Subgroups — #14772617

Order 2816.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	#13071571	C₂	1	2	yes	maximal
3	#13069387	C₂	1	4	yes	maximal
4	#13069379	C₂	1	2	yes	maximal
5	#13068779	C₂	1	2	yes	maximal
6	#13068776	C₂	1	1	yes	maximal, characteristic
7	#107×Dic₁₁	C₂	1	2	yes	maximal
8	C₄×C₁₁×#107	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×#66882	C₂²	1	2	yes	—
15	C₂×#66882	C₂²	1	2	yes	—
16	C₂×Dic₁₁⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
18	#10622590	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₁₁	C₂²	1	4	yes	—
20	C₂×#66872	C₂²	1	4	yes	—
21	#10622548	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₁₁	—	2	2	no	—
23	#10622542	—	2	2	no	—
24	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₂	C₄	1	8	yes	complemented
26	#10622268	—	2	4	no	—
27	C₁₁×#340	C₂²	1	2	yes	—
28	#10622167	C₂²	1	4	yes	—
29	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
33	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₁	—	2	2	no	—
35	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₁	—	2	2	no	—
37	C₂×C₄⋊Dic₁₁	—	2	2	no	—
38	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₁	C₂³	1	4	yes	—
40	C₂×Dic₁₁⋊C₄	D₄	1	2	yes	—
41	C₂×C₄×Dic₁₁	—	2	4	no	—
42	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
43	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
44	C₄×Dic₂₂	C₂×C₄	1	4	yes	—
45	C₄×Dic₂₂	Q₈	1	4	yes	—
46	C₁₁×#107	—	2	2	no	—
47	C₁₁×#107	Q₈	1	4	yes	complemented
48	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
49	C₁₁×#95	—	2	2	no	—
50	C₁₁⋊#95	—	2	4	no	—
51	C₁₁⋊#95	—	2	4	no	—
52	C₁₁⋊#95	—	2	4	no	—
53	#66882	—	2	4	no	—
54	#66882	—	4	2	no	—
55	Dic₁₁⋊C₈	C₂×C₄	1	8	yes	—
56	C₈×Dic₁₁	C₂×C₄	1	8	yes	—
57	#66872	—	4	4	no	—
58	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
59	#22727	—	11	1	no	maximal, Sylow
60	C₂²×C₄×C₁₁	—	2	2	no	—
61	C₂²×C₄×C₁₁	—	2	2	no	—
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₁	—	2	4	no	—
66	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
67	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
68	C₂×Dic₂₂	—	2	2	no	—
69	C₂×Dic₂₂	—	2	2	no	—
70	C₂×Dic₂₂	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₁₁	—	2	1	no	—
72	C₂×C₈×C₁₁	—	2	1	no	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₁	#41	1	2	yes	—
77	C₁₁×C₄⋊C₄	—	2	2	no	—
78	C₁₁×C₄⋊C₄	—	2	2	no	—
79	C₁₁×C₄⋊C₄	—	2	2	no	—
80	C₁₁×C₄⋊C₄	C₂×Q₈	1	4	yes	—
81	C₄²×C₁₁	—	4	4	no	—
82	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
84	C₄⋊Dic₁₁	—	2	4	no	—
85	C₄⋊Dic₁₁	—	2	4	no	—
86	C₄⋊Dic₁₁	—	4	4	no	—
87	C₄⋊Dic₁₁	—	8	2	no	—
88	C₄⋊Dic₁₁	C₄⋊C₄	1	4	yes	—
89	Dic₁₁⋊C₄	C₄⋊C₄	1	8	yes	—
90	C₄×Dic₁₁	—	8	4	no	—
91	C₄×Dic₁₁	C₂²×C₄	1	4	yes	—
92	C₄×Dic₁₁	C₂×Q₈	1	4	yes	—
93	C₂×C₁₁⋊C₈	—	2	2	no	—
94	C₂×C₁₁⋊C₈	—	2	2	no	—
95	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₈×Q₈	—	11	1	no	—
97	#2355	—	11	2	no	—
98	#1949	—	11	4	no	—
99	#1948	—	11	2	no	—
100	#1846	—	11	2	no	—
101	#1845	—	11	1	no	—
102	C₄×#107	—	11	2	no	—
103	C₄×#107	D₁₁	1	1	yes	characteristic
