Group #46846887 — subgroups

Subgroups — #46846887

Order 34688.

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

normal subgroups only

This group has 348 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	#31774450	C₂	1	2	yes	maximal
2	#31774305	C₂	1	4	yes	maximal, complemented
3	#31774304	C₂	1	4	yes	maximal, complemented
4	#31773935	C₂	1	1	yes	maximal, characteristic
5	#31773933	C₂	1	2	yes	maximal
6	C₂×#22576070	C₂	1	1	yes	maximal, characteristic
7	#427×C₂₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#22576159	C₂²	1	4	yes	—
10	#22576151	C₂²	1	4	yes	—
11	#22576137	—	2	4	no	—
12	C₄×Dic₅₄₂	—	2	4	no	—
13	#22576119	C₂²	1	2	yes	—
14	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
15	#22576094	C₂²	1	2	yes	—
16	#22576093	C₂²	1	4	yes	—
17	#22576092	C₂²	1	2	yes	—
18	#22576091	C₂²	1	4	yes	—
19	#22576070	C₂²	1	2	yes	—
20	#22576070	C₂²	1	2	yes	—
21	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
22	#117×C₂₇₁	C₂²	1	2	yes	—
23	#107×C₂₇₁	C₂²	1	2	yes	—
24	#106×C₂₇₁	C₂²	1	2	yes	—
25	C₂×Dic₅₄₂	—	2	4	no	—
26	C₂²×Dic₂₇₁	D₄	1	2	yes	—
27	C₂×C₂₇₁⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₂₇₁	—	2	4	no	—
30	C₄⋊Dic₂₇₁	C₂³	1	4	yes	—
31	Dic₂₇₁⋊C₄	—	2	4	no	—
32	Dic₂₇₁⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₂₇₁	—	2	2	no	—
34	C₄×Dic₂₇₁	—	2	2	no	—
35	C₄×Dic₂₇₁	D₄	1	2	yes	—
36	C₄×Dic₂₇₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₂₇₁	Q₈	1	4	yes	complemented
39	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₇₁	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
43	C₄²×C₂₇₁	—	2	2	no	—
44	Dic₅₄₂	—	4	8	no	—
45	C₂×Dic₂₇₁	—	2	4	no	—
46	C₂×Dic₂₇₁	—	4	2	no	—
47	C₂×Dic₂₇₁	C₂×D₄	1	4	yes	—
48	C₂₇₁⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₇₁	—	2	4	no	—
51	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
55	C₈×C₂₇₁	C₂×Q₈	1	2	yes	—
56	C₈×C₂₇₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₂₇₁	—	2	2	no	—
58	C₂²×C₂₇₁	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₂₇₁	—	4	4	no	—
61	C₄×C₂₇₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
64	Dic₂₇₁	—	2	4	no	—
65	Dic₂₇₁	—	4	4	no	—
66	C₂×C₂₇₁	—	2	2	no	—
67	C₂×C₂₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₂₇₁	#575	1	2	yes	—
69	C₂₇₁	#3162	1	1	yes	characteristic, complemented, Sylow
70	#3162	—	271	1	no	maximal, Sylow
71	#533	—	271	2	no	—
72	#501	—	271	4	no	—
73	#501	—	271	4	no	—
74	#427	—	271	1	no	—
75	#427	—	271	2	no	—
76	#427	D₂₇₁	1	1	yes	characteristic
77	C₂×#97	—	271	1	no	—
78	C₂²×C₈	—	271	1	no	—
79	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	271	4	no	—
81	#125	—	271	4	no	—
82	C₂²⋊Q₈	—	542	4	no	—
83	C₄×Q₈	—	542	4	no	—
84	#117	—	271	2	no	—
85	#117	C₂×D₂₇₁	1	2	yes	—
86	C₂×C₄²	—	271	1	no	—
87	#107	—	271	2	no	—
88	#107	—	271	4	no	—
89	#107	C₂×D₂₇₁	1	2	yes	—
90	#106	—	271	2	no	—
91	#106	—	271	4	no	—
92	#106	C₂×D₂₇₁	1	2	yes	—
93	#97	—	271	2	no	—
94	#97	—	271	2	no	—
95	C₂×Q₈	—	542	4	no	—
96	C₂²×C₄	—	271	2	no	—
97	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
98	C₂×C₈	—	271	4	no	—
99	C₂×C₈	—	271	2	no	—
100	C₂×C₈	Dic₅₄₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
103	C₄⋊C₄	—	271	4	no	—
104	C₄⋊C₄	—	542	4	no	—
105	C₄⋊C₄	—	542	4	no	—
106	C₄⋊C₄	—	542	4	no	—
107	C₄⋊C₄	C₂²×D₂₇₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	542	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	271	2	no	—
112	C₄²	—	271	2	no	—
113	C₄²	—	542	2	no	—
114	C₂³	D₄×D₂₇₁	1	1	yes	characteristic
115	Q₈	—	1084	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	271	4	no	—
118	C₂×C₄	—	542	4	no	—
119	C₂×C₄	—	1084	2	no	—
120	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₂₇₁	1	2	yes	—
122	C₂×C₄	C₂×Dic₅₄₂	1	2	yes	—
123	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
124	C₈	—	271	4	no	—
125	C₈	C₂×Dic₅₄₂	1	2	yes	—
126	C₈	C₂×Dic₅₄₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#22576159	1	2	yes	—
129	C₂²	C₂×D₄×D₂₇₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	542	4	no	—
132	C₄	—	1084	4	no	—
133	C₄	#22576159	1	2	yes	—
134	C₄	C₂×Q₈×D₂₇₁	1	1	yes	characteristic
135	C₄	C₂²×Dic₅₄₂	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#22576159	1	1	yes	characteristic
138	C₂	#31774694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31774450

