Subgroups — #13077728

Order 1408.

388 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4341	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622741	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622731	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622724	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622624	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622450	C₂	1	1	yes	maximal, characteristic, complemented
9	#10622441	C₂	1	1	yes	maximal, characteristic, complemented
10	#10622343	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622342	C₂	1	1	yes	maximal, characteristic, complemented
12	#10622332	C₂	1	1	yes	maximal, characteristic, complemented
13	#10622331	C₂	1	1	yes	maximal, characteristic, complemented
14	#10622318	C₂	1	1	yes	maximal, characteristic, complemented
15	#10622314	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#4341	C₄	1	1	yes	characteristic
17	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₁	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
25	C₂×#4318	C₂²	1	1	yes	characteristic
26	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×#4313	C₂²	1	1	yes	characteristic
28	C₂×#4313	C₂²	1	1	yes	characteristic
29	#66945	C₄	1	1	yes	characteristic
30	#66940	C₄	1	1	yes	characteristic
31	(C₂²⋊D₁₁)⋊C₄	C₄	1	1	yes	characteristic
32	(C₂²⋊D₁₁)⋊C₄	C₄	1	1	yes	characteristic
33	#66930	C₄	1	1	yes	characteristic
34	#66930	C₄	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₁×#97	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	#66879	C₂²	1	1	yes	characteristic
46	#66879	C₂²	1	1	yes	characteristic
47	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₁₁⋊C₈	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	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₁	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₁	C₂²	1	1	yes	characteristic
57	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
58	#66867	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
68	#4341	—	2	4	no	—
69	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
72	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
73	C₁₁×M₁₆	—	4	1	no	—
74	C₂×C₈×C₁₁	—	4	1	no	—
75	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₁₁×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₁₁×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₁	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	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
88	Dic₁₁⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₁₁⋊C₄	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	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₁	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	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
101	#4318	—	4	1	no	—
102	C₂×C₁₁⋊C₈	—	4	1	no	—
103	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
107	#4313	—	4	1	no	—
108	#4313	—	4	1	no	—
109	#2989	—	11	1	no	maximal, Sylow
110	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	2	1	no	—
118	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₁	—	2	2	no	—
