Group #34100255 — subgroups

Subgroups — #34100255

Order 20096.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
2	#23757119	C₂	1	1	yes	maximal, characteristic
3	#23757027	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
7	C₄⋊#11929434	—	2	2	no	—
8	#19097678	C₄	1	2	yes	—
9	#19097678	C₄	1	2	yes	—
10	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₅₇	—	2	2	no	—
13	C₂²×Dic₁₅₇	C₂×C₄	1	4	yes	—
14	C₂×#11929434	—	2	2	no	—
15	C₂×#11929434	—	2	4	no	—
16	C₄×Dic₁₅₇	—	2	2	no	—
17	C₄×Dic₁₅₇	—	2	2	no	—
18	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₅₇	—	2	2	no	—
22	C₂×C₈×C₁₅₇	—	2	4	no	—
23	C₄²×C₁₅₇	D₄	1	2	yes	—
24	C₄²×C₁₅₇	Q₈	1	2	yes	—
25	C₂×Dic₁₅₇	—	2	4	no	—
26	C₂×Dic₁₅₇	—	2	4	no	—
27	C₂×Dic₁₅₇	—	2	4	no	—
28	#11929434	—	4	2	no	—
29	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	—	2	2	no	—
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	#41	1	2	yes	—
37	C₂×C₄×C₁₅₇	#41	1	2	yes	—
38	C₈×C₁₅₇	—	4	2	no	—
39	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₅₇	#117	1	2	yes	—
41	C₂²×C₁₅₇	#131	1	2	yes	—
42	C₂²×C₁₅₇	#131	1	2	yes	—
43	C₄×C₁₅₇	—	2	2	no	—
44	C₄×C₁₅₇	—	2	2	no	—
45	C₄×C₁₅₇	#124	1	1	yes	characteristic
46	C₄×C₁₅₇	#124	1	1	yes	characteristic
47	C₄×C₁₅₇	#125	1	2	yes	—
48	Dic₁₅₇	—	4	4	no	—
49	C₂×C₁₅₇	#382	1	1	yes	characteristic
50	C₂×C₁₅₇	#424	1	1	yes	characteristic
51	C₂×C₁₅₇	#424	1	1	yes	characteristic
52	C₂×C₁₅₇	#433	1	4	yes	—
53	C₁₅₇	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	157	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	157	1	no	—
56	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
57	#376	—	157	1	no	—
58	#336	—	157	4	no	—
59	C₂²×C₈	—	157	2	no	—
60	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
61	C₂×C₄²	—	157	2	no	—
62	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	314	2	no	—
65	#95	—	157	2	no	—
66	#95	—	157	2	no	—
67	C₂²×C₄	—	157	4	no	—
68	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	314	2	no	—
74	C₂×C₈	—	314	4	no	—
75	C₄²	—	314	2	no	—
76	C₄²	—	314	2	no	—
77	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
78	C₄²	Dic₃₁₄	1	2	yes	—
79	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	314	4	no	—
83	C₂×C₄	—	314	4	no	—
84	C₂×C₄	—	314	4	no	—
85	C₂×C₄	#14578933	1	2	yes	—
86	C₂×C₄	#14578933	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
89	C₂×C₄	#14578953	1	2	yes	—
90	C₂×C₄	#14578955	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	628	2	no	—
93	C₂²	C₂×#14578933	1	1	yes	characteristic
94	C₂²	#19097729	1	2	yes	—
95	C₂²	#19097777	1	2	yes	—
96	C₂²	#19097779	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	628	4	no	—
100	C₄	#19097758	1	1	yes	characteristic
101	C₄	#19097760	1	1	yes	characteristic
102	C₄	#19097762	1	2	yes	—
103	C₂	#23757138	1	1	yes	characteristic
104	C₂	#23757299	1	1	yes	characteristic
105	C₂	#23757301	1	1	yes	characteristic
106	C₂	#23757333	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

#23757119

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₄⋊#11929434

—

#19097678

C₄

yes

—

#19097678

C₄

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#117

yes

—

C₂²×C₁₅₇

#131

yes

—

C₂²×C₁₅₇

#131

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₁₅₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

#376

—

157

—

#336

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

314

—

#95

—

157

—

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

#14578933

yes

—

C₂×C₄

#14578933

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

C₂×C₄

#14578953

yes

—

C₂×C₄

#14578955

yes

—

C₈

—

C₈

—

628

—

C₂²

C₂×#14578933

yes

characteristic

C₂²

#19097729

yes

—

C₂²

#19097777

yes

—

C₂²

#19097779

yes

—

C₄

—

C₄

—

C₄

—

628

—

100

C₄

#19097758

yes

characteristic

101

C₄

#19097760

yes

characteristic

102

C₄

#19097762

yes

—

103

C₂

#23757138

yes

characteristic

104

C₂

#23757299

yes

characteristic

105

C₂

#23757301

yes

characteristic

106

C₂

#23757333

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator, socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1832
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₇

commutator, socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₅₇

C₂

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₁₅₇	#1832	semidirect

H ≅ G/N

type

53. C₁₅₇

#1832

semidirect

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

subgroup	index
C₂×C₄⋊#11929434	2
#23757119	2
#23757027	2
C₂×C₄⋊C₈×C₁₅₇	2
#1832	157

subgroup

index

C₂×C₄⋊C₈×C₁₅₇

157

Subgroups — #34100255

Conjugacy classes of subgroups

Complements

Maximal subgroups