Group #46809556 — subgroups

Subgroups — #46809556

Order 34432.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#22575453	C₂	1	1	yes	maximal, characteristic
2	#31770629	C₂	1	1	yes	maximal, characteristic
3	#31770350	C₂	1	4	yes	maximal, complemented
4	#431×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₃₈	—	2	1	no	—
6	C₂×#16864386	—	2	1	no	—
7	#22575453	—	2	1	no	—
8	#22575453	—	2	1	no	—
9	#22575453	C₄	1	2	yes	—
10	#22575453	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
12	C₂×C₄⋊Dic₂₆₉	C₄	1	2	yes	—
13	C₄⋊#14524130	—	2	1	no	—
14	C₄⋊#14524130	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₆₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₂₆₉	—	2	1	no	—
18	C₄×C₈×C₂₆₉	—	2	1	no	—
19	C₂×Dic₅₃₈	—	2	2	no	—
20	C₂×Dic₅₃₈	—	2	2	no	—
21	C₂×Dic₅₃₈	—	4	2	no	—
22	C₂×Dic₅₃₈	—	4	2	no	—
23	C₂²×Dic₂₆₉	—	2	2	no	—
24	#16864386	—	4	2	no	—
25	C₂×#14524130	—	2	2	no	—
26	C₄⋊Dic₂₆₉	—	2	2	no	—
27	C₄⋊Dic₂₆₉	—	2	2	no	—
28	C₄⋊Dic₂₆₉	—	2	2	no	—
29	C₄⋊Dic₂₆₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₆₉	—	2	2	no	—
34	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
36	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
37	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
38	Dic₅₃₈	—	4	4	no	—
39	Dic₅₃₈	—	4	2	no	—
40	Dic₅₃₈	—	4	2	no	—
41	C₂×Dic₂₆₉	—	2	4	no	—
42	C₂×Dic₂₆₉	—	4	4	no	—
43	#14524130	—	4	2	no	—
44	C₂³×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	—	2	2	no	—
46	C₂×C₄×C₂₆₉	—	2	2	no	—
47	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₆₉	D₈	1	2	yes	—
51	C₂×C₄×C₂₆₉	SD₁₆	1	2	yes	—
52	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₂₆₉	—	4	2	no	—
56	C₂²×C₂₆₉	—	2	1	no	—
57	C₂²×C₂₆₉	—	2	1	no	—
58	C₂²×C₂₆₉	#102	1	2	yes	—
59	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₆₉	—	2	1	no	—
61	C₄×C₂₆₉	—	2	1	no	—
62	C₄×C₂₆₉	#101	1	2	yes	—
63	C₄×C₂₆₉	#102	1	2	yes	—
64	C₄×C₂₆₉	C₂×D₈	1	1	yes	characteristic
65	C₄×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₂₆₉	#137	1	2	yes	—
67	Dic₂₆₉	—	4	4	no	—
68	C₂×C₂₆₉	—	2	2	no	—
69	C₂×C₂₆₉	C₂×#101	1	1	yes	characteristic
70	C₂×C₂₆₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₂₆₉	#419	1	1	yes	characteristic
72	C₂₆₉	#1546	1	1	yes	characteristic, complemented, Sylow
73	#1546	—	269	1	no	maximal, Sylow
74	C₂×C₄⋊Q₈	—	269	1	no	—
75	#431	D₂₆₉	1	1	yes	characteristic
76	#423	—	269	1	no	—
77	#332	—	269	4	no	—
78	C₂²×Q₈	—	538	1	no	—
79	C₂×M₁₆	—	538	1	no	—
80	C₄⋊Q₈	—	269	2	no	—
81	C₄⋊Q₈	—	269	2	no	—
82	C₄⋊Q₈	—	538	1	no	—
83	C₄⋊Q₈	—	538	1	no	—
84	C₂×C₄⋊C₄	—	269	2	no	—
85	C₂×C₄⋊C₄	—	538	1	no	—
86	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
87	C₄⋊C₈	—	269	2	no	—
88	C₄⋊C₈	—	538	1	no	—
89	C₄⋊C₈	C₂×D₂₆₉	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	538	2	no	—
93	C₂×Q₈	—	538	2	no	—
94	C₂×Q₈	—	1076	2	no	—
95	C₂×Q₈	—	1076	2	no	—
96	C₂²×C₄	—	538	2	no	—
97	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
99	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
100	M₁₆	—	1076	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	538	2	no	—
103	C₄⋊C₄	—	269	4	no	—
104	C₄⋊C₄	—	538	2	no	—
105	C₄⋊C₄	—	538	2	no	—
106	C₄⋊C₄	—	538	2	no	—
107	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
108	C₄²	C₂²⋊D₂₆₉	1	1	yes	characteristic
109	C₄²	D₁₀₇₆	1	1	yes	characteristic
110	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
111	C₂³	#16864379	1	1	yes	characteristic
112	Q₈	—	1076	4	no	—
113	Q₈	—	1076	2	no	—
114	Q₈	—	1076	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	538	4	no	—
118	C₂×C₄	—	1076	4	no	—
119	C₂×C₄	#16864379	1	1	yes	characteristic
120	C₂×C₄	#16864379	1	1	yes	characteristic
121	C₂×C₄	#16864379	1	1	yes	characteristic
122	C₂×C₄	D₂₁₅₂	1	2	yes	—
123	C₂×C₄	#16864391	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	1076	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#22575375	1	2	yes	—
132	C₂²	C₂×#16864379	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	1076	4	no	—
136	C₄	#22575371	1	2	yes	—
137	C₄	#22575375	1	2	yes	—
138	C₄	C₂×D₂₁₅₂	1	1	yes	characteristic
139	C₄	C₂×#16864391	1	1	yes	characteristic
140	C₄	#22575483	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#22575371	1	1	yes	characteristic
143	C₂	C₂×#22575375	1	1	yes	characteristic
144	C₂	#31770613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575453

