Group #46821388 — subgroups

Subgroups — C₂×#31771006 #46821388

Order 34432.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22575444	C₂	1	2	yes	maximal
2	#31771006	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22575387	C₂	1	1	yes	maximal, characteristic
4	C₂×#22575384	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#14524130	C₂²	1	1	yes	characteristic
8	#22575444	—	2	4	no	—
9	#22575444	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₂₆₉	—	2	2	no	—
11	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
12	C₂×#16864381	—	2	2	no	—
13	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
14	#22575387	C₂²	1	4	yes	—
15	#22575384	C₂²	1	8	yes	—
16	C₈×Dic₂₆₉	C₂²	1	4	yes	—
17	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
19	#107×C₂₆₉	C₂²	1	4	yes	—
20	C₂²×Dic₂₆₉	—	2	2	no	—
21	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
22	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
23	C₂×#14524130	Q₈	1	4	yes	—
24	C₂×#14524130	C₂³	1	2	yes	—
25	C₄⋊Dic₂₆₉	—	2	8	no	—
26	C₄⋊Dic₂₆₉	—	2	2	no	—
27	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
28	#16864381	—	2	8	no	—
29	C₄×Dic₂₆₉	D₄	1	2	yes	—
30	C₄×Dic₂₆₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₆₉	—	2	2	no	—
32	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₂₆₉	Q₈	1	4	yes	—
34	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₆₉	—	2	2	no	—
36	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
37	C₂×Dic₂₆₉	—	2	2	no	—
38	C₂×Dic₂₆₉	—	2	2	no	—
39	C₂×Dic₂₆₉	—	2	4	no	—
40	C₂×Dic₂₆₉	—	4	4	no	—
41	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
42	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
43	#14524130	C₂×Q₈	1	4	yes	—
44	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	—	2	4	no	—
46	C₂×C₄×C₂₆₉	—	4	4	no	—
47	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₂₆₉	C₂×Q₈	1	4	yes	—
52	C₂²×C₂₆₉	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₂₆₉	#135	1	2	yes	—
54	C₂²×C₂₆₉	#135	1	2	yes	—
55	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₂₆₉	—	4	4	no	—
57	C₄×C₂₆₉	C₄⋊Q₈	1	2	yes	—
58	C₄×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
60	Dic₂₆₉	—	2	2	no	—
61	Dic₂₆₉	—	2	2	no	—
62	Dic₂₆₉	—	4	4	no	—
63	C₂×C₂₆₉	#499	1	4	yes	complemented
64	C₂×C₂₆₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₂₆₉	C₂×#135	1	1	yes	characteristic
66	C₂×C₂₆₉	C₂×#135	1	1	yes	characteristic
67	C₂₆₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
68	C₂×#499	—	269	1	no	maximal, Sylow
69	C₂×#125	—	269	2	no	—
70	#499	—	269	8	no	—
71	C₂×#107	—	269	1	no	—
72	C₂×#107	D₂₆₉	1	1	yes	characteristic
73	C₂×#106	—	269	2	no	—
74	C₂×C₄×C₈	—	269	1	no	—
75	C₂²×C₈	—	269	1	no	—
76	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
77	#125	—	269	8	no	—
78	#125	—	538	4	no	—
79	C₂×C₄⋊C₄	—	269	2	no	—
80	C₂×C₄⋊C₄	—	538	2	no	—
81	C₂×C₄⋊C₄	—	538	2	no	—
82	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
83	C₂×C₄²	—	269	1	no	—
84	#107	—	269	4	no	—
85	#107	C₂×D₂₆₉	1	4	yes	—
86	#106	—	269	8	no	—
87	C₄×C₈	—	269	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	269	1	no	—
90	C₂²×C₄	—	269	1	no	—
91	C₂²×C₄	—	538	2	no	—
92	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
93	C₂×C₈	—	269	4	no	—
94	C₂×C₈	—	269	2	no	—
95	C₂×C₈	Dic₅₃₈	1	4	yes	—
96	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	269	4	no	—
99	C₄⋊C₄	—	538	8	no	—
100	C₄⋊C₄	—	538	8	no	—
101	C₄⋊C₄	—	538	2	no	—
102	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
103	C₄²	—	269	2	no	—
104	C₄²	—	269	2	no	—
105	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	269	2	no	—
109	C₂×C₄	—	269	2	no	—
110	C₂×C₄	—	538	2	no	—
111	C₂×C₄	—	538	2	no	—
112	C₂×C₄	—	538	4	no	—
113	C₂×C₄	—	1076	4	no	—
114	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
116	C₂×C₄	C₂×Dic₅₃₈	1	2	yes	—
117	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
118	C₈	—	269	4	no	—
119	C₈	C₂×Dic₅₃₈	1	4	yes	—
120	C₂²	#22575452	1	2	yes	—
121	C₂²	#22575472	1	2	yes	—
122	C₂²	#22575476	1	2	yes	—
123	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	538	2	no	—
126	C₄	—	538	2	no	—
127	C₄	—	1076	4	no	—
128	C₄	#22575452	1	2	yes	—
129	C₄	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
130	C₄	C₂²×Dic₅₃₈	1	1	yes	characteristic
131	C₂	#31771006	1	4	yes	direct product factor
132	C₂	C₂×#22575452	1	1	yes	characteristic
133	C₂	C₂×#22575472	1	1	yes	characteristic
134	C₂	C₂×#22575476	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575444