104	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	4	no	—
108	C₂×C₄×C₁₁	—	4	2	no	—
109	C₂×C₄×C₁₁	—	4	2	no	—
110	C₂×C₄×C₁₁	—	4	2	no	—
111	C₂×C₄×C₁₁	—	4	2	no	—
112	C₂×C₄×C₁₁	—	4	2	no	—
113	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₁	#135	1	2	yes	—
116	C₂×C₄×C₁₁	#135	1	2	yes	—
117	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₁₁	—	2	2	no	—
124	C₂×Dic₁₁	—	2	2	no	—
125	C₂×Dic₁₁	—	2	2	no	—
126	C₂×Dic₁₁	—	2	2	no	—
127	C₂×Dic₁₁	—	4	4	no	—
128	C₂×Dic₁₁	—	4	4	no	—
129	C₂×Dic₁₁	—	4	4	no	—
130	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
132	Dic₂₂	—	2	8	no	—
133	C₈×C₁₁	—	2	2	no	—
134	C₈×C₁₁	C₄×Q₈	1	4	yes	—
135	C₁₁⋊C₈	—	2	4	no	—
136	C₂×C₄×Q₈	—	11	1	no	—
137	C₈×Q₈	—	11	8	no	—
138	C₂×#107	—	11	2	no	—
139	C₂×#107	—	11	2	no	—
140	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
141	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
142	C₂×#106	—	11	4	no	—
143	C₂×C₄⋊C₈	—	11	2	no	—
144	C₂×C₄⋊C₈	—	11	1	no	—
145	C₂×C₄×C₈	—	11	2	no	—
146	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
147	#391	—	22	2	no	—
148	#387	—	22	4	no	—
149	#384	—	11	2	no	—
150	#382	—	11	4	no	—
151	C₄×C₄⋊C₄	—	11	4	no	—
152	C₄×C₄⋊C₄	—	22	2	no	—
153	C₄×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
154	#340	—	11	4	no	—
155	#340	C₂×D₁₁	1	2	yes	—
156	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₁	D₄×Q₈	1	1	yes	characteristic
158	C₂²×C₁₁	#550	1	1	yes	characteristic
159	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
160	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
161	C₂²×C₁₁	#576	1	1	yes	characteristic
162	C₂²×C₁₁	#578	1	1	yes	characteristic
163	C₄×C₁₁	—	2	2	no	—
164	C₄×C₁₁	—	4	4	no	—
165	C₄×C₁₁	—	8	2	no	—
166	C₄×C₁₁	#427	1	4	yes	—
167	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₁₁	#520	1	1	yes	characteristic
169	C₄×C₁₁	#557	1	1	yes	characteristic
170	C₄×C₁₁	Q₈²	1	1	yes	characteristic
171	Dic₁₁	—	2	4	no	—
172	Dic₁₁	—	2	4	no	—
173	Dic₁₁	—	8	4	no	—
174	C₂²×Q₈	—	11	1	no	—
175	C₂²×C₈	—	11	2	no	—
176	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
178	C₄×Q₈	—	11	4	no	—
179	C₄×Q₈	—	11	4	no	—
180	C₂×C₄⋊C₄	—	11	1	no	—
181	C₂×C₄⋊C₄	—	11	2	no	—
182	C₂×C₄⋊C₄	—	11	4	no	—
183	C₂×C₄⋊C₄	—	22	2	no	—
184	C₂×C₄⋊C₄	—	22	2	no	—
185	C₂×C₄⋊C₄	C₂²×D₁₁	1	2	yes	—
186	C₂×C₄⋊C₄	C₂²×D₁₁	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	11	2	no	—
189	C₂×C₄²	—	22	4	no	—
190	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
191	#107	—	2	2	no	—
192	#107	—	22	4	no	—
193	#107	—	44	2	no	—
194	#107	Dic₂₂	1	4	yes	complemented
195	#106	—	44	4	no	—
196	C₄⋊C₈	—	11	8	no	—
197	C₄⋊C₈	—	11	4	no	—
198	C₄×C₈	—	11	8	no	—
199	C₄×C₈	C₄×D₁₁	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	22	4	no	—
202	#95	—	22	4	no	—
203	#95	—	22	4	no	—
204	C₂×C₁₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₁₁	C₂×#427	1	1	yes	characteristic
206	C₂×C₁₁	#2913	1	1	yes	characteristic
207	C₂×C₁₁	#3380	1	1	yes	characteristic
208	C₂×C₁₁	#3383	1	1	yes	characteristic
209	C₂×C₁₁	#3393	1	1	yes	characteristic
