Group #46810752 — subgroups

Subgroups — #46810752

Order 34432.

272 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 157 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×C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575363	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#31770359	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14524130	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
9	C₄⋊#14524130	C₄	1	4	yes	complemented
10	#22575363	—	2	4	no	—
11	#22575360	C₄	1	2	yes	—
12	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₆₉	C₄	1	4	yes	complemented
16	C₂²×Dic₂₆₉	C₂×C₄	1	4	yes	—
17	C₂×#14524130	—	2	2	no	—
18	C₂×#14524130	—	2	2	no	—
19	C₂×#14524130	C₂×C₄	1	4	yes	—
20	C₄⋊Dic₂₆₉	—	2	4	no	—
21	C₄×Dic₂₆₉	—	2	2	no	—
22	C₄×Dic₂₆₉	—	2	2	no	—
23	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₆₉	—	2	1	no	—
27	C₂×C₈×C₂₆₉	—	2	1	no	—
28	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
31	C₂×C₈×C₂₆₉	Q₈	1	2	yes	—
32	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
33	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₆₉	—	2	4	no	—
35	C₂×Dic₂₆₉	—	2	4	no	—
36	C₂×Dic₂₆₉	—	2	4	no	—
37	#14524130	—	2	4	no	—
38	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₆₉	—	2	2	no	—
40	C₂×C₄×C₂₆₉	C₄²	1	2	yes	—
41	C₂×C₄×C₂₆₉	C₄²	1	2	yes	—
42	C₂×C₄×C₂₆₉	M₁₆	1	4	yes	—
43	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
49	C₈×C₂₆₉	—	2	2	no	—
50	C₈×C₂₆₉	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₂₆₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₂₆₉	C₂×M₁₆	1	2	yes	—
55	C₂²×C₂₆₉	#131	1	2	yes	—
56	C₄×C₂₆₉	—	2	2	no	—
57	C₄×C₂₆₉	#97	1	4	yes	complemented
58	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₂₆₉	#117	1	1	yes	characteristic
60	C₄×C₂₆₉	C₄×D₄	1	1	yes	characteristic
61	C₄×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
62	Dic₂₆₉	—	4	4	no	—
63	C₂×C₂₆₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₂₆₉	C₂×#97	1	1	yes	characteristic
65	C₂×C₂₆₉	#405	1	1	yes	characteristic
66	C₂×C₂₆₉	#436	1	2	yes	—
67	C₂×C₂₆₉	#446	1	2	yes	—
68	C₂₆₉	#1770	1	1	yes	characteristic, complemented, Sylow
69	#1770	—	269	1	no	maximal, Sylow
70	C₂×C₄⋊C₈	—	269	1	no	—
71	C₂×#97	—	269	2	no	—
72	C₂×C₄×C₈	D₂₆₉	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	269	1	no	—
74	#336	—	269	2	no	—
75	C₂²×C₈	—	269	2	no	—
76	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
78	C₂×C₄⋊C₄	—	269	2	no	—
79	C₂×C₄²	—	269	2	no	—
80	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
81	C₄⋊C₈	—	269	4	no	—
82	#97	—	538	4	no	—
83	C₄×C₈	Dic₂₆₉	1	4	yes	complemented
84	#95	—	269	2	no	—
85	C₂²×C₄	—	269	4	no	—
86	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	269	4	no	—
92	C₂×C₈	—	538	2	no	—
93	C₂×C₈	—	538	2	no	—
94	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
95	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
96	C₂×C₈	D₁₀₇₆	1	2	yes	—
97	C₂×C₈	Dic₅₃₈	1	2	yes	—
98	C₄⋊C₄	—	538	4	no	—
99	C₄²	—	538	2	no	—
100	C₄²	—	538	2	no	—
101	C₄²	C₄×D₂₆₉	1	2	yes	—
102	C₄²	C₂×Dic₂₆₉	1	2	yes	—
103	C₂³	C₂×C₄×D₂₆₉	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	538	4	no	—
106	C₂×C₄	—	538	4	no	—
107	C₂×C₄	—	538	4	no	—
108	C₂×C₄	C₄×Dic₂₆₉	1	2	yes	—
109	C₂×C₄	C₄×Dic₂₆₉	1	2	yes	—
110	C₂×C₄	#16864385	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
113	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
115	C₂×C₄	#16864402	1	1	yes	characteristic
116	C₂×C₄	#16864402	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	538	4	no	—
119	C₈	C₄⋊Dic₂₆₉	1	4	yes	complemented
120	C₂²	C₂×C₄×Dic₂₆₉	1	1	yes	characteristic
121	C₂²	C₄×D₁₀₇₆	1	1	yes	characteristic
122	C₂²	C₄×Dic₅₃₈	1	1	yes	characteristic
123	C₂²	C₂×#16864385	1	2	yes	—
124	C₂²	#22575459	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	1076	4	no	—
127	C₄	#22575363	1	4	yes	complemented
128	C₄	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
129	C₄	#22575412	1	1	yes	characteristic
130	C₄	C₄×D₁₀₇₆	1	1	yes	characteristic
131	C₄	C₄×Dic₅₃₈	1	1	yes	characteristic
132	C₂	C₄×C₄⋊Dic₂₆₉	1	1	yes	characteristic
133	C₂	C₂×#22575363	1	1	yes	characteristic
134	C₂	#31770560	1	1	yes	characteristic
135	C₂	#31770684	1	2	yes	—
136	C₂	#31770735	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14524130

