Group #31780455 — subgroups

Subgroups — #31780455

Order 17536.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22577780	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577780	C₂	1	2	yes	maximal, complemented
3	#22577776	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577750	C₂	1	2	yes	maximal, complemented
5	#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555499	—	2	2	no	—
7	C₂×D₁₀₉₆	—	2	4	no	—
8	C₂×D₁₀₉₆	—	2	2	no	—
9	#17984088	C₂²	1	1	yes	characteristic
10	#17984085	C₂²	1	2	yes	—
11	#17984085	C₂²	1	1	yes	characteristic
12	#17984010	—	2	4	no	—
13	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₄	—	2	1	no	—
16	C₂×D₅₄₈	—	2	2	no	—
17	C₂×D₅₄₈	—	2	2	no	—
18	C₂×D₅₄₈	—	2	2	no	—
19	C₂×D₅₄₈	—	2	2	no	—
20	C₂×D₅₄₈	—	2	1	no	—
21	#14555499	—	4	4	no	—
22	D₁₀₉₆	—	4	8	no	—
23	D₁₀₉₆	—	4	4	no	—
24	C₄⋊Dic₁₃₇	—	2	2	no	—
25	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
26	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
27	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×D₁₃₇	—	4	2	no	—
29	C₂²×D₁₃₇	—	4	1	no	—
30	Dic₂₇₄	—	4	2	no	—
31	D₅₄₈	—	4	4	no	—
32	D₅₄₈	—	4	2	no	—
33	D₅₄₈	—	4	4	no	—
34	D₅₄₈	—	4	4	no	—
35	D₅₄₈	—	4	4	no	—
36	C₂×Dic₁₃₇	—	4	1	no	—
37	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
39	C₈×C₁₃₇	—	2	4	no	—
40	C₈×C₁₃₇	D₈	1	4	yes	complemented
41	C₂²×C₁₃₇	#127	1	1	yes	characteristic
42	C₂×D₁₃₇	—	8	2	no	—
43	C₂×D₁₃₇	—	8	1	no	—
44	C₂×D₁₃₇	—	8	2	no	—
45	C₂×D₁₃₇	—	8	2	no	—
46	C₂×D₁₃₇	—	8	2	no	—
47	C₄×C₁₃₇	C₂×D₈	1	2	yes	—
48	C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₃₇	—	8	1	no	—
50	C₂×C₁₃₇	#493	1	1	yes	characteristic
51	C₂×C₁₃₇	#496	1	2	yes	—
52	D₁₃₇	—	16	2	no	—
53	D₁₃₇	—	16	1	no	—
54	C₁₃₇	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	137	1	no	maximal, Sylow
56	#493	—	137	2	no	—
57	#493	—	137	1	no	—
58	#492	—	137	1	no	—
59	#486	—	137	2	no	—
60	#322	D₁₃₇	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	274	2	no	—
62	C₂×D₈	—	274	4	no	—
63	C₂×D₈	—	274	2	no	—
64	C₄⋊Q₈	—	137	1	no	—
65	#127	—	137	2	no	—
66	#127	—	137	1	no	—
67	#102	—	274	4	no	—
68	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
69	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
70	C₂×Q₈	—	274	1	no	—
71	C₂×D₄	—	274	2	no	—
72	C₂×D₄	—	274	2	no	—
73	C₂×D₄	—	274	2	no	—
74	C₂×D₄	—	274	2	no	—
75	C₂×D₄	—	274	1	no	—
76	SD₁₆	—	548	4	no	—
77	D₈	—	548	8	no	—
78	D₈	—	548	4	no	—
79	C₂×C₈	D₅₄₈	1	4	yes	—
80	C₂×C₈	D₅₄₈	1	2	yes	—
81	C₄⋊C₄	—	274	2	no	—
82	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
83	C₂³	—	548	2	no	—
84	C₂³	—	548	1	no	—
85	Q₈	—	548	2	no	—
86	D₄	—	548	4	no	—
87	D₄	—	548	2	no	—
88	D₄	—	548	4	no	—
89	D₄	—	548	4	no	—
90	D₄	—	548	4	no	—
91	C₂×C₄	—	548	1	no	—
92	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₁₀₉₆	1	4	yes	complemented
96	C₂²	—	1096	2	no	—
97	C₂²	—	1096	1	no	—
98	C₂²	—	1096	2	no	—
99	C₂²	—	1096	2	no	—
100	C₂²	—	1096	2	no	—
101	C₂²	#17984085	1	1	yes	characteristic
102	C₄	—	1096	1	no	—
103	C₄	C₂×D₁₀₉₆	1	2	yes	—
104	C₄	#17984116	1	4	yes	—
105	C₂	—	2192	2	no	—
106	C₂	—	2192	1	no	—
107	C₂	#22577780	1	1	yes	characteristic
108	C₂	#22577792	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577780

