Group #46819752 — subgroups

Subgroups — #46819752

Order 34432.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic
2	#31771082	C₂	1	2	yes	maximal
3	#31770711	C₂	1	4	yes	maximal, complemented
4	#31770699	C₂	1	4	yes	maximal, complemented
5	#31770613	C₂	1	2	yes	maximal
6	#31770560	C₂	1	1	yes	maximal, characteristic
7	#427×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
9	C₂×Dic₁₀₇₆	C₄	1	8	yes	complemented
10	C₂×Dic₁₀₇₆	C₂²	1	2	yes	—
11	C₂×Dic₁₀₇₆	C₂²	1	2	yes	—
12	C₂×#16864386	C₂²	1	1	yes	characteristic
13	C₄×Dic₅₃₈	—	2	4	no	—
14	#22575418	C₂²	1	8	yes	—
15	#22575412	C₂²	1	1	yes	characteristic
16	#22575409	—	2	4	no	—
17	#22575376	C₂²	1	8	yes	—
18	#22575363	C₂²	1	2	yes	—
19	C₈×Dic₂₆₉	C₂²	1	2	yes	—
20	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	#117×C₂₆₉	C₂²	1	2	yes	—
22	#107×C₂₆₉	C₂²	1	2	yes	—
23	#106×C₂₆₉	C₂²	1	2	yes	—
24	C₂×Dic₅₃₈	—	2	2	no	—
25	C₂×Dic₅₃₈	—	2	4	no	—
26	C₂×Dic₅₃₈	C₂×C₄	1	8	yes	—
27	C₂×Dic₅₃₈	C₂³	1	4	yes	—
28	C₂²×Dic₂₆₉	—	2	2	no	—
29	Dic₁₀₇₆	C₂×C₄	1	8	yes	—
30	Dic₁₀₇₆	C₂×C₄	1	8	yes	—
31	#16864386	—	2	2	no	—
32	C₂×#14524130	C₂³	1	2	yes	—
33	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
34	#16864381	—	2	8	no	—
35	C₂²⋊Dic₂₆₉	D₄	1	2	yes	—
36	C₄×Dic₂₆₉	—	2	8	no	—
37	C₄×Dic₂₆₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₆₉	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₂₆₉	—	2	2	no	—
44	C₄²×C₂₆₉	—	2	2	no	—
45	Dic₅₃₈	—	2	8	no	—
46	Dic₅₃₈	—	2	4	no	—
47	Dic₅₃₈	C₂²×C₄	1	8	yes	—
48	C₂×Dic₂₆₉	—	2	8	no	—
49	C₂×Dic₂₆₉	—	2	4	no	—
50	C₂×Dic₂₆₉	C₂×D₄	1	4	yes	—
51	#14524130	—	2	2	no	—
52	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	—	2	4	no	—
54	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₆₉	#41	1	2	yes	—
57	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
58	C₈×C₂₆₉	C₂²×C₄	1	2	yes	—
59	C₈×C₂₆₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₂₆₉	—	2	2	no	—
61	C₂²×C₂₆₉	C₄×D₄	1	2	yes	—
62	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₂₆₉	—	4	4	no	—
64	C₄×C₂₆₉	C₄×D₄	1	2	yes	—
65	C₄×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
67	Dic₂₆₉	—	2	8	no	—
68	Dic₂₆₉	—	2	4	no	—
69	C₂×C₂₆₉	—	2	2	no	—
70	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₂₆₉	#578	1	2	yes	—