C₂

yes

maximal

#31774305

C₂

yes

maximal, complemented

#31774304

C₂

yes

maximal, complemented

#31773935

C₂

yes

maximal, characteristic

#31773933

C₂

yes

maximal

C₂×#22576070

C₂

yes

maximal, characteristic

#427×C₂₇₁

C₂

yes

maximal, characteristic

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

#22576159

C₂²

yes

—

#22576151

C₂²

yes

—

#22576137

—

C₄×Dic₅₄₂

—

#22576119

C₂²

yes

—

C₂×C₄×Dic₂₇₁

C₂²

yes

characteristic

#22576094

C₂²

yes

—

#22576093

C₂²

yes

—

#22576092

C₂²

yes

—

#22576091

C₂²

yes

—

#22576070

C₂²

yes

—

#22576070

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

#117×C₂₇₁

C₂²

yes

—

#107×C₂₇₁

C₂²

yes

—

#106×C₂₇₁

C₂²

yes

—

C₂×Dic₅₄₂

—

C₂²×Dic₂₇₁

D₄

yes

—

C₂×C₂₇₁⋊C₈

Q₈

yes

complemented

C₂×C₂₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

C₂³

yes

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

C₂²⋊Dic₂₇₁

—

C₄×Dic₂₇₁

—

C₄×Dic₂₇₁

D₄

yes

—

C₄×Dic₂₇₁

C₂³

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₂×C₈×C₂₇₁

Q₈

yes

complemented

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₁

C₂³

yes

—

C₂²⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

—

Dic₅₄₂

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

C₂×D₄

yes

—

C₂₇₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂⁴

yes

characteristic

C₈×C₂₇₁

C₂×Q₈

yes

—

C₈×C₂₇₁

C₂×Q₈

yes

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₄⋊Q₈

yes

—

C₂²×C₂₇₁

C₂²×D₄

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

C₄⋊Q₈

yes

—

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₇₁

#575

yes

—

C₂₇₁

#3162

yes

characteristic, complemented, Sylow

#3162

—

271

maximal, Sylow

#533

—

271

—

#501

—

271

—

#501

—

271

—

#427

—

271

—

#427

—

271

—

#427

D₂₇₁

yes

characteristic

C₂×#97

—

271

—

C₂²×C₈

—

271

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

C₄⋊Q₈

—

271

—

#125

—

271

—

C₂²⋊Q₈

—

542

—

C₄×Q₈

—

542

—

#117

—

271

—

#117

C₂×D₂₇₁

yes

—

C₂×C₄²

—

271

—

#107

—

271

—

#107

—

271

—

#107

C₂×D₂₇₁

yes

—

#106

—

271

—

#106

—

271

—

#106

C₂×D₂₇₁

yes

—

#97

—

271

—

#97

—

271

—

C₂×Q₈

—

542

—

C₂²×C₄

—

271

