Group #41012901 — subgroups

Subgroups — #41012901

Order 27008.

146 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 55 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	#27213438	C₂	1	1	yes	maximal, characteristic
2	#27213353	C₂	1	1	yes	maximal, characteristic
3	#27213276	C₂	1	1	yes	maximal, characteristic
4	#27212899	C₂	1	1	yes	maximal, characteristic
5	#27212895	C₂	1	1	yes	maximal, characteristic
6	#27212892	C₂	1	1	yes	maximal, characteristic
7	#498×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16809952	—	2	1	no	—
9	C₄×Dic₄₂₂	C₂²	1	1	yes	characteristic
10	#20269910	—	2	1	no	—
11	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₂₁₁⋊C₈	—	2	1	no	—
15	#20269901	—	2	1	no	—
16	#20269900	—	2	1	no	—
17	#20269887	—	2	1	no	—
18	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	#106×C₂₁₁	—	2	1	no	—
21	#106×C₂₁₁	—	2	1	no	—
22	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₄₂₂	D₄	1	1	yes	characteristic
24	#16809952	—	4	2	no	—
25	C₂×C₂₁₁⋊C₈	—	2	1	no	—
26	C₂×C₂₁₁⋊C₈	—	2	1	no	—
27	C₂×C₂₁₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₂₁₁	D₄	1	1	yes	characteristic
29	Dic₂₁₁⋊C₄	—	2	1	no	—
30	C₄×Dic₂₁₁	—	2	1	no	—
31	C₂×Q₈×C₂₁₁	—	2	1	no	—
32	C₂×Q₈×C₂₁₁	—	2	1	no	—
33	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₁₁	—	2	1	no	—
36	C₄⋊C₄×C₂₁₁	—	2	1	no	—
37	C₄⋊C₄×C₂₁₁	—	2	1	no	—
38	C₄⋊C₄×C₂₁₁	—	2	1	no	—
39	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
40	Dic₄₂₂	—	2	1	no	—
41	Dic₄₂₂	—	2	1	no	—
42	C₂×Dic₂₁₁	—	2	1	no	—
43	C₂×Dic₂₁₁	—	2	1	no	—
44	C₂₁₁⋊C₈	—	4	1	no	—
45	C₂₁₁⋊C₈	—	4	1	no	—
46	C₂₁₁⋊C₈	—	4	1	no	—
47	Q₈×C₂₁₁	—	4	1	no	—
48	Q₈×C₂₁₁	—	4	1	no	—
49	Q₈×C₂₁₁	—	8	1	no	—
50	C₂×C₄×C₂₁₁	—	4	1	no	—
51	C₂×C₄×C₂₁₁	—	4	1	no	—
52	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
55	C₈×C₂₁₁	—	2	1	no	—
56	C₈×C₂₁₁	Q₁₆	1	2	yes	complemented
57	C₂²×C₂₁₁	#121	1	1	yes	characteristic
58	C₄×C₂₁₁	—	2	1	no	—
59	C₄×C₂₁₁	—	8	1	no	—
60	C₄×C₂₁₁	—	8	1	no	—
61	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₄×C₂₁₁	#137	1	1	yes	characteristic
64	C₄×C₂₁₁	#137	1	1	yes	characteristic
65	Dic₂₁₁	—	4	1	no	—
66	Dic₂₁₁	—	4	1	no	—
67	C₂×C₂₁₁	#462	1	1	yes	characteristic
68	C₂×C₂₁₁	#468	1	1	yes	characteristic
69	C₂×C₂₁₁	#471	1	1	yes	characteristic
70	C₂₁₁	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	211	1	no	maximal, Sylow
72	#498	D₂₁₁	1	1	yes	characteristic
73	#477	—	211	1	no	—
74	#462	—	211	1	no	—
75	#446	—	211	1	no	—
76	#335	—	211	1	no	—
77	#333	—	211	1	no	—
78	#332	—	211	1	no	—
79	C₂×Q₁₆	—	422	1	no	—
80	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
82	C₄×Q₈	—	211	1	no	—
83	#107	—	422	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	422	1	no	—
87	C₄⋊C₈	—	211	1	no	—
88	C₄⋊C₈	—	211	1	no	—
89	C₄⋊C₈	—	211	1	no	—
90	#103	—	422	1	no	—
91	#103	—	422	1	no	—
92	#97	—	422	1	no	—
93	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	211	1	no	—
97	Q₁₆	—	844	2	no	—
98	C₂×C₈	—	422	1	no	—
99	C₂×C₈	—	422	1	no	—
100	C₂×C₈	—	422	1	no	—
101	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	422	1	no	—
108	C₄⋊C₄	—	211	1	no	—
109	C₄²	—	422	1	no	—
110	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	422	1	no	—
115	Q₈	—	422	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	422	1	no	—
119	C₂×C₄	—	422	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	—	2	1	no	—
124	C₈	—	844	1	no	—
125	C₈	—	844	1	no	—
126	C₈	—	844	1	no	—
127	C₈	#16809952	1	2	yes	complemented
128	C₂²	#20269948	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	844	1	no	—
133	C₄	—	844	1	no	—
134	C₄	C₂×#16809952	1	1	yes	characteristic
135	C₄	#20270004	1	1	yes	characteristic
136	C₄	#20270006	1	1	yes	characteristic
137	C₄	#20270008	1	1	yes	characteristic
138	C₂	#27213353	1	1	yes	characteristic
139	C₂	#27213389	1	1	yes	characteristic
140	C₂	#27213406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213438

