Group #31792588 — subgroups

Subgroups — #31792588

Order 17536.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22578204	C₂	1	4	yes	maximal, complemented
2	#22577670	C₂	1	4	yes	maximal, complemented
3	#22577570	C₂	1	4	yes	maximal, complemented
4	#22577521	C₂	1	2	yes	maximal, complemented
5	#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984133	C₂²	1	4	yes	complemented
7	#17984114	C₂²	1	8	yes	complemented
8	#17984112	—	2	8	no	—
9	#17984110	C₂²	1	2	yes	—
10	#17984107	—	2	4	no	—
11	D₈×D₁₃₇	—	2	4	no	—
12	#17984094	C₂²	1	4	yes	complemented
13	#17984024	C₂²	1	2	yes	complemented
14	#17984016	C₂²	1	4	yes	complemented
15	#17984005	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₃₇	C₂²	1	4	yes	complemented
17	#127×C₁₃₇	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
19	#14555510	C₂³	1	4	yes	—
20	#14555509	—	2	8	no	—
21	C₂×C₂²⋊D₁₃₇	—	4	4	no	—
22	D₄×D₁₃₇	—	2	8	no	—
23	Dic₅₄₈	D₄	1	2	yes	complemented
24	#14555499	D₄	1	4	yes	complemented
25	#14555498	—	2	8	no	—
26	D₁₀₉₆	D₄	1	2	yes	complemented
27	#14555495	—	2	8	no	—
28	#14555494	C₂³	1	4	yes	—
29	#14555493	—	2	4	no	—
30	C₈×D₁₃₇	—	2	4	no	—
31	C₂×D₄×C₁₃₇	—	4	2	no	—
32	C₂×D₄×C₁₃₇	C₂³	1	4	yes	—
33	D₈×C₁₃₇	—	2	4	no	—
34	D₈×C₁₃₇	—	2	4	no	—
35	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
36	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×D₁₃₇	—	4	4	no	—
38	Dic₂₇₄	C₂×D₄	1	4	yes	—
39	D₅₄₈	C₂×D₄	1	4	yes	—
40	C₂²⋊D₁₃₇	—	2	4	no	—
41	C₂²⋊D₁₃₇	—	4	8	no	—
42	C₂×Dic₁₃₇	—	4	4	no	—
43	C₄×D₁₃₇	—	2	4	no	—
44	#11764040	—	2	4	no	—
45	C₂³×C₁₃₇	—	2	4	no	—
46	D₄×C₁₃₇	—	2	8	no	—
47	D₄×C₁₃₇	—	8	4	no	—
48	C₂×C₄×C₁₃₇	—	2	1	no	—
49	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
51	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₃₇	—	4	4	no	—
53	C₂²×C₁₃₇	—	8	4	no	—
54	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₃₇	—	2	4	no	—
56	C₂×D₁₃₇	—	16	4	no	—
57	C₄×C₁₃₇	—	2	2	no	—
58	C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
59	Dic₁₃₇	—	2	4	no	—
60	C₂×C₁₃₇	—	2	1	no	—
61	C₂×C₁₃₇	—	8	4	no	—
62	C₂×C₁₃₇	D₄²	1	1	yes	characteristic
63	D₁₃₇	—	4	4	no	—
64	C₁₃₇	#3292	1	1	yes	characteristic, complemented, Sylow
65	#3292	—	137	1	no	maximal, Sylow
66	#576	—	137	4	no	—
67	#493	D₁₃₇	1	1	yes	characteristic, complemented
68	#472	—	137	4	no	—
69	#453	—	137	4	no	—
70	#443	—	137	2	no	—
71	H(5,𝔽₂)	—	137	4	no	—
72	AGL(1,ℤ/8)	—	274	8	no	—
73	AGL(1,ℤ/8)	—	137	8	no	—
74	#135	—	274	4	no	—
75	#135	—	137	2	no	—
76	C₂×D₈	—	274	4	no	—
77	C₂×D₈	C₂×D₁₃₇	1	4	yes	complemented
78	#131	—	137	4	no	—
79	#127	C₂×D₁₃₇	1	2	yes	complemented
80	#108	—	137	2	no	—
81	#104	—	137	4	no	—
82	#100	—	137	4	no	—
83	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
84	#41	—	274	8	no	—
85	#41	—	137	4	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	274	8	no	—
88	C₂×D₄	—	548	4	no	—
89	C₂×D₄	C₂²×D₁₃₇	1	4	yes	—
90	Q₁₆	—	137	2	no	—
91	SD₁₆	—	274	8	no	—
92	SD₁₆	—	137	4	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	274	8	no	—
96	D₈	—	137	2	no	—
97	M₁₆	—	274	4	no	—
98	M₁₆	—	137	4	no	—
99	C₂×C₈	—	274	4	no	—
100	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
101	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	548	4	no	—
104	Q₈	—	137	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	274	4	no	—
108	D₄	—	548	8	no	—
109	D₄	—	137	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	274	4	no	—
112	C₂×C₄	—	548	4	no	—
113	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
114	C₈	—	274	4	no	—
115	C₈	D₄×D₁₃₇	1	2	yes	—
116	C₈	D₄×D₁₃₇	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	274	4	no	—
120	C₂²	—	2192	4	no	—
121	C₂²	#17984133	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	274	4	no	—
124	C₄	C₂×D₄×D₁₃₇	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	548	4	no	—
128	C₂	#22578038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578204