C₂

yes

maximal, characteristic

#31770629

C₂

yes

maximal, characteristic

#31770350

C₂

yes

maximal, complemented

#431×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

—

—

—

—

C₄

yes

—

#22575453

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₄

yes

—

C₄⋊#14524130

—

C₄⋊#14524130

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

—

C₂²⋊C₈×C₂₆₉

—

C₄×C₈×C₂₆₉

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂²×Dic₂₆₉

—

#16864386

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

—

C₄²×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

D₈

yes

—

C₂×C₄×C₂₆₉

SD₁₆

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#102

yes

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#101

yes

—

C₄×C₂₆₉

#102

yes

—

C₄×C₂₆₉

C₂×D₈

yes

characteristic

C₄×C₂₆₉

C₂×SD₁₆

yes

characteristic

C₄×C₂₆₉

#137

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1546

—

269

maximal, Sylow

C₂×C₄⋊Q₈

—

269

—

#431

D₂₆₉

yes

characteristic

#423

—

269

—

#332

—

269

—

C₂²×Q₈

—

538

—

C₂×M₁₆

—

538

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

538

—

C₄⋊Q₈

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

269

—

C₄⋊C₈

—

538

—

C₄⋊C₈

C₂×D₂₆₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

538

—

C₂×Q₈

—

538

—

C₂×Q₈

—

1076

—

C₂×Q₈

—

1076

—

C₂²×C₄

—

538

—

C₂²×C₄

C₄×D₂₆₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₆₉

yes

characteristic

C₂²×C₄

D₁₀₇₆

yes

characteristic

100

M₁₆

—

1076

—

101

C₂×C₈

—

102

C₂×C₈

—

538

—

103

C₄⋊C₄

—

269

—

104

C₄⋊C₄

—

538

—

105

C₄⋊C₄

—

538

—

106

C₄⋊C₄

—

538

—

107

C₄²

C₄×D₂₆₉

yes

characteristic

108

C₄²

C₂²⋊D₂₆₉

yes

characteristic

109

C₄²

D₁₀₇₆

yes

characteristic

110

C₄²

C₂²×D₂₆₉

yes

characteristic

111

C₂³

#16864379

yes

characteristic

112

Q₈

—

1076

—

113

Q₈

—

1076

—

114

Q₈

—

1076

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

538

—

118

C₂×C₄

—

1076

—

119

C₂×C₄

#16864379

yes

characteristic

120

C₂×C₄

#16864379

yes

characteristic

121

C₂×C₄

#16864379

yes

characteristic

122

C₂×C₄

D₂₁₅₂

yes

—

123

C₂×C₄

#16864391

yes

—

124

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

126

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

127

C₈

—

128

C₈

—

1076

—

129

C₂²

—

130

C₂²

—

131

C₂²

#22575375

yes

—

132

C₂²

C₂×#16864379

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

1076

—

136

C₄

#22575371

yes

—

137

C₄

#22575375

yes

—

138

C₄

C₂×D₂₁₅₂

yes

characteristic

139

C₄

C₂×#16864391

yes

characteristic

140

C₄

#22575483

yes

—

141

C₂

—

142

C₂

C₂×#22575371

yes

characteristic

143

C₂

C₂×#22575375

yes

characteristic

144

C₂

#31770613

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#16864379
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₆₉
Fitting	#431×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₆₉	C₂²⋊C₄
cosocle	C₄²×C₂₆₉	C₂³
hypercenter	#431	D₂₆₉
hypocenter	C₂₆₉	#1546
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#16864379

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₆₉

Fitting

#431×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₆₉

C₂²⋊C₄

cosocle

C₄²×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #31770350	C₂	semidirect
72. C₂₆₉	#1546	semidirect

H ≅ G/N

type

3. #31770350

C₂

semidirect

72. C₂₆₉

#1546

semidirect

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

subgroup	index
C₂×#22575453	2
#31770629	2
#31770350	2
#431×C₂₆₉	2
#1546	269

subgroup

index

#431×C₂₆₉

269

Subgroups — #46809556

Conjugacy classes of subgroups

Complements

Maximal subgroups