Subgroups — #14907474

Order 2816.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077591	C₂	1	1	yes	maximal, characteristic
2	#13070522	C₂	1	2	yes	maximal, complemented
3	#13070443	C₂	1	2	yes	maximal, complemented
4	#13070438	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₁₁)⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#13070422	C₂	1	2	yes	maximal, complemented
7	#13070373	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	#10623010	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	#10622724	C₄	1	2	yes	—
13	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
14	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#10622539	C₄	1	1	yes	characteristic
17	C₂×#66862	C₂²	1	2	yes	—
18	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
19	#10622303	C₄	1	2	yes	—
20	#10622302	C₂²	1	2	yes	—
21	#10622301	C₄	1	1	yes	characteristic
22	C₈×D₄₄	C₂²	1	1	yes	characteristic
23	#10622292	C₄	1	1	yes	characteristic
24	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
25	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₁₁×#346	C₂²	1	2	yes	—
29	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
31	#10622140	C₂²	1	2	yes	—
32	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
33	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
34	C₁₆×Dic₁₁	C₂²	1	2	yes	—
35	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
36	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
40	#66983	C₈	1	1	yes	characteristic
41	#66982	C₈	1	2	yes	—
42	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
43	#66977	C₈	1	1	yes	characteristic
44	C₂×#4313	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₁	C₂³	1	2	yes	—
46	#66929	C₈	1	2	yes	—
47	#66928	C₈	1	1	yes	characteristic
48	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
49	#66925	C₂×C₄	1	2	yes	—
50	#66923	C₈	1	1	yes	characteristic
51	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
53	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
59	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
60	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
61	#66879	C₂×C₄	1	2	yes	—
62	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₁	C₂³	1	2	yes	—
65	C₂×C₁₁⋊C₁₆	—	4	1	no	—
66	C₂×C₁₁⋊C₁₆	—	4	1	no	—
67	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
68	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
69	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	#66867	C₂×C₄	1	2	yes	—
72	#66862	—	4	2	no	—
73	C₁₆×D₁₁	—	4	2	no	—
74	#33506	—	11	1	no	maximal, Sylow
75	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
79	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₁₁	—	2	1	no	—
83	C₂×C₈×C₁₁	—	2	1	no	—
84	C₂×C₈×C₁₁	—	4	1	no	—
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
89	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
94	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
95	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
96	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
97	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
99	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
100	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
101	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
102	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
103	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
105	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
107	#4313	—	4	2	no	—
108	C₈×D₁₁	—	2	2	no	—
109	C₈×D₁₁	—	2	2	no	—
110	C₁₁×C₁₆	—	2	2	no	—
111	C₁₁×C₁₆	—	2	2	no	—
112	C₁₁⋊C₁₆	—	2	2	no	—
113	C₁₁⋊C₁₆	—	2	2	no	—
114	#2966	—	11	1	no	—
115	#2183	—	11	2	no	—
116	#2168	—	11	2	no	—
117	#2168	—	11	2	no	—
118	D₄×C₁₆	—	11	2	no	—
119	#2164	—	11	2	no	—
120	#2163	D₁₁	1	1	yes	characteristic, complemented
121	#2151	—	11	2	no	—
122	C₂×C₄⋊C₁₆	—	11	1	no	—
123	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
124	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	4	1	no	—
129	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
135	C₂²⋊D₁₁	—	2	4	no	—
136	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
137	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
138	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
139	D₄₄	—	4	1	no	—
140	C₄×D₁₁	—	2	2	no	—
141	C₄×D₁₁	—	2	2	no	—
142	Dic₂₂	—	4	1	no	—
143	C₈×C₁₁	—	2	1	no	—
144	C₈×C₁₁	—	2	1	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₁₁⋊C₈	—	2	2	no	—
149	C₁₁⋊C₈	—	2	2	no	—
150	#529	—	11	1	no	—
151	C₂×M₃₂	—	11	2	no	—
152	C₂²×C₁₆	—	11	1	no	—
153	C₂²×C₁₆	—	11	1	no	—
154	C₂²×C₁₆	—	11	2	no	—
155	#446	—	11	1	no	—
156	C₈×Q₈	—	11	1	no	—
157	#436	—	11	1	no	—
158	#435	—	11	2	no	—
159	C₈×D₄	—	11	2	no	—
160	C₈×D₄	—	11	1	no	—
161	#433	—	11	2	no	—