C₂

yes

maximal, characteristic

#27213353

C₂

yes

maximal, characteristic

#27213276

C₂

yes

maximal, characteristic

#27212899

C₂

yes

maximal, characteristic

#27212895

C₂

yes

maximal, characteristic

#27212892

C₂

yes

maximal, characteristic

#498×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809952

—

C₄×Dic₄₂₂

C₂²

yes

characteristic

#20269910

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

Dic₂₁₁⋊C₈

—

#20269901

—

#20269900

—

#20269887

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

#106×C₂₁₁

—

#106×C₂₁₁

—

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

D₄

yes

characteristic

#16809952

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

D₄

yes

characteristic

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×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

—

C₈×C₂₁₁

Q₁₆

yes

complemented

C₂²×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×Q₁₆

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1692

—

211

maximal, Sylow

#498

D₂₁₁

yes

characteristic

#477

—

211

—

#462

—

211

—

#446

—

211

—

#335

—

211

—

#333

—

211

—

#332

—

211

—

C₂×Q₁₆

—

422

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

C₄×Q₈

—

211

—

#107

—

422

—

#106

—

#106

—

C₄⋊C₈

—

422

—

C₄⋊C₈

—

211

—

C₄⋊C₈

—

211

—

C₄⋊C₈

—

211

—

#103

—

422

—

#103

—

422

—

#97

—

422

—

C₄×C₈

C₂×D₂₁₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

211

—

Q₁₆

—

844

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

100

C₂×C₈

—

422

—

101

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

422

—

108

C₄⋊C₄

—

211

—

109

C₄²

—

422

—

110

C₄²

C₂²×D₂₁₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

422

—

115

Q₈

—

422

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

422

—

119

C₂×C₄

—

422

—

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

—

844

—

127

C₈

#16809952

yes

complemented

128

C₂²

#20269948

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

844

—

133

C₄

—

844

—

134

C₄

C₂×#16809952

yes

characteristic

135

C₄

#20270004

yes

characteristic

136

C₄

#20270006

yes

characteristic

137

C₄

#20270008

yes

characteristic

138

C₂

#27213353

yes

characteristic

139

C₂

#27213389

yes

characteristic

140

C₂

#27213406

yes

characteristic

	subgroup	G/H
center	C₂²	#20269948
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#498×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#121
hypercenter	#498	D₂₁₁
hypocenter	C₂₁₁	#1692
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#20269948

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#498×C₂₁₁

C₂

C₂²×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
56. C₈×C₂₁₁	Q₁₆	semidirect
70. C₂₁₁	#1692	semidirect
127. C₈	#16809952	semidirect

H ≅ G/N

type

56. C₈×C₂₁₁

Q₁₆

semidirect

70. C₂₁₁

#1692

semidirect

127. C₈

#16809952

semidirect

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

subgroup	index
#27213438	2
#27213353	2
#27213276	2
#27212899	2
#27212895	2
#27212892	2
#498×C₂₁₁	2
#1692	211

subgroup

index

#498×C₂₁₁

211

Subgroups — #41012901

Conjugacy classes of subgroups

Complements

Maximal subgroups