Subgroups — #14773527

Order 2816.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13071573	C₂	1	2	yes	maximal
2	#13069478	C₂	1	2	yes	maximal
3	#13069259	C₂	1	4	yes	maximal
4	#13068776	C₂	1	1	yes	maximal, characteristic
5	#13068621	C₂	1	2	yes	maximal
6	#107×Dic₁₁	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₁	C₂²	1	2	yes	—
11	C₄×C₁₁×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#103	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	#10622843	C₂²	1	4	yes	—
16	C₂×#66899	C₂²	1	4	yes	—
17	C₂×#66882	C₂²	1	2	yes	—
18	C₂×#66882	C₂²	1	2	yes	—
19	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
20	#10622594	—	2	4	no	—
21	C₄⋊C₄×Dic₁₁	C₂²	1	4	yes	—
22	#10622549	—	2	2	no	—
23	#10622548	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₁₁	—	2	2	no	—
25	C₄×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
27	#10622188	C₂²	1	2	yes	—
28	#10622167	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₁	C₂³	1	2	yes	—
30	C₂×C₁₁×Q₁₆	—	2	2	no	—
31	C₂×C₁₁×Q₁₆	—	2	2	no	—
32	C₂×C₁₁×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₁	—	2	4	no	—
36	C₄×Q₈×C₁₁	C₂×C₄	1	8	yes	—
37	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₁	—	2	2	no	—
40	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₁	—	2	2	no	—
42	C₂×C₄⋊Dic₁₁	—	2	2	no	—
43	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₁	C₂³	1	4	yes	—
46	C₂×C₄×Dic₁₁	—	2	4	no	—
47	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
48	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
49	C₁₁×#107	C₂×C₄	1	4	yes	—
50	C₁₁×#103	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
52	#66899	—	4	4	no	—
53	C₁₁⋊#95	—	2	4	no	—
54	C₁₁⋊#95	—	2	4	no	—
55	C₁₁⋊#95	—	2	4	no	—
56	C₁₁⋊#95	D₄	1	2	yes	—
57	#66882	—	2	4	no	—
58	#66882	—	4	2	no	—
59	C₈×Dic₁₁	—	2	4	no	—
60	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
61	#22808	—	11	1	no	maximal, Sylow
62	C₂×Q₈×C₁₁	—	2	2	no	—
63	C₂×Q₈×C₁₁	—	2	2	no	—
64	C₂×Q₈×C₁₁	—	2	2	no	—
65	C₂×Q₈×C₁₁	—	2	2	no	—
66	C₂×Q₈×C₁₁	—	2	2	no	—
67	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₁	—	2	2	no	—
69	C₂²×C₄×C₁₁	—	2	2	no	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₁	—	2	4	no	—
74	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
75	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
76	C₁₁×Q₁₆	—	2	8	no	—
77	C₂×C₈×C₁₁	—	2	1	no	—
78	C₂×C₈×C₁₁	—	2	1	no	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₁	#41	1	2	yes	—
82	C₂×C₈×C₁₁	#41	1	2	yes	—
83	C₁₁×C₄⋊C₄	—	2	8	no	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₁	—	2	8	no	—
87	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₁	#41	1	2	yes	—
89	C₄⋊Dic₁₁	—	2	2	no	—
90	C₄⋊Dic₁₁	—	2	4	no	—
91	C₄⋊Dic₁₁	—	2	4	no	—
92	C₄⋊Dic₁₁	—	4	4	no	—
93	C₄⋊Dic₁₁	—	8	2	no	—
94	C₄⋊Dic₁₁	Q₁₆	1	4	yes	complemented
95	C₄×Dic₁₁	—	2	2	no	—
96	C₄×Dic₁₁	—	2	2	no	—
97	C₄×Dic₁₁	—	8	4	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	C₂×C₄×Q₁₆	D₁₁	1	1	yes	characteristic
102	#2355	—	11	2	no	—
103	#1969	—	11	2	no	—
104	#1928	—	11	4	no	—
105	#1845	—	11	1	no	—
106	#1803	—	11	2	no	—
107	C₄×#107	—	11	2	no	—
108	C₈×C₄⋊C₄	—	11	1	no	—
109	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₁₁	—	2	4	no	—
