Group #34099765 — subgroups

Subgroups — #34099765

Order 20096.

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	#23757650	C₂	1	1	yes	maximal, characteristic
2	#23757647	C₂	1	2	yes	maximal, complemented
3	#23757621	C₂	1	2	yes	maximal, complemented
4	#23756994	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578941	—	2	4	no	—
7	C₂×D₁₂₅₆	—	2	2	no	—
8	#19097768	C₂²	1	1	yes	characteristic
9	#19097768	C₂²	1	1	yes	characteristic
10	#19097691	—	2	4	no	—
11	C₄×#11929434	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
13	#106×C₁₅₇	—	2	2	no	—
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×D₆₂₈	—	2	2	no	—
16	C₂×D₆₂₈	—	2	1	no	—
17	C₂×D₆₂₈	—	2	1	no	—
18	C₂×D₆₂₈	—	2	2	no	—
19	#14578941	—	4	8	no	—
20	D₁₂₅₆	—	4	4	no	—
21	C₂×#11929434	D₄	1	4	yes	—
22	C₂×Q₈×C₁₅₇	—	2	2	no	—
23	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₅₇	—	2	2	no	—
25	C₄⋊C₄×C₁₅₇	—	2	2	no	—
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₅₇	—	4	1	no	—
28	C₂²×D₁₅₇	—	4	1	no	—
29	D₆₂₈	—	4	2	no	—
30	D₆₂₈	—	4	2	no	—
31	D₆₂₈	—	4	4	no	—
32	D₆₂₈	—	4	4	no	—
33	#11929434	—	2	4	no	—
34	Q₈×C₁₅₇	—	4	4	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
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	—	8	2	no	—
45	C₄×C₁₅₇	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₅₇	AGL(1,ℤ/8)	1	4	yes	—
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	D₁₅₇	1	1	yes	characteristic, complemented
54	#493	—	157	1	no	—
55	#492	—	157	2	no	—
56	#486	—	157	2	no	—
57	#322	—	157	1	no	—
58	C₂×SD₁₆	—	314	4	no	—
59	C₂×D₈	—	314	2	no	—
60	C₄⋊Q₈	C₂×D₁₅₇	1	2	yes	—
61	#127	—	157	1	no	—
62	#127	—	157	1	no	—
63	#106	—	2	2	no	—
64	#102	—	314	4	no	—
65	C₄×C₈	—	157	2	no	—
66	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	314	2	no	—
69	C₂×D₄	—	314	2	no	—
70	C₂×D₄	—	314	1	no	—
71	C₂×D₄	—	314	1	no	—
72	SD₁₆	—	628	8	no	—
73	D₈	—	628	4	no	—
74	C₂×C₈	—	157	4	no	—
75	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
79	C₂³	—	628	1	no	—
80	C₂³	—	628	1	no	—
81	Q₈	—	4	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	—	4	2	no	—
87	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
89	C₈	—	314	4	no	—
90	C₈	#14578941	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	#19097767	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#14578941	1	2	yes	—
98	C₄	#19097798	1	4	yes	—
99	C₂	—	2512	1	no	—
100	C₂	—	2512	1	no	—
101	C₂	#23757647	1	1	yes	characteristic
102	C₂	#23757659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757650

C₂

yes

maximal, characteristic

#23757647

C₂

yes

maximal, complemented

#23757621

C₂

yes

maximal, complemented

#23756994

C₂

yes

maximal, characteristic, complemented

#498×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578941

—

C₂×D₁₂₅₆

—

#19097768

C₂²

yes

characteristic

#19097768

C₂²

yes

characteristic

#19097691

—

C₄×#11929434

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

#106×C₁₅₇

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

#14578941

—

D₁₂₅₆

—

C₂×#11929434

D₄

yes

—

C₂×Q₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

#11929434

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₈×C₁₅₇

SD₁₆

yes

complemented

C₂²×C₁₅₇

#127

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×SD₁₆

yes

—

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₂×C₁₅₇

#492

yes

characteristic

C₂×C₁₅₇

#496

yes

—

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#1715

yes

characteristic, complemented, Sylow

#1715

—

157

maximal, Sylow

#498

D₁₅₇

yes

characteristic, complemented

#493

—

157

—

#492

—

157

—

#486

—

157

—

#322

—

157

—

C₂×SD₁₆

—

314

—

C₂×D₈

—

314

—

C₄⋊Q₈

C₂×D₁₅₇

yes

—

#127

—

157

—

#127

—

157

—

#106

—

#102

—

314

—

C₄×C₈

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

SD₁₆

—

628

—

D₈

—

628

—

C₂×C₈

—

157

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

—

628

—

C₂³

—

628

—

Q₈

—

D₄

—

628

—

D₄

—

628

—

D₄

—

628

—

D₄

—

628

—

C₂×C₄

—

C₂×C₄

D₄×D₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₈

—

314

—

C₈

#14578941

yes

complemented

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

—

1256

—

C₂²

#19097767

yes

characteristic

C₄

—

C₄

C₂×#14578941

yes

—

C₄

#19097798

yes

—

C₂

—

2512

—

100

C₂

—

2512

—

101

C₂

#23757647

yes

characteristic

102

C₂

#23757659

yes

—

	subgroup	G/H
center	C₂²	#19097767
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#498×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#127
prime radical	#23757650	C₂
hypercenter	#498	D₁₅₇
hypocenter	C₁₅₇	#1715
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#19097767

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#498×C₁₅₇

C₂

C₂²×C₁₅₇

C₂

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #23757647	C₂	semidirect
3. #23757621	C₂	semidirect
4. #23756994	C₂	semidirect
5. #498×C₁₅₇	C₂	semidirect
38. C₈×C₁₅₇	SD₁₆	semidirect
51. C₁₅₇	#1715	semidirect
53. #498	D₁₅₇	semidirect
90. C₈	#14578941	semidirect

H ≅ G/N

type

2. #23757647

C₂

semidirect

3. #23757621

C₂

semidirect

4. #23756994

C₂

semidirect

5. #498×C₁₅₇

C₂

semidirect

38. C₈×C₁₅₇

SD₁₆

semidirect

51. C₁₅₇

#1715

semidirect

53. #498

D₁₅₇

semidirect

90. C₈

#14578941

semidirect

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

subgroup	index
#23757650	2
#23757647	2
#23757621	2
#23756994	2
#498×C₁₅₇	2
#1715	157

subgroup

index

#498×C₁₅₇

157

Subgroups — #34099765

Conjugacy classes of subgroups

Complements

Maximal subgroups