Group #34108499 — subgroups

Subgroups — #34108499

Order 20096.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#19097763	C₂	1	2	yes	maximal
2	#23757171	C₂	1	8	yes	maximal
3	#23757143	C₂	1	2	yes	maximal
4	#23757139	C₂	1	2	yes	maximal
5	C₄³×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097763	—	2	8	no	—
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	8	yes	—
9	C₂×#14578933	C₂²	1	8	yes	—
10	#19097678	C₂²	1	8	yes	—
11	C₂×C₄²×C₁₅₇	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	C₂³	1	8	yes	—
16	C₄⋊Dic₁₅₇	—	2	8	no	—
17	C₄⋊Dic₁₅₇	—	2	16	no	—
18	#14578933	—	2	16	no	—
19	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
20	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
23	C₄²×C₁₅₇	—	2	8	no	—
24	C₄²×C₁₅₇	D₄	1	4	yes	—
25	C₄²×C₁₅₇	Q₈	1	4	yes	—
26	C₄²×C₁₅₇	Q₈	1	4	yes	—
27	C₂×Dic₁₅₇	—	2	8	no	—
28	C₂×Dic₁₅₇	—	2	16	no	—
29	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	—	2	4	no	—
31	C₂×C₄×C₁₅₇	—	2	4	no	—
32	C₂×C₄×C₁₅₇	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
36	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₅₇	#41	1	8	yes	—
39	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
40	C₂²×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₁₅₇	C₂×#41	1	4	yes	—
43	C₄×C₁₅₇	—	2	4	no	—
44	C₄×C₁₅₇	—	2	4	no	—
45	C₄×C₁₅₇	#125	1	8	yes	complemented
46	C₄×C₁₅₇	C₄⋊Q₈	1	4	yes	—
47	Dic₁₅₇	—	4	8	no	—
48	C₂×C₁₅₇	C₂×#125	1	2	yes	—
49	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₅₇	#532	1	2	yes	—
51	C₂×C₁₅₇	#533	1	2	yes	—
52	C₁₅₇	#2859	1	1	yes	characteristic, complemented, Sylow
53	#2859	—	157	1	no	maximal, Sylow
54	C₂×#125	—	157	2	no	—
55	#389	—	157	8	no	—
56	#384	—	157	2	no	—
57	#382	—	157	2	no	—
58	C₄³	D₁₅₇	1	1	yes	characteristic
59	#125	—	314	8	no	—
60	C₂×C₄⋊C₄	—	157	8	no	—
61	C₂×C₄⋊C₄	—	157	4	no	—
62	C₂×C₄⋊C₄	—	157	8	no	—
63	C₂×C₄²	C₂×D₁₅₇	1	4	yes	—
64	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
67	#95	—	157	8	no	—
68	C₂²×C₄	—	157	8	no	—
69	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
71	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
73	C₄⋊C₄	—	314	16	no	—
74	C₄⋊C₄	—	314	8	no	—
75	C₄⋊C₄	—	314	16	no	—
76	C₄²	—	2	8	no	—
77	C₄²	D₆₂₈	1	4	yes	—
78	C₄²	Dic₃₁₄	1	4	yes	—
79	C₄²	Dic₃₁₄	1	4	yes	—
80	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	314	8	no	—
84	C₂×C₄	—	314	16	no	—
85	C₂×C₄	C₂×D₆₂₈	1	4	yes	—
86	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
87	C₂×C₄	C₂×Dic₃₁₄	1	4	yes	—
88	C₂×C₄	C₂×Dic₃₁₄	1	4	yes	—
89	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
90	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
91	C₂×C₄	#14578954	1	8	yes	—
92	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
93	C₂²	C₂²×Dic₃₁₄	1	1	yes	characteristic
94	C₂²	C₂²×Dic₃₁₄	1	1	yes	characteristic
95	C₂²	C₂×#14578954	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	628	8	no	—
99	C₄	#19097763	1	8	yes	complemented
100	C₄	#19097771	1	4	yes	—
101	C₂	C₂×#19097763	1	2	yes	—
102	C₂	C₂×#19097771	1	1	yes	characteristic
103	C₂	#23757823	1	2	yes	—
104	C₂	#23757830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097763

C₂

yes

maximal

#23757171

C₂

yes

maximal

#23757143

C₂

yes

maximal

#23757139

C₂

yes

maximal

C₄³×C₁₅₇

C₂

yes

maximal, characteristic

#19097763

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×#14578933

C₂²

yes

—

#19097678

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

#14578933

—

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₂²×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₅₇

C₂×#41

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#125

yes

complemented

C₄×C₁₅₇

C₄⋊Q₈

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

C₂×#125

yes

—

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

#532

yes

—

C₂×C₁₅₇

#533

yes

—

C₁₅₇

#2859

yes

characteristic, complemented, Sylow

#2859

—

157

maximal, Sylow

C₂×#125

—

157

—

#389

—

157

—

#384

—

157

—

#382

—

157

—

C₄³

D₁₅₇

yes

characteristic

#125

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

—

C₄²

D₆₂₈

yes

—

C₄²

Dic₃₁₄

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂³×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂×D₆₂₈

yes

—

C₂×C₄

C₂×D₆₂₈

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

—

C₂×C₄

#14578954

yes

—

C₂²

C₂²×D₆₂₈

yes

characteristic

C₂²

C₂²×Dic₃₁₄

yes

characteristic

C₂²

C₂²×Dic₃₁₄

yes

characteristic

C₂²

C₂×#14578954

yes

—

C₄

—

C₄

—

C₄

—

628

—

C₄

#19097763

yes

complemented

100

C₄

#19097771

yes

—

101

C₂

C₂×#19097763

yes

—

102

C₂

C₂×#19097771

yes

characteristic

103

C₂

#23757823

yes

—

104

C₂

#23757830

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator, socle, prime radical, cosocle	C₂³×C₁₅₇	C₂⁴
Fitting	C₄³×C₁₅₇	C₂
hypercenter	C₄³	D₁₅₇
hypocenter	C₁₅₇	#2859
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₅₇

commutator, socle, prime radical, cosocle

C₂³×C₁₅₇

C₂⁴

Fitting

C₄³×C₁₅₇

C₂

hypercenter

C₄³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
45. C₄×C₁₅₇	#125	semidirect
52. C₁₅₇	#2859	semidirect
99. C₄	#19097763	semidirect

H ≅ G/N

type

45. C₄×C₁₅₇

#125

semidirect

52. C₁₅₇

#2859

semidirect

99. C₄

#19097763

semidirect

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

subgroup	index
C₂×#19097763	2
#23757171	2
#23757143	2
#23757139	2
C₄³×C₁₅₇	2
#2859	157

subgroup

index

C₄³×C₁₅₇

157

Subgroups — #34108499

Conjugacy classes of subgroups

Complements

Maximal subgroups