Group #31797206 — subgroups

Subgroups — #31797206

Order 17536.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22578324	C₂	1	1	yes	maximal, characteristic
2	#22578324	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14555513	C₂	1	1	yes	maximal, characteristic
4	#22578292	C₂	1	1	yes	maximal, characteristic
5	#22578291	C₂	1	1	yes	maximal, characteristic
6	#22578289	C₂	1	1	yes	maximal, characteristic
7	#22577527	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555513	C₂²	1	1	yes	characteristic
9	C₂×#14555513	C₂²	1	1	yes	characteristic
10	C₂×#14555513	C₂²	1	1	yes	characteristic
11	C₂×#14555513	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
13	#17984017	C₄	1	1	yes	characteristic
14	#17984017	C₄	1	1	yes	characteristic
15	C₄×#11764040	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
19	#14555513	—	2	1	no	—
20	#14555513	—	2	1	no	—
21	#14555513	—	2	1	no	—
22	#14555513	Q₈	1	2	yes	complemented
23	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
24	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
25	C₂×#11764040	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
27	#14555489	C₈	1	1	yes	characteristic
28	#14555489	C₈	1	1	yes	characteristic
29	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	Dic₂₇₄	—	2	2	no	—
35	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
39	#11764040	—	2	1	no	—
40	#11764040	C₂×Q₈	1	1	yes	characteristic
41	#11764040	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₃₇	—	2	1	no	—
46	C₈×C₁₃₇	—	2	1	no	—
47	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₃₇	—	2	1	no	—
49	C₄×C₁₃₇	—	2	1	no	—
50	C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₃₇	#131	1	1	yes	characteristic
52	Dic₁₃₇	—	2	1	no	—
53	Dic₁₃₇	—	2	1	no	—
54	Dic₁₃₇	—	2	1	no	—
55	Dic₁₃₇	M₃₂	1	2	yes	—
56	C₂×C₁₃₇	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₃₇	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₃₇	#504	1	1	yes	characteristic
59	C₁₃₇	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	137	1	no	maximal, Sylow
61	C₈×Q₈	—	137	1	no	—
62	C₄⋊C₁₆	—	137	1	no	—
63	C₄⋊C₁₆	—	137	1	no	—
64	C₄⋊C₁₆	—	137	1	no	—
65	#346	—	137	1	no	—
66	#346	—	137	1	no	—
67	C₄×C₁₆	—	137	1	no	—
68	C₄×Q₈	—	137	1	no	—
69	C₂×C₁₆	—	137	1	no	—
70	C₂×C₁₆	—	137	1	no	—
71	C₂×C₁₆	—	137	1	no	—
72	C₂×C₁₆	—	137	1	no	—
73	C₄⋊C₈	—	137	1	no	—
74	C₄⋊C₈	—	137	1	no	—
75	C₄⋊C₈	#10587739	1	1	yes	characteristic
76	C₄×C₈	—	137	1	no	—
77	C₄×C₈	—	137	1	no	—
78	C₄×C₈	—	137	1	no	—
79	C₂×Q₈	—	137	1	no	—
80	C₂×C₈	—	137	1	no	—
81	C₂×C₈	—	137	1	no	—
82	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
83	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
84	C₄⋊C₄	—	137	1	no	—
85	C₄⋊C₄	—	137	1	no	—
86	C₄⋊C₄	—	137	1	no	—
87	C₄²	—	137	1	no	—
88	C₄²	—	137	1	no	—
89	C₄²	C₂×#10587739	1	1	yes	characteristic
90	C₁₆	—	274	1	no	—
91	C₁₆	—	274	1	no	—
92	C₁₆	—	274	1	no	—
93	C₁₆	—	137	2	no	—
94	Q₈	—	274	2	no	—
95	C₂×C₄	—	137	1	no	—
96	C₂×C₄	—	137	1	no	—
97	C₂×C₄	—	137	1	no	—
98	C₂×C₄	—	137	1	no	—
99	C₂×C₄	#14555518	1	1	yes	characteristic
100	C₂×C₄	#14555518	1	1	yes	characteristic
101	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	274	1	no	—
105	C₈	—	137	1	no	—
106	C₈	—	137	1	no	—
107	C₂²	C₂×#14555518	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	274	1	no	—
111	C₄	—	274	1	no	—
112	C₄	—	274	1	no	—
113	C₄	—	137	2	no	—
114	C₄	Q₈×#10587739	1	1	yes	characteristic
115	C₄	#17984175	1	1	yes	characteristic
116	C₂	#22578452	1	1	yes	characteristic
117	C₂	#22578459	1	1	yes	characteristic
118	C₂	#22578462	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578324

