Group #31795639 — subgroups

Subgroups — #31795639

Order 17536.

164 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 44 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	#22578426	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
3	#22577458	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10587739	—	2	1	no	—
5	C₂×C₄×#10587739	C₄	1	2	yes	—
6	#17984155	C₄	1	2	yes	complemented
7	#17984145	—	2	1	no	—
8	#17984143	—	2	1	no	—
9	(C₂×#10587739)⋊C₄	—	2	1	no	—
10	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
11	#17984017	—	2	1	no	—
12	#17984000	—	2	1	no	—
13	#17983999	C₄	1	1	yes	characteristic, complemented
14	#97×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×#10587739	—	2	2	no	—
16	C₂×#11764047	—	2	2	no	—
17	C₄⋊#10587739	—	2	4	no	—
18	C₄×#10587739	—	4	2	no	—
19	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
22	C₂×#11764040	—	2	1	no	—
23	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
25	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	—	2	1	no	—
27	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#10587739	—	2	4	no	—
29	C₂×#10587739	—	4	2	no	—
30	C₂×#10587739	—	4	2	no	—
31	#11764047	—	4	2	no	—
32	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄×D₁₃₇	—	2	1	no	—
37	C₄×D₁₃₇	—	2	1	no	—
38	C₄×D₁₃₇	—	2	2	no	—
39	C₄×D₁₃₇	—	4	1	no	—
40	#11764040	—	4	1	no	—
41	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
44	C₈×C₁₃₇	—	4	1	no	—
45	C₂²×C₁₃₇	#95	1	1	yes	characteristic
46	C₂×D₁₃₇	—	2	1	no	—
47	C₂×D₁₃₇	—	2	1	no	—
48	C₂×D₁₃₇	#100	1	1	yes	characteristic
49	C₂×D₁₃₇	#101	1	1	yes	characteristic
50	#10587739	—	4	4	no	—
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	—	2	1	no	—
53	C₄×C₁₃₇	—	2	1	no	—
54	Dic₁₃₇	—	2	1	no	—
55	Dic₁₃₇	—	2	1	no	—
56	Dic₁₃₇	#104	1	2	yes	—
57	C₂×C₁₃₇	H(3,ℤ/4)	1	1	yes	characteristic
58	C₂×C₁₃₇	#339	1	1	yes	characteristic
59	C₂×C₁₃₇	#343	1	1	yes	characteristic
60	D₁₃₇	—	2	1	no	—
61	D₁₃₇	—	2	1	no	—
62	C₁₃₇	#1285	1	1	yes	characteristic, complemented, Sylow
63	#1285	—	137	1	no	maximal, Sylow
64	#432	—	137	1	no	—
65	#432	—	137	1	no	—
66	C₄×C₄⋊C₄	—	137	1	no	—
67	C₂×C₄⋊C₄	—	274	1	no	—
68	C₂×C₄²	—	137	2	no	—
69	C₂×C₄²	—	137	1	no	—
70	C₄⋊C₈	—	274	1	no	—
71	C₄⋊C₈	—	137	2	no	—
72	C₂²⋊C₈	—	274	1	no	—
73	C₂²⋊C₈	—	274	1	no	—
74	#97	—	274	1	no	—
75	#97	—	137	1	no	—
76	#97	#10587739	1	1	yes	characteristic, complemented
77	#95	—	274	1	no	—
78	C₂²×C₄	—	274	2	no	—
79	C₂²×C₄	—	137	1	no	—
80	C₂²×C₄	—	137	1	no	—
81	C₂²×C₄	—	137	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	274	2	no	—
84	C₂×C₈	—	274	1	no	—
85	C₄⋊C₄	—	274	4	no	—
86	C₄²	—	548	2	no	—
87	C₄²	—	137	1	no	—
88	C₄²	—	137	1	no	—
89	C₄²	—	137	1	no	—
90	C₄²	C₂×#10587739	1	1	yes	characteristic
91	C₂³	—	137	1	no	—
92	C₂×C₄	—	274	1	no	—
93	C₂×C₄	—	274	1	no	—
94	C₂×C₄	—	274	2	no	—
95	C₂×C₄	—	274	4	no	—
96	C₂×C₄	—	548	1	no	—
97	C₂×C₄	—	548	2	no	—
98	C₂×C₄	—	548	2	no	—
99	C₂×C₄	—	137	1	no	—
100	C₂×C₄	—	137	1	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	548	1	no	—
107	C₈	—	548	2	no	—
108	C₂²	—	274	1	no	—
109	C₂²	—	274	1	no	—
110	C₂²	—	137	1	no	—
111	C₂²	—	137	1	no	—
112	C₂²	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	274	1	no	—
117	C₄	—	274	1	no	—
118	C₄	—	548	4	no	—
119	C₄	—	137	2	no	—
120	C₂	—	274	1	no	—
121	C₂	—	274	1	no	—
122	C₂	#22578270	1	1	yes	characteristic
123	C₂	#22578275	1	1	yes	characteristic
124	C₂	#22578282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578426