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

C₂×C₈

—

271

—

C₂×C₈

—

271

—

100

C₂×C₈

Dic₅₄₂

yes

complemented

101

C₂×C₈

C₂²×D₂₇₁

yes

characteristic

102

C₂×C₈

C₂²×D₂₇₁

yes

characteristic

103

C₄⋊C₄

—

271

—

104

C₄⋊C₄

—

542

—

105

C₄⋊C₄

—

542

—

106

C₄⋊C₄

—

542

—

107

C₄⋊C₄

C₂²×D₂₇₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

542

—

110

C₄²

—

111

C₄²

—

271

—

112

C₄²

—

271

—

113

C₄²

—

542

—

114

C₂³

D₄×D₂₇₁

yes

characteristic

115

Q₈

—

1084

—

116

C₂×C₄

—

117

C₂×C₄

—

271

—

118

C₂×C₄

—

542

—

119

C₂×C₄

—

1084

—

120

C₂×C₄

D₄×D₂₇₁

yes

characteristic

121

C₂×C₄

Q₈×D₂₇₁

yes

—

122

C₂×C₄

C₂×Dic₅₄₂

yes

—

123

C₂×C₄

C₂³×D₂₇₁

yes

characteristic

124

C₈

—

271

—

125

C₈

C₂×Dic₅₄₂

yes

—

126

C₈

C₂×Dic₅₄₂

yes

—

127

C₂²

—

128

C₂²

#22576159

yes

—

129

C₂²

C₂×D₄×D₂₇₁

yes

characteristic

130

C₄

—

131

C₄

—

542

—

132

C₄

—

1084

—

133

C₄

#22576159

yes

—

134

C₄

C₂×Q₈×D₂₇₁

yes

characteristic

135

C₄

C₂²×Dic₅₄₂

yes

characteristic

136

C₂

—

137

C₂

C₂×#22576159

yes

characteristic

138

C₂

#31774694

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#427×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂²×D₄
prime radical	C₂³×C₂₇₁	C₂×D₄
hypercenter	#427	D₂₇₁
hypocenter	C₂₇₁	#3162
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₇₁

commutator, cosocle

C₂×C₄×C₂₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₇₁

Fitting

#427×C₂₇₁

C₂

socle

C₂²×C₂₇₁

C₂²×D₄

prime radical

C₂³×C₂₇₁

C₂×D₄

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #31774305	C₂	semidirect
3. #31774304	C₂	semidirect
27. C₂×C₂₇₁⋊C₈	Q₈	semidirect
38. C₂×C₈×C₂₇₁	Q₈	semidirect
69. C₂₇₁	#3162	semidirect
100. C₂×C₈	Dic₅₄₂	semidirect

H ≅ G/N

type

2. #31774305

C₂

semidirect

3. #31774304

C₂

semidirect

27. C₂×C₂₇₁⋊C₈

Q₈

semidirect

38. C₂×C₈×C₂₇₁

Q₈

semidirect

69. C₂₇₁

#3162

semidirect

100. C₂×C₈

Dic₅₄₂

semidirect

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

subgroup	index
#31774450	2
#31774305	2
#31774304	2
#31773935	2
#31773933	2
C₂×#22576070	2
#427×C₂₇₁	2
#3162	271

subgroup

index

#427×C₂₇₁

271

Subgroups — #46846887

Conjugacy classes of subgroups

Complements

Maximal subgroups