Group #31780466 — subgroups

Subgroups — #31780466

Order 17536.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22577808	C₂	1	1	yes	maximal, characteristic
2	#22577808	C₂	1	1	yes	maximal, characteristic
3	#22577801	C₂	1	1	yes	maximal, characteristic
4	#22577783	C₂	1	1	yes	maximal, characteristic
5	#22577754	C₂	1	1	yes	maximal, characteristic
6	#22577754	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₄₈	—	2	2	no	—
9	#17984088	C₂²	1	1	yes	characteristic
10	#17984088	C₂²	1	1	yes	characteristic
11	#17984088	C₂²	1	1	yes	characteristic
12	#17984088	C₂²	1	1	yes	characteristic
13	#17984022	—	2	2	no	—
14	#17984022	—	2	2	no	—
15	#17984020	—	2	2	no	—
16	#17984013	—	2	2	no	—
17	#17984013	—	2	2	no	—
18	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₄	—	2	1	no	—
22	C₂×Dic₂₇₄	—	2	1	no	—
23	C₂×Dic₂₇₄	—	2	1	no	—
24	C₂×Dic₂₇₄	—	2	1	no	—
25	Dic₅₄₈	—	4	4	no	—
26	C₄⋊Dic₁₃₇	—	2	1	no	—
27	C₄⋊Dic₁₃₇	—	2	1	no	—
28	C₄⋊Dic₁₃₇	—	2	1	no	—
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₄⋊Dic₁₃₇	—	2	1	no	—
31	C₄⋊Dic₁₃₇	—	2	1	no	—
32	C₄⋊Dic₁₃₇	—	2	1	no	—
33	C₄⋊Dic₁₃₇	—	2	1	no	—
34	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
35	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
36	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
37	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
38	Dic₂₇₄	—	4	2	no	—
39	Dic₂₇₄	—	4	2	no	—
40	Dic₂₇₄	—	4	2	no	—
41	Dic₂₇₄	—	4	2	no	—
42	C₂×Dic₁₃₇	—	4	1	no	—
43	C₂×Dic₁₃₇	—	4	1	no	—
44	C₂×Dic₁₃₇	—	4	1	no	—
45	C₂×Dic₁₃₇	—	4	1	no	—
46	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	2	no	—
50	C₈×C₁₃₇	—	2	2	no	—
51	C₈×C₁₃₇	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₃₇	#127	1	1	yes	characteristic
53	C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₃₇	#137	1	2	yes	—
56	Dic₁₃₇	—	8	1	no	—
57	Dic₁₃₇	—	8	1	no	—
58	Dic₁₃₇	—	8	1	no	—
59	Dic₁₃₇	—	8	1	no	—
60	C₂×C₁₃₇	#494	1	1	yes	characteristic
61	C₂×C₁₃₇	#496	1	1	yes	characteristic
62	C₂×C₁₃₇	#497	1	1	yes	characteristic
63	C₁₃₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	137	1	no	maximal, Sylow
65	#500	—	137	1	no	—
66	#500	—	137	1	no	—
67	#498	—	137	1	no	—
68	#494	—	137	1	no	—
69	#487	—	137	1	no	—
70	#487	—	137	1	no	—
71	#322	D₁₃₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	274	2	no	—
73	C₄⋊Q₈	—	137	1	no	—
74	C₄⋊Q₈	—	137	1	no	—
75	C₄⋊Q₈	—	137	1	no	—
76	C₄⋊Q₈	—	137	1	no	—
77	#107	—	274	2	no	—
78	#107	—	274	2	no	—
79	#106	—	274	2	no	—
80	#103	—	274	2	no	—
81	#103	—	274	2	no	—
82	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
85	C₂×Q₈	—	274	1	no	—
86	C₂×Q₈	—	274	1	no	—
87	C₂×Q₈	—	274	1	no	—
88	C₂×Q₈	—	274	1	no	—
89	Q₁₆	—	548	4	no	—
90	C₂×C₈	D₅₄₈	1	2	yes	—
91	C₂×C₈	D₅₄₈	1	2	yes	—
92	C₂×C₈	D₅₄₈	1	2	yes	—
93	C₄⋊C₄	—	274	1	no	—
94	C₄⋊C₄	—	274	1	no	—
95	C₄⋊C₄	—	274	1	no	—
96	C₄⋊C₄	—	274	1	no	—
97	C₄⋊C₄	—	274	1	no	—
98	C₄⋊C₄	—	274	1	no	—
99	C₄⋊C₄	—	274	1	no	—
100	C₄⋊C₄	—	274	1	no	—
101	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
102	Q₈	—	548	2	no	—
103	Q₈	—	548	2	no	—
104	Q₈	—	548	2	no	—
105	Q₈	—	548	2	no	—
106	C₂×C₄	—	548	1	no	—
107	C₂×C₄	—	548	1	no	—
108	C₂×C₄	—	548	1	no	—
109	C₂×C₄	—	548	1	no	—
110	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
111	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
112	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₅₄₈	1	4	yes	complemented
116	C₂²	#17984085	1	1	yes	characteristic
117	C₄	—	1096	1	no	—
118	C₄	—	1096	1	no	—
119	C₄	—	1096	1	no	—
120	C₄	—	1096	1	no	—
121	C₄	C₂×Dic₅₄₈	1	2	yes	—
122	C₄	#17984116	1	2	yes	—
123	C₄	#17984121	1	2	yes	—
124	C₂	#22577783	1	1	yes	characteristic
125	C₂	#22577792	1	1	yes	characteristic
126	C₂	#22577797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577808