C₂

yes

maximal, characteristic, complemented

#22577780

C₂

yes

maximal, complemented

#22577776

C₂

yes

maximal, characteristic, complemented

#22577750

C₂

yes

maximal, complemented

#322×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#14555499

—

C₂×D₁₀₉₆

—

C₂×D₁₀₉₆

—

#17984088

C₂²

yes

characteristic

#17984085

C₂²

yes

—

#17984085

C₂²

yes

characteristic

#17984010

—

C₄×C₈×C₁₃₇

C₂²

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×D₅₄₈

—

C₂×D₅₄₈

—

C₂×D₅₄₈

—

C₂×D₅₄₈

—

C₂×D₅₄₈

—

#14555499

—

D₁₀₉₆

—

D₁₀₉₆

—

C₄⋊Dic₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

C₂×Dic₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

D₈

yes

complemented

C₂²×C₁₃₇

#127

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

C₂×D₈

yes

—

C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#493

yes

characteristic

C₂×C₁₃₇

#496

yes

—

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1721

yes

characteristic, complemented, Sylow

#1721

—

137

maximal, Sylow

#493

—

137

—

#493

—

137

—

#492

—

137

—

#486

—

137

—

#322

D₁₃₇

yes

characteristic, complemented

C₂×SD₁₆

—

274

—

C₂×D₈

—

274

—

C₂×D₈

—

274

—

C₄⋊Q₈

—

137

—

#127

—

137

—

#127

—

137

—

#102

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

SD₁₆

—

548

—

D₈

—

548

—

D₈

—

548

—

C₂×C₈

D₅₄₈

yes

—

C₂×C₈

D₅₄₈

yes

—

C₄⋊C₄

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

548

—

C₂³

—

548

—

Q₈

—

548

—

D₄

—

548

—

D₄

—

548

—

D₄

—

548

—

D₄

—

548

—

D₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₂×C₄

C₂×D₅₄₈

yes

—

C₈

—

C₈

D₁₀₉₆

yes

complemented

C₂²

—

1096

—

C₂²

—

1096

—

C₂²

—

1096

—

C₂²

—

1096

—

100

C₂²

—

1096

—

101

C₂²

#17984085

yes

characteristic

102

C₄

—

1096

—

103

C₄

C₂×D₁₀₉₆

yes

—

104

C₄

#17984116

yes

—

105

C₂

—

2192

—

106

C₂

—

2192

—

107

C₂

#22577780

yes

characteristic

108

C₂

#22577792

yes

—

	subgroup	G/H
center	C₂²	#17984085
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#322×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#127
hypercenter	#322	D₁₃₇
hypocenter	C₁₃₇	#1721
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17984085

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#322×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #22577780	C₂	semidirect
2. #22577780	C₂	semidirect
3. #22577776	C₂	semidirect
4. #22577750	C₂	semidirect
5. #322×C₁₃₇	C₂	semidirect
40. C₈×C₁₃₇	D₈	semidirect
54. C₁₃₇	#1721	semidirect
60. #322	D₁₃₇	semidirect
95. C₈	D₁₀₉₆	semidirect

H ≅ G/N

type

1. #22577780

C₂

semidirect

2. #22577780

C₂

semidirect

3. #22577776

C₂

semidirect

4. #22577750

C₂

semidirect

5. #322×C₁₃₇

C₂

semidirect

40. C₈×C₁₃₇

D₈

semidirect

54. C₁₃₇

#1721

semidirect

60. #322

D₁₃₇

semidirect

95. C₈

D₁₀₉₆

semidirect

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

subgroup	index
#22577780	2
#22577776	2
#22577750	2
#322×C₁₃₇	2
#1721	137

subgroup

index

#322×C₁₃₇

137

Subgroups — #31780455

Conjugacy classes of subgroups

Complements

Maximal subgroups