Group #31780415 — subgroups

Subgroups — #31780415

Order 17536.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22577801	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577775	C₂	1	2	yes	maximal, complemented
3	#22577749	C₂	1	2	yes	maximal, complemented
4	#22577124	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555498	—	2	4	no	—
7	#17984088	C₂²	1	2	yes	—
8	#17984020	—	2	2	no	—
9	#17984009	—	2	4	no	—
10	C₄×#11764040	C₂²	1	2	yes	—
11	C₂×D₈×C₁₃₇	—	2	2	no	—
12	#127×C₁₃₇	C₂²	1	1	yes	characteristic
13	#127×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₄	—	2	2	no	—
16	#14555498	—	4	8	no	—
17	C₂×#11764040	D₄	1	4	yes	—
18	C₄⋊Dic₁₃₇	—	2	2	no	—
19	C₄⋊Dic₁₃₇	—	2	2	no	—
20	C₂×D₄×C₁₃₇	—	2	2	no	—
21	C₂×D₄×C₁₃₇	—	2	2	no	—
22	C₂×D₄×C₁₃₇	—	2	1	no	—
23	C₂×D₄×C₁₃₇	—	2	1	no	—
24	D₈×C₁₃₇	—	4	4	no	—
25	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
26	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
27	Dic₂₇₄	—	4	4	no	—
28	C₂×Dic₁₃₇	—	4	2	no	—
29	#11764040	—	2	4	no	—
30	C₂³×C₁₃₇	—	4	1	no	—
31	C₂³×C₁₃₇	—	4	1	no	—
32	D₄×C₁₃₇	—	4	2	no	—
33	D₄×C₁₃₇	—	4	2	no	—
34	D₄×C₁₃₇	—	4	4	no	—
35	D₄×C₁₃₇	—	4	4	no	—
36	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₃₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₃₇	—	8	1	no	—
40	C₂²×C₁₃₇	—	8	1	no	—
41	C₂²×C₁₃₇	—	8	2	no	—
42	C₂²×C₁₃₇	—	8	2	no	—
43	C₂²×C₁₃₇	#127	1	1	yes	characteristic
44	C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₃₇	—	8	2	no	—
47	C₂×C₁₃₇	—	16	1	no	—
48	C₂×C₁₃₇	—	16	1	no	—
49	C₂×C₁₃₇	#492	1	1	yes	characteristic
50	C₂×C₁₃₇	#496	1	2	yes	—
51	C₁₃₇	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	137	1	no	maximal, Sylow
53	#498	—	137	1	no	—
54	#493	D₁₃₇	1	1	yes	characteristic
55	#492	—	137	2	no	—
56	#486	—	137	2	no	—
57	#322	—	137	1	no	—
58	C₂×SD₁₆	—	274	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	137	2	no	—
61	#127	C₂×D₁₃₇	1	1	yes	characteristic
62	#127	C₂×D₁₃₇	1	1	yes	characteristic
63	#106	—	274	2	no	—
64	#102	—	274	4	no	—
65	C₄×C₈	—	137	2	no	—
66	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
67	C₂×Q₈	—	274	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	548	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	137	4	no	—
75	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
76	C₄⋊C₄	—	274	2	no	—
77	C₄⋊C₄	—	274	2	no	—
78	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	548	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	548	2	no	—
87	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
89	C₈	—	274	4	no	—
90	C₈	#14555498	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#17984084	1	1	yes	characteristic
96	C₄	—	1096	2	no	—
97	C₄	C₂×#14555498	1	2	yes	—
98	C₄	#17984112	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#22577775	1	1	yes	characteristic
102	C₂	#22577791	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577801

C₂

yes

maximal, characteristic, complemented

#22577775

C₂

yes

maximal, complemented

#22577749

C₂

yes

maximal, complemented

#22577124

C₂

yes

maximal, characteristic, complemented

#493×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555498

—

#17984088

C₂²

yes

—

#17984020

—

#17984009

—

C₄×#11764040

C₂²

yes

—

C₂×D₈×C₁₃₇

—

#127×C₁₃₇

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

#14555498

—

C₂×#11764040

D₄

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

D₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

SD₁₆

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#127

yes

characteristic

C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#492

yes

characteristic

C₂×C₁₃₇

#496

yes

—

C₁₃₇

#1715

yes

characteristic, complemented, Sylow

#1715

—

137

maximal, Sylow

#498

—

137

—

#493

D₁₃₇

yes

characteristic

#492

—

137

—

#486

—

137

—

#322

—

137

—

C₂×SD₁₆

—

274

—

C₂×D₈

—

C₄⋊Q₈

—

137

—

#127

C₂×D₁₃₇

yes

characteristic

#127

C₂×D₁₃₇

yes

characteristic

#106

—

274

—

#102

—

274

—

C₄×C₈

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

548

—

D₈

—

C₂×C₈

—

137

—

C₂×C₈

C₂²⋊D₁₃₇

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

548

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

548

—

C₂×C₄

D₄×D₁₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₈

—

274

—

C₈

#14555498

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#17984084

yes

characteristic

C₄

—

1096

—

C₄

C₂×#14555498

yes

—

C₄

#17984112

yes

—

C₂

—

100

C₂

—

101

C₂

#22577775

yes

characteristic

102

C₂

#22577791

yes

—

	subgroup	G/H
center	C₂²	#17984084
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting, prime radical	#493×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#127
hypercenter	#493	D₁₃₇
hypocenter	C₁₃₇	#1715
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#17984084

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting, prime radical

#493×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22577801	C₂	semidirect
2. #22577775	C₂	semidirect
3. #22577749	C₂	semidirect
4. #22577124	C₂	semidirect
38. C₈×C₁₃₇	SD₁₆	semidirect
51. C₁₃₇	#1715	semidirect
90. C₈	#14555498	semidirect

H ≅ G/N

type

1. #22577801

C₂

semidirect

2. #22577775

C₂

semidirect

3. #22577749

C₂

semidirect

4. #22577124

C₂

semidirect

38. C₈×C₁₃₇

SD₁₆

semidirect

51. C₁₃₇

#1715

semidirect

90. C₈

#14555498

semidirect

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

subgroup	index
#22577801	2
#22577775	2
#22577749	2
#22577124	2
#493×C₁₃₇	2
#1715	137

subgroup

index

#493×C₁₃₇

137

Subgroups — #31780415

Conjugacy classes of subgroups

Complements

Maximal subgroups