C₂

yes

maximal, characteristic

C₂×#22575363

C₂

yes

maximal

C₄×C₄⋊Dic₂₆₉

C₂

yes

maximal, characteristic

#31770359

C₂

yes

maximal

C₂×C₄×C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₄

yes

—

C₂×C₄×Dic₂₆₉

C₂²

yes

—

C₄⋊#14524130

C₄

yes

complemented

#22575363

—

#22575360

C₄

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₄

yes

complemented

C₂²×Dic₂₆₉

C₂×C₄

yes

—

C₂×#14524130

—

C₂×#14524130

—

C₂×#14524130

C₂×C₄

yes

—

C₄⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₂×C₈×C₂₆₉

Q₈

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄²

yes

—

C₂×C₄×C₂₆₉

C₄²

yes

—

C₂×C₄×C₂₆₉

M₁₆

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

C₂×C₄×C₂₆₉

yes

C₂×C₄×C₂₆₉

yes

C₈×C₂₆₉

—

C₈×C₂₆₉

C₄⋊C₄

yes

complemented

C₂²×C₂₆₉

C₂×C₄²

yes

characteristic

C₂²×C₂₆₉

C₄×D₄

yes

characteristic

C₂²×C₂₆₉

C₄×Q₈

yes

characteristic

C₂²×C₂₆₉

C₂×M₁₆

yes

—

C₂²×C₂₆₉

#131

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#97

yes

complemented

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

#117

yes

characteristic

C₄×C₂₆₉

C₄×D₄

yes

characteristic

C₄×C₂₆₉

C₄×Q₈

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₆₉

C₂×#97

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂×C₂₆₉

#446

yes

—

C₂₆₉

#1770

yes

characteristic, complemented, Sylow

#1770

—

269

maximal, Sylow

C₂×C₄⋊C₈

—

269

—

C₂×#97

—

269

—

C₂×C₄×C₈

D₂₆₉

yes

characteristic

C₄×C₄⋊C₄

—

269

—

#336

—

269

—

C₂²×C₈

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂×C₄⋊C₄

—

269

—

C₂×C₄²

—

269

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

269

—

#97

—

538

—

C₄×C₈

Dic₂₆₉

yes

complemented

#95

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

269

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

C₂×Dic₂₆₉

yes

—

C₂×C₈

C₂×Dic₂₆₉

yes

—

C₂×C₈

D₁₀₇₆

yes

—

C₂×C₈

Dic₅₃₈

yes

—

C₄⋊C₄

—

538

—

C₄²

—

538

—

100

C₄²

—

538

—

101

C₄²

C₄×D₂₆₉

yes

—

102

C₄²

C₂×Dic₂₆₉

yes

—

103

C₂³

C₂×C₄×D₂₆₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

538

—

106

C₂×C₄

—

538

—

107

C₂×C₄

—

538

—

108

C₂×C₄

C₄×Dic₂₆₉

yes

—

109

C₂×C₄

C₄×Dic₂₆₉

yes

—

110

C₂×C₄

#16864385

yes

—

111

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

112

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

113

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

114

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

115

C₂×C₄

#16864402

yes

characteristic

116

C₂×C₄

#16864402

yes

characteristic

117

C₈

—

118

C₈

—

538

—

119

C₈

C₄⋊Dic₂₆₉

yes

complemented

120

C₂²

C₂×C₄×Dic₂₆₉

yes

characteristic

121

C₂²

C₄×D₁₀₇₆

yes

characteristic

122

C₂²

C₄×Dic₅₃₈

yes

characteristic

123

C₂²

C₂×#16864385

yes

—

124

C₂²

#22575459

yes

—

125

C₄

—

126

C₄

—

1076

—

127

C₄

#22575363

yes

complemented

128

C₄

C₂×C₄⋊Dic₂₆₉

yes

characteristic

129

C₄

#22575412

yes

characteristic

130

C₄

C₄×D₁₀₇₆

yes

characteristic

131

C₄

C₄×Dic₅₃₈

yes

characteristic

132

C₂

C₄×C₄⋊Dic₂₆₉

yes

characteristic

133

C₂

C₂×#22575363

yes

characteristic

134

C₂

#31770560

yes

characteristic

135

C₂

#31770684

yes

—

136

C₂

#31770735

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₆₉
commutator	C₂²×C₂₆₉	C₂×C₄²
Fitting	C₂×C₄×C₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₆₉
hypocenter	C₂₆₉	#1770
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₆₉

commutator

C₂²×C₂₆₉

C₂×C₄²

Fitting

C₂×C₄×C₈×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂²×C₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₄⋊#14524130	C₄	semidirect
15. C₄×C₈×C₂₆₉	C₄	semidirect
50. C₈×C₂₆₉	C₄⋊C₄	semidirect
57. C₄×C₂₆₉	#97	semidirect
68. C₂₆₉	#1770	semidirect
83. C₄×C₈	Dic₂₆₉	semidirect
119. C₈	C₄⋊Dic₂₆₉	semidirect
127. C₄	#22575363	semidirect

H ≅ G/N

type

9. C₄⋊#14524130

C₄

semidirect

15. C₄×C₈×C₂₆₉

C₄

semidirect

50. C₈×C₂₆₉

C₄⋊C₄

semidirect

57. C₄×C₂₆₉

#97

semidirect

68. C₂₆₉

#1770

semidirect

83. C₄×C₈

Dic₂₆₉

semidirect

119. C₈

C₄⋊Dic₂₆₉

semidirect

127. C₄

#22575363

semidirect

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

subgroup	index
C₂×C₄⋊#14524130	2
C₂×#22575363	2
C₄×C₄⋊Dic₂₆₉	2
#31770359	2
C₂×C₄×C₈×C₂₆₉	2
#1770	269

subgroup

index

C₄×C₄⋊Dic₂₆₉

C₂×C₄×C₈×C₂₆₉

269

Subgroups — #46810752

Conjugacy classes of subgroups

Complements

Maximal subgroups