Group #41013142 — subgroups

Subgroups — #41013142

Order 27008.

296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₁₁	C₂	1	2	yes	maximal
2	C₄²×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212901	C₂	1	1	yes	maximal, characteristic
4	#27212900	C₂	1	2	yes	maximal
5	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₁₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₂₁₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₈×Dic₂₁₁	C₄	1	8	yes	complemented
12	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
15	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₂₁₁	—	2	4	no	—
21	C₄×Dic₂₁₁	—	2	1	no	—
22	C₄×Dic₂₁₁	—	2	1	no	—
23	C₄×Dic₂₁₁	C₈	1	8	yes	complemented
24	C₄×Dic₂₁₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₄²×C₂₁₁	—	2	2	no	—
31	C₂×Dic₂₁₁	—	2	2	no	—
32	C₂×Dic₂₁₁	—	2	2	no	—
33	C₂×Dic₂₁₁	C₄²	1	2	yes	—
34	C₂×Dic₂₁₁	C₄²	1	2	yes	—
35	C₂×Dic₂₁₁	C₂×C₈	1	4	yes	—
36	C₂×Dic₂₁₁	C₂×C₈	1	4	yes	—
37	C₂×Dic₂₁₁	M₁₆	1	4	yes	—
38	C₂₁₁⋊C₈	—	2	4	no	—
39	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	#41	1	2	yes	—
45	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
47	C₈×C₂₁₁	—	2	4	no	—
48	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₂₁₁	#117	1	1	yes	characteristic
50	C₂²×C₂₁₁	#117	1	1	yes	characteristic
51	C₂²×C₂₁₁	C₂²×C₈	1	2	yes	—
52	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
53	C₄×C₂₁₁	—	2	4	no	—
54	C₄×C₂₁₁	#117	1	2	yes	—
55	C₄×C₂₁₁	#117	1	2	yes	—
56	Dic₂₁₁	—	2	4	no	—
57	Dic₂₁₁	C₄×C₈	1	4	yes	—
58	Dic₂₁₁	#97	1	4	yes	—
59	C₂×C₂₁₁	#376	1	1	yes	characteristic
60	C₂×C₂₁₁	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₂₁₁	C₂×#97	1	1	yes	characteristic
62	C₂×C₂₁₁	#431	1	2	yes	—
63	C₂×C₂₁₁	#431	1	2	yes	—
64	C₂₁₁	#1744	1	1	yes	characteristic, complemented, Sylow
65	#1744	—	211	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	211	2	no	—
67	C₄³	—	211	1	no	—
68	#336	—	211	2	no	—
69	#336	—	211	1	no	—
70	#336	D₂₁₁	1	1	yes	characteristic
71	C₂²×C₈	—	211	2	no	—
72	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
73	C₂×C₄²	—	211	2	no	—
74	C₂×C₄²	—	211	1	no	—
75	C₂×C₄²	—	211	2	no	—
76	C₂×C₄²	—	211	1	no	—
77	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
78	C₄×C₈	—	211	8	no	—
79	C₂²×C₄	—	211	2	no	—
80	C₂²×C₄	—	211	1	no	—
81	C₂²×C₄	—	211	1	no	—
82	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	422	2	no	—
87	C₂×C₈	—	422	2	no	—
88	C₂×C₈	—	211	4	no	—
89	C₂×C₈	C₄×D₂₁₁	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	422	4	no	—
92	C₄²	—	422	1	no	—
93	C₄²	—	422	1	no	—
94	C₄²	—	211	8	no	—
95	C₄²	—	211	4	no	—
96	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	422	2	no	—
100	C₂×C₄	—	422	2	no	—
101	C₂×C₄	—	211	4	no	—
102	C₂×C₄	—	211	2	no	—
103	C₂×C₄	—	211	2	no	—
104	C₂×C₄	—	211	4	no	—
105	C₂×C₄	—	211	4	no	—
106	C₂×C₄	C₈×D₂₁₁	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
108	C₂×C₄	#16809961	1	1	yes	characteristic
109	C₂×C₄	#16809961	1	2	yes	—
110	C₂×C₄	#16809963	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	422	4	no	—
113	C₂²	C₄²×D₂₁₁	1	1	yes	characteristic
114	C₂²	#20269933	1	1	yes	characteristic
115	C₂²	#20269935	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₂₁₁	1	2	yes	—
117	C₂²	M₁₆×D₂₁₁	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	422	4	no	—
120	C₄	—	211	4	no	—
121	C₄	—	211	4	no	—
122	C₄	#20269933	1	2	yes	—
123	C₄	#20269935	1	2	yes	—
124	C₂	#27212991	1	1	yes	characteristic
125	C₂	C₄×C₈×D₂₁₁	1	1	yes	characteristic
126	C₂	#97×D₂₁₁	1	1	yes	characteristic
127	C₂	#27213196	1	2	yes	—
128	C₂	#27213198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₁₁