72	C₂₆₉	#2950	1	1	yes	characteristic, complemented, Sylow
73	#2950	—	269	1	no	maximal, Sylow
74	C₂²×Q₁₆	—	269	1	no	—
75	#519	—	269	2	no	—
76	#441	—	269	4	no	—
77	C₄×Q₁₆	—	269	4	no	—
78	#427	D₂₆₉	1	1	yes	characteristic
79	#419	—	269	2	no	—
80	#405	—	269	1	no	—
81	C₂²×Q₈	—	269	2	no	—
82	C₂×Q₁₆	—	269	8	no	—
83	C₂×Q₁₆	—	269	2	no	—
84	C₂×Q₁₆	—	269	2	no	—
85	C₂×M₁₆	—	269	1	no	—
86	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
87	C₄×Q₈	—	269	8	no	—
88	C₄×Q₈	—	538	4	no	—
89	#117	—	269	1	no	—
90	#117	—	538	4	no	—
91	#117	C₂×D₂₆₉	1	2	yes	—
92	#107	C₂×D₂₆₉	1	2	yes	—
93	#106	C₂×D₂₆₉	1	2	yes	—
94	#103	—	269	8	no	—
95	#97	—	269	2	no	—
96	C₄×C₈	—	269	2	no	—
97	C₂×Q₈	—	269	8	no	—
98	C₂×Q₈	—	269	4	no	—
99	C₂×Q₈	—	538	2	no	—
100	C₂×Q₈	—	538	4	no	—
101	C₂²×C₄	—	538	2	no	—
102	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
103	Q₁₆	—	269	8	no	—
104	Q₁₆	—	269	8	no	—
105	M₁₆	—	538	2	no	—
106	C₂×C₈	—	269	2	no	—
107	C₂×C₈	C₄×D₂₆₉	1	4	yes	—
108	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
110	C₄⋊C₄	—	269	2	no	—
111	C₄⋊C₄	—	538	8	no	—
112	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	269	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	269	2	no	—
117	C₄²	—	538	8	no	—
118	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
119	Q₈	—	269	8	no	—
120	Q₈	—	538	8	no	—
121	Q₈	—	538	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	269	4	no	—
124	C₂×C₄	—	538	8	no	—
125	C₂×C₄	—	538	4	no	—
126	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
127	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
128	C₂×C₄	#16864403	1	2	yes	—
129	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
130	C₈	—	538	2	no	—
131	C₈	C₂×C₄×D₂₆₉	1	2	yes	—
132	C₈	C₂×C₄×D₂₆₉	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#22575415	1	2	yes	—
135	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	538	4	no	—
138	C₄	—	538	8	no	—
139	C₄	#22575415	1	2	yes	—
140	C₄	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
141	C₄	C₂×#16864403	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#22575415	1	1	yes	characteristic
144	C₂	#31771419	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₀₇₆