C₂

yes

maximal

#31771006

C₂

yes

maximal, direct product factor

C₂×#22575387

C₂

yes

maximal, characteristic

C₂×#22575384

C₂

yes

maximal

C₂×C₈×Dic₂₆₉

C₂

yes

maximal, characteristic

C₂×#107×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

characteristic

#22575444

—

#22575444

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×#16864381

—

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

#22575387

C₂²

yes

—

#22575384

C₂²

yes

—

C₈×Dic₂₆₉

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

—

#107×C₂₆₉

C₂²

yes

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

D₄

yes

characteristic

C₂²×Dic₂₆₉

D₄

yes

characteristic

C₂×#14524130

Q₈

yes

—

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

C₄×Dic₂₆₉

D₄

yes

—

C₄×Dic₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

Q₈

yes

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

#14524130

C₂×Q₈

yes

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂×Q₈

yes

—

C₂²×C₂₆₉

C₄⋊Q₈

yes

—

C₂²×C₂₆₉

#135

yes

—

C₂²×C₂₆₉

#135

yes

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

C₄⋊Q₈

yes

—

C₄×C₂₆₉

C₂²×Q₈

yes

characteristic

C₄×C₂₆₉

C₂²×Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

#499

yes

complemented

C₂×C₂₆₉

C₂×C₄⋊Q₈

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

C₂×#499

—

269

maximal, Sylow

C₂×#125

—

269

—

#499

—

269

—

C₂×#107

—

269

—

C₂×#107

D₂₆₉

yes

characteristic

C₂×#106

—

269

—

C₂×C₄×C₈

—

269

—

C₂²×C₈

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#125

—

269

—

#125

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂×C₄²

—

269

—

#107

—

269

—

#107

C₂×D₂₆₉

yes

—

#106

—

269

—

C₄×C₈

—

269

—

C₂²×C₄

—

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₂×C₈

Dic₅₃₈

yes

—

C₂×C₈

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

100

C₄⋊C₄

—

538

—

101

C₄⋊C₄

—

538

—

102

C₄⋊C₄

C₂²×D₂₆₉

yes

—

103

C₄²

—

269

—

104

C₄²

—

269

—

105

C₂³

D₄×D₂₆₉

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

269

—

109

C₂×C₄

—

269

—

110

C₂×C₄

—

538

—

111

C₂×C₄

—

538

—

112

C₂×C₄

—

538

—

113

C₂×C₄

—

1076

—

114

C₂×C₄

D₄×D₂₆₉

yes

characteristic

115

C₂×C₄

Q₈×D₂₆₉

yes

—

116

C₂×C₄

C₂×Dic₅₃₈

yes

—

117

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

118

C₈

—

269

—

119

C₈

C₂×Dic₅₃₈

yes

—

120

C₂²

#22575452

yes

—

121

C₂²

#22575472

yes

—

122

C₂²

#22575476

yes

—

123

C₂²

C₂×D₄×D₂₆₉

yes

characteristic

124

C₄

—

125

C₄

—

538

—

126

C₄

—

538

—

127

C₄

—

1076

—

128

C₄

#22575452

yes

—

129

C₄

C₂×Q₈×D₂₆₉

yes

characteristic

130

C₄

C₂²×Dic₅₃₈

yes

characteristic

131

C₂

#31771006

yes

direct product factor

132

C₂

C₂×#22575452

yes

characteristic

133

C₂

C₂×#22575472

yes

characteristic

134

C₂

C₂×#22575476

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×#107×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	C₂×#107	D₂₆₉
hypocenter	C₂₆₉	C₂×#499
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₆₉

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

C₂×#107×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #31771006	C₂	direct (N × H)
63. C₂×C₂₆₉	#499	semidirect
67. C₂₆₉	C₂×#499	semidirect
131. C₂	#31771006	direct (N × H)

H ≅ G/N

type

2. #31771006

C₂

direct (N × H)

63. C₂×C₂₆₉

#499

semidirect

67. C₂₆₉

C₂×#499

semidirect

131. C₂

#31771006

direct (N × H)

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

subgroup	index
C₂×#22575444	2
#31771006	2
C₂×#22575387	2
C₂×#22575384	2
C₂×C₈×Dic₂₆₉	2
C₂×#107×C₂₆₉	2
C₂×#499	269

subgroup

index

C₂×C₈×Dic₂₆₉

C₂×#107×C₂₆₉

C₂×#499

269

Subgroups — C₂×#31771006 #46821388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#31771006 #46821388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771006 #46821388