Subgroups — #14723262

Order 2816.

714 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	#13077725	C₂	1	2	yes	maximal, complemented
2	C₄×#66957	C₂	1	1	yes	maximal, characteristic
3	#13067439	C₂	1	2	yes	maximal, complemented
4	#13067434	C₂	1	2	yes	maximal, complemented
5	#13067415	C₂	1	4	yes	maximal, complemented
6	#13067001	C₂	1	2	yes	maximal, complemented
7	#13066899	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#334	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×#334	C₂²	1	2	yes	—
29	C₁₁×#334	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	2	yes	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	#17858	—	11	1	no	maximal, Sylow
76	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₁	—	2	2	no	—
80	#4339	—	2	4	no	—
81	#4339	C₂×C₈	1	8	yes	complemented
82	C₂×D₄₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₁	—	2	2	no	—
84	C₂×C₄×D₁₁	—	2	2	no	—
85	C₂×C₄×D₁₁	—	2	2	no	—
86	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	4	no	—
89	C₂×C₈×C₁₁	—	8	2	no	—
90	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
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	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₁	#41	1	1	yes	characteristic
98	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₁₁	—	2	2	no	—
100	(C₂×D₁₁)⋊C₄	—	2	2	no	—
101	(C₂×D₁₁)⋊C₄	—	2	2	no	—
102	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
104	Dic₁₁⋊C₄	—	2	2	no	—
105	Dic₁₁⋊C₄	—	2	2	no	—
106	C₄×Dic₁₁	—	2	2	no	—
107	C₄×Dic₁₁	—	2	2	no	—
108	C₄×Dic₁₁	—	2	2	no	—
109	C₂×C₁₁⋊C₈	—	2	2	no	—
110	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
111	Dic₄₄	C₂×C₈	1	4	yes	complemented
112	D₈₈	C₂×C₈	1	4	yes	complemented
113	#4314	C₂×C₈	1	8	yes	complemented
114	C₈×D₁₁	—	8	4	no	—
115	#2988	—	11	2	no	—
116	C₄×#135	—	11	1	no	—
117	#1580	—	11	2	no	—
118	#1579	—	11	2	no	—
119	C₈×SD₁₆	—	11	4	no	—
120	C₂×#334	D₁₁	1	1	yes	characteristic, complemented
121	#1477	—	11	2	no	—
122	#1450	—	11	1	no	—
123	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂²×D₁₁	—	2	2	no	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	4	1	no	—
127	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₁	#131	1	2	yes	—
132	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
135	C₂²⋊D₁₁	—	2	8	no	—
136	C₂×Dic₁₁	—	2	2	no	—
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	D₄₄	—	2	2	no	—
140	D₄₄	C₂²×C₈	1	4	yes	—
141	C₄×D₁₁	—	2	8	no	—
142	C₄×D₁₁	—	4	2	no	—
143	C₄×D₁₁	—	4	2	no	—
144	C₄×D₁₁	—	8	2	no	—
145	Dic₂₂	—	2	2	no	—
146	Dic₂₂	C₂²×C₈	1	4	yes	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	—	4	4	no	—
150	C₈×C₁₁	C₂²×C₈	1	2	yes	—
151	C₈×C₁₁	C₂²×C₈	1	2	yes	—
152	C₁₁⋊C₈	—	2	4	no	—
153	C₁₁⋊C₈	—	4	2	no	—
154	C₂×#135	—	11	1	no	—
155	C₄×#41	—	11	2	no	—
156	C₈×Q₈	—	11	4	no	—
157	C₄×Q₁₆	—	11	1	no	—
158	C₄×Q₁₆	—	11	1	no	—
159	C₄×SD₁₆	—	11	2	no	—
160	C₄×SD₁₆	—	11	2	no	—
161	C₄×D₈	—	11	1	no	—
162	C₄×D₈	—	11	1	no	—
163	C₈×D₄	—	11	4	no	—
164	C₈×D₄	—	22	4	no	—
165	#431	—	11	1	no	—
166	#431	—	11	1	no	—
167	#431	—	22	2	no	—
168	#427	—	11	1	no	—
169	C₂×C₄⋊C₈	—	22	2	no	—
170	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
171	#416	—	11	2	no	—
172	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
173	#334	C₂×D₁₁	1	2	yes	—
174	#334	C₂×D₁₁	1	2	yes	—
175	Q₈⋊C₈	—	11	4	no	—
176	D₄⋊C₈	—	11	4	no	—
177	#322	—	11	2	no	—
178	C₈²	—	11	2	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	C₈×D₄	1	2	yes	—
182	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂×D₁₁	—	2	4	no	—
184	C₂×D₁₁	—	4	2	no	—
185	C₂×D₁₁	—	4	2	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	C₈×D₄	1	2	yes	—
190	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
191	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
192	C₄×C₁₁	#577	1	2	yes	—
193	Dic₁₁	—	2	4	no	—
194	Dic₁₁	—	4	2	no	—
195	Dic₁₁	—	4	2	no	—
196	C₂×#41	—	11	2	no	—
197	#135	—	11	8	no	—
198	C₂×Q₁₆	—	11	1	no	—
199	C₂×SD₁₆	—	11	2	no	—
200	C₂×D₈	—	11	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	22	4	no	—
203	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
204	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
205	C₄×Q₈	—	11	2	no	—
206	C₄×Q₈	—	11	2	no	—
