Group #34099046 — subgroups

Subgroups — #34099046

Order 20096.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 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	#23757403	C₂	1	2	yes	maximal
2	C₄×Dic₆₂₈	C₂	1	1	yes	maximal, characteristic
3	#23757024	C₂	1	1	yes	maximal, characteristic
4	#23757005	C₂	1	2	yes	maximal
5	#322×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₂₈	C₄	1	1	yes	characteristic
7	C₄×Dic₃₁₄	C₂²	1	2	yes	—
8	#19097705	C₄	1	1	yes	characteristic
9	C₄⋊#11929434	C₂²	1	2	yes	—
10	#19097700	—	2	2	no	—
11	#19097696	C₄	1	2	yes	—
12	#19097681	—	2	2	no	—
13	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₄	C₂×C₄	1	2	yes	—
17	Dic₆₂₈	—	2	2	no	—
18	C₂×#11929434	—	2	2	no	—
19	C₄⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
20	#14578933	—	2	2	no	—
21	C₄×Dic₁₅₇	—	2	2	no	—
22	C₂×C₈×C₁₅₇	—	2	1	no	—
23	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	Dic₃₁₄	—	2	2	no	—
28	Dic₃₁₄	—	2	2	no	—
29	C₂×Dic₁₅₇	—	2	2	no	—
30	C₂×Dic₁₅₇	—	2	2	no	—
31	#11929434	—	4	2	no	—
32	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
35	C₈×C₁₅₇	—	2	1	no	—
36	C₈×C₁₅₇	—	2	2	no	—
37	C₈×C₁₅₇	—	4	1	no	—
38	C₈×C₁₅₇	M₁₆	1	2	yes	—
39	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₅₇	—	2	1	no	—
41	C₄×C₁₅₇	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₅₇	#131	1	1	yes	characteristic
43	C₄×C₁₅₇	#137	1	2	yes	—
44	Dic₁₅₇	—	4	2	no	—
45	Dic₁₅₇	—	4	2	no	—
46	C₂×C₁₅₇	#436	1	1	yes	characteristic
47	C₂×C₁₅₇	#441	1	1	yes	characteristic
48	C₂×C₁₅₇	#444	1	1	yes	characteristic
49	C₁₅₇	#1593	1	1	yes	characteristic, complemented, Sylow
50	#1593	—	157	1	no	maximal, Sylow
51	#446	—	157	2	no	—
52	C₄×Q₁₆	—	157	1	no	—
53	#334	—	157	1	no	—
54	Q₈⋊C₈	—	157	2	no	—
55	#322	D₁₅₇	1	1	yes	characteristic
56	C₂×Q₁₆	—	157	1	no	—
57	C₄×Q₈	—	157	2	no	—
58	#107	—	157	1	no	—
59	C₄⋊C₈	—	314	2	no	—
60	C₄⋊C₈	—	157	2	no	—
61	#103	—	157	2	no	—
62	#97	—	314	2	no	—
63	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
66	C₂×Q₈	—	157	2	no	—
67	Q₁₆	—	314	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	314	2	no	—
70	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
71	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
72	C₂×C₈	D₆₂₈	1	2	yes	—
73	C₄⋊C₄	—	314	2	no	—
74	C₄⋊C₄	—	157	2	no	—
75	C₄²	—	314	2	no	—
76	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
77	Q₈	—	314	2	no	—
78	Q₈	—	314	2	no	—
79	C₂×C₄	—	314	2	no	—
80	C₂×C₄	—	314	2	no	—
81	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
82	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
83	C₂×C₄	#14578954	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	628	2	no	—
88	C₈	#14578937	1	2	yes	—
89	C₂²	C₄×D₆₂₈	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	628	2	no	—
92	C₄	—	628	2	no	—
93	C₄	C₂×#14578937	1	1	yes	characteristic
94	C₄	#19097777	1	1	yes	characteristic
95	C₄	#19097804	1	2	yes	—
96	C₂	#23757352	1	1	yes	characteristic
97	C₂	#23757381	1	1	yes	characteristic
98	C₂	#23757396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757403

C₂

yes

maximal

C₄×Dic₆₂₈

C₂

yes

maximal, characteristic

#23757024

C₂

yes

maximal, characteristic

#23757005

C₂

yes

maximal

#322×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₆₂₈

C₄

yes

characteristic

C₄×Dic₃₁₄

C₂²

yes

—

#19097705

C₄

yes

characteristic

C₄⋊#11929434

C₂²

yes

—

#19097700

—

#19097696

C₄

yes

—

#19097681

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

C₂×C₄

yes

—

Dic₆₂₈

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

#14578933

—

C₄×Dic₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

M₁₆

yes

—

C₂²×C₁₅₇

C₄×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×M₁₆

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1593

—

157

maximal, Sylow

#446

—

157

—

C₄×Q₁₆

—

157

—

#334

—

157

—

Q₈⋊C₈

—

157

—

#322

D₁₅₇

yes

characteristic

C₂×Q₁₆

—

157

—

C₄×Q₈

—

157

—

#107

—

157

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

#103

—

157

—

#97

—

314

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

157

—

Q₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

C₄×D₁₅₇

yes

characteristic

C₂×C₈

C₄×D₁₅₇

yes

characteristic

C₂×C₈

D₆₂₈

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₄²

—

314

—

C₄²

C₂²×D₁₅₇

yes

characteristic

Q₈

—

314

—

Q₈

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

#14578954

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

628

—

C₈

#14578937

yes

—

C₂²

C₄×D₆₂₈

yes

characteristic

C₄

—

C₄

—

628

—

C₄

—

628

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#322×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₄×D₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	#322	D₁₅₇
hypocenter	C₁₅₇	#1593
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₆₂₈

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#322×C₁₅₇

C₂

socle, prime radical

C₂²×C₁₅₇

C₄×D₄

cosocle

C₄²×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
49. C₁₅₇	#1593	semidirect

H ≅ G/N

type

49. C₁₅₇

#1593

semidirect

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

subgroup	index
#23757403	2
C₄×Dic₆₂₈	2
#23757024	2
#23757005	2
#322×C₁₅₇	2
#1593	157

subgroup

index

C₄×Dic₆₂₈

#322×C₁₅₇

157

Subgroups — #34099046

Conjugacy classes of subgroups

Complements

Maximal subgroups