Subgroups — #14907356

Order 2816.

662 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₄	C₂	1	1	yes	maximal, characteristic
2	C₁₆×D₄₄	C₂	1	4	yes	maximal, complemented
3	#13070435	C₂	1	4	yes	maximal, complemented
4	#13070415	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₁×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	C₁₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₁	C₂²	1	2	yes	—
8	C₂×C₄×D₄₄	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₁)⋊C₈	C₄	1	2	yes	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×C₁₆×D₁₁	C₂²	1	8	yes	complemented
13	C₈×D₄₄	C₄	1	2	yes	—
14	C₈×D₄₄	C₄	1	2	yes	—
15	C₈×D₄₄	C₂²	1	2	yes	—
16	C₈×D₄₄	C₂²	1	2	yes	—
17	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₄×C₁₁×C₁₆	C₂²	1	2	yes	complemented
20	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
21	(C₂×D₁₁)⋊C₁₆	C₂²	1	8	yes	complemented
22	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	complemented
23	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	complemented
24	C₂²×D₄₄	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×(C₂×D₁₁)⋊C₄	C₈	1	2	yes	—
30	C₂×C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
31	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₁₁	—	4	4	no	—
33	C₂×C₈×D₁₁	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₁	C₂³	1	4	yes	—
35	C₄×D₄₄	C₈	1	2	yes	—
36	C₄×D₄₄	C₈	1	2	yes	—
37	C₄×D₄₄	C₂×C₄	1	2	yes	—
38	C₄×D₄₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
40	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
45	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
46	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
47	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
48	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
49	C₁₆×D₁₁	—	4	8	no	—
50	#33486	—	11	1	no	maximal, Sylow
51	C₂³×D₁₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
56	C₂×D₄₄	C₁₆	1	8	yes	complemented
57	C₂×D₄₄	C₂×C₈	1	2	yes	—
58	C₂×D₄₄	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₁₁	—	4	2	no	—
60	C₂×C₄×D₁₁	—	4	2	no	—
61	C₂×C₄×D₁₁	C₂×C₈	1	4	yes	—
62	C₂×C₄×D₁₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	—	4	1	no	—
65	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
69	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
74	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
75	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
76	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
77	C₂×C₁₁⋊C₈	—	4	2	no	—
78	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
79	C₈×D₁₁	—	2	8	no	—
80	C₈×D₁₁	—	4	4	no	—
81	C₁₁×C₁₆	—	2	4	no	—
82	C₁₁⋊C₁₆	—	2	4	no	—
83	C₂×C₈×D₄	—	11	1	no	—
84	D₄×C₁₆	—	11	4	no	—
85	D₄×C₁₆	—	11	4	no	—
86	#2162	—	11	1	no	—
87	#2162	D₁₁	1	1	yes	characteristic, complemented
88	C₂×C₂²⋊C₁₆	—	11	4	no	—
89	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
90	C₂²×D₁₁	—	4	2	no	—
91	C₂²×D₁₁	—	4	2	no	—
92	C₂²×D₁₁	C₂×C₁₆	1	8	yes	complemented
93	C₂²×D₁₁	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	4	1	no	—
97	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	complemented
98	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₁₁	—	4	2	no	—
105	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
106	D₄₄	C₂×C₁₆	1	8	yes	complemented
107	D₄₄	C₂×C₁₆	1	8	yes	complemented
108	C₄×D₁₁	—	2	4	no	—
109	C₄×D₁₁	—	2	4	no	—
110	C₄×D₁₁	—	4	4	no	—
111	C₈×C₁₁	—	2	1	no	—
112	C₈×C₁₁	—	2	1	no	—
113	C₈×C₁₁	—	2	1	no	—
114	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
115	C₁₁⋊C₈	—	2	4	no	—
116	C₂³×C₈	—	11	2	no	—
117	C₂×C₄×D₄	—	11	1	no	—
118	C₂²×C₁₆	—	11	8	no	—
119	C₈×D₄	—	11	2	no	—
120	C₈×D₄	—	11	2	no	—
121	C₈×D₄	—	11	2	no	—
122	C₈×D₄	—	11	2	no	—
123	C₂×C₄⋊C₈	—	11	1	no	—
124	C₂×C₂²⋊C₈	—	11	2	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
126	C₄⋊C₁₆	—	11	2	no	—
127	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	complemented
128	C₂²⋊C₁₆	—	11	2	no	—
129	C₂²⋊C₁₆	—	11	8	no	—
130	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
131	C₄×C₁₆	—	11	2	no	—
132	C₄×C₁₆	C₂×D₁₁	1	2	yes	complemented
133	C₂²×C₁₁	—	2	1	no	—
134	C₂²×C₁₁	—	2	1	no	—
135	C₂²×C₁₁	C₂²×C₁₆	1	2	yes	—
136	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₁₁	—	2	4	no	—
138	C₂×D₁₁	—	2	4	no	—
139	C₂×D₁₁	—	4	8	no	—
140	C₂×D₁₁	C₂²×C₁₆	1	8	yes	—
