Group #31796323 — subgroups

Subgroups — #31796323

Order 17536.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	2	yes	maximal
2	#22578263	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984155	C₂²	1	4	yes	—
5	#17984145	—	2	2	no	—
6	#17984143	—	2	2	no	—
7	C₂×Q₈×D₁₃₇	C₄	1	2	yes	—
8	#17984088	C₄	1	1	yes	characteristic
9	#17984087	—	2	2	no	—
10	#17984086	C₂²	1	2	yes	—
11	C₄⋊C₄×D₁₃₇	—	2	2	no	—
12	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×#11764047	—	2	4	no	—
15	C₂×Dic₂₇₄	—	2	2	no	—
16	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
17	Q₈×D₁₃₇	—	4	4	no	—
18	C₂×C₄×D₁₃₇	—	2	2	no	—
19	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
21	C₄⋊Dic₁₃₇	—	2	2	no	—
22	#14555489	—	2	4	no	—
23	C₄×Dic₁₃₇	D₄	1	2	yes	—
24	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	#11764047	—	4	4	no	—
29	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	Dic₂₇₄	—	4	4	no	—
31	Dic₂₇₄	—	4	2	no	—
32	C₂×Dic₁₃₇	—	2	2	no	—
33	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
35	C₄×D₁₃₇	—	2	2	no	—
36	C₄×D₁₃₇	—	2	4	no	—
37	C₄×D₁₃₇	—	4	4	no	—
38	Q₈×C₁₃₇	—	4	2	no	—
39	C₂×C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₁₃₇	—	2	2	no	—
44	C₂×D₁₃₇	#100	1	2	yes	—
45	C₄×C₁₃₇	—	2	1	no	—
46	C₄×C₁₃₇	—	2	2	no	—
47	C₄×C₁₃₇	—	4	2	no	—
48	Dic₁₃₇	—	2	1	no	—
49	Dic₁₃₇	—	4	2	no	—
50	Dic₁₃₇	#137	1	4	yes	—
51	C₂×C₁₃₇	C₂×#100	1	1	yes	characteristic
52	C₂×C₁₃₇	#419	1	2	yes	—
53	D₁₃₇	—	2	2	no	—
54	C₁₃₇	#1558	1	1	yes	characteristic, complemented, Sylow
55	#1558	—	137	1	no	maximal, Sylow
56	C₂×C₄⋊Q₈	—	137	1	no	—
57	#432	—	137	2	no	—
58	#333	—	137	4	no	—
59	C₂²×Q₈	—	137	2	no	—
60	C₄⋊Q₈	—	274	2	no	—
61	C₄⋊Q₈	—	137	2	no	—
62	C₄⋊Q₈	—	137	1	no	—
63	C₄⋊Q₈	#10587739	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	274	2	no	—
65	C₂×C₄²	—	137	1	no	—
66	C₄⋊C₈	—	137	4	no	—
67	C₂²⋊C₈	—	274	2	no	—
68	#97	—	274	2	no	—
69	C₂×Q₈	—	274	2	no	—
70	C₂×Q₈	—	548	4	no	—
71	C₂×Q₈	—	137	2	no	—
72	C₂×Q₈	C₂×#10587739	1	2	yes	—
73	C₂²×C₄	—	274	2	no	—
74	C₂²×C₄	—	137	2	no	—
75	C₂²×C₄	—	137	1	no	—
76	C₂×C₈	—	274	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	274	4	no	—
79	C₄⋊C₄	—	274	2	no	—
80	C₄²	—	137	2	no	—
81	C₄²	—	137	1	no	—
82	C₄²	C₂×#10587739	1	1	yes	characteristic
83	C₂³	—	137	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	548	2	no	—
86	Q₈	—	548	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	274	2	no	—
89	C₂×C₄	—	274	4	no	—
90	C₂×C₄	—	274	2	no	—
91	C₂×C₄	—	548	4	no	—
92	C₂×C₄	—	137	1	no	—
93	C₂×C₄	—	137	2	no	—
94	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
95	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
96	C₈	—	548	4	no	—
97	C₂²	—	274	2	no	—
98	C₂²	—	137	2	no	—
99	C₂²	C₂×C₂²⋊#10587739	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	274	1	no	—
104	C₄	—	548	2	no	—
105	C₄	—	137	4	no	—
106	C₂	—	274	2	no	—
107	C₂	D₁₃₇⋊#100	1	1	yes	characteristic
108	C₂	#22578396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578426

C₂

yes

maximal

#22578263

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₃₇

C₂

yes

maximal, characteristic

#17984155

C₂²

yes

—

#17984145

—

#17984143

—

C₂×Q₈×D₁₃₇

C₄

yes

—

#17984088

C₄

yes

characteristic

#17984087

—

#17984086

C₂²

yes

—

C₄⋊C₄×D₁₃₇

—

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₄

yes

characteristic

C₂×#11764047

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

—

C₄⋊Dic₁₃₇

—

#14555489

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

C₂²×D₁₃₇

C₂²⋊C₄

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×D₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

#100

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#137

yes

—

C₂×C₁₃₇

C₂×#100

yes

characteristic

C₂×C₁₃₇

#419

yes

—

D₁₃₇

—

C₁₃₇

#1558

yes

characteristic, complemented, Sylow

#1558

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

#432

—

137

—

#333

—

137

—

C₂²×Q₈

—

137

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

#10587739

yes

characteristic

C₂×C₄⋊C₄

—

274

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

274

—

#97

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

548

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×#10587739

yes

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

Q₈

—

Q₈

—

548

—

Q₈

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²⋊#10587739

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

548

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂×C₂²⋊#10587739

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

274

—

104

C₄

—

548

—

105

C₄

—

137

—

106

C₂

—

274

—

107

C₂

D₁₃₇⋊#100

yes

characteristic

108

C₂

#22578396

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₇	C₂²⋊C₄
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1558
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄⋊Q₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₇

C₂²⋊C₄

cosocle

C₄×Dic₁₃₇

C₂³

C₄⋊Q₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #22578263	C₂	semidirect
54. C₁₃₇	#1558	semidirect

H ≅ G/N

type

2. #22578263

C₂

semidirect

54. C₁₃₇

#1558

semidirect

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

subgroup	index
#22578426	2
#22578263	2
C₄⋊Q₈×D₁₃₇	2
#1558	137

subgroup

index

#22578426

#22578263

C₄⋊Q₈×D₁₃₇

#1558

137

Subgroups — #31796323

Conjugacy classes of subgroups

Complements

Maximal subgroups