Subgroups — C₂²×Q₈×C₁₁⋊C₈ #15618125

Order 2816.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₁⋊C₈	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	3	yes	maximal
3	C₂²×C₄×C₁₁⋊C₈	C₂	1	3	yes	maximal
4	C₂²×C₄×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₁	C₄	1	12	yes	—
7	C₂×C₄×Q₈×C₁₁	C₂²	1	12	yes	—
8	C₂²×C₁₁×C₄⋊C₄	C₄	1	3	yes	—
9	C₂²×C₄²×C₁₁	C₂²	1	3	yes	—
10	C₂³×C₁₁⋊C₈	C₂²	1	3	yes	—
11	C₂³×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	Q₈×C₁₁⋊C₈	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	36	yes	—
14	C₂×C₄×C₁₁⋊C₈	C₂²	1	36	yes	—
15	C₂²×Q₈×C₁₁	C₈	1	16	yes	complemented
16	C₂²×Q₈×C₁₁	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₁₁	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₁₁	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
20	C₄×Q₈×C₁₁	C₂×C₄	1	48	yes	—
21	C₄×Q₈×C₁₁	C₂³	1	16	yes	—
22	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	36	yes	—
23	C₂×C₄²×C₁₁	C₂×C₄	1	18	yes	—
24	C₂×C₄²×C₁₁	C₂³	1	18	yes	—
25	C₂²×C₁₁⋊C₈	—	2	12	no	—
26	C₂²×C₁₁⋊C₈	Q₈	1	8	yes	direct product factor
27	C₂²×C₁₁⋊C₈	C₂³	1	18	yes	—
28	C₂²×C₁₁⋊C₈	C₂³	1	6	yes	—
29	C₄⋊(C₁₁⋊C₈)	C₂³	1	48	yes	—
30	C₄×C₁₁⋊C₈	C₂³	1	48	yes	—
31	C₂²×C₈×Q₈	—	11	1	no	maximal, Sylow
32	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₁	C₂×C₈	1	96	yes	complemented
34	C₂×Q₈×C₁₁	C₂²×C₄	1	16	yes	—
35	C₂²×C₄×C₁₁	—	2	12	no	—
36	C₂²×C₄×C₁₁	C₂×C₈	1	24	yes	—
37	C₂²×C₄×C₁₁	C₂²×C₄	1	18	yes	—
38	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₁₁	C₂²×C₄	1	18	yes	—
40	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₁₁	#41	1	4	yes	—
42	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
43	C₁₁×C₄⋊C₄	C₂²×C₄	1	48	yes	—
44	C₄²×C₁₁	C₂²×C₄	1	36	yes	—
45	C₄²×C₁₁	C₂⁴	1	12	yes	—
46	C₂×C₁₁⋊C₈	—	2	36	no	—
47	C₂×C₁₁⋊C₈	C₂×Q₈	1	24	yes	direct product factor
48	C₂×C₁₁⋊C₈	C₂⁴	1	12	yes	—
49	C₂×C₁₁⋊C₈	C₂⁴	1	4	yes	—
50	C₂²×C₄×Q₈	D₁₁	1	1	yes	characteristic
51	C₂×C₈×Q₈	—	11	24	no	—
52	C₂²×C₄⋊C₈	—	11	3	no	—
53	C₂²×C₄×C₈	—	11	3	no	—
54	C₂³×C₁₁	C₄×Q₈	1	4	yes	—
55	C₂³×C₁₁	C₂²×C₈	1	4	yes	—
56	C₂³×C₁₁	#131	1	4	yes	—
57	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
58	Q₈×C₁₁	C₂²×C₈	1	64	yes	complemented
59	C₂×C₄×C₁₁	—	2	36	no	—