141	C₄×C₁₁	—	2	1	no	—
142	C₄×C₁₁	—	2	1	no	—
143	C₄×C₁₁	—	2	1	no	—
144	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
145	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
146	C₄×C₁₁	#518	1	1	yes	characteristic
147	C₄×C₁₁	#568	1	1	yes	characteristic
148	Dic₁₁	—	2	4	no	—
149	C₂²×D₄	—	11	1	no	—
150	C₂³×C₄	—	11	2	no	—
151	C₂²×C₈	—	11	2	no	—
152	C₂²×C₈	—	11	4	no	—
153	C₂²×C₈	—	11	4	no	—
154	C₂²×C₈	—	44	4	no	—
155	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
157	C₄×D₄	—	11	2	no	—
158	C₄×D₄	—	11	2	no	—
159	C₄×D₄	—	11	2	no	—
160	C₄×D₄	—	11	2	no	—
161	C₂×C₄⋊C₄	—	11	1	no	—
162	C₂×C₂²⋊C₄	—	11	2	no	—
163	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
164	C₂×C₁₆	—	11	4	no	—
165	C₂×C₁₆	—	44	8	no	—
166	C₂×C₁₆	C₂²×D₁₁	1	4	yes	—
167	C₄⋊C₈	—	11	2	no	—
168	C₄⋊C₈	—	11	2	no	—
169	C₂²⋊C₈	—	11	4	no	—
170	C₂²⋊C₈	—	11	4	no	—
171	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
172	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
175	C₂×C₁₁	—	2	2	no	—
176	C₂×C₁₁	#2972	1	1	yes	characteristic
177	C₂×C₁₁	C₂³×C₁₆	1	1	yes	characteristic
178	C₂×C₁₁	#3407	1	1	yes	characteristic
179	D₁₁	—	2	8	no	—
180	C₂⁴	—	11	2	no	—
181	C₂×D₄	—	11	8	no	—
182	C₂×D₄	—	11	2	no	—
183	C₂×D₄	—	11	2	no	—
184	C₂²×C₄	—	11	2	no	—
185	C₂²×C₄	—	11	4	no	—
186	C₂²×C₄	—	11	4	no	—
187	C₂²×C₄	—	44	2	no	—
188	C₂²×C₄	—	44	2	no	—
189	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
190	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	11	4	no	—
195	C₂×C₈	—	22	8	no	—
196	C₂×C₈	—	44	4	no	—
197	C₂×C₈	—	44	2	no	—
198	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
201	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
202	C₄⋊C₄	—	11	2	no	—
203	C₄⋊C₄	—	11	2	no	—
204	C₂²⋊C₄	—	11	4	no	—
205	C₂²⋊C₄	—	11	4	no	—
206	C₄²	C₈×D₁₁	1	1	yes	characteristic
207	C₄²	C₈×D₁₁	1	1	yes	characteristic
208	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
209	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
210	C₁₆	—	2	4	no	—
211	C₁₆	—	22	4	no	—
212	C₁₁	#33486	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	11	4	no	—
214	C₂³	—	11	8	no	—
215	C₂³	—	44	2	no	—
216	C₂³	—	44	2	no	—
217	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
218	D₄	—	11	8	no	—
219	D₄	—	11	8	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	11	4	no	—
224	C₂×C₄	—	22	4	no	—
225	C₂×C₄	—	22	4	no	—
226	C₂×C₄	—	44	4	no	—
227	C₂×C₄	—	44	2	no	—
228	C₂×C₄	C₁₆×D₁₁	1	4	yes	complemented
229	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
230	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
232	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	22	4	no	—
239	C₈	#67042	1	2	yes	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	11	8	no	—
243	C₂²	—	22	4	no	—
244	C₂²	—	22	4	no	—
245	C₂²	—	44	8	no	—
246	C₂²	C₂×C₁₆×D₁₁	1	2	yes	—
247	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	22	4	no	—
252	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
253	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
254	C₄	#10623052	1	1	yes	characteristic
255	C₄	#10623276	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	22	8	no	—
258	C₂	#13077639	1	1	yes	characteristic
259	C₂	C₂²×C₁₆×D₁₁	1	1	yes	characteristic
260	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator	C₂×C₁₁	C₂³×C₁₆
Fitting	C₁₁×#2162	C₂
socle	C₂²×C₁₁	C₂³×C₈
prime radical	C₂²×D₄₄	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2162	D₁₁
hypocenter	C₁₁	#33486
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. C₁₆×D₄₄	C₂	semidirect
3. #13070435	C₂	semidirect
4. #13070415	C₂	semidirect
5. D₁₁×C₂²⋊C₁₆	C₂	semidirect
6. C₁₁×#2162	C₂	semidirect
12. C₂×C₁₆×D₁₁	C₂²	semidirect
17. C₁₁×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₁×C₁₆	C₂²	semidirect
21. (C₂×D₁₁)⋊C₁₆	C₂²	semidirect
22. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
23. C₄×C₁₁⋊C₁₆	C₂²	semidirect
56. C₂×D₄₄	C₁₆	semidirect
87. #2162	D₁₁	semidirect
92. C₂²×D₁₁	C₂×C₁₆	semidirect
97. C₂×C₄×C₁₁	C₂×C₁₆	semidirect
106. D₄₄	C₂×C₁₆	semidirect
107. D₄₄	C₂×C₁₆	semidirect
127. C₄⋊C₁₆	C₂×D₁₁	semidirect
132. C₄×C₁₆	C₂×D₁₁	semidirect
212. C₁₁	#33486	semidirect
228. C₂×C₄	C₁₆×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄₄	2
C₁₆×D₄₄	2
#13070435	2
#13070415	2
D₁₁×C₂²⋊C₁₆	2
C₁₁×#2162	2
#33486	11