Group #31795667 — subgroups

Subgroups — #31795667

Order 17536.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#17984155	C₂	1	2	yes	maximal
2	#22577272	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764047	—	2	2	no	—
4	#17984155	—	2	2	no	—
5	#17984155	C₄	1	4	yes	complemented
6	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#14555489	—	2	2	no	—
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
10	C₂×#11764047	—	2	4	no	—
11	C₂×#11764047	—	4	2	no	—
12	C₂×#11764047	—	4	2	no	—
13	C₂²×Dic₁₃₇	—	2	1	no	—
14	C₂²×Dic₁₃₇	D₄	1	2	yes	—
15	C₄⋊Dic₁₃₇	—	4	1	no	—
16	#14555489	—	4	2	no	—
17	#14555489	—	4	2	no	—
18	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
20	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	—	2	1	no	—
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₇	—	4	1	no	—
24	#11764047	—	4	4	no	—
25	C₂×Dic₁₃₇	—	4	1	no	—
26	C₂×Dic₁₃₇	—	4	2	no	—
27	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
28	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×Dic₁₃₇	D₈	1	2	yes	—
30	C₂×Dic₁₃₇	SD₁₆	1	2	yes	—
31	C₂×Dic₁₃₇	SD₁₆	1	2	yes	—
32	C₂×Dic₁₃₇	Q₁₆	1	2	yes	—
33	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	—	2	1	no	—
35	C₂×C₄×C₁₃₇	—	2	1	no	—
36	C₂×C₄×C₁₃₇	—	4	1	no	—
37	C₂×C₄×C₁₃₇	—	4	2	no	—
38	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₃₇	#95	1	1	yes	characteristic
41	C₂²×C₁₃₇	#100	1	1	yes	characteristic
42	C₂²×C₁₃₇	#101	1	1	yes	characteristic
43	C₂²×C₁₃₇	#102	1	2	yes	—
44	C₂²×C₁₃₇	#103	1	2	yes	—
45	C₄×C₁₃₇	—	2	1	no	—
46	C₄×C₁₃₇	—	2	1	no	—
47	C₄×C₁₃₇	—	8	1	no	—
48	Dic₁₃₇	—	8	1	no	—
49	Dic₁₃₇	#102	1	2	yes	—
50	Dic₁₃₇	#103	1	2	yes	—
51	Dic₁₃₇	#106	1	2	yes	—
52	Dic₁₃₇	#107	1	2	yes	—
53	C₂×C₁₃₇	#331	1	1	yes	characteristic
54	C₂×C₁₃₇	#332	1	2	yes	—
55	C₂×C₁₃₇	#333	1	1	yes	characteristic
56	C₂×C₁₃₇	#340	1	2	yes	—
57	C₂×C₁₃₇	#343	1	1	yes	characteristic
58	C₁₃₇	#1296	1	1	yes	characteristic, complemented, Sylow
59	#1296	—	137	1	no	maximal, Sylow
60	C₂×C₄⋊C₈	—	137	2	no	—
61	#384	—	137	1	no	—
62	C₂²×C₈	—	274	2	no	—
63	C₂×C₄⋊C₄	—	274	2	no	—
64	C₂×C₄⋊C₄	—	137	1	no	—
65	C₂×C₄⋊C₄	#10587739	1	1	yes	characteristic
66	C₂×C₄²	—	137	1	no	—
67	C₄⋊C₈	—	274	2	no	—
68	C₄⋊C₈	—	137	4	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	274	1	no	—
71	C₂²×C₄	—	137	2	no	—
72	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
73	C₂×C₈	—	274	4	no	—
74	C₂×C₈	—	548	2	no	—
75	C₂×C₈	—	548	2	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	548	1	no	—
78	C₄⋊C₄	—	548	2	no	—
79	C₄⋊C₄	—	548	2	no	—
80	C₄²	—	137	2	no	—
81	C₄²	—	137	1	no	—
82	C₄²	—	137	1	no	—
83	C₂³	C₂²⋊#10587739	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	548	1	no	—
89	C₂×C₄	—	548	2	no	—
90	C₂×C₄	—	137	2	no	—
91	C₂×C₄	—	137	2	no	—
92	C₂×C₄	—	137	2	no	—
93	C₂×C₄	—	137	2	no	—
94	C₂×C₄	—	137	2	no	—
95	C₂×C₄	—	137	2	no	—
96	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
98	C₈	—	548	4	no	—
99	C₂²	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
100	C₂²	#17984148	1	1	yes	characteristic
101	C₂²	#17984149	1	1	yes	characteristic
102	C₂²	#17984150	1	2	yes	—
103	C₂²	#17984151	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	1096	1	no	—
108	C₄	—	137	2	no	—
109	C₄	—	137	2	no	—
110	C₄	—	137	2	no	—
111	C₄	—	137	2	no	—
112	C₂	#22578261	1	1	yes	characteristic
113	C₂	#22578262	1	2	yes	—
114	C₂	#22578263	1	1	yes	characteristic
115	C₂	#22578276	1	2	yes	—
116	C₂	#22578283	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984155