C₂

yes

maximal, characteristic

#22577808

C₂

yes

maximal, characteristic

#22577801

C₂

yes

maximal, characteristic

#22577783

C₂

yes

maximal, characteristic

#22577754

C₂

yes

maximal, characteristic

#22577754

C₂

yes

maximal, characteristic

#322×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Dic₅₄₈

—

#17984088

C₂²

yes

characteristic

#17984088

C₂²

yes

characteristic

#17984088

C₂²

yes

characteristic

#17984088

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

Dic₅₄₈

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

Q₁₆

yes

complemented

C₂²×C₁₃₇

#127

yes

characteristic

C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1723

—

137

maximal, Sylow

#500

—

137

—

#500

—

137

—

#498

—

137

—

#494

—

137

—

#487

—

137

—

#487

—

137

—

#322

D₁₃₇

yes

characteristic

C₂×Q₁₆

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

#107

—

274

—

#107

—

274

—

#106

—

274

—

#103

—

274

—

#103

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

Q₁₆

—

548

—

C₂×C₈

D₅₄₈

yes

—

C₂×C₈

D₅₄₈

yes

—

C₂×C₈

D₅₄₈

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

100

C₄⋊C₄

—

274

—

101

C₄²

C₂²×D₁₃₇

yes

characteristic

102

Q₈

—

548

—

103

Q₈

—

548

—

104

Q₈

—

548

—

105

Q₈

—

548

—

106

C₂×C₄

—

548

—

107

C₂×C₄

—

548

—

108

C₂×C₄

—

548

—

109

C₂×C₄

—

548

—

110

C₂×C₄

C₂×D₅₄₈

yes

characteristic

111

C₂×C₄

C₂×D₅₄₈

yes

characteristic

112

C₂×C₄

C₂×D₅₄₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₅₄₈

yes

complemented

116

C₂²

#17984085

yes

characteristic

117

C₄

—

1096

—

118

C₄

—

1096

—

119

C₄

—

1096

—

120

C₄

—

1096

—

121

C₄

C₂×Dic₅₄₈

yes

—

122

C₄

#17984116

yes

—

123

C₄

#17984121

yes

—

124

C₂

#22577783

yes

characteristic

125

C₂

#22577792

yes

characteristic

126

C₂

#22577797

yes

characteristic

	subgroup	G/H
center	C₂²	#17984085
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#322×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#127
hypercenter	#322	D₁₃₇
hypocenter	C₁₃₇	#1723
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17984085

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#322×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₃₇	Q₁₆	semidirect
63. C₁₃₇	#1723	semidirect
115. C₈	Dic₅₄₈	semidirect

H ≅ G/N

type

51. C₈×C₁₃₇

Q₁₆

semidirect

63. C₁₃₇

#1723

semidirect

115. C₈

Dic₅₄₈

semidirect

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

subgroup	index
#22577808	2
#22577801	2
#22577783	2
#22577754	2
#322×C₁₃₇	2
#1723	137

subgroup

index

#322×C₁₃₇

137

Subgroups — #31780466

Conjugacy classes of subgroups

Complements

Maximal subgroups