Group #31771359 — subgroups

Subgroups — #31771359

Order 17216.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22575448	C₂	1	2	yes	maximal, complemented
3	#22575448	C₂	1	2	yes	maximal, complemented
4	#22575445	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575441	C₂	1	1	yes	maximal, characteristic
6	#22575438	C₂	1	2	yes	maximal
7	#22575433	C₂	1	2	yes	maximal, complemented
8	#22575431	C₂	1	2	yes	maximal, complemented
9	#125×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₅₃₈	C₂²	1	2	yes	—
11	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
12	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
14	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
16	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
17	#16864381	C₂²	1	2	yes	—
18	#16864381	C₂²	1	2	yes	—
19	#16864381	C₂²	1	2	yes	—
20	#16864379	C₂²	1	2	yes	—
21	#16864379	C₂²	1	2	yes	—
22	#16864379	C₂²	1	2	yes	—
23	#16864379	C₂²	1	2	yes	—
24	#16864379	C₂²	1	2	yes	—
25	C₄×Dic₂₆₉	C₂²	1	2	yes	—
26	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
28	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
30	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
31	C₂²×D₂₆₉	C₂³	1	2	yes	—
32	Dic₅₃₈	—	4	2	no	—
33	Dic₅₃₈	—	4	1	no	—
34	D₁₀₇₆	—	4	1	no	—
35	C₂×Dic₂₆₉	C₂³	1	2	yes	—
36	C₂×Dic₂₆₉	C₂³	1	2	yes	—
37	C₂×Dic₂₆₉	C₂³	1	2	yes	—
38	C₄×D₂₆₉	—	4	2	no	—
39	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
43	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
44	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
45	C₂×D₂₆₉	—	2	2	no	—
46	C₂×D₂₆₉	—	2	2	no	—
47	C₂×D₂₆₉	—	2	2	no	—
48	C₄×C₂₆₉	—	2	2	no	—
49	C₄×C₂₆₉	—	2	1	no	—
50	C₄×C₂₆₉	—	2	1	no	—
51	C₄×C₂₆₉	—	2	2	no	—
52	C₄×C₂₆₉	—	2	1	no	—
53	Dic₂₆₉	—	2	2	no	—
54	Dic₂₆₉	—	2	2	no	—
55	Dic₂₆₉	—	2	2	no	—
56	C₂×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
59	D₂₆₉	—	4	2	no	—
60	C₂₆₉	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	269	1	no	maximal, Sylow
62	C₄⋊Q₈	—	269	2	no	—
63	#126	—	269	2	no	—
64	#126	—	269	2	no	—
65	#125	—	269	1	no	—
66	#125	D₂₆₉	1	1	yes	characteristic, complemented
67	#124	—	269	1	no	—
68	#123	—	269	2	no	—
69	C₂²⋊Q₈	—	269	2	no	—
70	C₂²⋊Q₈	—	269	2	no	—
71	C₂×Q₈	—	269	2	no	—
72	C₂×Q₈	—	269	1	no	—
73	C₂×D₄	—	269	1	no	—
74	C₂²×C₄	—	269	2	no	—
75	C₄⋊C₄	—	269	1	no	—
76	C₄⋊C₄	—	269	2	no	—
77	C₄⋊C₄	—	269	2	no	—
78	C₄⋊C₄	—	269	2	no	—
79	C₄⋊C₄	—	269	2	no	—
80	C₄⋊C₄	—	269	1	no	—
81	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
83	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
84	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
85	C₂²⋊C₄	—	269	2	no	—
86	C₂²⋊C₄	—	269	2	no	—
87	C₂²⋊C₄	—	269	2	no	—
88	C₂²⋊C₄	—	269	2	no	—
89	C₂²⋊C₄	—	269	2	no	—
90	C₄²	—	269	2	no	—
91	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
92	C₂³	—	269	2	no	—
93	Q₈	—	1076	2	no	—
94	Q₈	—	1076	1	no	—
95	D₄	—	1076	1	no	—
96	C₂×C₄	—	269	2	no	—
97	C₂×C₄	—	269	2	no	—
98	C₂×C₄	—	269	2	no	—
99	C₂×C₄	—	1076	2	no	—
100	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
101	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
105	C₂²	—	538	2	no	—
106	C₂²	—	538	2	no	—
107	C₂²	—	538	2	no	—
108	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	538	2	no	—
115	C₄	—	538	2	no	—
116	C₄	—	538	2	no	—
117	C₂	—	1076	2	no	—
118	C₂	#22575499	1	1	yes	characteristic
119	C₂	#22575500	1	1	yes	characteristic
120	C₂	#22575501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22575452

