Subgroups — #14723253

Order 2816.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077725	C₂	1	2	yes	maximal, complemented
2	C₄×#66957	C₂	1	1	yes	maximal, characteristic
3	#13067427	C₂	1	4	yes	maximal, complemented
4	#13067422	C₂	1	2	yes	maximal, complemented
5	#13067410	C₂	1	2	yes	maximal, complemented
6	#13067001	C₂	1	2	yes	maximal, complemented
7	#13066899	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66957	C₄	1	1	yes	characteristic
10	C₄×#4339	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622726	C₄	1	2	yes	—
14	C₈×C₂²⋊D₁₁	—	2	4	no	—
15	#10622719	C₄	1	1	yes	characteristic
16	#10622538	C₂²	1	1	yes	characteristic
17	#10622538	C₂²	1	1	yes	characteristic
18	#10622447	C₂²	1	4	yes	—
19	#10622446	—	2	2	no	—
20	D₁₁×C₄⋊C₈	—	2	2	no	—
21	#10622443	C₂²	1	4	yes	—
22	C₄×Dic₄₄	C₄	1	1	yes	characteristic
23	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
24	C₄×D₈₈	C₄	1	1	yes	characteristic
25	C₄×D₈₈	C₂²	1	1	yes	characteristic
26	C₄×#4314	C₄	1	2	yes	—
27	C₄×#4314	C₂²	1	2	yes	—
28	C₁₁×#335	C₂²	1	2	yes	—
29	C₁₁×#335	C₂²	1	2	yes	—
30	#10622122	C₂²	1	4	yes	—
31	#10622121	C₂²	1	4	yes	—
32	#10622093	C₂²	1	2	yes	—
33	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
34	C₂×#4339	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₁	—	2	2	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₁	—	2	2	no	—
40	C₄×C₂²⋊D₁₁	—	2	2	no	—
41	#66979	C₂×C₄	1	2	yes	—
42	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
43	#66957	C₈	1	8	yes	complemented
44	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
45	C₂×#4314	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₁	—	2	4	no	—
47	C₄×D₄₄	C₂×C₄	1	2	yes	—
48	C₄×D₄₄	C₂³	1	2	yes	—
49	#66925	—	2	2	no	—
50	C₄²×D₁₁	—	2	2	no	—
51	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₂	C₂³	1	2	yes	—
53	C₁₁×C₄⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
61	#66885	C₂×C₄	1	2	yes	—
62	#66885	C₂×C₄	1	2	yes	—
63	(C₂×D₁₁)⋊C₈	—	2	4	no	—
64	#66882	C₂×C₄	1	1	yes	characteristic
65	#66882	C₂×C₄	1	1	yes	characteristic
66	#66881	C₂×C₄	1	2	yes	—
67	#66880	C₂×C₄	1	2	yes	—
68	#66880	C₂×C₄	1	2	yes	—
69	Dic₁₁⋊C₈	—	2	4	no	—
70	C₈×Dic₁₁	—	2	4	no	—
71	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
72	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
73	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
76	#17857	—	11	1	no	maximal, Sylow
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₁	—	2	2	no	—
81	#4339	—	2	4	no	—
82	#4339	C₂×C₈	1	8	yes	complemented
83	C₂×D₄₄	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₁₁	—	2	2	no	—
85	C₂×C₄×D₁₁	—	2	2	no	—
86	C₂×C₄×D₁₁	—	2	2	no	—
87	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	4	no	—
90	C₂×C₈×C₁₁	—	8	2	no	—
91	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₁	#41	1	1	yes	characteristic
99	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₁	—	2	2	no	—
101	(C₂×D₁₁)⋊C₄	—	2	2	no	—
102	(C₂×D₁₁)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
105	Dic₁₁⋊C₄	—	2	2	no	—
106	Dic₁₁⋊C₄	—	2	2	no	—
107	C₄×Dic₁₁	—	2	2	no	—
108	C₄×Dic₁₁	—	2	2	no	—
109	C₄×Dic₁₁	—	2	2	no	—
110	C₂×C₁₁⋊C₈	—	2	2	no	—
111	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
112	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
113	Dic₄₄	C₂×C₈	1	4	yes	complemented
114	D₈₈	C₂×C₈	1	4	yes	complemented
115	#4314	C₂×C₈	1	8	yes	complemented
116	C₈×D₁₁	—	8	4	no	—
