Group #41014888 — subgroups

Subgroups — #41014888

Order 27008.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20269911	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27213045	C₂	1	2	yes	maximal
4	#27212908	C₂	1	2	yes	maximal
5	#340×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
10	#20269911	—	2	2	no	—
11	Dic₂₁₁⋊C₈	—	2	2	no	—
12	C₂₁₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂²×Dic₂₁₁	—	2	2	no	—
16	C₂²×Dic₂₁₁	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₁₁	D₄	1	1	yes	characteristic
18	C₂×C₂₁₁⋊C₈	—	2	1	no	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₁₁	—	2	2	no	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	4	4	no	—
23	C₄×Dic₂₁₁	—	2	1	no	—
24	C₄×Dic₂₁₁	—	2	1	no	—
25	C₂²×C₄×C₂₁₁	—	2	2	no	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₁₁	—	2	2	no	—
30	C₄⋊C₄×C₂₁₁	Q₈	1	4	yes	—
31	C₂×Dic₂₁₁	—	2	1	no	—
32	C₂×Dic₂₁₁	—	2	1	no	—
33	C₂×Dic₂₁₁	—	2	1	no	—
34	C₂×Dic₂₁₁	—	2	2	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	4	2	no	—
37	C₂×Dic₂₁₁	—	4	2	no	—
38	C₂×Dic₂₁₁	D₈	1	2	yes	—
39	C₂₁₁⋊C₈	—	4	1	no	—
40	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	—	2	4	no	—
42	C₂×C₄×C₂₁₁	—	4	2	no	—
43	C₂×C₄×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₁₁	#41	1	2	yes	—
47	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
48	C₈×C₂₁₁	—	2	2	no	—
49	C₂²×C₂₁₁	#121	1	1	yes	characteristic
50	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₂₁₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₂₁₁	#135	1	1	yes	characteristic
53	C₂²×C₂₁₁	#137	1	2	yes	—
54	C₄×C₂₁₁	—	4	4	no	—
55	C₄×C₂₁₁	#123	1	1	yes	characteristic
56	C₄×C₂₁₁	#125	1	2	yes	—
57	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₁₁	—	2	2	no	—
59	Dic₂₁₁	—	4	1	no	—
60	Dic₂₁₁	—	8	2	no	—
61	C₂×C₂₁₁	#398	1	1	yes	characteristic
62	C₂×C₂₁₁	#466	1	1	yes	characteristic
63	C₂×C₂₁₁	#470	1	1	yes	characteristic
64	C₂×C₂₁₁	#474	1	2	yes	—
65	C₂×C₂₁₁	#479	1	2	yes	—
66	C₂₁₁	#2086	1	1	yes	characteristic, complemented, Sylow
67	#2086	—	211	1	no	maximal, Sylow
68	C₂×#107	—	211	1	no	—
69	C₂×C₄⋊C₈	—	211	1	no	—
70	#389	—	211	2	no	—
71	#340	—	211	2	no	—
72	#340	D₂₁₁	1	1	yes	characteristic
73	C₂²×C₈	—	211	1	no	—
74	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	422	2	no	—
76	C₂×C₄⋊C₄	—	211	2	no	—
77	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
78	C₂×C₄²	—	211	1	no	—
79	#107	—	422	2	no	—
80	C₄⋊C₈	—	422	2	no	—
81	#95	—	422	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	422	2	no	—
84	C₂²×C₄	—	211	1	no	—
85	C₂²×C₄	—	211	1	no	—
86	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	422	1	no	—
89	C₂×C₈	—	422	2	no	—
90	C₂×C₈	D₈₄₄	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	844	4	no	—
93	C₄⋊C₄	—	422	2	no	—
94	C₄⋊C₄	—	422	2	no	—
95	C₄⋊C₄	Dic₄₂₂	1	4	yes	—
96	C₄²	—	422	1	no	—
97	C₄²	—	422	1	no	—
98	C₂³	D₄×D₂₁₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	844	2	no	—
103	C₂×C₄	—	844	2	no	—
104	C₂×C₄	—	844	2	no	—
105	C₂×C₄	—	422	1	no	—
106	C₂×C₄	—	422	2	no	—
107	C₂×C₄	—	422	1	no	—
108	C₂×C₄	—	422	1	no	—
109	C₂×C₄	—	211	2	no	—
110	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
112	C₂×C₄	#16809961	1	2	yes	—
113	C₂×C₄	#16809963	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	844	1	no	—
116	C₂²	#20269952	1	1	yes	characteristic
117	C₂²	#20269954	1	2	yes	—
118	C₂²	D₈×D₂₁₁	1	1	yes	characteristic
119	C₂²	#20270000	1	1	yes	characteristic
120	C₂²	#20270006	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	844	1	no	—
123	C₄	—	1688	2	no	—
124	C₄	—	422	2	no	—
125	C₄	#20269961	1	1	yes	characteristic
126	C₄	#20269968	1	2	yes	—
127	C₄	#20269977	1	1	yes	characteristic
128	C₂	#27213078	1	1	yes	characteristic
129	C₂	#27213383	1	1	yes	characteristic
130	C₂	#27213405	1	1	yes	characteristic
131	C₂	#27213424	1	2	yes	—
132	C₂	#27213455	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269911

