Group #41012782 — subgroups

Subgroups — #41012782

Order 27008.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27213542	C₂	1	1	yes	maximal, characteristic
2	#27213356	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27212896	C₂	1	1	yes	maximal, characteristic
5	#27212895	C₂	1	1	yes	maximal, characteristic
6	#27212893	C₂	1	1	yes	maximal, characteristic
7	#477×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16809952	—	2	1	no	—
9	#20269977	C₂²	1	1	yes	characteristic
10	#20269908	—	2	1	no	—
11	#20269908	—	2	1	no	—
12	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
13	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#20269900	—	2	1	no	—
16	C₄×C₂₁₁⋊C₈	—	2	1	no	—
17	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	#107×C₂₁₁	—	2	1	no	—
21	C₄⋊C₈×C₂₁₁	C₂²	1	1	yes	characteristic
22	#103×C₂₁₁	—	2	1	no	—
23	C₂×Dic₄₂₂	—	2	1	no	—
24	#16809952	—	4	2	no	—
25	C₂×C₂₁₁⋊C₈	—	2	1	no	—
26	C₂×C₂₁₁⋊C₈	—	2	1	no	—
27	C₂×C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₂₁₁	—	2	1	no	—
30	C₄⋊Dic₂₁₁	—	2	1	no	—
31	C₂×Q₈×C₂₁₁	—	2	1	no	—
32	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₁₁	—	2	1	no	—
34	C₄⋊C₄×C₂₁₁	—	2	1	no	—
35	C₄⋊C₄×C₂₁₁	—	2	1	no	—
36	C₄⋊C₄×C₂₁₁	—	2	1	no	—
37	C₄⋊C₄×C₂₁₁	D₄	1	1	yes	characteristic
38	C₄²×C₂₁₁	—	2	1	no	—
39	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
40	Dic₄₂₂	—	4	1	no	—
41	Dic₄₂₂	—	4	1	no	—
42	C₂×Dic₂₁₁	—	4	1	no	—
43	C₂₁₁⋊C₈	—	2	1	no	—
44	C₂₁₁⋊C₈	—	4	1	no	—
45	C₂₁₁⋊C₈	—	4	1	no	—
46	C₂₁₁⋊C₈	Q₁₆	1	2	yes	complemented
47	Q₈×C₂₁₁	—	2	1	no	—
48	Q₈×C₂₁₁	—	8	1	no	—
49	Q₈×C₂₁₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₂₁₁	—	2	1	no	—
51	C₂×C₄×C₂₁₁	—	2	1	no	—
52	C₂×C₄×C₂₁₁	—	4	1	no	—
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
56	C₈×C₂₁₁	—	4	1	no	—
57	C₂²×C₂₁₁	#121	1	1	yes	characteristic
58	C₄×C₂₁₁	—	2	2	no	—
59	C₄×C₂₁₁	—	2	1	no	—
60	C₄×C₂₁₁	—	4	1	no	—
61	C₄×C₂₁₁	—	8	1	no	—
62	C₄×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₂₁₁	#135	1	1	yes	characteristic
65	C₄×C₂₁₁	#137	1	1	yes	characteristic
66	Dic₂₁₁	—	8	1	no	—
67	C₂×C₂₁₁	#462	1	1	yes	characteristic
68	C₂×C₂₁₁	#465	1	1	yes	characteristic
69	C₂×C₂₁₁	#471	1	1	yes	characteristic
70	C₂₁₁	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	211	1	no	maximal, Sylow
72	#498	—	211	1	no	—
73	#477	D₂₁₁	1	1	yes	characteristic
74	#462	—	211	1	no	—
75	C₈×Q₈	—	211	1	no	—
76	#334	—	211	1	no	—
77	#333	—	211	1	no	—
78	#332	—	211	1	no	—
79	C₂×Q₁₆	—	422	1	no	—
80	C₄⋊Q₈	—	211	1	no	—
81	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
83	#107	—	2	1	no	—
84	#106	—	422	1	no	—
85	#106	—	422	1	no	—
86	C₄⋊C₈	—	422	1	no	—
87	C₄⋊C₈	—	211	1	no	—
88	C₄⋊C₈	—	211	1	no	—
89	C₄⋊C₈	C₂×D₂₁₁	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	422	1	no	—
92	C₄×C₈	—	422	1	no	—
93	C₄×C₈	—	211	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	422	1	no	—
96	C₂×Q₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
97	Q₁₆	—	844	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	422	1	no	—
100	C₂×C₈	—	422	1	no	—
101	C₂×C₈	—	211	1	no	—
102	C₂×C₈	—	211	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	422	1	no	—
107	C₄⋊C₄	—	422	1	no	—
108	C₄⋊C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	844	1	no	—
114	Q₈	—	844	1	no	—
115	Q₈	#16809949	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	844	1	no	—
120	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
122	C₂×C₄	#16809961	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	844	1	no	—
125	C₈	—	844	1	no	—
126	C₈	—	422	1	no	—
127	C₈	—	211	2	no	—
128	C₂²	#20269949	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1688	1	no	—
134	C₄	C₂×#16809949	1	1	yes	characteristic
135	C₄	Q₁₆×D₂₁₁	1	1	yes	characteristic
136	C₄	#20269997	1	1	yes	characteristic
137	C₄	#20270009	1	1	yes	characteristic
138	C₂	#27213354	1	1	yes	characteristic
139	C₂	#27213377	1	1	yes	characteristic
140	C₂	#27213407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213542

