Subgroups — #14908701

Order 2816.

610 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×D₈₈	C₂	1	1	yes	maximal, characteristic
2	#13070811	C₂	1	2	yes	maximal, complemented
3	#13070760	C₂	1	2	yes	maximal, complemented
4	#13070658	C₂	1	2	yes	maximal, complemented
5	#13070544	C₂	1	2	yes	maximal
6	C₄×#66892	C₂	1	4	yes	maximal, complemented
7	#13070415	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#533	—	2	1	no	—
10	C₂²×D₈₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄₄	—	2	1	no	—
12	C₁₁×#500	C₂²	1	1	yes	characteristic, complemented
13	C₁₁×#494	C₂²	1	1	yes	characteristic, complemented
14	C₁₁×#487	—	2	1	no	—
15	C₁₁×#467	—	2	1	no	—
16	C₁₁×#467	C₂²	1	2	yes	—
17	C₄×C₁₁×Q₁₆	—	2	1	no	—
18	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	complemented
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66892	C₂²	1	4	yes	—
21	C₂×#66885	—	2	1	no	—
22	C₂×#66882	C₂²	1	1	yes	characteristic
23	C₄×D₈₈	—	2	1	no	—
24	C₄×D₈₈	—	2	1	no	—
25	C₄×D₈₈	C₂²	1	2	yes	—
26	C₄×D₈₈	C₂²	1	2	yes	—
27	#10622202	C₂²	1	4	yes	complemented
28	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
29	#10622129	C₂²	1	4	yes	—
30	#10622128	C₂²	1	4	yes	complemented
31	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	complemented
32	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	complemented
33	C₂²×D₄₄	—	2	1	no	—
34	C₂²×C₄×D₁₁	—	4	1	no	—
35	C₂×C₁₁×Q₁₆	—	2	1	no	—
36	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₁₁×C₄⋊Q₈	—	2	1	no	—
40	C₁₁×C₄⋊Q₈	—	2	1	no	—
41	C₁₁×#125	—	4	1	no	—
42	C₁₁×C₂²⋊Q₈	—	2	2	no	—
43	C₁₁×C₂²⋊Q₈	—	4	1	no	—
44	C₄×Q₈×C₁₁	—	2	2	no	—
45	C₄×Q₈×C₁₁	—	4	1	no	—
46	C₁₁×#117	—	4	1	no	—
47	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₁₁)⋊C₄	—	4	1	no	—
49	C₂×C₄⋊Dic₁₁	—	2	1	no	—
50	C₂×D₈₈	—	2	1	no	—
51	C₂×D₈₈	—	2	4	no	—
52	C₂×D₈₈	C₂³	1	2	yes	—
53	C₄×D₄₄	—	2	2	no	—
54	C₄×D₄₄	—	2	2	no	—
55	C₄×D₄₄	—	4	1	no	—
56	C₄×D₄₄	—	4	1	no	—
57	C₁₁×#107	—	2	1	no	—
58	C₁₁×#107	C₂³	1	2	yes	—
59	C₁₁×#103	—	2	2	no	—
60	C₁₁×#103	—	2	2	no	—
61	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	#66892	—	2	8	no	—
66	#66885	—	2	2	no	—
67	#66885	—	2	2	no	—
68	#66882	—	2	1	no	—
69	#66882	C₂³	1	2	yes	—
70	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
71	#33661	—	11	1	no	maximal, Sylow
72	C₂³×D₁₁	—	4	1	no	—
73	C₂×Q₈×C₁₁	—	2	2	no	—
74	C₂×Q₈×C₁₁	—	4	1	no	—
75	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₁	—	4	1	no	—
79	C₂×D₄₄	—	2	4	no	—
80	C₂×D₄₄	—	2	2	no	—
81	C₂×D₄₄	—	4	2	no	—
82	C₂×D₄₄	—	4	1	no	—
83	C₂×C₄×D₁₁	—	4	2	no	—
84	C₂×C₄×D₁₁	—	4	2	no	—
85	C₂×C₄×D₁₁	—	8	4	no	—
86	C₁₁×Q₁₆	—	2	4	no	—
87	C₁₁×Q₁₆	—	4	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	4	1	no	—
90	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₁₁×C₄⋊C₄	—	2	2	no	—
95	C₁₁×C₄⋊C₄	—	4	1	no	—
96	C₁₁×C₄⋊C₄	—	4	1	no	—
97	C₁₁×C₄⋊C₄	—	4	2	no	—
98	C₁₁×C₄⋊C₄	—	4	2	no	—
99	C₁₁×C₄⋊C₄	—	4	1	no	—
100	C₁₁×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₁	—	4	2	no	—
102	C₄²×C₁₁	D₈	1	1	yes	characteristic
103	C₄²×C₁₁	D₈	1	1	yes	characteristic