C₂

yes

maximal, complemented

#22577670

C₂

yes

maximal, complemented

#22577570

C₂

yes

maximal, complemented

#22577521

C₂

yes

maximal, complemented

#493×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#17984133

C₂²

yes

complemented

#17984114

C₂²

yes

complemented

#17984112

—

#17984110

C₂²

yes

—

#17984107

—

D₈×D₁₃₇

—

#17984094

C₂²

yes

complemented

#17984024

C₂²

yes

complemented

#17984016

C₂²

yes

complemented

#17984005

C₂²

yes

complemented

C₂×D₈×C₁₃₇

C₂²

yes

complemented

#127×C₁₃₇

C₂²

yes

complemented

C₄×C₈×C₁₃₇

C₂²

yes

characteristic, complemented

#14555510

C₂³

yes

—

#14555509

—

C₂×C₂²⋊D₁₃₇

—

D₄×D₁₃₇

—

Dic₅₄₈

D₄

yes

complemented

#14555499

D₄

yes

complemented

#14555498

—

D₁₀₉₆

D₄

yes

complemented

#14555495

—

#14555494

C₂³

yes

—

#14555493

—

C₈×D₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

C₂³

yes

—

D₈×C₁₃₇

—

D₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

—

Dic₂₇₄

C₂×D₄

yes

—

D₅₄₈

C₂×D₄

yes

—

C₂²⋊D₁₃₇

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

C₂×D₄

yes

—

C₈×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×D₄

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

D₄²

yes

characteristic

D₁₃₇

—

C₁₃₇

#3292

yes

characteristic, complemented, Sylow

#3292

—

137

maximal, Sylow

#576

—

137

—

#493

D₁₃₇

yes

characteristic, complemented

#472

—

137

—

#453

—

137

—

#443

—

137

—

H(5,𝔽₂)

—

137

—

AGL(1,ℤ/8)

—

274

—

AGL(1,ℤ/8)