117	#2988	—	11	2	no	—
118	C₄×#135	—	11	1	no	—
119	SD₁₆⋊C₈	—	11	4	no	—
120	C₈×Q₁₆	—	11	2	no	—
121	C₈×D₈	—	11	2	no	—
122	C₂×#335	D₁₁	1	1	yes	characteristic, complemented
123	#1477	—	11	2	no	—
124	#1450	—	11	1	no	—
125	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
126	C₂²×D₁₁	—	2	2	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	4	1	no	—
130	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₁	#131	1	2	yes	—
135	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
138	C₂²⋊D₁₁	—	2	8	no	—
139	C₂×Dic₁₁	—	2	2	no	—
140	C₂×Dic₁₁	—	2	2	no	—
141	C₂×Dic₁₁	—	2	2	no	—
142	D₄₄	—	2	2	no	—
143	D₄₄	C₂²×C₈	1	4	yes	—
144	C₄×D₁₁	—	2	8	no	—
145	C₄×D₁₁	—	4	2	no	—
146	C₄×D₁₁	—	4	2	no	—
147	C₄×D₁₁	—	8	2	no	—
148	Dic₂₂	—	2	2	no	—
149	Dic₂₂	C₂²×C₈	1	4	yes	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	—	4	4	no	—
153	C₈×C₁₁	C₂²×C₈	1	2	yes	—
154	C₈×C₁₁	C₂²×C₈	1	2	yes	—
155	C₁₁⋊C₈	—	2	2	no	—
156	C₁₁⋊C₈	—	2	2	no	—
157	C₁₁⋊C₈	—	4	2	no	—
158	C₂×#135	—	11	1	no	—
159	C₄×#41	—	11	2	no	—
160	C₈×Q₈	—	11	4	no	—
161	C₄×Q₁₆	—	11	1	no	—
162	C₄×Q₁₆	—	11	1	no	—
163	C₄×SD₁₆	—	11	2	no	—
164	C₄×SD₁₆	—	11	2	no	—
165	C₄×D₈	—	11	1	no	—
166	C₄×D₈	—	11	1	no	—
167	C₈×D₄	—	11	4	no	—
168	C₈×D₄	—	22	4	no	—
169	#431	—	11	1	no	—
170	#431	—	11	1	no	—
171	#431	—	22	2	no	—
172	#427	—	11	1	no	—
173	C₂×C₄⋊C₈	—	22	2	no	—
174	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
175	#416	—	11	2	no	—
176	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
177	#335	C₂×D₁₁	1	2	yes	—
178	#335	C₂×D₁₁	1	2	yes	—
179	Q₈⋊C₈	—	11	4	no	—
180	D₄⋊C₈	—	11	4	no	—
181	#322	—	11	2	no	—
182	C₈²	—	11	2	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	—	2	1	no	—
185	C₂²×C₁₁	C₈×D₄	1	2	yes	—
186	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂×D₁₁	—	2	4	no	—
188	C₂×D₁₁	—	4	2	no	—
189	C₂×D₁₁	—	4	2	no	—
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	—	2	1	no	—
192	C₄×C₁₁	—	2	1	no	—
193	C₄×C₁₁	C₈×D₄	1	2	yes	—
194	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
195	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
196	C₄×C₁₁	#576	1	1	yes	characteristic
197	C₄×C₁₁	#578	1	1	yes	characteristic
198	Dic₁₁	—	2	4	no	—
199	Dic₁₁	—	4	2	no	—
200	Dic₁₁	—	4	2	no	—
201	C₂×#41	—	11	2	no	—
202	#135	—	11	8	no	—
203	C₂×Q₁₆	—	11	1	no	—
204	C₂×SD₁₆	—	11	2	no	—
205	C₂×D₈	—	11	1	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	22	4	no	—
208	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
210	C₄×Q₈	—	11	2	no	—
211	C₄×Q₈	—	11	2	no	—
212	C₄×D₄	—	11	2	no	—
213	C₄×D₄	—	11	2	no	—
214	C₄×D₄	—	22	2	no	—
215	C₄×D₄	—	22	2	no	—
216	#117	—	11	2	no	—
217	#117	—	22	2	no	—
218	C₂×C₄²	—	22	2	no	—
219	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
220	#107	—	11	1	no	—
221	#107	—	11	1	no	—
222	#106	—	11	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	11	1	no	—
225	C₄⋊C₈	—	11	1	no	—
226	C₄⋊C₈	—	22	4	no	—
227	C₄⋊C₈	—	22	1	no	—
228	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
229	#103	—	11	2	no	—
230	#103	—	11	2	no	—
231	#102	—	11	2	no	—
232	#102	—	11	2	no	—
233	C₂²⋊C₈	—	11	2	no	—
234	C₂²⋊C₈	—	22	4	no	—
235	C₂²⋊C₈	—	22	1	no	—
236	C₄×C₈	—	11	2	no	—
237	C₄×C₈	—	11	2	no	—
238	C₄×C₈	—	22	4	no	—
239	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
240	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
243	C₂×C₁₁	—	2	2	no	—
244	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
