Group #31780923 — subgroups

Subgroups — #31780923

Order 17536.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#22577273	C₂	1	1	yes	maximal, characteristic
2	#22577269	C₂	1	1	yes	maximal, characteristic
3	#22577166	C₂	1	4	yes	maximal
4	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
6	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
8	#17983995	—	2	2	no	—
9	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#97×C₁₃₇	C₄	1	4	yes	complemented
12	C₂²×Dic₁₃₇	—	2	2	no	—
13	C₂²×Dic₁₃₇	—	2	2	no	—
14	C₄⋊Dic₁₃₇	—	2	4	no	—
15	C₄⋊Dic₁₃₇	—	2	4	no	—
16	C₄⋊Dic₁₃₇	—	4	4	no	—
17	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
18	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₃₇	—	2	2	no	—
20	C₂×C₈×C₁₃₇	—	2	2	no	—
21	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₇	Q₈	1	2	yes	—
24	C₂×Dic₁₃₇	—	4	2	no	—
25	C₂×Dic₁₃₇	—	4	2	no	—
26	C₂×Dic₁₃₇	—	4	4	no	—
27	C₂×Dic₁₃₇	—	4	2	no	—
28	C₂×Dic₁₃₇	—	4	2	no	—
29	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	2	no	—
31	C₂×C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	#41	1	2	yes	—
36	C₂×C₄×C₁₃₇	#41	1	2	yes	—
37	C₈×C₁₃₇	—	2	4	no	—
38	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₃₇	#124	1	1	yes	characteristic
40	C₂²×C₁₃₇	#124	1	1	yes	characteristic
41	C₂²×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₃₇	#137	1	2	yes	—
43	C₄×C₁₃₇	—	2	4	no	—
44	C₄×C₁₃₇	#117	1	2	yes	—
45	C₄×C₁₃₇	#125	1	2	yes	—
46	Dic₁₃₇	—	8	2	no	—
47	Dic₁₃₇	—	8	2	no	—
48	C₂×C₁₃₇	#382	1	1	yes	characteristic
49	C₂×C₁₃₇	#428	1	2	yes	—
50	C₂×C₁₃₇	#489	1	2	yes	—
51	C₂×C₁₃₇	#490	1	1	yes	characteristic
52	C₂×C₁₃₇	#491	1	1	yes	characteristic
53	C₁₃₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	137	1	no	maximal, Sylow
55	C₂×#97	D₁₃₇	1	1	yes	characteristic
56	#384	—	137	1	no	—
57	#382	—	137	1	no	—
58	#340	—	137	4	no	—
59	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
60	C₂×C₄⋊C₄	—	274	2	no	—
61	C₂×C₄⋊C₄	—	137	2	no	—
62	C₂×C₄⋊C₄	—	137	2	no	—
63	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
64	#97	Dic₁₃₇	1	4	yes	complemented
65	#95	—	274	2	no	—
66	C₂²×C₄	—	274	2	no	—
67	C₂²×C₄	—	274	2	no	—
68	C₂²×C₄	D₅₄₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
73	C₄⋊C₄	—	274	4	no	—
74	C₄⋊C₄	—	274	4	no	—
75	C₄⋊C₄	—	548	4	no	—
76	C₄²	C₂×Dic₁₃₇	1	2	yes	—
77	C₄²	Dic₂₇₄	1	2	yes	—
78	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	548	2	no	—
82	C₂×C₄	—	548	2	no	—
83	C₂×C₄	—	548	4	no	—
84	C₂×C₄	—	548	2	no	—
85	C₂×C₄	—	548	2	no	—
86	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
89	C₂×C₄	#14555510	1	2	yes	—
90	C₂×C₄	#14555510	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
93	C₂²	#17984076	1	1	yes	characteristic
94	C₂²	#17984076	1	1	yes	characteristic
95	C₂²	#17984116	1	2	yes	—
96	C₂²	#17984121	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	1096	2	no	—
99	C₄	—	1096	2	no	—
100	C₄	#17984047	1	2	yes	—
101	C₄	#17984080	1	2	yes	—
102	C₂	#22577269	1	1	yes	characteristic
103	C₂	#22577445	1	2	yes	—
104	C₂	#22577763	1	2	yes	—
105	C₂	#22577768	1	1	yes	characteristic
106	C₂	#22577772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577273

C₂

yes

maximal, characteristic

#22577269

C₂

yes

maximal, characteristic

#22577166

C₂

yes

maximal

C₂×#97×C₁₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17983995

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₄

yes

complemented

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₂²×C₁₃₇

#137

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

C₄×C₁₃₇

#125

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#382

yes

characteristic

C₂×C₁₃₇

#428

yes

—

C₂×C₁₃₇

#489

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1836

—

137

maximal, Sylow

C₂×#97

D₁₃₇

yes

characteristic

#384

—

137

—

#382

—

137

—

#340

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#97

Dic₁₃₇

yes

complemented

#95

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

D₅₄₈

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃₇

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

548

—

C₄²

C₂×Dic₁₃₇

yes

—

C₄²

Dic₂₇₄

yes

—

C₂³

C₂×D₅₄₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

C₄⋊Dic₁₃₇

yes

—

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

C₂×C₄

#14555510

yes

—

C₂×C₄

#14555510

yes

—

C₈

—

C₂²

C₂×C₄⋊Dic₁₃₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#17984121

yes

—

C₄

—

C₄

—

1096

—

C₄

—

1096

—

100

C₄

#17984047

yes

—

101

C₄

#17984080

yes

—

102

C₂

#22577269

yes

characteristic

103

C₂

#22577445

yes

—

104

C₂

#22577763

yes

—

105

C₂

#22577768

yes

characteristic

106

C₂

#22577772

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#97×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#1836
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₄₈

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×#97×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

hypercenter

C₂×#97

D₁₃₇

hypocenter

C₁₃₇

#1836

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₁₃₇	C₄	semidirect
53. C₁₃₇	#1836	semidirect
64. #97	Dic₁₃₇	semidirect

H ≅ G/N

type

11. #97×C₁₃₇

C₄

semidirect

53. C₁₃₇

#1836

semidirect

64. #97

Dic₁₃₇

semidirect

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

subgroup	index
#22577273	2
#22577269	2
#22577166	2
C₂×#97×C₁₃₇	2
#1836	137

subgroup

index

C₂×#97×C₁₃₇

137

Subgroups — #31780923

Conjugacy classes of subgroups

Complements

Maximal subgroups