Subgroups — C₂³×C₄⋊(C₁₁⋊C₈) #15617381

Order 2816.

3870 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₁⋊C₈	C₂	1	2	yes	maximal
2	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₁	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₁	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁	C₄	1	14	yes	—
8	C₂²×C₄²×C₁₁	C₂²	1	14	yes	—
9	C₂³×C₁₁⋊C₈	—	2	16	no	—
10	C₂³×C₁₁⋊C₈	C₂²	1	28	yes	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	112	yes	direct product factor
12	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₁₁	—	2	8	no	—
14	C₂³×C₄×C₁₁	C₈	1	16	yes	complemented
15	C₂³×C₄×C₁₁	C₂×C₄	1	14	yes	—
16	C₂³×C₄×C₁₁	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₁₁	C₂×C₄	1	14	yes	—
18	C₂³×C₄×C₁₁	D₄	1	8	yes	—
19	C₂³×C₄×C₁₁	Q₈	1	8	yes	—
20	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
21	C₂×C₄²×C₁₁	C₂×C₄	1	84	yes	—
22	C₂×C₄²×C₁₁	C₂³	1	28	yes	—
23	C₂²×C₁₁⋊C₈	—	2	112	no	—
24	C₂²×C₁₁⋊C₈	C₂³	1	56	yes	—
25	C₄⋊(C₁₁⋊C₈)	C₂³	1	64	yes	direct product factor
26	C₂³×C₄⋊C₈	—	11	1	no	maximal, Sylow
27	C₂⁴×C₁₁	C₄⋊C₄	1	8	yes	—
28	C₂⁴×C₁₁	C₂×C₈	1	8	yes	—
29	C₂⁴×C₁₁	M₁₆	1	8	yes	—
30	C₂⁴×C₁₁	C₂²×C₄	1	7	yes	—
31	C₂²×C₄×C₁₁	—	2	56	no	—
32	C₂²×C₄×C₁₁	C₄⋊C₄	1	56	yes	—
33	C₂²×C₄×C₁₁	C₂×C₈	1	112	yes	complemented
34	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
35	C₂²×C₄×C₁₁	C₂²×C₄	1	21	yes	—
36	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
37	C₂²×C₄×C₁₁	C₂×D₄	1	28	yes	—
38	C₂²×C₄×C₁₁	C₂×Q₈	1	28	yes	—
39	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
40	C₄²×C₁₁	C₂²×C₄	1	56	yes	—
41	C₄²×C₁₁	C₂⁴	1	8	yes	—
42	C₂×C₁₁⋊C₈	—	2	112	no	—
43	C₂×C₁₁⋊C₈	C₂⁴	1	16	yes	—
44	C₂⁴×C₈	—	11	2	no	—
45	C₂³×C₄²	D₁₁	1	1	yes	characteristic
46	C₂²×C₄⋊C₈	—	11	28	no	—
47	C₂³×C₁₁	C₄⋊C₈	1	64	yes	complemented
48	C₂³×C₁₁	C₂×C₄⋊C₄	1	28	yes	—
49	C₂³×C₁₁	C₂²×C₈	1	28	yes	—
50	C₂³×C₁₁	C₂×M₁₆	1	28	yes	—
51	C₂³×C₁₁	C₂³×C₄	1	7	yes	—
52	C₂×C₄×C₁₁	—	2	56	no	—
53	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	84	yes	—
54	C₂×C₄×C₁₁	C₂²×C₈	1	112	yes	complemented
55	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
56	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
57	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
58	C₂×C₄×C₁₁	C₂²×D₄	1	14	yes	—
59	C₂×C₄×C₁₁	C₂²×Q₈	1	14	yes	—
60	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
61	C₁₁⋊C₈	—	2	16	no	—
62	C₂⁴×C₄	Dic₁₁	1	1	yes	characteristic
63	C₂⁴×C₄	Dic₁₁	1	1	yes	characteristic
64	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
65	C₂³×C₈	—	11	28	no	—
66	C₂³×C₈	—	22	16	no	—
67	C₂²×C₄²	Dic₁₁	1	14	yes	—
68	C₂²×C₄²	C₂×D₁₁	1	14	yes	—
69	C₂×C₄⋊C₈	—	11	112	no	—
70	C₂²×C₁₁	C₂×C₄⋊C₈	1	112	yes	complemented
71	C₂²×C₁₁	C₂²×C₄⋊C₄	1	14	yes	—
72	C₂²×C₁₁	C₂³×C₈	1	14	yes	—
73	C₂²×C₁₁	C₂²×M₁₆	1	14	yes	—
74	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₁₁	—	2	8	no	—
76	C₄×C₁₁	C₂²×C₄⋊C₄	1	14	yes	—
77	C₄×C₁₁	C₂³×C₈	1	16	yes	complemented
78	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
79	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
80	C₂⁵	C₂×Dic₁₁	1	1	yes	characteristic
81	C₂³×C₄	—	2	8	no	—
82	C₂³×C₄	C₁₁⋊C₈	1	16	yes	complemented
83	C₂³×C₄	Dic₂₂	1	8	yes	—
84	C₂³×C₄	D₄₄	1	8	yes	—
85	C₂³×C₄	C₂×Dic₁₁	1	14	yes	—
86	C₂³×C₄	C₂×Dic₁₁	1	14	yes	—