111	Q₈×C₁₁	—	2	4	no	—
112	Q₈×C₁₁	—	4	2	no	—
113	Q₈×C₁₁	—	4	2	no	—
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	4	no	—
117	C₂×C₄×C₁₁	—	2	4	no	—
118	C₂×C₄×C₁₁	—	4	2	no	—
119	C₂×C₄×C₁₁	—	4	2	no	—
120	C₂×C₄×C₁₁	—	4	2	no	—
121	C₂×C₄×C₁₁	—	4	2	no	—
122	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₁₁	#135	1	2	yes	—
126	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×Dic₁₁	—	2	2	no	—
133	C₂×Dic₁₁	—	2	2	no	—
134	C₂×Dic₁₁	—	2	2	no	—
135	C₂×Dic₁₁	—	2	2	no	—
136	C₂×Dic₁₁	—	2	2	no	—
137	C₂×Dic₁₁	—	4	4	no	—
138	C₂×Dic₁₁	—	4	4	no	—
139	C₂×Dic₁₁	—	4	4	no	—
140	C₂×Dic₁₁	C₂×Q₁₆	1	4	yes	—
141	C₈×C₁₁	—	2	2	no	—
142	C₈×C₁₁	#117	1	4	yes	—
143	C₁₁⋊C₈	—	4	2	no	—
144	C₂²×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×D₁₁	1	2	yes	—
146	C₄×Q₁₆	Dic₁₁	1	8	yes	complemented
147	C₂×#107	—	11	2	no	—
148	C₂×#107	—	11	2	no	—
149	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	11	1	no	—
151	C₂×#103	—	11	4	no	—
152	C₂×#103	C₂×D₁₁	1	2	yes	—
153	C₂×C₄×C₈	—	11	2	no	—
154	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
155	#391	—	11	4	no	—
156	#387	—	22	4	no	—
157	#384	—	11	2	no	—
158	#382	—	22	2	no	—
159	C₄×C₄⋊C₄	—	11	1	no	—
160	C₄×C₄⋊C₄	—	11	4	no	—
161	C₄×C₄⋊C₄	—	22	2	no	—
162	#340	—	11	4	no	—
163	#336	—	11	2	no	—
164	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₁	#547	1	1	yes	characteristic
166	C₂²×C₁₁	#550	1	1	yes	characteristic
167	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
168	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
169	C₂²×C₁₁	#576	1	1	yes	characteristic
170	C₂²×C₁₁	#578	1	1	yes	characteristic
171	C₄×C₁₁	—	2	2	no	—
172	C₄×C₁₁	—	4	4	no	—
173	C₄×C₁₁	—	4	4	no	—
174	C₄×C₁₁	C₄×Q₁₆	1	4	yes	—
175	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
176	C₄×C₁₁	#520	1	1	yes	characteristic
177	C₄×C₁₁	#555	1	1	yes	characteristic
178	C₄×C₁₁	#556	1	1	yes	characteristic
179	Dic₁₁	—	2	4	no	—
180	Dic₁₁	—	4	2	no	—
181	Dic₁₁	—	8	4	no	—
182	C₂²×Q₈	C₂²×D₁₁	1	2	yes	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	C₂×Dic₁₁	1	4	yes	—
186	C₂²×C₈	—	11	2	no	—
187	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	C₂×Dic₁₁	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	11	1	no	—
193	C₂×C₄⋊C₄	—	11	1	no	—
194	C₂×C₄⋊C₄	—	11	4	no	—
195	C₂×C₄⋊C₄	—	22	2	no	—
196	C₂×C₄⋊C₄	—	22	2	no	—
197	C₂×C₄⋊C₄	C₂²×D₁₁	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	11	2	no	—
200	C₂×C₄²	—	22	4	no	—
201	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
202	#107	—	22	4	no	—
203	#107	—	44	2	no	—
204	#107	C₂×Dic₁₁	1	4	yes	—
205	C₄⋊C₈	—	22	2	no	—
206	#103	—	44	4	no	—
207	#103	C₂×Dic₁₁	1	8	yes	—
208	C₄×C₈	—	22	4	no	—
209	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
210	#95	—	11	2	no	—
211	#95	—	22	4	no	—
212	#95	—	22	4	no	—
213	#95	—	22	4	no	—
214	C₂×C₁₁	#2354	1	1	yes	characteristic
215	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
216	C₂×C₁₁	#2951	1	1	yes	characteristic
217	C₂×C₁₁	#3338	1	1	yes	characteristic
218	C₂×C₁₁	#3342	1	1	yes	characteristic
219	C₂×C₁₁	#3395	1	1	yes	characteristic
220	C₂×C₁₁	#3396	1	1	yes	characteristic