C₂

yes

maximal

C₄²×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212901

C₂

yes

maximal, characteristic

#27212900

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₄

yes

—

C₂×C₄×Dic₂₁₁

C₄

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₈×Dic₂₁₁

C₄

yes

complemented

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

C₈

yes

complemented

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄²

yes

—

C₂×Dic₂₁₁

C₄²

yes

—

C₂×Dic₂₁₁

C₂×C₈

yes

—

C₂×Dic₂₁₁

C₂×C₈

yes

—

C₂×Dic₂₁₁

M₁₆

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄²

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂²×C₈

yes

—

C₂²×C₂₁₁

C₂×M₁₆

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#117

yes

—

C₄×C₂₁₁

#117

yes

—

Dic₂₁₁

—

Dic₂₁₁

C₄×C₈

yes

—

Dic₂₁₁

#97

yes

—

C₂×C₂₁₁

#376

yes

characteristic

C₂×C₂₁₁

C₂×C₄×C₈

yes

characteristic

C₂×C₂₁₁

C₂×#97

yes

characteristic

C₂×C₂₁₁

#431

yes

—

C₂×C₂₁₁

#431

yes

—

C₂₁₁

#1744

yes

characteristic, complemented, Sylow

#1744

—

211

maximal, Sylow

C₂×C₄×C₈

—

211

—

C₄³

—

211

—

#336

—

211

—

#336

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄×C₈

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₄×D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄²

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

422

—

100

C₂×C₄

—

422

—

101

C₂×C₄

—

211

—

102

C₂×C₄

—

211

—

103

C₂×C₄

—

211

—

104

C₂×C₄

—

211

—

105

C₂×C₄

—

211

—

106

C₂×C₄

C₈×D₂₁₁

yes

—

107

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

108

C₂×C₄

#16809961

yes

characteristic

109

C₂×C₄

#16809961

yes

—

110

C₂×C₄

#16809963

yes

characteristic

111

C₈

—

112

C₈

—

422

—

113

C₂²

C₄²×D₂₁₁

yes

characteristic

114

C₂²

#20269933

yes

characteristic

115

C₂²

#20269935

yes

characteristic

116

C₂²

C₂×C₈×D₂₁₁

yes

—

117

C₂²

M₁₆×D₂₁₁

yes

—

118

C₄

—

119

C₄

—

422

—

120

C₄

—

211

—

121

C₄

—

211

—

122

C₄

#20269933

yes

—

123

C₄

#20269935

yes

—

124

C₂

#27212991

yes

characteristic

125

C₂

C₄×C₈×D₂₁₁

yes

characteristic

126

C₂

#97×D₂₁₁

yes

characteristic

127

C₂

#27213196

yes

—

128

C₂

#27213198

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂×C₂₁₁	C₂×C₄×C₈
Fitting	#336×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#336	D₂₁₁
hypocenter	C₂₁₁	#1744
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator

C₂×C₂₁₁

C₂×C₄×C₈

Fitting

#336×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. C₈×Dic₂₁₁	C₄	semidirect
23. C₄×Dic₂₁₁	C₈	semidirect
64. C₂₁₁	#1744	semidirect

H ≅ G/N

type

11. C₈×Dic₂₁₁

C₄

semidirect

23. C₄×Dic₂₁₁

C₈

semidirect

64. C₂₁₁

#1744

semidirect

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

subgroup	index
C₂×C₈×Dic₂₁₁	2
C₄²×Dic₂₁₁	2
#27212901	2
#27212900	2
#336×C₂₁₁	2
#1744	211

subgroup

index

C₂×C₈×Dic₂₁₁

C₄²×Dic₂₁₁

#27212901

#27212900

#336×C₂₁₁

#1744

211

Subgroups — #41013142

Conjugacy classes of subgroups

Complements

Maximal subgroups