Group #34099764 — subgroups

Subgroups — #34099764

Order 20096.

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

normal subgroups only

This group has 220 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	#23757671	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757650	C₂	1	1	yes	maximal, characteristic
3	#23757646	C₂	1	2	yes	maximal, complemented
4	#23757620	C₂	1	2	yes	maximal, complemented
5	#322×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578943	—	2	4	no	—
7	C₂×D₁₂₅₆	—	2	2	no	—
8	#19097771	C₂²	1	2	yes	—
9	#19097768	C₂²	1	1	yes	characteristic
10	#19097768	C₂²	1	1	yes	characteristic
11	#19097703	—	2	2	no	—
12	#19097693	—	2	4	no	—
13	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₄	—	2	2	no	—
16	C₂×D₆₂₈	—	2	2	no	—
17	C₂×D₆₂₈	—	2	1	no	—
18	C₂×D₆₂₈	—	2	1	no	—
19	C₂×D₆₂₈	—	2	2	no	—
20	#14578943	—	4	8	no	—
21	D₁₂₅₆	—	4	4	no	—
22	C₄⋊Dic₁₅₇	—	2	2	no	—
23	C₄⋊Dic₁₅₇	—	2	2	no	—
24	C₂×C₈×C₁₅₇	D₄	1	4	yes	—
25	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₅₇	—	4	1	no	—
28	C₂²×D₁₅₇	—	4	1	no	—
29	Dic₃₁₄	—	4	4	no	—
30	D₆₂₈	—	4	2	no	—
31	D₆₂₈	—	4	2	no	—
32	D₆₂₈	—	4	4	no	—
33	D₆₂₈	—	4	4	no	—
34	C₂×Dic₁₅₇	—	4	2	no	—
35	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
37	C₈×C₁₅₇	—	2	4	no	—
38	C₈×C₁₅₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₅₇	#127	1	1	yes	characteristic
40	C₂×D₁₅₇	—	8	1	no	—
41	C₂×D₁₅₇	—	8	1	no	—
42	C₂×D₁₅₇	—	8	2	no	—
43	C₂×D₁₅₇	—	8	2	no	—
44	C₄×C₁₅₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₅₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₅₇	—	8	2	no	—
47	C₂×C₁₅₇	#492	1	1	yes	characteristic
48	C₂×C₁₅₇	#496	1	2	yes	—
49	D₁₅₇	—	16	1	no	—
50	D₁₅₇	—	16	1	no	—
51	C₁₅₇	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	157	1	no	maximal, Sylow
53	#498	—	157	1	no	—
54	#493	—	157	1	no	—
55	#492	—	157	2	no	—
56	#486	—	157	2	no	—
57	#322	D₁₅₇	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	314	4	no	—
59	C₂×D₈	—	314	2	no	—
60	C₄⋊Q₈	—	157	2	no	—
61	#127	—	157	1	no	—
62	#127	—	157	1	no	—
63	#106	—	314	2	no	—
64	#102	—	314	4	no	—
65	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
66	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
67	C₂×Q₈	—	314	2	no	—
68	C₂×D₄	—	314	2	no	—
69	C₂×D₄	—	314	1	no	—
70	C₂×D₄	—	314	1	no	—
71	C₂×D₄	—	314	2	no	—
72	SD₁₆	—	628	8	no	—
73	D₈	—	628	4	no	—
74	C₂×C₈	D₆₂₈	1	4	yes	—
75	C₂×C₈	D₆₂₈	1	2	yes	—
76	C₄⋊C₄	—	314	2	no	—
77	C₄⋊C₄	—	314	2	no	—
78	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
79	C₂³	—	628	1	no	—
80	C₂³	—	628	1	no	—
81	Q₈	—	628	4	no	—
82	D₄	—	628	2	no	—
83	D₄	—	628	2	no	—
84	D₄	—	628	4	no	—
85	D₄	—	628	4	no	—
86	C₂×C₄	—	628	2	no	—
87	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#14578943	1	4	yes	complemented
91	C₂²	—	1256	1	no	—
92	C₂²	—	1256	1	no	—
93	C₂²	—	1256	2	no	—
94	C₂²	—	1256	2	no	—
95	C₂²	#19097768	1	1	yes	characteristic
96	C₄	—	1256	2	no	—
97	C₄	C₂×#14578943	1	2	yes	—
98	C₄	#19097799	1	4	yes	—
99	C₂	—	2512	1	no	—
100	C₂	—	2512	1	no	—
101	C₂	#23757646	1	1	yes	characteristic
102	C₂	#23757662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757671

