Subgroups — #14909306

Order 2816.

602 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079195	C₂	1	1	yes	maximal, characteristic, complemented
2	#13070889	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070865	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070841	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070839	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070700	C₂	1	4	yes	maximal, complemented
7	C₄×#66890	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁×#532	—	2	1	no	—
11	#10623008	—	2	1	no	—
12	C₁₁×#500	C₂²	1	1	yes	characteristic
13	C₁₁×#493	C₂²	1	1	yes	characteristic
14	C₁₁×#486	—	2	1	no	—
15	C₁₁×#466	—	2	1	no	—
16	C₁₁×#466	C₂²	1	2	yes	—
17	C₄×C₁₁×D₈	—	2	1	no	—
18	C₄×C₁₁×D₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66890	C₂²	1	4	yes	—
21	#10622728	—	2	1	no	—
22	#10622728	C₂²	1	2	yes	—
23	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
24	#10622308	C₂²	1	1	yes	characteristic
25	#10622307	—	2	1	no	—
26	C₄×D₈₈	—	2	1	no	—
27	C₄×D₈₈	C₂²	1	2	yes	—
28	#10622295	C₂²	1	1	yes	characteristic
29	#10622200	C₂²	1	4	yes	—
30	#10622129	C₂²	1	4	yes	—
31	#10622127	C₂²	1	4	yes	—
32	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₁×#41	—	4	1	no	—
36	C₂×#4339	—	4	1	no	—
37	C₂×C₁₁×D₈	—	2	1	no	—
38	C₂×C₁₁×D₈	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	C₁₁×C₄⋊Q₈	—	2	1	no	—
42	C₁₁×#127	—	2	1	no	—
43	C₁₁×#124	—	4	1	no	—
44	C₁₁×#121	—	2	2	no	—
45	C₁₁×#121	—	4	1	no	—
46	C₄×D₄×C₁₁	—	2	2	no	—
47	C₄×D₄×C₁₁	—	4	1	no	—
48	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
49	#66983	—	2	2	no	—
50	#66983	—	4	1	no	—
51	C₂×D₈₈	—	2	1	no	—
52	C₂×D₈₈	C₂³	1	2	yes	—
53	#66928	—	4	1	no	—
54	#66927	—	2	1	no	—
55	C₄×D₄₄	—	2	2	no	—
56	C₄×D₄₄	—	4	1	no	—
57	#66922	—	2	1	no	—
58	C₁₁×#107	—	2	1	no	—
59	C₁₁×#107	C₂³	1	2	yes	—
60	C₁₁×#102	—	2	2	no	—
61	C₁₁×#102	—	2	2	no	—
62	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
66	#66890	—	2	8	no	—
67	#66885	—	2	2	no	—
68	#66885	—	2	2	no	—
69	#66882	—	2	1	no	—
70	#66882	C₂³	1	2	yes	—
71	C₂×C₁₁⋊C₁₆	—	2	4	no	—
72	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
73	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
74	#33733	—	11	1	no	maximal, Sylow
75	C₁₁×#41	—	8	4	no	—
76	C₂×Q₈×C₁₁	—	4	1	no	—
77	C₂×D₄×C₁₁	—	2	2	no	—
78	C₂×D₄×C₁₁	—	4	2	no	—
79	C₂×D₄×C₁₁	—	4	1	no	—
80	C₂×D₄×C₁₁	—	4	1	no	—
81	C₂²×C₄×C₁₁	—	4	2	no	—
82	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₁	—	4	2	no	—
86	#4339	—	8	4	no	—
87	C₂×D₄₄	—	2	2	no	—
88	C₂×D₄₄	—	4	1	no	—
89	C₂×D₄₄	—	4	1	no	—
90	C₂×C₄×D₁₁	—	4	2	no	—
91	C₂×Dic₂₂	—	4	1	no	—
92	C₁₁×D₈	—	2	4	no	—
93	C₁₁×D₈	—	4	2	no	—
94	C₂×C₈×C₁₁	—	2	2	no	—
95	C₂×C₈×C₁₁	—	2	2	no	—
96	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
100	C₁₁×C₄⋊C₄	—	2	2	no	—
101	C₁₁×C₄⋊C₄	—	4	1	no	—
102	C₁₁×C₂²⋊C₄	—	4	2	no	—
103	C₁₁×C₂²⋊C₄	—	4	2	no	—
104	C₄²×C₁₁	D₈	1	1	yes	characteristic
105	C₄²×C₁₁	D₈	1	1	yes	characteristic
106	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
108	(C₂×D₁₁)⋊C₄	—	4	2	no	—
109	(C₂×D₁₁)⋊C₄	—	4	2	no	—
110	C₄⋊Dic₁₁	—	2	2	no	—