C₂

yes

maximal, characteristic

C₄×C₄⋊#10587739

C₂

yes

maximal, characteristic

#22577458

C₂

yes

maximal, characteristic

C₂×C₄⋊#10587739

—

C₂×C₄×#10587739

C₄

yes

—

#17984155

C₄

yes

—

—

—

C₄²×D₁₃₇

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#97×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×#11764040

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

—

C₂²×D₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

#95

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

yes

C₂×D₁₃₇

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#104

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1285

yes

characteristic, complemented, Sylow

#1285

—

137

maximal, Sylow

#432

—

137

—

#432

—

137

—

C₄×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

274

—

C₂²⋊C₈

—

274

—

#97

—

274

—

#97

—

137

—

#97

#10587739

yes

characteristic, complemented

#95

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₄⋊C₄

—

274

—

C₄²

—

548

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

100

C₂×C₄

—

137

—

101

C₂×C₄

—

137

—

102

C₂×C₄

C₄×#10587739

yes

characteristic

103

C₂×C₄

C₂²⋊#10587739

yes

characteristic

104

C₂×C₄

C₄⋊#10587739

yes

characteristic

105

C₈

—

106

C₈

—

548

—

107

C₈

—

548

—

108

C₂²

—

274

—

109

C₂²

—

274

—

110

C₂²

—

137

—

111

C₂²

—

137

—

112

C₂²

(C₂×#10587739)⋊C₄

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

274

—

117

C₄

—

274

—

118

C₄

—

548

—

119

C₄

—

137

—

120

C₂

—

274

—

121

C₂

—

274

—

122

C₂

#22578270

yes

characteristic

123

C₂

#22578275

yes

characteristic

124

C₂

#22578282

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10587739)⋊C₄
commutator	C₂×C₄×C₁₃₇	C₄²
Frattini	C₄²	C₂×#10587739
Fitting	#97×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#95
prime radical	C₂²×D₁₃₇	C₂²⋊C₄
cosocle	C₄²×D₁₃₇	C₂²
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#1285
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

(C₂×#10587739)⋊C₄

commutator

C₂×C₄×C₁₃₇

C₄²

Frattini

C₄²

C₂×#10587739

Fitting

#97×C₁₃₇

C₄

socle

C₂²×C₁₃₇

#95

prime radical

C₂²×D₁₃₇

C₂²⋊C₄

cosocle

C₄²×D₁₃₇

C₂²

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
6. #17984155	C₄	semidirect
13. #17983999	C₄	semidirect
14. #97×C₁₃₇	C₄	semidirect
62. C₁₃₇	#1285	semidirect
76. #97	#10587739	semidirect

H ≅ G/N

type

6. #17984155

C₄

semidirect

13. #17983999

C₄

semidirect

14. #97×C₁₃₇

C₄

semidirect

62. C₁₃₇

#1285

semidirect

76. #97

#10587739

semidirect

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

subgroup	index
#22578426	2
C₄×C₄⋊#10587739	2
#22577458	2
#1285	137

subgroup

index

137

Subgroups — #31795639

Conjugacy classes of subgroups

Complements

Maximal subgroups