Group #31771234 — subgroups

Subgroups — #31771234

Order 17216.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22575452	C₂	1	2	yes	maximal, complemented
2	#22575451	C₂	1	2	yes	maximal, complemented
3	#22575443	C₂	1	2	yes	maximal, complemented
4	#22575443	C₂	1	2	yes	maximal, complemented
5	#22575432	C₂	1	1	yes	maximal, characteristic
6	#22575430	C₂	1	2	yes	maximal
7	#22575409	C₂	1	2	yes	maximal
8	C₂×#16864381	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₅₃₈	C₂²	1	2	yes	—
11	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
15	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
16	#16864381	C₂²	1	2	yes	—
17	#16864381	C₂²	1	2	yes	—
18	#16864381	C₂²	1	2	yes	—
19	#16864381	C₂²	1	2	yes	—
20	#16864381	C₂²	1	2	yes	—
21	#16864381	C₂²	1	2	yes	—
22	#16864381	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
24	C₄×Dic₂₆₉	C₂²	1	2	yes	—
25	C₄×Dic₂₆₉	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
31	Dic₅₃₈	—	4	2	no	—
32	Dic₅₃₈	—	4	1	no	—
33	C₂×Dic₂₆₉	—	4	1	no	—
34	C₂×Dic₂₆₉	Q₈	1	4	yes	complemented
35	C₂×Dic₂₆₉	C₂³	1	2	yes	—
36	C₂×Dic₂₆₉	C₂³	1	2	yes	—
37	C₂×Dic₂₆₉	C₂³	1	2	yes	—
38	C₂×Dic₂₆₉	C₂³	1	2	yes	—
39	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
40	Q₈×C₂₆₉	—	4	1	no	—
41	C₂×C₄×C₂₆₉	—	4	1	no	—
42	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
43	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
44	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
46	C₂²×C₂₆₉	—	2	1	no	—
47	C₂²×C₂₆₉	—	2	1	no	—
48	C₂²×C₂₆₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₆₉	—	2	1	no	—
51	C₄×C₂₆₉	—	2	1	no	—
52	C₄×C₂₆₉	—	2	2	no	—
53	C₄×C₂₆₉	—	2	2	no	—
54	Dic₂₆₉	—	2	2	no	—
55	Dic₂₆₉	—	2	2	no	—
56	Dic₂₆₉	—	2	2	no	—
57	Dic₂₆₉	C₂×Q₈	1	4	yes	—
58	C₂×C₂₆₉	—	2	2	no	—
59	C₂×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂₆₉	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	269	1	no	maximal, Sylow
64	C₄⋊Q₈	—	269	2	no	—
65	C₄⋊Q₈	—	269	2	no	—
66	#125	—	269	2	no	—
67	#125	—	269	2	no	—
68	C₂²⋊Q₈	—	269	1	no	—
69	C₂²⋊Q₈	—	269	2	no	—
70	C₂²⋊Q₈	D₂₆₉	1	1	yes	characteristic
71	#117	—	269	2	no	—
72	C₂×C₄⋊C₄	—	269	1	no	—
73	C₂×Q₈	—	269	1	no	—
74	C₂×Q₈	—	269	2	no	—
75	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
76	C₂²×C₄	—	269	1	no	—
77	C₂²×C₄	—	269	1	no	—
78	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
79	C₄⋊C₄	—	269	2	no	—
80	C₄⋊C₄	—	269	2	no	—
81	C₄⋊C₄	—	269	2	no	—
82	C₄⋊C₄	—	269	2	no	—
83	C₄⋊C₄	—	269	2	no	—
84	C₄⋊C₄	—	269	1	no	—
85	C₄⋊C₄	—	269	2	no	—
86	C₄⋊C₄	—	269	2	no	—
87	C₄⋊C₄	—	269	2	no	—
88	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
90	C₂²⋊C₄	—	269	2	no	—
91	C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
92	C₄²	—	269	2	no	—
93	C₄²	—	269	2	no	—
94	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	1076	1	no	—
97	Q₈	—	1076	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	269	4	no	—
100	C₂×C₄	—	269	2	no	—
101	C₂×C₄	—	269	2	no	—
102	C₂×C₄	—	269	2	no	—
103	C₂×C₄	—	269	2	no	—
104	C₂×C₄	—	1076	1	no	—
105	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
106	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
107	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₂₆₉	1	2	yes	—
112	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	269	4	no	—
118	C₄	—	538	2	no	—
119	C₄	—	538	2	no	—
120	C₄	—	538	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
123	C₂	#22575498	1	1	yes	characteristic
124	C₂	#22575500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22575452