111	C₄⋊Dic₁₁	—	4	1	no	—
112	D₈₈	—	2	4	no	—
113	D₈₈	—	4	2	no	—
114	C₁₁⋊C₁₆	—	2	2	no	—
115	C₁₁⋊C₁₆	#41	1	4	yes	complemented
116	#3169	—	11	1	no	—
117	#3169	D₁₁	1	1	yes	characteristic, complemented
118	#2252	—	11	1	no	—
119	#2246	—	11	1	no	—
120	#2240	—	11	1	no	—
121	#2240	—	11	1	no	—
122	#2215	—	11	4	no	—
123	C₄×D₁₆	—	11	4	no	—
124	C₂×C₄×C₁₆	—	11	1	no	—
125	C₂³×C₁₁	—	4	2	no	—
126	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
127	C₂²×D₁₁	—	4	2	no	—
128	Q₈×C₁₁	—	8	2	no	—
129	D₄×C₁₁	—	4	2	no	—
130	D₄×C₁₁	—	4	2	no	—
131	D₄×C₁₁	—	8	4	no	—
132	D₄×C₁₁	—	8	2	no	—
133	D₄×C₁₁	—	8	2	no	—
134	C₂×C₄×C₁₁	—	2	1	no	—
135	C₂×C₄×C₁₁	—	2	1	no	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	4	2	no	—
138	C₂×C₄×C₁₁	—	8	4	no	—
139	C₂×C₄×C₁₁	D₁₆	1	4	yes	complemented
140	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₁₁	—	8	4	no	—
147	C₂×Dic₁₁	—	4	2	no	—
148	D₄₄	—	4	2	no	—
149	D₄₄	—	4	2	no	—
150	D₄₄	—	8	2	no	—
151	D₄₄	—	8	2	no	—
152	C₄×D₁₁	—	8	4	no	—
153	Dic₂₂	—	8	2	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	—	2	1	no	—
156	C₈×C₁₁	#135	1	2	yes	—
157	C₈×C₁₁	C₂×#41	1	2	yes	—
158	#532	—	2	1	no	—
159	#532	—	22	1	no	—
160	C₂×D₁₆	—	11	4	no	—
161	C₂²×C₁₆	—	11	2	no	—
162	#500	—	11	1	no	—
163	#500	C₂×D₁₁	1	1	yes	characteristic
164	#493	—	11	1	no	—
165	#493	C₂×D₁₁	1	1	yes	characteristic
166	#486	—	2	1	no	—
167	#486	—	22	1	no	—
168	#466	—	2	1	no	—
169	#466	—	11	2	no	—
170	#466	—	22	1	no	—
171	#466	C₂×D₁₁	1	2	yes	—
172	C₄×D₈	—	2	1	no	—
173	C₄×D₈	—	11	2	no	—
174	C₄×D₈	—	22	1	no	—
175	C₄×D₈	C₂×D₁₁	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
177	#366	—	11	4	no	—
178	#357	—	11	4	no	—
179	#357	—	11	4	no	—
180	C₄×C₁₆	—	11	2	no	—
181	C₄×C₁₆	—	11	1	no	—
182	C₄×C₁₆	—	11	1	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	—	2	1	no	—
185	C₂²×C₁₁	—	8	2	no	—
186	C₂²×C₁₁	—	8	2	no	—
187	C₂²×C₁₁	—	8	2	no	—
188	C₂²×C₁₁	C₂×D₁₆	1	2	yes	—
189	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
190	C₂×D₁₁	—	8	2	no	—
191	C₂×D₁₁	—	8	2	no	—
192	C₂×D₁₁	—	8	2	no	—
193	C₄×C₁₁	—	2	1	no	—
194	C₄×C₁₁	—	2	1	no	—
195	C₄×C₁₁	—	8	2	no	—
196	C₄×C₁₁	C₂×D₁₆	1	2	yes	—
197	C₄×C₁₁	C₂×D₁₆	1	2	yes	—
198	C₄×C₁₁	#508	1	2	yes	—
199	C₄×C₁₁	#532	1	1	yes	characteristic
200	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
201	Dic₁₁	—	8	2	no	—
202	C₂×#41	—	4	1	no	—
203	C₂×#41	—	44	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	11	2	no	—
206	C₂×D₈	—	22	1	no	—
207	C₂×D₈	C₂²×D₁₁	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	22	1	no	—
212	#127	—	2	1	no	—
213	#127	—	22	1	no	—
214	#124	—	4	1	no	—
215	#124	—	44	1	no	—
216	#121	—	2	2	no	—
217	#121	—	4	1	no	—
218	#121	—	22	2	no	—
219	#121	—	44	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	—	22	2	no	—
223	C₄×D₄	—	44	1	no	—
224	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
225	D₁₆	—	22	8	no	—
226	C₂×C₁₆	—	11	2	no	—
227	C₂×C₁₆	—	11	2	no	—
228	C₂×C₁₆	—	22	4	no	—
229	#107	—	2	1	no	—
230	#107	—	11	2	no	—
231	#107	—	22	1	no	—
232	#107	C₂²×D₁₁	1	2	yes	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	22	2	no	—