C₂

yes

maximal, characteristic

C₂×Dic₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27213045

C₂

yes

maximal

#27212908

C₂

yes

maximal

#340×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269911

—

Dic₂₁₁⋊C₈

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

D₄

yes

characteristic

C₂²×Dic₂₁₁

D₄

yes

characteristic

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

D₈

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₂²×C₂₁₁

C₂×D₈

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#123

yes

characteristic

C₄×C₂₁₁

#125

yes

—

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#479

yes

—

C₂₁₁

#2086

yes

characteristic, complemented, Sylow

#2086

—

211

maximal, Sylow

C₂×#107

—

211

—

C₂×C₄⋊C₈

—

211

—

#389

—

211

—

#340

—

211

—

#340

D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

#107

—

422

—

C₄⋊C₈

—

422

—

#95

—

422

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

D₈₄₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

844

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

Dic₄₂₂

yes

—

C₄²

—

422

—

C₄²

—

422

—

C₂³

D₄×D₂₁₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

844

—

103

C₂×C₄

—

844

—

104

C₂×C₄

—

844

—

105

C₂×C₄

—

422

—

106

C₂×C₄

—

422

—

107

C₂×C₄

—

422

—

108

C₂×C₄

—

422

—

109

C₂×C₄

—

211

—

110

C₂×C₄

C₂×D₈₄₄

yes

characteristic

111

C₂×C₄

C₂×Dic₄₂₂

yes

—

112

C₂×C₄

#16809961

yes

—

113

C₂×C₄

#16809963

yes

characteristic

114

C₈

—

115

C₈

—

844

—

116

C₂²

#20269952

yes

characteristic

117

C₂²

#20269954

yes

—

118

C₂²

D₈×D₂₁₁

yes

characteristic

119

C₂²

#20270000

yes

characteristic

120

C₂²

#20270006

yes

—

121

C₄

—

122

C₄

—

844

—

123

C₄

—

1688

—

124

C₄

—

422

—

125

C₄

#20269961

yes

characteristic

126

C₄

#20269968

yes

—

127

C₄

#20269977

yes

characteristic

128

C₂

#27213078

yes

characteristic

129

C₂

#27213383

yes

characteristic

130

C₂

#27213405

yes

characteristic

131

C₂

#27213424

yes

—

132

C₂

#27213455

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₁₁
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#340×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	#340	D₂₁₁
hypocenter	C₂₁₁	#2086
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₁₁

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#340×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
66. C₂₁₁	#2086	semidirect

H ≅ G/N

type

66. C₂₁₁

#2086

semidirect

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

subgroup	index
C₂×#20269911	2
C₂×Dic₂₁₁⋊C₈	2
#27213045	2
#27212908	2
#340×C₂₁₁	2
#2086	211

subgroup

index

C₂×Dic₂₁₁⋊C₈

#340×C₂₁₁

211

Subgroups — #41014888

Conjugacy classes of subgroups

Complements

Maximal subgroups