124	C₂²⋊D₁₁	—	2	2	no	—
125	C₂×Dic₁₁	—	2	2	no	—
126	C₂×Dic₁₁	—	4	1	no	—
127	C₂×Dic₁₁	—	4	1	no	—
128	C₂×Dic₁₁	M₁₆	1	4	yes	complemented
129	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
136	C₄×D₁₁	—	2	1	no	—
137	C₄×D₁₁	—	2	1	no	—
138	C₄×D₁₁	—	2	2	no	—
139	C₄×D₁₁	—	4	1	no	—
140	Dic₂₂	—	2	2	no	—
141	C₈×C₁₁	—	2	1	no	—
142	C₈×C₁₁	—	2	1	no	—
143	C₈×C₁₁	—	2	1	no	—
144	C₈×C₁₁	—	2	1	no	—
145	C₁₁⋊C₈	—	2	1	no	—
146	C₁₁⋊C₈	—	2	1	no	—
147	C₁₁⋊C₈	—	2	1	no	—
148	C₁₁⋊C₈	—	2	1	no	—
149	C₄×#41	—	11	1	no	—
150	#446	—	11	1	no	—
151	#446	—	11	1	no	—
152	#436	—	11	1	no	—
153	#436	—	11	1	no	—
154	#435	—	11	1	no	—
155	#435	—	11	1	no	—
156	#435	—	11	1	no	—
157	#435	D₁₁	1	1	yes	characteristic, complemented
158	#432	—	11	1	no	—
159	#432	—	11	1	no	—
160	#431	—	11	1	no	—
161	#423	—	11	1	no	—
162	#423	—	11	1	no	—
163	C₂×C₄⋊C₈	—	11	1	no	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	—	2	1	no	—
167	C₂²×C₁₁	—	2	1	no	—
168	C₂²×C₁₁	—	2	1	no	—
169	C₂²×C₁₁	C₂×M₁₆	1	2	yes	—
170	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×D₁₁	—	2	1	no	—
172	C₂×D₁₁	—	2	1	no	—
173	C₂×D₁₁	—	2	1	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	—	2	1	no	—
177	C₄×C₁₁	—	2	1	no	—
178	C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
180	Dic₁₁	—	2	1	no	—
181	Dic₁₁	—	2	1	no	—
182	Dic₁₁	—	2	1	no	—
183	Dic₁₁	—	2	1	no	—
184	Dic₁₁	—	2	1	no	—
185	Dic₁₁	C₂×M₁₆	1	2	yes	—
186	Dic₁₁	C₂×M₁₆	1	2	yes	—
187	C₂×#41	—	11	1	no	—
188	C₂×M₁₆	—	11	1	no	—
189	C₂×M₁₆	—	11	1	no	—
190	C₂×M₁₆	—	11	1	no	—
191	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
192	C₂²×C₈	—	11	1	no	—
193	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
194	C₄×Q₈	—	11	1	no	—
195	C₄×Q₈	—	11	1	no	—
196	C₄×D₄	—	11	1	no	—
197	C₄×D₄	—	11	1	no	—
198	C₄×D₄	—	11	1	no	—
199	C₄×D₄	—	11	1	no	—
200	C₄×D₄	—	11	1	no	—
201	C₄×D₄	C₂×D₁₁	1	1	yes	characteristic
202	#117	—	11	1	no	—
203	#117	—	11	1	no	—
204	#117	—	11	1	no	—
205	C₂×C₄²	—	11	1	no	—
206	C₂×C₄²	—	11	1	no	—
207	C₂×C₄²	—	11	1	no	—
208	C₄⋊C₈	—	11	1	no	—
209	C₄⋊C₈	—	11	1	no	—
210	C₄⋊C₈	—	11	1	no	—
211	C₄⋊C₈	—	11	1	no	—
212	C₄⋊C₈	—	11	1	no	—
213	C₄⋊C₈	—	11	1	no	—
214	C₄⋊C₈	—	11	1	no	—
215	C₄⋊C₈	—	11	1	no	—
216	C₄⋊C₈	—	11	1	no	—
217	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
218	C₂²⋊C₈	—	11	1	no	—
219	C₂²⋊C₈	—	11	1	no	—
220	C₂²⋊C₈	—	11	1	no	—
221	C₂²⋊C₈	—	11	1	no	—
222	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
223	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
224	#97	—	11	1	no	—
225	#97	—	11	1	no	—
226	#97	—	11	1	no	—
227	#97	C₂×D₁₁	1	1	yes	characteristic
228	C₄×C₈	—	11	1	no	—
229	C₄×C₈	—	11	1	no	—
230	C₂×C₁₁	—	2	2	no	—
231	C₂×C₁₁	—	4	1	no	—
232	C₂×C₁₁	#520	1	1	yes	characteristic
233	C₂×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
234	C₂×C₁₁	#568	1	1	yes	characteristic
235	D₁₁	—	4	1	no	—
236	#41	—	22	4	no	—
237	C₂×Q₈	—	11	1	no	—
238	C₂×D₄	—	11	1	no	—
239	C₂×D₄	—	11	1	no	—
240	C₂×D₄	C₄×D₁₁	1	1	yes	characteristic
241	C₂²×C₄	—	11	1	no	—
242	C₂²×C₄	—	11	1	no	—
243	C₂²×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	C₂²×D₁₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