210	C₂×C₁₁	#3396	1	1	yes	characteristic
211	C₂×Q₈	—	11	4	no	—
212	C₂×Q₈	—	22	2	no	—
213	C₂×Q₈	—	22	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	11	2	no	—
217	C₂²×C₄	—	11	2	no	—
218	C₂²×C₄	—	22	4	no	—
219	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
220	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
221	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	11	4	no	—
225	C₂×C₈	—	22	2	no	—
226	C₂×C₈	—	22	2	no	—
227	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
230	C₂×C₈	#4339	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	11	8	no	—
235	C₄⋊C₄	—	11	4	no	—
236	C₄⋊C₄	—	22	4	no	—
237	C₄⋊C₄	—	22	4	no	—
238	C₄⋊C₄	—	44	4	no	—
239	C₄⋊C₄	—	88	2	no	—
240	C₄⋊C₄	C₂×Dic₂₂	1	4	yes	—
241	C₄²	—	4	4	no	—
242	C₄²	—	11	4	no	—
243	C₄²	—	11	4	no	—
244	C₄²	—	88	4	no	—
245	C₄²	C₂×C₄×D₁₁	1	2	yes	—
246	C₄²	Q₈×D₁₁	1	2	yes	—
247	C₁₁	#22727	1	1	yes	characteristic, complemented, Sylow
248	C₂³	C₂×D₄×D₁₁	1	1	yes	characteristic
249	Q₈	—	22	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	11	4	no	—
259	C₂×C₄	—	11	4	no	—
260	C₂×C₄	—	22	2	no	—
261	C₂×C₄	—	22	2	no	—
262	C₂×C₄	—	22	2	no	—
263	C₂×C₄	—	22	2	no	—
264	C₂×C₄	—	44	4	no	—
265	C₂×C₄	—	44	4	no	—
266	C₂×C₄	—	44	4	no	—
267	C₂×C₄	C₄⋊C₄×D₁₁	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₁₁	1	2	yes	—
269	C₂×C₄	#66965	1	2	yes	—
270	C₂×C₄	#66972	1	2	yes	—
271	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
273	C₂×C₄	C₂×#4339	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
275	C₂×C₄	#67042	1	1	yes	characteristic
276	C₂×C₄	#67043	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	22	4	no	—
279	C₈	C₄×Dic₂₂	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
281	C₂²	D₄×Dic₂₂	1	1	yes	characteristic
282	C₂²	#10623084	1	1	yes	characteristic
283	C₂²	C₂×#66965	1	1	yes	characteristic
284	C₂²	C₂×#66972	1	1	yes	characteristic
285	C₂²	#10623300	1	1	yes	characteristic
286	C₂²	#10623302	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	22	4	no	—
291	C₄	—	22	4	no	—
292	C₄	—	88	4	no	—
293	C₄	#10622493	1	4	yes	—
294	C₄	C₂×C₄×Dic₂₂	1	1	yes	characteristic
295	C₄	#10623054	1	1	yes	characteristic
296	C₄	Q₈×Dic₂₂	1	1	yes	characteristic
297	C₄	#10623098	1	1	yes	characteristic
298	C₂	C₄⋊C₄×Dic₂₂	1	1	yes	characteristic
299	C₂	C₂×#10622493	1	1	yes	characteristic
300	C₂	#13077213	1	1	yes	characteristic
301	C₂	#13081481	1	1	yes	characteristic
302	C₂	#13081505	1	1	yes	characteristic
303	C₂	#13081602	1	1	yes	characteristic
304	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	C₄×C₁₁×#107	C₂
socle, prime radical	C₂³×C₁₁	C₂²×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄×#107	D₁₁
hypocenter	C₁₁	#22727
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
25. C₈×Dic₂₂	C₄	semidirect
47. C₁₁×#107	Q₈	semidirect
194. #107	Dic₂₂	semidirect
247. C₁₁	#22727	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₂	2
#13071571	2
#13069387	2
#13069379	2
#13068779	2
#13068776	2
#107×Dic₁₁	2
C₄×C₁₁×#107	2
#22727	11