C₂

yes

maximal, characteristic

#22578324

C₂

yes

maximal, characteristic

C₄⋊#14555513

C₂

yes

maximal, characteristic

#22578292

C₂

yes

maximal, characteristic

#22578291

C₂

yes

maximal, characteristic

#22578289

C₂

yes

maximal, characteristic

#22577527

C₂

yes

maximal, characteristic

C₂×#14555513

C₂²

yes

characteristic

C₂×#14555513

C₂²

yes

characteristic

C₂×#14555513

C₂²

yes

characteristic

C₂×#14555513

C₂²

yes

characteristic

C₄×Dic₂₇₄

C₄

yes

characteristic

#17984017

C₄

yes

characteristic

#17984017

C₄

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₄

yes

—

—

—

Q₈

yes

complemented

C₂×Dic₂₇₄

C₈

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂³

yes

characteristic

C₄⋊Dic₁₃₇

C₈

yes

characteristic

#14555489

C₈

yes

characteristic

#14555489

C₈

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

#11764040

—

#11764040

C₂×Q₈

yes

yes

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×Q₈

yes

characteristic

C₄×C₁₃₇

#131

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

M₃₂

yes

—

C₂×C₁₃₇

C₈×Q₈

yes

characteristic

C₂×C₁₃₇

C₂×M₃₂

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#2183

—

137

maximal, Sylow

C₈×Q₈

—

137

—

C₄⋊C₁₆

—

137

—

C₄⋊C₁₆

—

137

—

C₄⋊C₁₆

—

137

—

#346

—

137

—

#346

—

137

—

C₄×C₁₆

—

137

—

C₄×Q₈

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

#10587739

yes

characteristic

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₁₆

—

274

—

C₁₆

—

274

—

C₁₆

—

274

—

C₁₆

—

137

—

Q₈

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

#14555518

yes

characteristic

100

C₂×C₄

#14555518

yes

characteristic

101

C₂×C₄

C₂²×#10587739

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

274

—

105

C₈

—

137

—

106

C₈

—

137

—

107

C₂²

C₂×#14555518

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

274

—

111

C₄

—

274

—

112

C₄

—

274

—

113

C₄

—

137

—

114

C₄

Q₈×#10587739

yes

characteristic

115

C₄

#17984175

yes

characteristic

116

C₂

#22578452

yes

characteristic

117

C₂

#22578459

yes

characteristic

118

C₂

#22578462

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle, prime radical	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₄⋊C₈×C₁₃₇	C₄
cosocle	C₂×#11764040	C₂³
hypercenter	C₄⋊C₈	#10587739
hypocenter	C₁₃₇	#2183
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle, prime radical

C₂²×C₁₃₇

C₂²×C₈

Fitting

C₄⋊C₈×C₁₃₇

C₄

C₂³

C₄⋊C₈

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
22. #14555513	Q₈	semidirect
59. C₁₃₇	#2183	semidirect

H ≅ G/N

type

22. #14555513

Q₈

semidirect

59. C₁₃₇

#2183

semidirect

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

subgroup	index
#22578324	2
C₄⋊#14555513	2
#22578292	2
#22578291	2
#22578289	2
#22577527	2
#2183	137

subgroup

index

137

Subgroups — #31797206

Conjugacy classes of subgroups

Complements

Maximal subgroups