162	#432	—	11	1	no	—
163	#405	—	11	2	no	—
164	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
165	C₄⋊C₁₆	—	11	2	no	—
166	C₄⋊C₁₆	—	11	2	no	—
167	C₄⋊C₁₆	—	11	2	no	—
168	C₄⋊C₁₆	—	11	2	no	—
169	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
170	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
171	C₂²⋊C₁₆	—	11	2	no	—
172	C₂²⋊C₁₆	—	11	2	no	—
173	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
174	#346	—	11	2	no	—
175	#346	C₂×D₁₁	1	2	yes	—
176	C₄×C₁₆	—	11	2	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	#504	1	2	yes	—
180	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
181	C₂×D₁₁	—	2	2	no	—
182	C₂×D₁₁	—	2	2	no	—
183	C₂×D₁₁	—	2	2	no	—
184	C₄×C₁₁	—	2	1	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	#520	1	1	yes	characteristic
190	C₄×C₁₁	#568	1	1	yes	characteristic
191	Dic₁₁	—	2	2	no	—
192	Dic₁₁	—	2	2	no	—
193	Dic₁₁	—	2	2	no	—
194	C₂×M₁₆	—	11	2	no	—
195	C₂²×C₈	—	11	2	no	—
196	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
198	#126	—	11	2	no	—
199	#125	—	11	1	no	—
200	#124	—	11	1	no	—
201	#123	—	11	2	no	—
202	C₂²⋊Q₈	—	11	1	no	—
203	#121	—	11	1	no	—
204	C₄×Q₈	—	11	1	no	—
205	C₄×D₄	—	11	2	no	—
206	C₄×D₄	—	11	1	no	—
207	#117	—	11	2	no	—
208	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
209	M₃₂	—	44	2	no	—
210	C₂×C₁₆	—	11	2	no	—
211	C₂×C₁₆	—	11	2	no	—
212	C₂×C₁₆	—	44	2	no	—
213	C₂×C₁₆	—	44	1	no	—
214	C₂×C₁₆	—	44	1	no	—
215	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
216	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
217	C₄⋊C₈	—	11	2	no	—
218	C₄⋊C₈	—	11	2	no	—
219	C₄⋊C₈	—	11	1	no	—
220	C₄⋊C₈	—	11	1	no	—
221	C₂²⋊C₈	—	11	2	no	—
222	C₂²⋊C₈	—	11	2	no	—
223	C₂²⋊C₈	—	11	2	no	—
224	#97	—	11	2	no	—
225	#97	—	11	2	no	—
226	C₄×C₈	—	11	2	no	—
227	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
228	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
231	C₂×C₁₁	—	2	2	no	—
232	C₂×C₁₁	#2988	1	1	yes	characteristic
233	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
234	C₂×C₁₁	#3407	1	1	yes	characteristic
235	D₁₁	—	4	2	no	—
236	C₂×Q₈	—	11	1	no	—
237	C₂×D₄	—	11	2	no	—
238	C₂×D₄	—	11	1	no	—
239	C₂²×C₄	—	11	2	no	—
240	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
241	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
243	M₁₆	—	44	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	11	2	no	—
248	C₂×C₈	—	11	2	no	—
249	C₂×C₈	—	22	2	no	—
250	C₂×C₈	—	22	2	no	—
251	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
255	C₄⋊C₄	—	11	2	no	—
256	C₄⋊C₄	—	11	2	no	—
257	C₄⋊C₄	—	11	1	no	—
258	C₄⋊C₄	—	11	2	no	—
259	C₄⋊C₄	—	11	1	no	—
260	C₄⋊C₄	—	11	2	no	—
261	C₂²⋊C₄	—	11	2	no	—
262	C₂²⋊C₄	—	11	2	no	—
263	C₂²⋊C₄	—	11	2	no	—
264	C₂²⋊C₄	—	11	2	no	—
265	C₂²⋊C₄	—	11	2	no	—
266	C₄²	—	11	2	no	—
267	C₄²	C₈×D₁₁	1	1	yes	characteristic
268	C₄²	C₈×D₁₁	1	1	yes	characteristic
269	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	22	2	no	—
274	C₁₆	—	22	2	no	—
275	C₁₁	#33506	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	11	2	no	—
277	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
278	Q₈	—	44	1	no	—
279	D₄	—	22	4	no	—
280	D₄	—	44	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	11	2	no	—
286	C₂×C₄	—	11	2	no	—
287	C₂×C₄	—	11	2	no	—
288	C₂×C₄	—	22	2	no	—
289	C₂×C₄	—	22	2	no	—
290	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
291	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	22	2	no	—
300	C₈	—	22	2	no	—
301	C₈	#67042	1	1	yes	characteristic
302	C₈	#67043	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	22	2	no	—
306	C₂²	—	22	2	no	—
307	C₂²	—	22	2	no	—
308	C₂²	#10622520	1	2	yes	—
309	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	22	2	no	—
316	C₄	—	22	2	no	—
317	C₄	—	22	2	no	—
318	C₄	#10623054	1	1	yes	characteristic
319	C₄	#10623276	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	44	2	no	—
322	C₂	#13077724	1	1	yes	characteristic
323	C₂	C₂×#10622520	1	1	yes	characteristic
324	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2163	C₂
prime radical	#66983	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2163	D₁₁
hypocenter	C₁₁	#33506
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #13070522	C₂	semidirect
3. #13070443	C₂	semidirect
4. #13070438	C₂	semidirect
5. (C₂²⋊D₁₁)⋊C₁₆	C₂	semidirect
6. #13070422	C₂	semidirect
7. #13070373	C₂	semidirect
8. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
9. C₁₁×#2163	C₂	semidirect
120. #2163	D₁₁	semidirect
275. C₁₁	#33506	semidirect

Maximal subgroups

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

subgroup	index
#13077591	2
#13070522	2
#13070443	2
#13070438	2
(C₂²⋊D₁₁)⋊C₁₆	2
#13070422	2
#13070373	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₁₁×#2163	2
#33506	11