60	C₂×C₄×C₁₁	C₄×Q₈	1	12	yes	—
61	C₂×C₄×C₁₁	C₂²×C₈	1	72	yes	—
62	C₂×C₄×C₁₁	C₂³×C₄	1	12	yes	—
63	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
64	C₂×C₄×C₁₁	C₂³×C₄	1	12	yes	—
65	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
66	C₂×C₄×C₁₁	C₂×#41	1	6	yes	—
67	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
68	C₁₁⋊C₈	—	2	12	no	—
69	C₁₁⋊C₈	C₂²×Q₈	1	8	yes	direct product factor
70	C₂³×Q₈	Dic₁₁	1	1	yes	characteristic
71	C₂³×C₈	—	11	1	no	—
72	C₂³×C₈	—	11	3	no	—
73	C₂×C₄×Q₈	Dic₁₁	1	12	yes	—
74	C₂×C₄×Q₈	C₂×D₁₁	1	12	yes	—
75	C₂²×C₄⋊C₄	Dic₁₁	1	3	yes	—
76	C₂²×C₄²	C₂×D₁₁	1	3	yes	—
77	C₈×Q₈	—	11	64	no	—
78	C₂×C₄⋊C₈	—	11	36	no	—
79	C₂×C₄×C₈	—	11	36	no	—
80	C₂²×C₁₁	C₈×Q₈	1	16	yes	complemented
81	C₂²×C₁₁	C₂×C₄×Q₈	1	6	yes	—
82	C₂²×C₁₁	C₂³×C₈	1	6	yes	—
83	C₂²×C₁₁	C₂×#131	1	6	yes	—
84	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
85	C₄×C₁₁	—	2	12	no	—
86	C₄×C₁₁	C₂×C₄×Q₈	1	6	yes	—
87	C₄×C₁₁	C₂³×C₈	1	24	yes	—
88	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
89	C₄×C₁₁	C₂²×#41	1	1	yes	characteristic
90	C₂²×Q₈	C₁₁⋊C₈	1	16	yes	direct product factor
91	C₂²×Q₈	C₂×Dic₁₁	1	12	yes	—
92	C₂³×C₄	C₂×Dic₁₁	1	3	yes	—
93	C₂³×C₄	C₂×Dic₁₁	1	3	yes	—
94	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
95	C₂²×C₈	—	11	8	no	—
96	C₂²×C₈	—	11	6	no	—
97	C₂²×C₈	—	11	18	no	—
98	C₂²×C₈	—	22	12	no	—
99	C₄×Q₈	C₂×Dic₁₁	1	48	yes	—
100	C₄×Q₈	C₂²×D₁₁	1	16	yes	—
101	C₂×C₄⋊C₄	C₂×Dic₁₁	1	36	yes	—
102	C₂×C₄²	C₂×Dic₁₁	1	18	yes	—
103	C₂×C₄²	C₂²×D₁₁	1	18	yes	—
104	C₄⋊C₈	—	11	48	no	—
105	C₄×C₈	—	11	48	no	—
106	C₂×C₁₁	C₂×C₈×Q₈	1	12	yes	complemented
107	C₂×C₁₁	C₂²×C₄×Q₈	1	1	yes	characteristic
108	C₂×C₁₁	C₂⁴×C₈	1	1	yes	characteristic
109	C₂×C₁₁	C₂²×#131	1	1	yes	characteristic
110	C₂⁴	C₂²×Dic₁₁	1	1	yes	characteristic
111	C₂×Q₈	C₂×C₁₁⋊C₈	1	96	yes	direct product factor
112	C₂×Q₈	C₂²×Dic₁₁	1	16	yes	—
113	C₂²×C₄	—	2	12	no	—
114	C₂²×C₄	C₂×C₁₁⋊C₈	1	24	yes	—
115	C₂²×C₄	Q₈×D₁₁	1	4	yes	—
116	C₂²×C₄	#4343	1	4	yes	—
117	C₂²×C₄	C₂²×Dic₁₁	1	18	yes	—
118	C₂²×C₄	C₂²×Dic₁₁	1	18	yes	—
119	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
120	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
121	C₂×C₈	—	11	24	no	—
122	C₂×C₈	—	11	4	no	—
123	C₂×C₈	—	11	12	no	—