104	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₁₁)⋊C₄	—	4	2	no	—
107	(C₂×D₁₁)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₁₁	—	2	2	no	—
109	C₄⋊Dic₁₁	—	4	1	no	—
110	D₈₈	—	2	4	no	—
111	D₈₈	—	2	2	no	—
112	D₈₈	#41	1	4	yes	—
113	C₁₁⋊C₁₆	—	2	4	no	—
114	#3170	D₁₁	1	1	yes	characteristic, complemented
115	C₂×C₄×D₈	—	11	1	no	—
116	#2235	—	11	2	no	—
117	#2226	—	11	2	no	—
118	#2208	—	11	2	no	—
119	#2187	—	11	2	no	—
120	C₄×SD₃₂	—	11	4	no	—
121	#2162	—	11	1	no	—
122	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₁₁	—	4	4	no	—
124	C₂²×D₁₁	—	4	2	no	—
125	C₂²×D₁₁	—	8	2	no	—
126	C₂²×D₁₁	—	8	2	no	—
127	Q₈×C₁₁	—	4	2	no	—
128	Q₈×C₁₁	—	4	2	no	—
129	Q₈×C₁₁	—	8	2	no	—
130	C₂×C₄×C₁₁	—	2	1	no	—
131	C₂×C₄×C₁₁	—	2	1	no	—
132	C₂×C₄×C₁₁	—	4	1	no	—
133	C₂×C₄×C₁₁	—	4	2	no	—
134	C₂×C₄×C₁₁	—	4	2	no	—
135	C₂×C₄×C₁₁	SD₃₂	1	4	yes	complemented
136	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×Dic₁₁	—	4	2	no	—
143	C₂×Dic₁₁	—	8	2	no	—
144	D₄₄	—	2	4	no	—
145	D₄₄	—	4	4	no	—
146	D₄₄	—	4	2	no	—
147	C₄×D₁₁	—	8	4	no	—
148	C₄×D₁₁	—	8	4	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	C₂×#41	1	2	yes	—
153	C₂²×D₈	—	11	1	no	—
154	#533	—	2	1	no	—
155	C₂×C₄×D₄	—	22	1	no	—
156	C₂×SD₃₂	—	11	4	no	—
157	#500	C₂×D₁₁	1	1	yes	characteristic, complemented
158	#494	C₂×D₁₁	1	1	yes	characteristic, complemented
159	#487	—	2	1	no	—
160	#467	—	2	1	no	—
161	#467	C₂×D₁₁	1	2	yes	—
162	C₄×Q₁₆	—	2	1	no	—
163	C₄×Q₁₆	C₂×D₁₁	1	2	yes	complemented
164	C₄×D₈	—	11	2	no	—
165	C₄×D₈	—	11	2	no	—
166	C₄×D₈	—	22	1	no	—
167	C₄×D₈	—	22	1	no	—
168	C₂×#107	—	11	1	no	—
169	C₂×#102	—	22	1	no	—
170	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
171	#367	—	11	4	no	—
172	C₄⋊C₁₆	—	11	2	no	—
173	#358	—	11	4	no	—
174	#357	—	11	4	no	—
175	C₂²⋊C₁₆	—	11	2	no	—
176	C₄×C₁₆	—	11	2	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	C₂×SD₃₂	1	2	yes	—
180	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₁₁	—	4	4	no	—
182	C₂×D₁₁	—	8	8	no	—
183	C₂×D₁₁	—	8	2	no	—
184	C₂×D₁₁	—	8	2	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	8	2	no	—
189	C₄×C₁₁	—	8	2	no	—
190	C₄×C₁₁	C₂×SD₃₂	1	2	yes	—
191	C₄×C₁₁	C₂×SD₃₂	1	2	yes	—
192	C₄×C₁₁	#525	1	1	yes	characteristic
193	C₄×C₁₁	#575	1	1	yes	characteristic
194	Dic₁₁	—	8	2	no	—
195	C₂²×D₄	—	22	1	no	—
196	C₂³×C₄	—	44	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
199	C₂×D₈	—	11	2	no	—
200	C₂×D₈	—	22	4	no	—
201	C₂×D₈	—	22	1	no	—
202	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	#125	—	4	1	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×D₄	—	22	2	no	—
212	C₄×D₄	—	22	2	no	—
213	C₄×D₄	—	44	1	no	—
214	C₄×D₄	—	44	1	no	—
215	#117	—	4	1	no	—
216	C₂×C₄⋊C₄	—	22	1	no	—
217	C₂×C₂²⋊C₄	—	44	1	no	—
218	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
219	SD₃₂	—	22	8	no	—
220	C₂×C₁₆	—	11	4	no	—
221	#107	—	2	1	no	—
222	#107	—	11	2	no	—
223	#107	—	22	1	no	—
224	#107	C₂²×D₁₁	1	2	yes	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	22	2	no	—
228	#102	—	22	2	no	—
229	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
230	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