—

137

—

#135

—

274

—

#135

—

137

—

C₂×D₈

—

274

—

C₂×D₈

C₂×D₁₃₇

yes

complemented

#131

—

137

—

#127

C₂×D₁₃₇

yes

complemented

#108

—

137

—

#104

—

137

—

#100

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic, complemented

#41

—

274

—

#41

—

137

—

C₂×D₄

—

C₂×D₄

—

274

—

C₂×D₄

—

548

—

C₂×D₄

C₂²×D₁₃₇

yes

—

Q₁₆

—

137

—

SD₁₆

—

274

—

SD₁₆

—

137

—

D₈

—

D₈

—

D₈

—

274

—

D₈

—

137

—

M₁₆

—

274

—

M₁₆

—

137

—

C₂×C₈

—

274

—

100

C₂×C₈

C₂²×D₁₃₇

yes

—

101

C₄²

C₂²×D₁₃₇

yes

characteristic

102

C₂³

—

103

C₂³

—

548

—

104

Q₈

—

137

—

105

D₄

—

106

D₄

—

107

D₄

—

274

—

108

D₄

—

548

—

109

D₄

—

137

—

110

C₂×C₄

—

111

C₂×C₄

—

274

—

112

C₂×C₄

—

548

—

113

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

114

C₈

—

274

—

115

C₈

D₄×D₁₃₇

yes

—

116

C₈

D₄×D₁₃₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

274

—

120

C₂²

—

2192

—

121

C₂²

#17984133

yes

characteristic

122

C₄

—

123

C₄

—

274

—

124

C₄

C₂×D₄×D₁₃₇

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

548

—

128

C₂

#22578038

yes

characteristic

	subgroup	G/H
center	C₂	#22578038
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#493×C₁₃₇	C₂
socle	C₂×C₁₃₇	D₄²
hypercenter	#493	D₁₃₇
hypocenter	C₁₃₇	#3292
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#22578038

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#493×C₁₃₇

C₂

socle

C₂×C₁₃₇

D₄²

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #22578204	C₂	semidirect
2. #22577670	C₂	semidirect
3. #22577570	C₂	semidirect
4. #22577521	C₂	semidirect
5. #493×C₁₃₇	C₂	semidirect
6. #17984133	C₂²	semidirect
7. #17984114	C₂²	semidirect
12. #17984094	C₂²	semidirect
13. #17984024	C₂²	semidirect
14. #17984016	C₂²	semidirect
15. #17984005	C₂²	semidirect
16. C₂×D₈×C₁₃₇	C₂²	semidirect
17. #127×C₁₃₇	C₂²	semidirect
18. C₄×C₈×C₁₃₇	C₂²	semidirect
23. Dic₅₄₈	D₄	semidirect
24. #14555499	D₄	semidirect
26. D₁₀₉₆	D₄	semidirect
64. C₁₃₇	#3292	semidirect
67. #493	D₁₃₇	semidirect
77. C₂×D₈	C₂×D₁₃₇	semidirect
79. #127	C₂×D₁₃₇	semidirect
83. C₄×C₈	C₂×D₁₃₇	semidirect

H ≅ G/N

type

1. #22578204

C₂

semidirect

2. #22577670

C₂

semidirect

3. #22577570

C₂

semidirect

4. #22577521

C₂

semidirect

5. #493×C₁₃₇

C₂

semidirect

6. #17984133

C₂²

semidirect

7. #17984114

C₂²

semidirect

12. #17984094

C₂²

semidirect

13. #17984024

C₂²

semidirect

14. #17984016

C₂²

semidirect

15. #17984005

C₂²

semidirect

16. C₂×D₈×C₁₃₇

C₂²

semidirect

17. #127×C₁₃₇

C₂²

semidirect

18. C₄×C₈×C₁₃₇

C₂²

semidirect

23. Dic₅₄₈

D₄

semidirect

24. #14555499

D₄

semidirect

26. D₁₀₉₆

D₄

semidirect

64. C₁₃₇

#3292

semidirect

67. #493

D₁₃₇

semidirect

77. C₂×D₈

C₂×D₁₃₇

semidirect

79. #127

C₂×D₁₃₇

semidirect

83. C₄×C₈

C₂×D₁₃₇

semidirect

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

subgroup	index
#22578204	2
#22577670	2
#22577570	2
#22577521	2
#493×C₁₃₇	2
#3292	137

subgroup

index

#493×C₁₃₇

137

Subgroups — #31792588

Conjugacy classes of subgroups

Complements

Maximal subgroups