Subgroups — #14907444

Order 2816.

570 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×C₁₁⋊C₁₆	C₂	1	6	yes	maximal, complemented
3	#13070415	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	3	yes	maximal, complemented
5	C₈×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₁×#41	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₁₁	C₄	1	3	yes	—
10	C₈×D₄×C₁₁	C₂²	1	3	yes	—
11	C₁₁×#431	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
13	C₂²×C₁₁⋊C₁₆	C₂²	1	6	yes	—
14	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	3	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
17	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
18	C₄×C₁₁⋊C₁₆	C₂²	1	6	yes	—
19	C₂×C₁₁×#41	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
22	C₄×Q₈×C₁₁	C₈	1	1	yes	characteristic
23	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×D₄×C₁₁	C₈	1	3	yes	—
25	C₄×D₄×C₁₁	C₂×C₄	1	3	yes	—
26	C₁₁×#117	C₈	1	3	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
28	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
29	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₁₁×C₂²⋊C₈	C₂×C₄	1	6	yes	—
31	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₁	C₂³	1	3	yes	—
34	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×C₁₁⋊C₁₆	—	4	6	no	—
36	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
37	C₂×C₁₁⋊C₁₆	C₂³	1	6	yes	—
38	#33502	—	11	1	no	maximal, Sylow
39	C₁₁×#41	C₁₆	1	8	yes	complemented
40	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
41	C₂×D₄×C₁₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₁₁	—	2	3	no	—
46	C₂×C₈×C₁₁	—	2	3	no	—
47	C₂×C₈×C₁₁	—	4	3	no	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
54	C₁₁×C₂²⋊C₄	C₂×C₈	1	6	yes	—
55	C₄²×C₁₁	C₂×C₈	1	3	yes	—
56	C₄²×C₁₁	C₂×C₈	1	3	yes	—
57	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
59	C₁₁⋊C₁₆	—	2	6	no	—
60	C₁₁⋊C₁₆	—	2	2	no	—
61	C₈×#41	D₁₁	1	1	yes	characteristic
62	Q₈×C₁₆	—	11	2	no	—
63	D₄×C₁₆	—	11	6	no	—
64	#2162	—	11	3	no	—
65	C₂×C₄⋊C₁₆	—	11	3	no	—
66	C₂³×C₁₁	C₂²×C₈	1	3	yes	—
67	Q₈×C₁₁	C₂×C₁₆	1	4	yes	complemented
68	D₄×C₁₁	C₂×C₁₆	1	12	yes	complemented
69	C₂×C₄×C₁₁	—	2	3	no	—
70	C₂×C₄×C₁₁	—	2	3	no	—
71	C₂×C₄×C₁₁	—	4	3	no	—
72	C₂×C₄×C₁₁	C₂×C₁₆	1	12	yes	complemented
73	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₁₁	—	2	3	no	—
80	C₈×C₁₁	—	2	1	no	—
81	C₈×C₁₁	—	2	3	no	—
82	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
83	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×#41	Dic₁₁	1	1	yes	characteristic
85	C₂²×C₁₆	—	11	6	no	—
86	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
87	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
88	C₈×D₄	Dic₁₁	1	3	yes	—
89	C₈×D₄	C₂×D₁₁	1	3	yes	—
90	#431	Dic₁₁	1	3	yes	—
91	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
92	C₄⋊C₁₆	—	11	3	no	—
93	C₄⋊C₁₆	—	11	6	no	—
94	C₄⋊C₁₆	—	11	1	no	—
95	C₂²⋊C₁₆	—	11	6	no	—
96	C₄×C₁₆	—	11	6	no	—
97	C₂²×C₁₁	—	2	3	no	—
98	C₂²×C₁₁	—	2	3	no	—
99	C₂²×C₁₁	C₂²×C₁₆	1	6	yes	—
100	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₁₁	—	2	1	no	—
102	C₄×C₁₁	—	2	3	no	—
103	C₄×C₁₁	—	2	3	no	—
104	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
105	C₄×C₁₁	C₂²×C₁₆	1	6	yes	—
106	C₄×C₁₁	#520	1	1	yes	characteristic
107	C₄×C₁₁	#568	1	1	yes	characteristic
108	C₂×#41	C₁₁⋊C₈	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
110	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
111	C₄×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
112	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