C₂

yes

maximal, characteristic

#27213356

C₂

yes

maximal, characteristic

Q₈×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212896

C₂

yes

maximal, characteristic

#27212895

C₂

yes

maximal, characteristic

#27212893

C₂

yes

maximal, characteristic

#477×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809952

—

#20269977

C₂²

yes

characteristic

#20269908

—

#20269908

—

C₄⋊(C₂₁₁⋊C₈)

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

#20269900

—

C₄×C₂₁₁⋊C₈

—

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

C₄×Q₈×C₂₁₁

C₂²

yes

characteristic

#107×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

characteristic

#103×C₂₁₁

—

C₂×Dic₄₂₂

—

#16809952

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

characteristic

C₂×C₂₁₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

Q₁₆

yes

complemented

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₄×C₂₁₁

C₂×Q₁₆

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1672

—

211

maximal, Sylow

#498

—

211

—

#477

D₂₁₁

yes

characteristic

#462

—

211

—

C₈×Q₈

—

211

—

#334

—

211

—

#333

—

211

—

#332

—

211

—

C₂×Q₁₆

—

422

—

C₄⋊Q₈

—

211

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

C₄×Q₈

C₂×D₂₁₁

yes

characteristic

#107

—

#106

—

422

—

#106

—

422

—

C₄⋊C₈

—

422

—

C₄⋊C₈

—

211

—

C₄⋊C₈

—

211

—

C₄⋊C₈

C₂×D₂₁₁

yes

characteristic

#103

—

#103

—

422

—

C₄×C₈

—

422

—

C₄×C₈

—

211

—

C₂×Q₈

—

C₂×Q₈

—

422

—

C₂×Q₈

C₂²⋊D₂₁₁

yes

characteristic

Q₁₆

—

844

—

C₂×C₈

—

C₂×C₈

—

422

—

100

C₂×C₈

—

422

—

101

C₂×C₈

—

211

—

102

C₂×C₈

—

211

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

422

—

107

C₄⋊C₄

—

422

—

108

C₄⋊C₄

C₂²⋊D₂₁₁

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₂₁₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

844

—

114

Q₈

—

844

—

115

Q₈

#16809949

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

844

—

120

C₂×C₄

D₄×D₂₁₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

122

C₂×C₄

#16809961

yes

characteristic

123

C₈

—

124

C₈

—

844

—

125

C₈

—

844

—

126

C₈

—

422

—

127

C₈

—

211

—

128

C₂²

#20269949

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1688

—

134

C₄

C₂×#16809949

yes

characteristic

135

C₄

Q₁₆×D₂₁₁

yes

characteristic

136

C₄

#20269997

yes

characteristic

137

C₄

#20270009

yes

characteristic

138

C₂

#27213354

yes

characteristic

139

C₂

#27213377

yes

characteristic

140

C₂

#27213407

yes

characteristic

	subgroup	G/H
center	C₂²	#20269949
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#477×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#121
hypercenter	#477	D₂₁₁
hypocenter	C₂₁₁	#1672
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#20269949

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#477×C₂₁₁

C₂

C₂²×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
46. C₂₁₁⋊C₈	Q₁₆	semidirect
70. C₂₁₁	#1672	semidirect

H ≅ G/N

type

46. C₂₁₁⋊C₈

Q₁₆

semidirect

70. C₂₁₁

#1672

semidirect

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

subgroup	index
#27213542	2
#27213356	2
Q₈×C₂₁₁⋊C₈	2
#27212896	2
#27212895	2
#27212893	2
#477×C₂₁₁	2
#1672	211

subgroup

index

Q₈×C₂₁₁⋊C₈

#477×C₂₁₁

211

Subgroups — #41012782

Conjugacy classes of subgroups

Complements

Maximal subgroups