Group #31781538 — subgroups

Subgroups — #31781538

Order 17536.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22577430	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#22577162	C₂	1	2	yes	maximal
5	#427×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	C₄⋊#11764040	—	2	4	no	—
11	C₈×Dic₁₃₇	—	2	2	no	—
12	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#117×C₁₃₇	C₂²	1	2	yes	—
14	#107×C₁₃₇	C₄	1	2	yes	complemented
15	#106×C₁₃₇	C₄	1	2	yes	complemented
16	C₂²×Dic₁₃₇	—	2	1	no	—
17	#14555494	—	2	2	no	—
18	#14555494	—	2	2	no	—
19	C₂×#11764040	—	2	1	no	—
20	C₂×#11764040	—	2	2	no	—
21	C₂×#11764040	—	2	1	no	—
22	C₂×#11764040	D₄	1	2	yes	—
23	C₄×Dic₁₃₇	—	2	1	no	—
24	C₄×Dic₁₃₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
32	C₄²×C₁₃₇	—	2	2	no	—
33	C₂×Dic₁₃₇	—	2	4	no	—
34	C₂×Dic₁₃₇	—	2	2	no	—
35	#11764040	—	2	2	no	—
36	#11764040	—	4	2	no	—
37	C₂³×C₁₃₇	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
45	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₇	—	2	2	no	—
48	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₇	#125	1	1	yes	characteristic
51	C₄×C₁₃₇	—	4	4	no	—
52	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₃₇	#121	1	1	yes	characteristic
55	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₃₇	—	2	4	no	—
57	C₂×C₁₃₇	—	2	1	no	—
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	#389	1	1	yes	characteristic
60	C₂×C₁₃₇	#443	1	2	yes	—
61	C₁₃₇	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	137	1	no	maximal, Sylow
63	C₂×#108	—	137	1	no	—
64	#427	D₁₃₇	1	1	yes	characteristic
65	#424	—	137	2	no	—
66	C₂×C₄×C₈	—	137	1	no	—
67	#339	—	137	2	no	—
68	C₂×M₁₆	—	137	2	no	—
69	C₂²×C₈	—	137	1	no	—
70	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
71	#117	C₂×D₁₃₇	1	2	yes	—
72	C₂×C₄²	—	137	1	no	—
73	#108	—	274	2	no	—
74	#107	Dic₁₃₇	1	2	yes	complemented
75	#106	Dic₁₃₇	1	2	yes	complemented
76	C₄⋊C₈	—	274	4	no	—
77	C₄×C₈	—	274	2	no	—
78	C₂²×C₄	—	274	1	no	—
79	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
80	M₁₆	—	274	2	no	—
81	M₁₆	—	274	2	no	—
82	C₂×C₈	—	274	1	no	—
83	C₂×C₈	—	274	1	no	—
84	C₂×C₈	—	274	2	no	—
85	C₂×C₈	—	137	2	no	—
86	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
88	C₂×C₈	D₅₄₈	1	2	yes	—
89	C₂×C₈	Dic₂₇₄	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₃₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	274	1	no	—
94	C₄²	—	137	2	no	—
95	C₂³	#14555509	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	274	4	no	—
98	C₂×C₄	—	274	2	no	—
99	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₃₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
104	C₂×C₄	#14555511	1	1	yes	characteristic
105	C₈	—	274	2	no	—
106	C₈	—	548	2	no	—
107	C₈	C₄⋊Dic₁₃₇	1	2	yes	—
108	C₈	C₄⋊Dic₁₃₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₃₇	1	1	yes	characteristic
111	C₂²	#17984068	1	1	yes	characteristic
112	C₂²	#17984079	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	274	4	no	—
115	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₃₇	1	1	yes	characteristic
117	C₄	#17984063	1	1	yes	characteristic
118	C₄	#17984087	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#22577302	1	1	yes	characteristic
122	C₂	#22577520	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984024

C₂

yes

maximal, characteristic

#22577430

C₂

yes

maximal

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

#22577162

C₂

yes

maximal

#427×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984024

—

C₄⋊#11764040

—

C₈×Dic₁₃₇

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₂²

yes

—

#107×C₁₃₇

C₄

yes

complemented

#106×C₁₃₇

C₄

yes

complemented

C₂²×Dic₁₃₇

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

Q₈

yes

—

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

#41

yes

characteristic

C₂×C₄×C₁₃₇

—

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×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×D₄

yes

characteristic

C₂²×C₁₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₇

#125

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₄×Q₈

yes

characteristic

C₄×C₁₃₇

#121

yes

characteristic

C₄×C₁₃₇

C₄⋊Q₈

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#389

yes

characteristic

C₂×C₁₃₇

#443

yes

—

C₁₃₇

#1945

yes

characteristic, complemented, Sylow

#1945

—

137

maximal, Sylow

C₂×#108

—

137

—

#427

D₁₃₇

yes

characteristic

#424

—

137

—

C₂×C₄×C₈

—

137

—

#339

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

#117

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

#108

—

274

—

#107

Dic₁₃₇

yes

complemented

#106

Dic₁₃₇

yes

complemented

C₄⋊C₈

—

274

—

C₄×C₈

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₂×Dic₁₃₇

yes

characteristic

C₂×C₈

C₂×Dic₁₃₇

yes

characteristic

C₂×C₈

D₅₄₈

yes

—

C₂×C₈

Dic₂₇₄

yes

—

C₄⋊C₄

C₂×Dic₁₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

274

—

C₄²

—

137

—

C₂³

#14555509

yes

characteristic

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

100

C₂×C₄

D₄×D₁₃₇

yes

characteristic

101

C₂×C₄

C₂×D₅₄₈

yes

characteristic

102

C₂×C₄

Q₈×D₁₃₇

yes

characteristic

103

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

104

C₂×C₄

#14555511

yes

characteristic

105

C₈

—

274

—

106

C₈

—

548

—

107

C₈

C₄⋊Dic₁₃₇

yes

—

108

C₈

C₄⋊Dic₁₃₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₃₇

yes

characteristic

111

C₂²

#17984068

yes

characteristic

112

C₂²

#17984079

yes

characteristic

113

C₄

—

114

C₄

—

274

—

115

C₄

C₂×C₄⋊Dic₁₃₇

yes

characteristic

116

C₄

Q₈×Dic₁₃₇

yes

characteristic

117

C₄

#17984063

yes

characteristic

118

C₄

#17984087

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#22577302

yes

characteristic

122

C₂

#22577520

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#427×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₄×D₄
prime radical	C₂³×C₁₃₇	#41
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#427	D₁₃₇
hypocenter	C₁₃₇	#1945
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#427×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₄×D₄

prime radical

C₂³×C₁₃₇

#41

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₃₇	C₄	semidirect
15. #106×C₁₃₇	C₄	semidirect
61. C₁₃₇	#1945	semidirect
74. #107	Dic₁₃₇	semidirect
75. #106	Dic₁₃₇	semidirect

H ≅ G/N

type

14. #107×C₁₃₇

C₄

semidirect

15. #106×C₁₃₇

C₄

semidirect

61. C₁₃₇

#1945

semidirect

74. #107

Dic₁₃₇

semidirect

75. #106

Dic₁₃₇

semidirect

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

subgroup	index
C₂×#17984024	2
#22577430	2
C₂×C₈×Dic₁₃₇	2
#22577162	2
#427×C₁₃₇	2
#1945	137

subgroup

index

C₂×C₈×Dic₁₃₇

#427×C₁₃₇

137

Subgroups — #31781538

Conjugacy classes of subgroups

Complements

Maximal subgroups