C₂

yes

maximal, complemented

#22575448

C₂

yes

maximal, complemented

#22575448

C₂

yes

maximal, complemented

#22575445

C₂

yes

maximal, characteristic, complemented

#22575441

C₂

yes

maximal, characteristic

#22575438

C₂

yes

maximal

#22575433

C₂

yes

maximal, complemented

#22575431

C₂

yes

maximal, complemented

#125×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₃₈

C₂²

yes

—

C₂×Dic₅₃₈

C₂²

yes

characteristic

C₂×D₁₀₇₆

C₂²

yes

characteristic

C₂×C₄×D₂₆₉

C₂²

yes

—

C₄⋊Dic₂₆₉

C₂²

yes

characteristic

C₄⋊Dic₂₆₉

C₂²

yes

—

C₄⋊Dic₂₆₉

C₂²

yes

characteristic

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

C₄×Dic₂₆₉

C₂²

yes

—

C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₄²×C₂₆₉

C₂²

yes

characteristic

C₂²×D₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

D₁₀₇₆

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₄×D₂₆₉

—

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₂₆₉

C₂⁴

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

H(5,𝔽₂)

yes

characteristic

C₂×C₂₆₉

2₋¹⁺⁴

yes

characteristic

C₂×C₂₆₉

2₋¹⁺⁴

yes

characteristic

D₂₆₉

—

C₂₆₉

#563

yes

characteristic, complemented, Sylow

#563

—

269

maximal, Sylow

C₄⋊Q₈

—

269

—

#126

—

269

—

#126

—

269

—

#125

—

269

—

#125

D₂₆₉

yes

characteristic, complemented

#124

—

269

—

#123

—

269

—

C₂²⋊Q₈

—

269

—

C₂²⋊Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×D₄

—

269

—

C₂²×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

—

C₄⋊C₄

C₂×D₂₆₉

yes

—

C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₄²

—

269

—

C₄²

C₂×D₂₆₉

yes

characteristic

C₂³

—

269

—

Q₈

—

1076

—

Q₈

—

1076

—

D₄

—

1076

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

1076

—

100

C₂×C₄

C₂²×D₂₆₉

yes

—

101

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

102

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

103

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

104

C₂×C₄

C₂²×D₂₆₉

yes

—

105

C₂²

—

538

—

106

C₂²

—

538

—

107

C₂²

—

538

—

108

C₂²

C₂³×D₂₆₉

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

538

—

115

C₄

—

538

—

116

C₄

—

538

—

117

C₂

—

1076

—

118

C₂

#22575499

yes

characteristic

119

C₂

#22575500

yes

characteristic

120

C₂

#22575501

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	#125×C₂₆₉	C₂
prime radical	C₂×D₁₀₇₆	C₂²
hypercenter	#125	D₂₆₉
hypocenter	C₂₆₉	#563
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₆₉

commutator, socle, cosocle

C₂²×C₂₆₉

C₂⁴

Fitting

#125×C₂₆₉

C₂

prime radical

C₂×D₁₀₇₆

C₂²

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #22575452	C₂	semidirect
2. #22575448	C₂	semidirect
3. #22575448	C₂	semidirect
4. #22575445	C₂	semidirect
7. #22575433	C₂	semidirect
8. #22575431	C₂	semidirect
9. #125×C₂₆₉	C₂	semidirect
60. C₂₆₉	#563	semidirect
66. #125	D₂₆₉	semidirect

H ≅ G/N

type

1. #22575452

C₂

semidirect

2. #22575448

C₂

semidirect

3. #22575448

C₂

semidirect

4. #22575445

C₂

semidirect

7. #22575433

C₂

semidirect

8. #22575431

C₂

semidirect

9. #125×C₂₆₉

C₂

semidirect

60. C₂₆₉

#563

semidirect

66. #125

D₂₆₉

semidirect

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

subgroup	index
#22575452	2
#22575448	2
#22575445	2
#22575441	2
#22575438	2
#22575433	2
#22575431	2
#125×C₂₆₉	2
#563	269

subgroup

index

#125×C₂₆₉

269

Subgroups — #31771359

Conjugacy classes of subgroups

Complements

Maximal subgroups