87	C₂³×C₄	C₂×Dic₁₁	1	7	yes	—
88	C₂³×C₄	C₂²×D₁₁	1	7	yes	—
89	C₂²×C₈	—	11	56	no	—
90	C₂²×C₈	—	22	112	no	—
91	C₂×C₄²	C₂×Dic₁₁	1	84	yes	—
92	C₂×C₄²	C₂²×D₁₁	1	28	yes	—
93	C₄⋊C₈	—	11	64	no	—
94	C₂×C₁₁	C₂²×C₄⋊C₈	1	28	yes	complemented
95	C₂×C₁₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₁₁	C₂⁴×C₈	1	1	yes	characteristic
97	C₂×C₁₁	C₂³×M₁₆	1	1	yes	characteristic
98	C₂⁴	C₂×C₁₁⋊C₈	1	8	yes	—
99	C₂⁴	#4318	1	8	yes	—
100	C₂⁴	C₄⋊Dic₁₁	1	8	yes	—
101	C₂⁴	C₂²×Dic₁₁	1	7	yes	—
102	C₂²×C₄	—	2	56	no	—
103	C₂²×C₄	C₂×C₁₁⋊C₈	1	112	yes	complemented
104	C₂²×C₄	C₄⋊Dic₁₁	1	56	yes	—
105	C₂²×C₄	C₂×Dic₂₂	1	28	yes	—
106	C₂²×C₄	C₂×D₄₄	1	28	yes	—
107	C₂²×C₄	C₂²×Dic₁₁	1	28	yes	—
108	C₂²×C₄	C₂²×Dic₁₁	1	28	yes	—
109	C₂²×C₄	C₂²×Dic₁₁	1	21	yes	—
110	C₂²×C₄	C₂³×D₁₁	1	7	yes	—
111	C₂×C₈	—	11	16	no	—
112	C₂×C₈	—	22	112	no	—
113	C₄²	C₂²×Dic₁₁	1	56	yes	—
114	C₄²	C₂³×D₁₁	1	8	yes	—
115	C₁₁	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
116	C₂³	C₄⋊(C₁₁⋊C₈)	1	64	yes	direct product factor
117	C₂³	C₂²×C₁₁⋊C₈	1	28	yes	—
118	C₂³	C₂×#4318	1	28	yes	—
119	C₂³	C₂×C₄⋊Dic₁₁	1	28	yes	—
120	C₂³	C₂³×Dic₁₁	1	7	yes	—
121	C₂×C₄	—	2	56	no	—
122	C₂×C₄	C₂²×C₁₁⋊C₈	1	112	yes	complemented
123	C₂×C₄	C₂×C₄⋊Dic₁₁	1	84	yes	—
124	C₂×C₄	C₂²×Dic₂₂	1	14	yes	—
125	C₂×C₄	C₂²×D₄₄	1	14	yes	—
126	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
127	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
128	C₂×C₄	C₂³×Dic₁₁	1	7	yes	—
129	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
130	C₈	—	22	16	no	—
131	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	112	yes	direct product factor
132	C₂²	C₂³×C₁₁⋊C₈	1	14	yes	—
133	C₂²	C₂²×#4318	1	14	yes	—
134	C₂²	C₂²×C₄⋊Dic₁₁	1	14	yes	—
135	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	C₂³×C₁₁⋊C₈	1	16	yes	complemented
138	C₄	C₂²×C₄⋊Dic₁₁	1	14	yes	—
139	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
140	C₄	C₂³×D₄₄	1	1	yes	characteristic
141	C₂	C₂²×C₄⋊(C₁₁⋊C₈)	1	28	yes	direct product factor
142	C₂	C₂³×C₄⋊Dic₁₁	1	1	yes	characteristic
143	C₂	C₂⁴×C₁₁⋊C₈	1	1	yes	characteristic
144	C₂	C₂³×#4318	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₁
commutator	C₂×C₁₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂³×C₄²×C₁₁	C₂
socle, prime radical	C₂⁵×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂³×C₄²	D₁₁
hypocenter	C₁₁	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊(C₁₁⋊C₈)	C₂	direct (N × H)
11. C₂×C₄⋊(C₁₁⋊C₈)	C₂²	direct (N × H)
14. C₂³×C₄×C₁₁	C₈	semidirect
25. C₄⋊(C₁₁⋊C₈)	C₂³	direct (N × H)
33. C₂²×C₄×C₁₁	C₂×C₈	semidirect
47. C₂³×C₁₁	C₄⋊C₈	semidirect
54. C₂×C₄×C₁₁	C₂²×C₈	semidirect
70. C₂²×C₁₁	C₂×C₄⋊C₈	semidirect
77. C₄×C₁₁	C₂³×C₈	semidirect
82. C₂³×C₄	C₁₁⋊C₈	semidirect
94. C₂×C₁₁	C₂²×C₄⋊C₈	semidirect
103. C₂²×C₄	C₂×C₁₁⋊C₈	semidirect
115. C₁₁	C₂³×C₄⋊C₈	semidirect
116. C₂³	C₄⋊(C₁₁⋊C₈)	direct (N × H)
122. C₂×C₄	C₂²×C₁₁⋊C₈	semidirect
131. C₂²	C₂×C₄⋊(C₁₁⋊C₈)	direct (N × H)
137. C₄	C₂³×C₁₁⋊C₈	semidirect
141. C₂	C₂²×C₄⋊(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
C₂⁴×C₁₁⋊C₈	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂³×C₄²×C₁₁	2
C₂³×C₄⋊C₈	11

Subgroups — C23×C4⋊(C11⋊C8) #15617381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊(C₁₁⋊C₈) #15617381