D₁₁	—	4	1	no	—
210	C₂⁴	—	11	1	no	—
211	C₂⁴	C₄×D₁₁	1	1	yes	characteristic
212	C₂×D₄	—	11	1	no	—
213	C₂×D₄	—	11	1	no	—
214	C₂×D₄	—	11	1	no	—
215	C₂×D₄	—	11	1	no	—
216	C₂×D₄	—	22	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	11	1	no	—
219	C₂²×C₄	—	11	1	no	—
220	C₂²×C₄	—	11	1	no	—
221	C₂²×C₄	—	11	1	no	—
222	C₂²×C₄	—	11	1	no	—
223	C₂²×C₄	—	11	1	no	—
224	C₂²×C₄	—	11	1	no	—
225	C₂²×C₄	—	44	2	no	—
226	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
227	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
232	M₁₆	—	4	2	no	—
233	M₁₆	—	44	2	no	—
234	C₂×C₈	—	11	2	no	—
235	C₂×C₈	—	11	2	no	—
236	C₂×C₈	—	44	2	no	—
237	C₂×C₈	—	44	2	no	—
238	C₂×C₈	C₂²×D₁₁	1	2	yes	—
239	C₂×C₈	C₂²×D₁₁	1	2	yes	—
240	C₄⋊C₄	—	11	1	no	—
241	C₄⋊C₄	—	11	1	no	—
242	C₄⋊C₄	—	11	1	no	—
243	C₄⋊C₄	—	11	1	no	—
244	C₂²⋊C₄	—	11	1	no	—
245	C₂²⋊C₄	—	11	1	no	—
246	C₂²⋊C₄	—	11	1	no	—
247	C₂²⋊C₄	—	11	1	no	—
248	C₂²⋊C₄	—	11	1	no	—
249	C₂²⋊C₄	—	11	1	no	—
250	C₂²⋊C₄	—	11	1	no	—
251	C₂²⋊C₄	—	11	1	no	—
252	C₄²	—	11	1	no	—
253	C₄²	—	11	1	no	—
254	C₄²	—	11	1	no	—
255	C₄²	—	11	1	no	—
256	C₁₁	#2974	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	11	1	no	—
261	C₂³	—	11	1	no	—
262	C₂³	—	22	2	no	—
263	C₂³	—	22	2	no	—
264	C₂³	—	44	1	no	—
265	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
266	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
268	D₄	—	11	4	no	—
269	D₄	—	22	2	no	—
270	D₄	—	22	2	no	—
271	D₄	—	22	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	11	1	no	—
279	C₂×C₄	—	11	1	no	—
280	C₂×C₄	—	11	1	no	—
281	C₂×C₄	—	11	1	no	—
282	C₂×C₄	—	11	1	no	—
283	C₂×C₄	—	11	1	no	—
284	C₂×C₄	—	22	2	no	—
285	C₂×C₄	—	22	2	no	—
286	C₂×C₄	—	22	2	no	—
287	C₂×C₄	—	22	2	no	—
288	C₂×C₄	—	44	1	no	—
289	C₂×C₄	—	44	1	no	—
290	C₂×C₄	—	44	1	no	—
291	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	22	2	no	—
298	C₈	—	22	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	11	2	no	—
308	C₂²	—	22	1	no	—
309	C₂²	—	22	1	no	—
310	C₂²	—	22	1	no	—
311	C₂²	—	22	1	no	—
312	C₂²	—	22	1	no	—
313	C₂²	—	44	4	no	—
314	C₂²	M₁₆×D₁₁	1	2	yes	—
315	C₂²	#66960	1	2	yes	—
316	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	11	2	no	—
321	C₄	—	22	1	no	—
322	C₄	—	22	1	no	—
323	C₄	—	22	1	no	—
324	C₄	—	22	1	no	—
325	C₄	—	22	1	no	—
326	C₄	#67037	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	22	2	no	—
331	C₂	—	44	1	no	—
332	C₂	#10623017	1	1	yes	characteristic
333	C₂	C₂×M₁₆×D₁₁	1	1	yes	characteristic
334	C₂	C₂×#66960	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#407	C₂
prime radical	C₂²×C₂²⋊D₁₁	C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#407	D₁₁
hypocenter	C₁₁	#2974
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₁₁×#407	C₂	semidirect
3. C₂×C₂²⋊(C₁₁⋊C₈)	C₂	semidirect
4. #10622741	C₂	semidirect
5. #10622740	C₂	semidirect
6. #10622343	C₂	semidirect
7. #10622339	C₂	semidirect
8. (C₂²⋊D₁₁)⋊C₈	C₂	semidirect
9. D₁₁×C₂²⋊C₈	C₂	semidirect
10. #10622332	C₂	semidirect
11. #10622331	C₂	semidirect
15. C₂×C₁₁×M₁₆	C₂²	semidirect
28. C₂²×C₁₁⋊C₈	C₂²	semidirect
29. C₂×#4313	C₂²	semidirect
30. C₂×C₈×D₁₁	C₂²	semidirect
33. C₁₁×C₂²⋊C₈	C₂²	semidirect
36. C₂²⋊(C₁₁⋊C₈)	C₂²	semidirect
37. (C₂×D₁₁)⋊C₈	C₂²	semidirect
38. (C₂×D₁₁)⋊C₈	C₂²	semidirect
39. #66879	C₂²	semidirect
40. Dic₁₁⋊C₈	C₂²	semidirect
41. Dic₁₁⋊C₈	C₂²	semidirect
42. C₈×Dic₁₁	C₂²	semidirect
114. C₂²⋊D₁₁	M₁₆	semidirect
140. #407	D₁₁	semidirect
175. C₂×M₁₆	C₂×D₁₁	semidirect
199. C₂²⋊C₈	C₂×D₁₁	semidirect
256. C₁₁	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₁₁	2
C₁₁×#407	2
C₂×C₂²⋊(C₁₁⋊C₈)	2
#10622741	2
#10622740	2
#10622343	2
#10622339	2
(C₂²⋊D₁₁)⋊C₈	2
D₁₁×C₂²⋊C₈	2
#10622332	2
#10622331	2
#2974	11