Subgroups — #14907413

Order 2816.

526 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₁₆×Dic₂₂	C₂	1	4	yes	maximal, complemented
3	#13070518	C₂	1	4	yes	maximal, complemented
4	#13070415	C₂	1	1	yes	maximal, characteristic
5	#13070412	C₂	1	4	yes	maximal
6	C₁₁×#2162	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₂₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₁⋊C₈	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
12	C₈×Dic₂₂	C₄	1	2	yes	—
13	C₈×Dic₂₂	C₄	1	2	yes	—
14	C₈×Dic₂₂	C₂²	1	2	yes	—
15	C₈×Dic₂₂	C₂²	1	2	yes	—
16	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
19	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
20	Dic₁₁⋊C₁₆	C₂²	1	8	yes	—
21	C₁₆×Dic₁₁	C₂²	1	8	yes	—
22	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
23	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
24	C₂²×Dic₂₂	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
29	C₂×Dic₁₁⋊C₄	C₈	1	2	yes	—
30	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
31	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₂	C₈	1	2	yes	—
33	C₄×Dic₂₂	C₈	1	2	yes	—
34	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
36	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
37	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	Dic₁₁⋊C₈	C₂×C₄	1	4	yes	—
42	Dic₁₁⋊C₈	C₂×C₄	1	4	yes	—
43	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
44	C₈×Dic₁₁	C₂³	1	4	yes	—
45	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
46	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
48	#33496	—	11	1	no	maximal, Sylow
49	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
53	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
54	C₂×Dic₂₂	C₁₆	1	8	yes	complemented
55	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
56	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₁	—	2	2	no	—
58	C₂×C₈×C₁₁	—	4	1	no	—
59	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
63	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
68	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
69	Dic₁₁⋊C₄	C₂×C₈	1	4	yes	—
70	Dic₁₁⋊C₄	C₂×C₈	1	4	yes	—
71	C₄×Dic₁₁	C₂×C₈	1	4	yes	—
72	C₄×Dic₁₁	C₂²×C₄	1	4	yes	—
73	C₂×C₁₁⋊C₈	—	4	2	no	—
74	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
75	C₁₁×C₁₆	—	2	4	no	—
76	C₁₁⋊C₁₆	—	2	4	no	—
77	C₂×C₈×Q₈	—	11	1	no	—
78	Q₈×C₁₆	—	11	4	no	—
79	Q₈×C₁₆	—	11	4	no	—
80	#2162	—	11	1	no	—
81	#2162	—	11	4	no	—
82	#2162	D₁₁	1	1	yes	characteristic
83	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₁	—	2	1	no	—
85	C₂×C₄×C₁₁	—	2	1	no	—
86	C₂×C₄×C₁₁	—	4	1	no	—
87	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	—
88	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₁₁	—	4	2	no	—
95	C₂×Dic₁₁	C₂×C₁₆	1	8	yes	—
96	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
97	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
98	Dic₂₂	C₂×C₁₆	1	8	yes	—
99	Dic₂₂	C₂×C₁₆	1	8	yes	—
100	C₈×C₁₁	—	2	1	no	—
101	C₈×C₁₁	—	2	1	no	—
102	C₈×C₁₁	—	2	1	no	—
103	C₈×C₁₁	2₋¹⁺⁴	1	2	yes	—
104	C₁₁⋊C₈	—	2	4	no	—
105	C₂×C₄×Q₈	—	11	1	no	—
106	C₈×Q₈	—	11	2	no	—
107	C₈×Q₈	—	11	2	no	—
108	C₈×Q₈	—	11	2	no	—
109	C₈×Q₈	—	11	2	no	—
110	C₂×C₄⋊C₈	—	11	2	no	—
111	C₂×C₄⋊C₈	—	11	1	no	—
112	C₂×C₄×C₈	—	11	2	no	—
113	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
114	C₄⋊C₁₆	—	11	8	no	—
115	C₄⋊C₁₆	—	11	2	no	—
116	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