113	C₄×D₄	C₁₁⋊C₈	1	3	yes	—
114	C₄×D₄	C₂×Dic₁₁	1	3	yes	—
115	#117	C₁₁⋊C₈	1	3	yes	—
116	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
117	C₂×C₁₆	—	11	2	no	—
118	C₂×C₁₆	—	11	6	no	—
119	C₂×C₁₆	—	44	6	no	—
120	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
121	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
122	C₂²⋊C₈	C₂×Dic₁₁	1	6	yes	—
123	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
124	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
125	C₄×C₈	C₂²×D₁₁	1	3	yes	—
126	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
127	C₂×C₁₁	—	2	6	no	—
128	C₂×C₁₁	#2988	1	1	yes	characteristic
129	C₂×C₁₁	C₂³×C₁₆	1	1	yes	characteristic
130	C₂×C₁₁	#3407	1	1	yes	characteristic
131	#41	C₁₁⋊C₁₆	1	8	yes	complemented
132	C₂×Q₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
133	C₂×D₄	C₂×C₁₁⋊C₈	1	3	yes	—
134	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
136	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
137	C₂×C₈	—	2	3	no	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	—	4	3	no	—
140	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
141	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
142	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
143	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
144	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
145	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
146	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	6	yes	—
147	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
148	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
149	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
150	C₄²	C₂²×Dic₁₁	1	3	yes	—
151	C₁₆	—	22	6	no	—
152	C₁₆	—	22	2	no	—
153	C₁₁	#33502	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂²×C₁₁⋊C₈	1	3	yes	—
155	Q₈	C₂×C₁₁⋊C₁₆	1	4	yes	complemented
156	D₄	C₂×C₁₁⋊C₁₆	1	12	yes	complemented
157	C₂×C₄	—	2	3	no	—
158	C₂×C₄	—	2	3	no	—
159	C₂×C₄	—	4	3	no	—
160	C₂×C₄	C₂×C₁₁⋊C₁₆	1	12	yes	complemented
161	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
166	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	3	no	—
169	C₈	—	2	3	no	—
170	C₈	#67042	1	1	yes	characteristic
171	C₈	#67043	1	1	yes	characteristic
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	C₂²×C₁₁⋊C₁₆	1	6	yes	—
175	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	3	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂²×C₁₁⋊C₁₆	1	2	yes	—
180	C₄	C₂²×C₁₁⋊C₁₆	1	6	yes	—
181	C₄	#10623350	1	1	yes	characteristic
182	C₄	#10623358	1	1	yes	characteristic
183	C₂	—	2	6	no	—
184	C₂	#13077727	1	1	yes	characteristic
185	C₂	C₂³×C₁₁⋊C₁₆	1	1	yes	characteristic
186	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator	C₂×C₁₁	C₂³×C₁₆
Fitting	C₈×C₁₁×#41	C₂
socle	C₂²×C₁₁	C₂³×C₈
prime radical	C₂×C₁₁×#41	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₈×#41	D₁₁
hypocenter	C₁₁	#33502
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₁₁⋊C₁₆	C₂	semidirect
2. D₄×C₁₁⋊C₁₆	C₂	semidirect
3. #13070415	C₂	semidirect
4. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
39. C₁₁×#41	C₁₆	semidirect
67. Q₈×C₁₁	C₂×C₁₆	semidirect
68. D₄×C₁₁	C₂×C₁₆	semidirect
72. C₂×C₄×C₁₁	C₂×C₁₆	semidirect
131. #41	C₁₁⋊C₁₆	semidirect
153. C₁₁	#33502	semidirect
155. Q₈	C₂×C₁₁⋊C₁₆	semidirect
156. D₄	C₂×C₁₁⋊C₁₆	semidirect
160. C₂×C₄	C₂×C₁₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₁⋊C₁₆	2
D₄×C₁₁⋊C₁₆	2
#13070415	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₈×C₁₁×#41	2
#33502	11