C₂

yes

maximal, complemented

#22575451

C₂

yes

maximal, complemented

#22575443

C₂

yes

maximal, complemented

#22575443

C₂

yes

maximal, complemented

#22575432

C₂

yes

maximal, characteristic

#22575430

C₂

yes

maximal

#22575409

C₂

yes

maximal

C₂×#16864381

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂×Dic₅₃₈

C₂²

yes

—

C₂×Dic₅₃₈

C₂²

yes

characteristic

C₂²×Dic₂₆₉

C₂²

yes

characteristic

C₂²×Dic₂₆₉

C₂²

yes

characteristic

C₄⋊Dic₂₆₉

C₂²

yes

—

C₄⋊Dic₂₆₉

C₂²

yes

characteristic

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

C₂²⋊Dic₂₆₉

C₂²

yes

—

C₄×Dic₂₆₉

C₂²

yes

—

C₄×Dic₂₆₉

C₂²

yes

—

C₂×Q₈×C₂₆₉

C₂²

yes

characteristic

C₂²×C₄×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₂²⋊C₄×C₂₆₉

C₂²

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

Q₈

yes

complemented

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂³×C₂₆₉

C₂³

yes

characteristic

Q₈×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×Q₈

yes

—

C₂²×C₂₆₉

C₂⁴

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

C₂×Q₈

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×Q₈

yes

characteristic

C₂×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₂×C₂₆₉

2₋¹⁺⁴

yes

characteristic

C₂₆₉

#544

yes

characteristic, complemented, Sylow

#544

—

269

maximal, Sylow

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

#125

—

269

—

#125

—

269

—

C₂²⋊Q₈

—

269

—

C₂²⋊Q₈

—

269

—

C₂²⋊Q₈

D₂₆₉

yes

characteristic

#117

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

C₂×D₂₆₉

yes

characteristic

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂×D₂₆₉

yes

characteristic

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

C₂×D₂₆₉

yes

—

C₄²

—

269

—

C₄²

—

269

—

C₂³

C₂²×D₂₆₉

yes

characteristic

Q₈

—

Q₈

—

1076

—

Q₈

—

1076

—

C₂×C₄

—

C₂×C₄

—

269

—

100

C₂×C₄

—

269

—

101

C₂×C₄

—

269

—

102

C₂×C₄

—

269

—

103

C₂×C₄

—

269

—

104

C₂×C₄

—

1076

—

105

C₂×C₄

C₂²×D₂₆₉

yes

—

106

C₂×C₄

C₂²×D₂₆₉

yes

—

107

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

108

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₂₆₉

yes

—

112

C₂²

C₂³×D₂₆₉

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

269

—

118

C₄

—

538

—

119

C₄

—

538

—

120

C₄

—

538

—

121

C₂

—

122

C₂

C₂×Q₈×D₂₆₉

yes

characteristic

123

C₂

#22575498

yes

characteristic

124

C₂

#22575500

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	C₂²⋊Q₈×C₂₆₉	C₂
prime radical	C₂³×C₂₆₉	C₂³
hypercenter	C₂²⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	#544
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₆₉

commutator, socle, cosocle

C₂²×C₂₆₉

C₂⁴

Fitting

C₂²⋊Q₈×C₂₆₉

C₂

prime radical

C₂³×C₂₆₉

C₂³

hypercenter

C₂²⋊Q₈

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #22575452	C₂	semidirect
2. #22575451	C₂	semidirect
3. #22575443	C₂	semidirect
4. #22575443	C₂	semidirect
34. C₂×Dic₂₆₉	Q₈	semidirect
62. C₂₆₉	#544	semidirect

H ≅ G/N

type

1. #22575452

C₂

semidirect

2. #22575451

C₂

semidirect

3. #22575443

C₂

semidirect

4. #22575443

C₂

semidirect

34. C₂×Dic₂₆₉

Q₈

semidirect

62. C₂₆₉

#544

semidirect

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

subgroup	index
#22575452	2
#22575451	2
#22575443	2
#22575432	2
#22575430	2
#22575409	2
C₂×#16864381	2
C₂²⋊Q₈×C₂₆₉	2
#544	269

subgroup

index

C₂²⋊Q₈×C₂₆₉

269

Subgroups — #31771234

Conjugacy classes of subgroups

Complements

Maximal subgroups