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂²×Dic₁₁	1	4	yes	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	11	2	no	—
230	C₂²×C₄	—	11	2	no	—
231	C₂²×C₄	—	22	4	no	—
232	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
234	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
235	Q₁₆	—	2	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	22	2	no	—
239	C₂×C₈	—	22	2	no	—
240	C₂×C₈	—	22	2	no	—
241	C₂×C₈	#4339	1	2	yes	—
242	C₂×C₈	#4339	1	2	yes	—
243	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
244	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	11	4	no	—
248	C₄⋊C₄	—	22	2	no	—
249	C₄⋊C₄	—	22	4	no	—
250	C₄⋊C₄	—	22	4	no	—
251	C₄⋊C₄	—	44	4	no	—
252	C₄⋊C₄	—	88	2	no	—
253	C₄⋊C₄	C₂²×Dic₁₁	1	4	yes	—
254	C₄²	—	2	8	no	—
255	C₄²	—	22	2	no	—
256	C₄²	—	22	2	no	—
257	C₄²	—	88	4	no	—
258	C₄²	#4341	1	2	yes	—
259	C₄²	C₂²×Dic₁₁	1	2	yes	—
260	C₁₁	#22808	1	1	yes	characteristic, complemented, Sylow
261	C₂³	C₂×D₄×D₁₁	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	2	4	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	11	4	no	—
275	C₂×C₄	—	22	2	no	—
276	C₂×C₄	—	22	2	no	—
277	C₂×C₄	—	22	2	no	—
278	C₂×C₄	—	22	2	no	—
279	C₂×C₄	—	22	2	no	—
280	C₂×C₄	—	44	4	no	—
281	C₂×C₄	—	44	4	no	—
282	C₂×C₄	—	44	4	no	—
283	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
284	C₂×C₄	#66972	1	2	yes	—
285	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
286	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
287	C₂×C₄	C₂×#4339	1	1	yes	characteristic
288	C₂×C₄	C₂×#4339	1	1	yes	characteristic
289	C₂×C₄	C₂×#4341	1	1	yes	characteristic
290	C₂×C₄	#67042	1	1	yes	characteristic
291	C₂×C₄	#67043	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	44	2	no	—
295	C₈	#66979	1	4	yes	—
296	C₂²	#10623082	1	1	yes	characteristic
297	C₂²	#10623084	1	1	yes	characteristic
298	C₂²	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
299	C₂²	C₂×#66972	1	1	yes	characteristic
300	C₂²	#10623300	1	1	yes	characteristic
301	C₂²	#10623302	1	1	yes	characteristic
302	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	4	no	—
306	C₄	—	22	4	no	—
307	C₄	—	44	2	no	—
308	C₄	—	88	4	no	—
309	C₄	Q₁₆×Dic₁₁	1	4	yes	—
310	C₄	#10623073	1	1	yes	characteristic
311	C₄	#10623094	1	1	yes	characteristic
312	C₄	C₂×#66979	1	1	yes	characteristic
313	C₄	#10623358	1	1	yes	characteristic
314	C₂	#13071565	1	1	yes	characteristic
315	C₂	C₂×Q₁₆×Dic₁₁	1	1	yes	characteristic
316	C₂	#13077492	1	1	yes	characteristic
317	C₂	#13081010	1	1	yes	characteristic
318	C₂	#13081048	1	1	yes	characteristic
319	C₂	#13081621	1	1	yes	characteristic
320	C₂	#13081626	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×Q₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂²×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₁
hypocenter	C₁₁	#22808
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₁₁×Q₁₆	C₄	semidirect
94. C₄⋊Dic₁₁	Q₁₆	semidirect
146. C₄×Q₁₆	Dic₁₁	semidirect
260. C₁₁	#22808	semidirect

Maximal subgroups

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

subgroup	index
#13071573	2
#13069478	2
#13069259	2
#13068776	2
#13068621	2
#107×Dic₁₁	2
C₈×C₄⋊Dic₁₁	2
C₂×C₄×C₁₁×Q₁₆	2
#22808	11