Group #31780962 — subgroups

Subgroups — #31780962

Order 17536.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#17984024	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984023	C₂	1	2	yes	maximal
3	#22577426	C₂	1	2	yes	maximal
4	C₂×#17983998	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#17984024	—	2	2	no	—
10	#17984023	—	2	4	no	—
11	#17984017	—	2	4	no	—
12	#17983998	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	#108×C₁₃₇	—	2	2	no	—
16	C₂²×Dic₁₃₇	Q₈	1	2	yes	—
17	#14555494	—	2	2	no	—
18	#14555494	—	2	2	no	—
19	C₂×#11764040	—	2	2	no	—
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₂×#11764040	D₄	1	2	yes	—
22	C₂×#11764040	Q₈	1	2	yes	—
23	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₃₇	—	2	2	no	—
27	M₁₆×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	—	2	2	no	—
29	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
32	C₂×Dic₁₃₇	—	4	2	no	—
33	C₂×Dic₁₃₇	C₄⋊C₄	1	4	yes	—
34	#11764040	—	4	2	no	—
35	#11764040	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₃₇	—	4	2	no	—
44	C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₁₃₇	—	2	2	no	—
46	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₃₇	#127	1	1	yes	characteristic
52	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₃₇	—	2	4	no	—
54	C₂×C₁₃₇	—	2	1	no	—
55	C₂×C₁₃₇	—	2	1	no	—
56	C₂×C₁₃₇	#384	1	1	yes	characteristic
57	C₂×C₁₃₇	#430	1	2	yes	—
58	C₁₃₇	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	137	1	no	maximal, Sylow
60	C₂×#108	—	137	2	no	—
61	C₂×#108	—	137	1	no	—
62	C₂×#108	D₁₃₇	1	1	yes	characteristic
63	#423	—	137	2	no	—
64	C₂×#97	—	137	1	no	—
65	C₂×M₁₆	—	137	2	no	—
66	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
67	C₂²×C₈	—	137	1	no	—
68	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
69	C₂×C₄²	—	137	1	no	—
70	#108	—	2	2	no	—
71	#108	—	274	4	no	—
72	#108	—	274	2	no	—
73	C₄⋊C₈	—	274	4	no	—
74	#97	—	137	4	no	—
75	C₂²×C₄	—	137	2	no	—
76	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	274	2	no	—
80	M₁₆	—	274	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	274	2	no	—
83	C₂×C₈	—	137	2	no	—
84	C₂×C₈	—	137	2	no	—
85	C₂×C₈	—	137	2	no	—
86	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
88	C₂×C₈	Dic₂₇₄	1	2	yes	—
89	C₄²	—	137	2	no	—
90	C₄²	—	137	2	no	—
91	C₂³	Q₈×D₁₃₇	1	1	yes	characteristic
92	C₂×C₄	—	548	2	no	—
93	C₂×C₄	—	137	4	no	—
94	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
95	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
96	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₁₃₇	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	548	2	no	—
102	C₈	—	137	4	no	—
103	C₈	#14555489	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
106	C₂²	#17984086	1	1	yes	characteristic
107	C₂²	#17984087	1	1	yes	characteristic
108	C₄	—	274	4	no	—
109	C₄	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
110	C₄	C₂×#14555489	1	1	yes	characteristic
111	C₄	#17984084	1	1	yes	characteristic
112	C₄	#17984087	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#22577272	1	1	yes	characteristic
116	C₂	#22577450	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984024

C₂

yes

maximal, characteristic

C₂×#17984023

C₂

yes

maximal

#22577426

C₂

yes

maximal

C₂×#17983998

C₂

yes

maximal, characteristic

C₂×#108×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

#108×C₁₃₇

—

C₂²×Dic₁₃₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11764040

D₄

yes

—

C₂×#11764040

Q₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

#11764040

—

#11764040

C₄⋊C₄

yes

—

C₂³×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

#127

yes

characteristic

C₄×C₁₃₇

C₄⋊Q₈

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#384

yes

characteristic

C₂×C₁₃₇

#430

yes

—

C₁₃₇

#1849

yes

characteristic, complemented, Sylow

#1849

—

137

maximal, Sylow

C₂×#108

—

137

—

C₂×#108

—

137

—

C₂×#108

D₁₃₇

yes

characteristic

#423

—

137

—

C₂×#97

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂×C₄²

—

137

—

#108

—

#108

—

274

—

#108

—

274

—

C₄⋊C₈

—

274

—

#97

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

—

C₂×C₈

C₂²⋊D₁₃₇

yes

—

C₂×C₈

Dic₂₇₄

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₂×C₄

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

100

C₈

—

101

C₈

—

548

—

102

C₈

—

137

—

103

C₈

#14555489

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

106

C₂²

#17984086

yes

characteristic

107

C₂²

#17984087

yes

characteristic

108

C₄

—

274

—

109

C₄

C₄⋊C₄×D₁₃₇

yes

characteristic

110

C₄

C₂×#14555489

yes

characteristic

111

C₄

#17984084

yes

characteristic

112

C₄

#17984087

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#22577272

yes

characteristic

116

C₂

#22577450

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#108×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#108	D₁₃₇
hypocenter	C₁₃₇	#1849
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×#108×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₁₃₇	#1849	semidirect

H ≅ G/N

type

58. C₁₃₇

#1849

semidirect

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

subgroup	index
C₂×#17984024	2
C₂×#17984023	2
#22577426	2
C₂×#17983998	2
C₂×#108×C₁₃₇	2
#1849	137

subgroup

index

C₂×#108×C₁₃₇

137

Subgroups — #31780962

Conjugacy classes of subgroups

Complements

Maximal subgroups