C₂

yes

maximal

#22577272

C₂

yes

maximal, characteristic

C₂²×#11764047

—

#17984155

—

#17984155

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₇

C₄

yes

characteristic

C₂×#14555489

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₄

yes

—

—

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

D₄

yes

—

C₄⋊Dic₁₃₇

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₇

—

#11764047

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

D₈

yes

—

C₂×Dic₁₃₇

SD₁₆

yes

—

C₂×Dic₁₃₇

SD₁₆

yes

—

C₂×Dic₁₃₇

Q₁₆

yes

—

C₂³×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₂²×C₁₃₇

#103

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#102

yes

—

Dic₁₃₇

#103

yes

—

Dic₁₃₇

#106

yes

—

Dic₁₃₇

#107

yes

—

C₂×C₁₃₇

#331

yes

characteristic

C₂×C₁₃₇

#332

yes

—

C₂×C₁₃₇

#333

yes

characteristic

C₂×C₁₃₇

#340

yes

—

C₂×C₁₃₇

#343

yes

characteristic

C₁₃₇

#1296

yes

characteristic, complemented, Sylow

#1296

—

137

maximal, Sylow

C₂×C₄⋊C₈

—

137

—

#384

—

137

—

C₂²×C₈

—

274

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

#10587739

yes

characteristic

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

C₂×C₈

—

274

—

C₂×C₈

—

548

—

C₂×C₈

—

548

—

C₄⋊C₄

—

C₄⋊C₄

—

548

—

C₄⋊C₄

—

548

—

C₄⋊C₄

—

548

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

C₂²⋊#10587739

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

548

—

C₂²

(C₂×#10587739)⋊C₄

yes

characteristic

100

C₂²

#17984148

yes

characteristic

101

C₂²

#17984149

yes

characteristic

102

C₂²

#17984150

yes

—

103

C₂²

#17984151

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

1096

—

108

C₄

—

137

—

109

C₄

—

137

—

110

C₄

—

137

—

111

C₄

—

137

—

112

C₂

#22578261

yes

characteristic

113

C₂

#22578262

yes

—

114

C₂

#22578263

yes

characteristic

115

C₂

#22578276

yes

—

116

C₂

#22578283

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₄²
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂×C₄⋊C₄×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃₇	C₂²
hypercenter	C₂×C₄⋊C₄	#10587739
hypocenter	C₁₃₇	#1296
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₄²

Frattini

C₂²×C₄

C₂×#10587739

Fitting

C₂×C₄⋊C₄×C₁₃₇

C₄

socle, prime radical

C₂³×C₁₃₇

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₃₇

C₂²

C₂×C₄⋊C₄

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
5. #17984155	C₄	semidirect
58. C₁₃₇	#1296	semidirect

H ≅ G/N

type

5. #17984155

C₄

semidirect

58. C₁₃₇

#1296

semidirect

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

subgroup	index
C₂×#17984155	2
#22577272	2
#1296	137

subgroup

index

C₂×#17984155

#22577272

#1296

137

Subgroups — #31795667

Conjugacy classes of subgroups

Complements

Maximal subgroups