117	C₂²⋊C₁₆	—	11	2	no	—
118	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
119	C₄×C₁₆	—	11	8	no	—
120	C₄×C₁₆	—	11	2	no	—
121	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
122	C₂²×C₁₁	—	2	1	no	—
123	C₂²×C₁₁	—	2	1	no	—
124	C₂²×C₁₁	C₂²×C₁₆	1	2	yes	—
125	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₁₁	—	2	1	no	—
127	C₄×C₁₁	—	2	1	no	—
128	C₄×C₁₁	—	2	1	no	—
129	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
130	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
131	C₄×C₁₁	#519	1	1	yes	characteristic
132	C₄×C₁₁	#568	1	1	yes	characteristic
133	Dic₁₁	—	2	4	no	—
134	Dic₁₁	C₂²×C₁₆	1	8	yes	—
135	C₂²×Q₈	—	11	1	no	—
136	C₂²×C₈	—	11	2	no	—
137	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
139	C₄×Q₈	—	11	2	no	—
140	C₄×Q₈	—	11	2	no	—
141	C₄×Q₈	—	11	2	no	—
142	C₄×Q₈	—	11	2	no	—
143	C₂×C₄⋊C₄	—	11	2	no	—
144	C₂×C₄⋊C₄	—	11	1	no	—
145	C₂×C₄²	—	11	2	no	—
146	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
147	C₂×C₁₆	—	11	4	no	—
148	C₂×C₁₆	C₂²×D₁₁	1	4	yes	—
149	C₄⋊C₈	—	11	4	no	—
150	C₄⋊C₈	—	11	4	no	—
151	C₄⋊C₈	—	11	2	no	—
152	C₄⋊C₈	—	11	2	no	—
153	C₄×C₈	—	11	4	no	—
154	C₄×C₈	—	11	4	no	—
155	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
156	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
159	C₂×C₁₁	—	2	2	no	—
160	C₂×C₁₁	#2982	1	1	yes	characteristic
161	C₂×C₁₁	C₂³×C₁₆	1	1	yes	characteristic
162	C₂×C₁₁	#3407	1	1	yes	characteristic
163	C₂×Q₈	—	11	8	no	—
164	C₂×Q₈	—	11	2	no	—
165	C₂×Q₈	—	11	2	no	—
166	C₂²×C₄	—	11	2	no	—
167	C₂²×C₄	—	11	2	no	—
168	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
169	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	11	4	no	—
174	C₂×C₈	—	44	2	no	—
175	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
178	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
179	C₄⋊C₄	—	11	4	no	—
180	C₄⋊C₄	—	11	4	no	—
181	C₄⋊C₄	—	11	2	no	—
182	C₄⋊C₄	—	11	2	no	—
183	C₄²	—	11	4	no	—
184	C₄²	—	11	4	no	—
185	C₄²	C₈×D₁₁	1	1	yes	characteristic
186	C₄²	C₈×D₁₁	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
189	C₁₆	—	2	4	no	—
190	C₁₆	—	22	4	no	—
191	C₁₁	#33496	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
193	Q₈	—	11	8	no	—
194	Q₈	—	11	8	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	11	4	no	—
199	C₂×C₄	—	11	8	no	—
200	C₂×C₄	—	11	4	no	—
201	C₂×C₄	—	44	2	no	—
202	C₂×C₄	C₁₆×D₁₁	1	4	yes	—
203	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
205	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	22	4	no	—
213	C₈	#67043	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×C₁₆×D₁₁	1	2	yes	—
217	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	11	8	no	—
222	C₄	—	22	4	no	—
223	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
224	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
225	C₄	#10623047	1	1	yes	characteristic
226	C₄	#10623276	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#13077696	1	1	yes	characteristic
229	C₂	C₂²×C₁₆×D₁₁	1	1	yes	characteristic
230	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	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2162	D₁₁
hypocenter	C₁₁	#33496
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×Dic₂₂	C₂	semidirect
3. #13070518	C₂	semidirect
54. C₂×Dic₂₂	C₁₆	semidirect
191. C₁₁	#33496	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
C₁₆×Dic₂₂	2
#13070518	2
#13070415	2
#13070412	2
C₁₁×#2162	2
#33496	11