245	C₂×C₁₁	#2951	1	1	yes	characteristic
246	C₂×C₁₁	#2957	1	1	yes	characteristic
247	D₁₁	—	4	4	no	—
248	#41	—	11	8	no	—
249	#41	—	22	4	no	—
250	C₂×Q₈	—	11	2	no	—
251	C₂×D₄	—	11	2	no	—
252	C₂×D₄	—	22	2	no	—
253	C₂²×C₄	—	22	2	no	—
254	C₂²×C₄	—	22	2	no	—
255	C₂²×C₄	—	22	2	no	—
256	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
258	C₂²×C₄	#4343	1	1	yes	characteristic
259	Q₁₆	—	11	4	no	—
260	SD₁₆	—	11	8	no	—
261	D₈	—	11	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	8	2	no	—
265	C₂×C₈	—	11	2	no	—
266	C₂×C₈	—	11	2	no	—
267	C₂×C₈	—	22	2	no	—
268	C₂×C₈	—	88	4	no	—
269	C₂×C₈	C₈×D₁₁	1	4	yes	complemented
270	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
274	C₄⋊C₄	—	11	2	no	—
275	C₄⋊C₄	—	11	2	no	—
276	C₄⋊C₄	—	22	2	no	—
277	C₄⋊C₄	—	22	2	no	—
278	C₂²⋊C₄	—	22	2	no	—
279	C₂²⋊C₄	—	22	2	no	—
280	C₂²⋊C₄	—	22	2	no	—
281	C₄²	—	22	2	no	—
282	C₄²	—	22	2	no	—
283	C₄²	—	22	2	no	—
284	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
285	C₄²	D₄×D₁₁	1	1	yes	characteristic
286	C₄²	#4343	1	1	yes	characteristic
287	C₄²	C₂³×D₁₁	1	1	yes	characteristic
288	C₁₁	#17857	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	22	2	no	—
290	C₂³	#66946	1	1	yes	characteristic
291	Q₈	—	11	4	no	—
292	Q₈	—	22	2	no	—
293	D₄	—	11	4	no	—
294	D₄	—	22	8	no	—
295	D₄	—	22	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	22	2	no	—
300	C₂×C₄	—	22	8	no	—
301	C₂×C₄	—	22	2	no	—
302	C₂×C₄	—	22	2	no	—
303	C₂×C₄	—	44	2	no	—
304	C₂×C₄	—	44	2	no	—
305	C₂×C₄	—	88	2	no	—
306	C₂×C₄	#66946	1	1	yes	characteristic
307	C₂×C₄	#66946	1	1	yes	characteristic
308	C₂×C₄	#66946	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
310	C₂×C₄	#66960	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
313	C₂×C₄	C₂×#4343	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	4	no	—
317	C₈	—	22	2	no	—
318	C₈	—	22	2	no	—
319	C₈	—	44	2	no	—
320	C₈	C₂×C₈×D₁₁	1	2	yes	—
321	C₈	C₂×C₈×D₁₁	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	22	4	no	—
325	C₂²	—	44	2	no	—
326	C₂²	—	44	2	no	—
327	C₂²	#10622447	1	2	yes	—
328	C₂²	C₂×#66946	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	22	4	no	—
333	C₄	—	44	2	no	—
334	C₄	—	44	2	no	—
335	C₄	#10622447	1	2	yes	—
336	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
337	C₄	C₂×#66960	1	1	yes	characteristic
338	C₄	#10623288	1	1	yes	characteristic
339	C₄	#10623297	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	44	4	no	—
342	C₂	C₂×#10622447	1	1	yes	characteristic
343	C₂	#13077490	1	1	yes	characteristic
344	C₂	#13077532	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×#66957	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#17857
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. #13077725	C₂	semidirect
3. #13067427	C₂	semidirect
4. #13067422	C₂	semidirect
5. #13067410	C₂	semidirect
6. #13067001	C₂	semidirect
7. #13066899	C₂	semidirect
8. C₂×C₁₁×#335	C₂	semidirect
43. #66957	C₈	semidirect
82. #4339	C₂×C₈	semidirect
91. C₂×C₈×C₁₁	C₂×C₈	semidirect
113. Dic₄₄	C₂×C₈	semidirect
114. D₈₈	C₂×C₈	semidirect
115. #4314	C₂×C₈	semidirect
122. C₂×#335	D₁₁	semidirect
269. C₂×C₈	C₈×D₁₁	semidirect
288. C₁₁	#17857	semidirect

Maximal subgroups

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

subgroup	index
#13077725	2
C₄×#66957	2
#13067427	2
#13067422	2
#13067410	2
#13067001	2
#13066899	2
C₂×C₁₁×#335	2
#17857	11