236	#102	—	22	2	no	—
237	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
241	C₂×C₁₁	—	2	2	no	—
242	C₂×C₁₁	—	16	2	no	—
243	C₂×C₁₁	#3169	1	1	yes	characteristic
244	C₂×C₁₁	C₂²×D₁₆	1	1	yes	characteristic
245	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
246	D₁₁	—	16	2	no	—
247	#41	—	8	4	no	—
248	#41	—	88	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	44	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	22	2	no	—
256	C₂×D₄	—	44	2	no	—
257	C₂×D₄	—	44	1	no	—
258	C₂×D₄	—	44	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	44	2	no	—
261	C₂²×C₄	#4323	1	1	yes	characteristic
262	C₂²×C₄	#4323	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
264	D₈	—	2	4	no	—
265	D₈	—	4	2	no	—
266	D₈	—	22	4	no	—
267	D₈	—	44	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	22	2	no	—
277	C₄⋊C₄	—	44	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	44	2	no	—
281	C₂²⋊C₄	—	44	2	no	—
282	C₄²	#4323	1	1	yes	characteristic
283	C₄²	#4323	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
286	C₁₆	—	11	4	no	—
287	C₁₆	—	22	2	no	—
288	C₁₁	#33733	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	4	2	no	—
290	C₂³	—	44	2	no	—
291	C₂³	C₂×#4323	1	1	yes	characteristic
292	Q₈	—	8	2	no	—
293	Q₈	—	88	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	44	2	no	—
300	D₄	—	44	2	no	—
301	D₄	—	88	4	no	—
302	D₄	—	88	2	no	—
303	D₄	—	88	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	44	2	no	—
310	C₂×C₄	—	88	4	no	—
311	C₂×C₄	#66890	1	4	yes	complemented
312	C₂×C₄	C₂×#4323	1	1	yes	characteristic
313	C₂×C₄	C₂×#4323	1	1	yes	characteristic
314	C₂×C₄	C₂×#4323	1	1	yes	characteristic
315	C₂×C₄	C₂×#4323	1	1	yes	characteristic
316	C₂×C₄	C₂×#4323	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#67003	1	2	yes	—
321	C₈	#41×D₁₁	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	88	2	no	—
328	C₂²	—	88	2	no	—
329	C₂²	—	88	2	no	—
330	C₂²	C₂×#66890	1	2	yes	—
331	C₂²	C₂²×#4323	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	88	2	no	—
336	C₄	C₂×#66890	1	2	yes	—
337	C₄	C₂×#66890	1	2	yes	—
338	C₄	#10622807	1	2	yes	—
339	C₄	C₂×#67003	1	1	yes	characteristic
340	C₄	#10623363	1	1	yes	characteristic
341	C₂	—	2	2	no	—
342	C₂	—	16	2	no	—
343	C₂	—	176	2	no	—
344	C₂	#13079191	1	1	yes	characteristic
345	C₂	C₂²×#66890	1	1	yes	characteristic
346	C₂	C₂×#10622807	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×#3169	C₂
socle	C₂²×C₁₁	C₂²×D₈
hypercenter	#3169	D₁₁
hypocenter	C₁₁	#33733
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13079195	C₂	semidirect
2. #13070889	C₂	semidirect
3. #13070865	C₂	semidirect
4. #13070841	C₂	semidirect
5. #13070839	C₂	semidirect
6. #13070700	C₂	semidirect
7. C₄×#66890	C₂	semidirect
8. C₂×C₄×C₁₁⋊C₁₆	C₂	semidirect
9. C₁₁×#3169	C₂	semidirect
115. C₁₁⋊C₁₆	#41	semidirect
117. #3169	D₁₁	semidirect
139. C₂×C₄×C₁₁	D₁₆	semidirect
288. C₁₁	#33733	semidirect
311. C₂×C₄	#66890	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13070889	2
#13070865	2
#13070841	2
#13070839	2
#13070700	2
C₄×#66890	2
C₂×C₄×C₁₁⋊C₁₆	2
C₁₁×#3169	2
#33733	11