Group #34100289 — subgroups

Subgroups — #34100289

Order 20096.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#19097700	C₂	1	4	yes	maximal
3	#23757119	C₂	1	1	yes	maximal, characteristic
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	#19097700	—	2	8	no	—
9	#19097678	C₄	1	2	yes	—
10	#19097678	C₄	1	2	yes	—
11	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	—	2	2	no	—
14	C₂²×Dic₁₅₇	C₂×C₄	1	4	yes	—
15	C₂×#11929434	—	2	4	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₄×Dic₁₅₇	D₄	1	4	yes	—
18	C₄×Dic₁₅₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₅₇	—	2	2	no	—
23	C₂×C₈×C₁₅₇	—	2	4	no	—
24	C₄²×C₁₅₇	D₄	1	2	yes	—
25	C₄²×C₁₅₇	Q₈	1	2	yes	—
26	C₂×Dic₁₅₇	—	2	4	no	—
27	C₂×Dic₁₅₇	—	2	4	no	—
28	C₂×Dic₁₅₇	C₄⋊C₄	1	8	yes	—
29	#11929434	—	4	2	no	—
30	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₁₅₇	—	4	2	no	—
40	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₅₇	#131	1	2	yes	—
43	C₂²×C₁₅₇	#131	1	2	yes	—
44	C₄×C₁₅₇	—	2	2	no	—
45	C₄×C₁₅₇	—	2	2	no	—
46	C₄×C₁₅₇	#127	1	1	yes	characteristic
47	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
49	Dic₁₅₇	—	2	8	no	—
50	C₂×C₁₅₇	#384	1	1	yes	characteristic
51	C₂×C₁₅₇	#424	1	4	yes	—
52	C₂×C₁₅₇	#424	1	1	yes	characteristic
53	C₂×C₁₅₇	#424	1	1	yes	characteristic
54	C₁₅₇	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	157	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	157	4	no	—
57	C₂×C₄⋊C₈	—	157	1	no	—
58	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
59	#376	—	157	1	no	—
60	C₂²×C₈	—	157	2	no	—
61	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
62	C₂×C₄²	—	157	2	no	—
63	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	314	8	no	—
66	C₄⋊C₈	—	314	2	no	—
67	#95	—	157	2	no	—
68	#95	—	157	2	no	—
69	C₂²×C₄	—	157	4	no	—
70	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	314	4	no	—
76	C₂×C₈	—	314	2	no	—
77	C₄²	—	157	4	no	—
78	C₄²	—	157	4	no	—
79	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
80	C₄²	Dic₃₁₄	1	2	yes	—
81	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	314	4	no	—
85	C₂×C₄	—	314	4	no	—
86	C₂×C₄	—	157	8	no	—
87	C₂×C₄	#14578933	1	2	yes	—
88	C₂×C₄	#14578933	1	2	yes	—
89	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	628	2	no	—
95	C₂²	C₄⋊C₄×D₁₅₇	1	2	yes	—
96	C₂²	C₂×#14578933	1	1	yes	characteristic
97	C₂²	#19097777	1	2	yes	—
98	C₂²	#19097779	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	314	8	no	—
102	C₄	#19097767	1	1	yes	characteristic
103	C₄	#19097769	1	1	yes	characteristic
104	C₄	#19097770	1	2	yes	—
105	C₂	#23757142	1	1	yes	characteristic
106	C₂	#23757298	1	4	yes	—
107	C₂	#23757299	1	1	yes	characteristic
108	C₂	#23757301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

C₂×#19097700

C₂

yes

maximal

#23757119

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₄⋊#11929434

—

#19097700

—

#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

D₄

yes

—

C₄×Dic₁₅₇

Q₈

yes

—

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

C₄⋊C₄

yes

—

#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

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#131

yes

—

C₂²×C₁₅₇

#131

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#127

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#384

yes

characteristic

C₂×C₁₅₇

#424

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1843

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×C₄⋊C₈

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

#376

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

314

—

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

—

157

—

C₄²

—

157

—

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

—

157

—

C₂×C₄

#14578933

yes

—

C₂×C₄

#14578933

yes

—

C₂×C₄

D₄×D₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

Q₈×D₁₅₇

yes

—

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

C₈

—

C₈

—

628

—

C₂²

C₄⋊C₄×D₁₅₇

yes

—

C₂²

C₂×#14578933

yes

characteristic

C₂²

#19097777

yes

—

C₂²

#19097779

yes

—

C₄

—

100

C₄

—

101

C₄

—

314

—

102

C₄

#19097767

yes

characteristic

103

C₄

#19097769

yes

characteristic

104

C₄

#19097770

yes

—

105

C₂

#23757142

yes

characteristic

106

C₂

#23757298

yes

—

107

C₂

#23757299

yes

characteristic

108

C₂

#23757301

yes

characteristic

	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	#1843
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
54. C₁₅₇	#1843	semidirect

H ≅ G/N

type

54. C₁₅₇

#1843

semidirect

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

subgroup	index
C₂×C₄⋊#11929434	2
C₂×#19097700	2
#23757119	2
C₂×C₄⋊C₈×C₁₅₇	2
#1843	157

subgroup

index

C₂×C₄⋊C₈×C₁₅₇

157

Subgroups — #34100289

Conjugacy classes of subgroups

Complements

Maximal subgroups