C₂

yes

maximal, characteristic, complemented

#23757650

C₂

yes

maximal, characteristic

#23757646

C₂

yes

maximal, complemented

#23757620

C₂

yes

maximal, complemented

#322×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578943

—

C₂×D₁₂₅₆

—

#19097771

C₂²

yes

—

#19097768

C₂²

yes

characteristic

#19097768

C₂²

yes

characteristic

#19097703

—

#19097693

—

C₄×C₈×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

#14578943

—

D₁₂₅₆

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

C₂×Dic₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

SD₁₆

yes

complemented

C₂²×C₁₅₇

#127

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×SD₁₆

yes

—

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#492

yes

characteristic

C₂×C₁₅₇

#496

yes

—

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#1715

yes

characteristic, complemented, Sylow

#1715

—

157

maximal, Sylow

#498

—

157

—

#493

—

157

—

#492

—

157

—

#486

—

157

—

#322

D₁₅₇

yes

characteristic, complemented

C₂×SD₁₆

—

314

—

C₂×D₈

—

314

—

C₄⋊Q₈

—

157

—

#127

—

157

—

#127

—

157

—

#106

—

314

—

#102

—

314

—

C₄×C₈

C₂×D₁₅₇

yes

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

SD₁₆

—

628

—

D₈

—

628

—

C₂×C₈

D₆₂₈

yes

—

C₂×C₈

D₆₂₈

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

—

628

—

C₂³

—

628

—

Q₈

—

628

—

D₄

—

628

—

D₄

—

628

—

D₄

—

628

—

D₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

C₂×D₆₂₈

yes

—

C₈

—

C₈

#14578943

yes

complemented

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

#19097768

yes

characteristic

C₄

—

1256

—

C₄

C₂×#14578943

yes

—

C₄

#19097799

yes

—

C₂

—

2512

—

100

C₂

—

2512

—

101

C₂

#23757646

yes

characteristic

102

C₂

#23757662

yes

—

	subgroup	G/H
center	C₂²	#19097768
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#322×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#127
prime radical	#23757650	C₂
hypercenter	#322	D₁₅₇
hypocenter	C₁₅₇	#1715
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#19097768

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#322×C₁₅₇

C₂

C₂²×C₁₅₇

C₂

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #23757671	C₂	semidirect
3. #23757646	C₂	semidirect
4. #23757620	C₂	semidirect
5. #322×C₁₅₇	C₂	semidirect
38. C₈×C₁₅₇	SD₁₆	semidirect
51. C₁₅₇	#1715	semidirect
57. #322	D₁₅₇	semidirect
90. C₈	#14578943	semidirect

H ≅ G/N

type

1. #23757671

C₂

semidirect

3. #23757646

C₂

semidirect

4. #23757620

C₂

semidirect

5. #322×C₁₅₇

C₂

semidirect

38. C₈×C₁₅₇

SD₁₆

semidirect

51. C₁₅₇

#1715

semidirect

57. #322

D₁₅₇

semidirect

90. C₈

#14578943

semidirect

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

subgroup	index
#23757671	2
#23757650	2
#23757646	2
#23757620	2
#322×C₁₅₇	2
#1715	157

subgroup

index

#322×C₁₅₇

157

Subgroups — #34099764

Conjugacy classes of subgroups

Complements

Maximal subgroups