250	M₁₆	—	4	1	no	—
251	M₁₆	—	44	1	no	—
252	M₁₆	—	44	1	no	—
253	M₁₆	—	44	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	11	1	no	—
256	C₂×C₈	—	11	1	no	—
257	C₂×C₈	—	11	1	no	—
258	C₂×C₈	—	11	1	no	—
259	C₂×C₈	—	44	1	no	—
260	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
264	C₄⋊C₄	—	11	1	no	—
265	C₄⋊C₄	—	11	1	no	—
266	C₄⋊C₄	—	11	1	no	—
267	C₄⋊C₄	—	11	1	no	—
268	C₄⋊C₄	—	11	1	no	—
269	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	11	1	no	—
271	C₂²⋊C₄	—	11	1	no	—
272	C₂²⋊C₄	—	11	1	no	—
273	C₂²⋊C₄	—	11	1	no	—
274	C₂²⋊C₄	C₄×D₁₁	1	1	yes	characteristic
275	C₂²⋊C₄	C₄×D₁₁	1	1	yes	characteristic
276	C₄²	—	11	1	no	—
277	C₄²	—	11	1	no	—
278	C₄²	—	11	1	no	—
279	C₄²	—	11	1	no	—
280	C₄²	—	11	1	no	—
281	C₄²	—	11	1	no	—
282	C₄²	—	11	1	no	—
283	C₄²	—	11	1	no	—
284	C₄²	—	11	1	no	—
285	C₄²	C₂²×D₁₁	1	1	yes	characteristic
286	C₁₁	#2989	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	11	1	no	—
288	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
290	Q₈	—	22	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	22	2	no	—
293	D₄	—	22	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	11	4	no	—
299	C₂×C₄	—	11	1	no	—
300	C₂×C₄	—	11	1	no	—
301	C₂×C₄	—	11	1	no	—
302	C₂×C₄	—	11	1	no	—
303	C₂×C₄	—	11	1	no	—
304	C₂×C₄	—	11	1	no	—
305	C₂×C₄	—	11	1	no	—
306	C₂×C₄	—	22	2	no	—
307	C₂×C₄	—	22	2	no	—
308	C₂×C₄	—	22	1	no	—
309	C₂×C₄	—	22	1	no	—
310	C₂×C₄	—	44	1	no	—
311	C₂×C₄	—	44	1	no	—
312	C₂×C₄	—	44	1	no	—
313	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	22	1	no	—
323	C₈	—	22	1	no	—
324	C₈	—	22	1	no	—
325	C₈	—	22	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	22	1	no	—
332	C₂²	—	22	1	no	—
333	C₂²	—	22	1	no	—
334	C₂²	M₁₆×D₁₁	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	11	2	no	—
341	C₄	—	11	2	no	—
342	C₄	—	22	1	no	—
343	C₄	—	22	1	no	—
344	C₄	—	22	1	no	—
345	C₄	—	22	1	no	—
346	C₄	—	22	1	no	—
347	C₄	#67037	1	1	yes	characteristic
348	C₄	#67043	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	44	1	no	—
352	C₂	#10623077	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₁	1	1	yes	characteristic
354	C₂	#10623281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#435	C₂
prime radical	C₂×#4341	C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#435	D₁₁
hypocenter	C₁₁	#2989
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. C₁₁×#435	C₂	semidirect
3. #10622741	C₂	semidirect
4. #10622731	C₂	semidirect
5. #10622724	C₂	semidirect
6. C₂×Dic₁₁⋊C₈	C₂	semidirect
7. #10622624	C₂	semidirect
8. #10622450	C₂	semidirect
9. #10622441	C₂	semidirect
10. #10622343	C₂	semidirect
11. #10622342	C₂	semidirect
12. #10622332	C₂	semidirect
13. #10622331	C₂	semidirect
14. #10622318	C₂	semidirect
15. #10622314	C₂	semidirect
128. C₂×Dic₁₁	M₁₆	semidirect
157. #435	D₁₁	semidirect
286. C₁₁	#2989	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4341	2
C₁₁×#435	2
#10622741	2
#10622731	2
#10622724	2
C₂×Dic₁₁⋊C₈	2
#10622624	2
#10622450	2
#10622441	2
#10622343	2
#10622342	2
#10622332	2
#10622331	2
#10622318	2
#10622314	2
#2989	11