124	C₂×C₈	—	22	36	no	—
125	C₄⋊C₄	C₂²×Dic₁₁	1	48	yes	—
126	C₄²	C₂²×Dic₁₁	1	36	yes	—
127	C₄²	C₂³×D₁₁	1	12	yes	—
128	C₁₁	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₂²×C₁₁⋊C₈	1	4	yes	—
130	C₂³	Q₈×Dic₁₁	1	4	yes	—
131	C₂³	#67001	1	4	yes	—
132	C₂³	C₂³×Dic₁₁	1	3	yes	—
133	Q₈	C₂²×C₁₁⋊C₈	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	C₂²×C₁₁⋊C₈	1	72	yes	—
136	C₂×C₄	Q₈×Dic₁₁	1	12	yes	—
137	C₂×C₄	C₂×Q₈×D₁₁	1	6	yes	—
138	C₂×C₄	C₂×#4343	1	6	yes	—
139	C₂×C₄	C₂³×Dic₁₁	1	12	yes	—
140	C₂×C₄	C₂³×Dic₁₁	1	12	yes	—
141	C₂×C₄	C₂³×Dic₁₁	1	3	yes	—
142	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
143	C₈	—	11	8	no	—
144	C₈	—	22	12	no	—
145	C₂²	Q₈×C₁₁⋊C₈	1	16	yes	direct product factor
146	C₂²	C₂³×C₁₁⋊C₈	1	6	yes	—
147	C₂²	C₂×Q₈×Dic₁₁	1	6	yes	—
148	C₂²	C₂×#67001	1	6	yes	—
149	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂³×C₁₁⋊C₈	1	24	yes	—
152	C₄	C₂×Q₈×Dic₁₁	1	6	yes	—
153	C₄	C₂²×Q₈×D₁₁	1	1	yes	characteristic
154	C₄	C₂²×#4343	1	1	yes	characteristic
155	C₂	C₂×Q₈×C₁₁⋊C₈	1	12	yes	direct product factor
156	C₂	C₂²×Q₈×Dic₁₁	1	1	yes	characteristic
157	C₂	C₂⁴×C₁₁⋊C₈	1	1	yes	characteristic
158	C₂	C₂²×#67001	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₁
commutator	C₂×C₁₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂²×C₄×Q₈×C₁₁	C₂
socle, prime radical	C₂⁴×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₁
hypocenter	C₁₁	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₁₁⋊C₈	C₂	direct (N × H)
12. Q₈×C₁₁⋊C₈	C₂²	direct (N × H)
15. C₂²×Q₈×C₁₁	C₈	semidirect
26. C₂²×C₁₁⋊C₈	Q₈	direct (N × H)
33. C₂×Q₈×C₁₁	C₂×C₈	semidirect
47. C₂×C₁₁⋊C₈	C₂×Q₈	direct (N × H)
58. Q₈×C₁₁	C₂²×C₈	semidirect
69. C₁₁⋊C₈	C₂²×Q₈	direct (N × H)
80. C₂²×C₁₁	C₈×Q₈	semidirect
90. C₂²×Q₈	C₁₁⋊C₈	direct (N × H)
106. C₂×C₁₁	C₂×C₈×Q₈	semidirect
111. C₂×Q₈	C₂×C₁₁⋊C₈	direct (N × H)
128. C₁₁	C₂²×C₈×Q₈	semidirect
133. Q₈	C₂²×C₁₁⋊C₈	direct (N × H)
145. C₂²	Q₈×C₁₁⋊C₈	direct (N × H)
155. C₂	C₂×Q₈×C₁₁⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₁⋊C₈	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂²×C₄×C₁₁⋊C₈	2
C₂²×C₄×Q₈×C₁₁	2
C₂²×C₈×Q₈	11

Subgroups — C22×Q8×C11⋊C8 #15618125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×C₁₁⋊C₈ #15618125