233	C₂×C₁₁	—	2	2	no	—
234	C₂×C₁₁	#3100	1	1	yes	characteristic
235	C₂×C₁₁	C₂²×SD₃₂	1	1	yes	characteristic
236	C₂×C₁₁	#3414	1	1	yes	characteristic
237	D₁₁	—	8	4	no	—
238	C₂⁴	—	44	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×D₄	—	22	4	no	—
242	C₂×D₄	—	22	2	no	—
243	C₂×D₄	—	44	2	no	—
244	C₂×D₄	—	44	1	no	—
245	C₂²×C₄	—	44	1	no	—
246	C₂²×C₄	—	44	2	no	—
247	C₂²×C₄	—	44	2	no	—
248	C₂²×C₄	—	88	4	no	—
249	C₂²×C₄	#4323	1	1	yes	characteristic
250	C₂²×C₄	#4323	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	11	4	no	—
255	D₈	—	22	4	no	—
256	D₈	—	22	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	22	2	no	—
270	C₄⋊C₄	—	44	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	44	2	no	—
273	C₂²⋊C₄	—	44	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	#4323	1	1	yes	characteristic
276	C₄²	#4323	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
279	C₁₆	—	22	4	no	—
280	C₁₁	#33661	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	44	4	no	—
282	C₂³	—	44	2	no	—
283	C₂³	—	88	2	no	—
284	C₂³	—	88	2	no	—
285	C₂³	C₂×#4323	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	22	4	no	—
290	D₄	—	44	4	no	—
291	D₄	—	44	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	44	2	no	—
298	C₂×C₄	—	88	2	no	—
299	C₂×C₄	—	88	4	no	—
300	C₂×C₄	—	88	4	no	—
301	C₂×C₄	#66892	1	4	yes	complemented
302	C₂×C₄	C₂×#4323	1	1	yes	characteristic
303	C₂×C₄	C₂×#4323	1	1	yes	characteristic
304	C₂×C₄	C₂×#4323	1	1	yes	characteristic
305	C₂×C₄	C₂×#4323	1	1	yes	characteristic
306	C₂×C₄	C₂×#4323	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₁₁	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	44	4	no	—
315	C₂²	—	88	8	no	—
316	C₂²	—	88	2	no	—
317	C₂²	—	88	2	no	—
318	C₂²	C₂×#66892	1	2	yes	—
319	C₂²	C₂²×#4323	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	88	2	no	—
326	C₄	C₂×#66892	1	2	yes	—
327	C₄	C₂×#66892	1	2	yes	—
328	C₄	#10623353	1	1	yes	characteristic
329	C₄	#10623359	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	88	4	no	—
332	C₂	#13078612	1	1	yes	characteristic
333	C₂	C₂²×#66892	1	1	yes	characteristic
334	C₂	#13081750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3170	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂²×D₈₈	C₂²
hypercenter	#3170	D₁₁
hypocenter	C₁₁	#33661
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13070811	C₂	semidirect
3. #13070760	C₂	semidirect
4. #13070658	C₂	semidirect
6. C₄×#66892	C₂	semidirect
7. #13070415	C₂	semidirect
8. C₁₁×#3170	C₂	semidirect
12. C₁₁×#500	C₂²	semidirect
13. C₁₁×#494	C₂²	semidirect
18. C₄×C₁₁×Q₁₆	C₂²	semidirect
27. #10622202	C₂²	semidirect
30. #10622128	C₂²	semidirect
31. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
32. C₄×C₁₁⋊C₁₆	C₂²	semidirect
114. #3170	D₁₁	semidirect
135. C₂×C₄×C₁₁	SD₃₂	semidirect
157. #500	C₂×D₁₁	semidirect
158. #494	C₂×D₁₁	semidirect
163. C₄×Q₁₆	C₂×D₁₁	semidirect
280. C₁₁	#33661	semidirect
301. C₂×C₄	#66892	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈₈	2
#13070811	2
#13070760	2
#13070658	2
#13070544	2
C₄×#66892	2
#13070415	2
C₁₁×#3170	2
#33661	11