C₂

yes

maximal, characteristic

#31771082

C₂

yes

maximal

#31770711

C₂

yes

maximal, complemented

#31770699

C₂

yes

maximal, complemented

#31770613

C₂

yes

maximal

#31770560

C₂

yes

maximal, characteristic

#427×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

C₂²

yes

—

C₂×Dic₁₀₇₆

C₄

yes

complemented

C₂×Dic₁₀₇₆

C₂²

yes

—

C₂×Dic₁₀₇₆

C₂²

yes

—

C₂×#16864386

C₂²

yes

characteristic

C₄×Dic₅₃₈

—

#22575418

C₂²

yes

—

#22575412

C₂²

yes

characteristic

#22575409

—

#22575376

C₂²

yes

—

#22575363

C₂²

yes

—

C₈×Dic₂₆₉

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

#117×C₂₆₉

C₂²

yes

—

#107×C₂₆₉

C₂²

yes

—

#106×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

C₂×Dic₅₃₈

C₂³

yes

—

C₂²×Dic₂₆₉

—

Dic₁₀₇₆

C₂×C₄

yes

—

Dic₁₀₇₆

C₂×C₄

yes

—

#16864386

—

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

D₄

yes

—

#16864381

—

C₂²⋊Dic₂₆₉

D₄

yes

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂³

yes

—

C₂²⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

C₂²×C₄

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

#14524130

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂²×C₄

yes

—

C₈×C₂₆₉

C₂²×C₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄×D₄

yes

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

C₄×D₄

yes

—

C₄×C₂₆₉

C₂³×C₄

yes

characteristic

C₄×C₂₆₉

C₂×#41

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂×C₄×D₄

yes

characteristic

C₂×C₂₆₉

#578

yes

—

C₂₆₉

#2950

yes

characteristic, complemented, Sylow

#2950

—

269

maximal, Sylow

C₂²×Q₁₆

—

269

—

#519

—

269

—

#441

—

269

—

C₄×Q₁₆

—

269

—

#427

D₂₆₉

yes

characteristic

#419

—

269

—

#405

—

269

—

C₂²×Q₈

—

269

—

C₂×Q₁₆

—

269

—

C₂×Q₁₆

—

269

—

C₂×Q₁₆

—

269

—

C₂×M₁₆

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₄×Q₈

—

269

—

C₄×Q₈

—

538

—

#117

—

269

—

#117

—

538

—

#117

C₂×D₂₆₉

yes

—

#107

C₂×D₂₆₉

yes

—

#106

C₂×D₂₆₉

yes

—

#103

—

269

—

#97

—

269

—

C₄×C₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

100

C₂×Q₈

—

538

—

101

C₂²×C₄

—

538

—

102

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

103

Q₁₆

—

269

—

104

Q₁₆

—

269

—

105

M₁₆

—

538

—

106

C₂×C₈

—

269

—

107

C₂×C₈

C₄×D₂₆₉

yes

—

108

C₂×C₈

C₂²×D₂₆₉

yes

characteristic

109

C₂×C₈

C₂²×D₂₆₉

yes

characteristic

110

C₄⋊C₄

—

269

—

111

C₄⋊C₄

—

538

—

112

C₄⋊C₄

C₂²×D₂₆₉

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

269

—

115

C₄²

—

116

C₄²

—

269

—

117

C₄²

—

538

—

118

C₂³

D₄×D₂₆₉

yes

characteristic

119

Q₈

—

269

—

120

Q₈

—

538

—

121

Q₈

—

538

—

122

C₂×C₄

—

123

C₂×C₄

—

269

—

124

C₂×C₄

—

538

—

125

C₂×C₄

—

538

—

126

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

127

C₂×C₄

D₄×D₂₆₉

yes

characteristic

128

C₂×C₄

#16864403

yes

—

129

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

130

C₈

—

538

—

131

C₈

C₂×C₄×D₂₆₉

yes

—

132

C₈

C₂×C₄×D₂₆₉

yes

—

133

C₂²

—

134

C₂²

#22575415

yes

—

135

C₂²

C₂×D₄×D₂₆₉

yes

characteristic

136

C₄

—

137

C₄

—

538

—

138

C₄

—

538

—

139

C₄

#22575415

yes

—

140

C₄

C₂²×C₄×D₂₆₉

yes

characteristic

141

C₄

C₂×#16864403

yes

characteristic

142

C₂

—

143

C₂

C₂×#22575415

yes

characteristic

144

C₂

#31771419

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₆₉
commutator	C₄×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#427×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#427	D₂₆₉
hypocenter	C₂₆₉	#2950
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₂₆₉

commutator

C₄×C₂₆₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#427×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂×C₄×C₂₆₉

C₂⁴

D₂₆₉

C₂₆₉

G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #31770711	C₂	semidirect
4. #31770699	C₂	semidirect
9. C₂×Dic₁₀₇₆	C₄	semidirect
72. C₂₆₉	#2950	semidirect

H ≅ G/N

type

3. #31770711

C₂

semidirect

4. #31770699

C₂

semidirect

9. C₂×Dic₁₀₇₆

C₄

semidirect

72. C₂₆₉

#2950

semidirect

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

subgroup	index
C₂²×Dic₁₀₇₆	2
#31771082	2
#31770711	2
#31770699	2
#31770613	2
#31770560	2
#427×C₂₆₉	2
#2950	269

subgroup

index

C₂²×Dic₁₀₇₆

#427×C₂₆₉

269

Subgroups — #46819752

Conjugacy classes of subgroups

Complements

Maximal subgroups