207	C₄×D₄	—	11	2	no	—
208	C₄×D₄	—	11	2	no	—
209	C₄×D₄	—	22	2	no	—
210	C₄×D₄	—	22	2	no	—
211	#117	—	11	2	no	—
212	#117	—	22	2	no	—
213	C₂×C₄²	—	22	2	no	—
214	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
215	#107	—	11	1	no	—
216	#107	—	11	1	no	—
217	#106	—	11	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	11	2	no	—
220	C₄⋊C₈	—	22	4	no	—
221	C₄⋊C₈	—	22	1	no	—
222	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
223	#103	—	11	2	no	—
224	#103	—	11	2	no	—
225	#102	—	11	2	no	—
226	#102	—	11	2	no	—
227	C₂²⋊C₈	—	11	2	no	—
228	C₂²⋊C₈	—	22	4	no	—
229	C₂²⋊C₈	—	22	1	no	—
230	C₄×C₈	—	11	2	no	—
231	C₄×C₈	—	11	2	no	—
232	C₄×C₈	—	22	4	no	—
233	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
234	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
237	C₂×C₁₁	—	2	2	no	—
238	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
239	C₂×C₁₁	#2952	1	1	yes	characteristic
240	C₂×C₁₁	#2957	1	1	yes	characteristic
241	D₁₁	—	4	4	no	—
242	#41	—	11	8	no	—
243	#41	—	22	4	no	—
244	C₂×Q₈	—	11	2	no	—
245	C₂×D₄	—	11	2	no	—
246	C₂×D₄	—	22	2	no	—
247	C₂²×C₄	—	22	2	no	—
248	C₂²×C₄	—	22	2	no	—
249	C₂²×C₄	—	22	2	no	—
250	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
252	C₂²×C₄	#4343	1	1	yes	characteristic
253	Q₁₆	—	11	4	no	—
254	SD₁₆	—	11	8	no	—
255	D₈	—	11	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	8	2	no	—
259	C₂×C₈	—	11	4	no	—
260	C₂×C₈	—	22	2	no	—
261	C₂×C₈	—	88	4	no	—
262	C₂×C₈	C₈×D₁₁	1	4	yes	complemented
263	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₄⋊C₄	—	11	2	no	—
268	C₄⋊C₄	—	11	2	no	—
269	C₄⋊C₄	—	22	2	no	—
270	C₄⋊C₄	—	22	2	no	—
271	C₂²⋊C₄	—	22	2	no	—
272	C₂²⋊C₄	—	22	2	no	—
273	C₂²⋊C₄	—	22	2	no	—
274	C₄²	—	22	2	no	—
275	C₄²	—	22	2	no	—
276	C₄²	—	22	2	no	—
277	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
278	C₄²	D₄×D₁₁	1	1	yes	characteristic
279	C₄²	#4343	1	1	yes	characteristic
280	C₄²	C₂³×D₁₁	1	1	yes	characteristic
281	C₁₁	#17858	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	22	2	no	—
283	C₂³	#66946	1	1	yes	characteristic
284	Q₈	—	11	4	no	—
285	Q₈	—	22	2	no	—
286	D₄	—	11	4	no	—
287	D₄	—	22	8	no	—
288	D₄	—	22	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	22	2	no	—
292	C₂×C₄	—	22	8	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	22	2	no	—
295	C₂×C₄	—	44	2	no	—
296	C₂×C₄	—	44	2	no	—
297	C₂×C₄	—	88	2	no	—
298	C₂×C₄	#66946	1	1	yes	characteristic
299	C₂×C₄	#66946	1	1	yes	characteristic
300	C₂×C₄	#66946	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
302	C₂×C₄	#66960	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#4343	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	4	no	—
309	C₈	—	22	4	no	—
310	C₈	—	44	2	no	—
311	C₈	C₂×C₈×D₁₁	1	2	yes	—
312	C₈	C₂×C₈×D₁₁	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	22	4	no	—
316	C₂²	—	44	2	no	—
317	C₂²	—	44	2	no	—
318	C₂²	#10622447	1	2	yes	—
319	C₂²	C₂×#66946	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	22	4	no	—
324	C₄	—	44	2	no	—
325	C₄	—	44	2	no	—
326	C₄	#10622447	1	2	yes	—
327	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
328	C₄	C₂×#66960	1	1	yes	characteristic
329	C₄	#10623293	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	44	4	no	—
332	C₂	C₂×#10622447	1	1	yes	characteristic
333	C₂	#13077499	1	1	yes	characteristic
334	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×#334	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×#66957	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#334	D₁₁
hypocenter	C₁₁	#17858
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. #13077725	C₂	semidirect
3. #13067439	C₂	semidirect
4. #13067434	C₂	semidirect
5. #13067415	C₂	semidirect
6. #13067001	C₂	semidirect
7. #13066899	C₂	semidirect
8. C₂×C₁₁×#334	C₂	semidirect
43. #66957	C₈	semidirect
81. #4339	C₂×C₈	semidirect
90. C₂×C₈×C₁₁	C₂×C₈	semidirect
111. Dic₄₄	C₂×C₈	semidirect
112. D₈₈	C₂×C₈	semidirect
113. #4314	C₂×C₈	semidirect
120. C₂×#334	D₁₁	semidirect
262. C₂×C₈	C₈×D₁₁	semidirect
281. C₁₁	#17858	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
#13067439	2
#13067434	2
#13067415	2
#13067001	2
#13066899	2
C₂×C₁₁×#334	2
#17858	11