Group #34098019 — subgroups

Subgroups — #34098019

Order 20096.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	2	yes	maximal
2	#384×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11929434	—	2	2	no	—
4	C₄⋊#11929434	—	2	2	no	—
5	C₄⋊#11929434	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₅₇	C₄	1	2	yes	—
8	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂×#11929434	—	2	4	no	—
10	C₂×#11929434	—	4	2	no	—
11	C₂×#11929434	—	4	2	no	—
12	C₂²×C₄×C₁₅₇	—	2	2	no	—
13	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₅₇	—	4	2	no	—
16	C₄⋊C₄×C₁₅₇	—	4	2	no	—
17	C₄⋊C₄×C₁₅₇	—	4	2	no	—
18	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₅₇	Q₈	1	1	yes	characteristic
21	#11929434	—	4	4	no	—
22	C₂³×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	—	2	1	no	—
24	C₂×C₄×C₁₅₇	—	2	1	no	—
25	C₂×C₄×C₁₅₇	—	4	2	no	—
26	C₂×C₄×C₁₅₇	—	4	4	no	—
27	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
32	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
34	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	—
35	C₂²×C₁₅₇	#95	1	1	yes	characteristic
36	C₂²×C₁₅₇	#100	1	1	yes	characteristic
37	C₂²×C₁₅₇	#101	1	1	yes	characteristic
38	C₂²×C₁₅₇	#102	1	2	yes	—
39	C₂²×C₁₅₇	#103	1	2	yes	—
40	C₄×C₁₅₇	—	2	1	no	—
41	C₄×C₁₅₇	—	2	1	no	—
42	C₄×C₁₅₇	—	8	2	no	—
43	C₄×C₁₅₇	#102	1	2	yes	—
44	C₄×C₁₅₇	#103	1	2	yes	—
45	C₄×C₁₅₇	#106	1	2	yes	—
46	C₄×C₁₅₇	#107	1	2	yes	—
47	C₂×C₁₅₇	#331	1	1	yes	characteristic
48	C₂×C₁₅₇	#332	1	2	yes	—
49	C₂×C₁₅₇	#333	1	1	yes	characteristic
50	C₂×C₁₅₇	#340	1	2	yes	—
51	C₂×C₁₅₇	#343	1	1	yes	characteristic
52	C₁₅₇	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	157	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	157	2	no	—
55	#384	D₁₅₇	1	1	yes	characteristic
56	C₂²×C₈	—	314	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₅₇	1	2	yes	—
59	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
60	C₄⋊C₈	—	314	2	no	—
61	C₄⋊C₈	—	157	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
65	C₂×C₈	—	314	4	no	—
66	C₂×C₈	—	628	2	no	—
67	C₂×C₈	—	628	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₁₅₇	1	2	yes	—
72	C₄²	D₆₂₈	1	1	yes	characteristic
73	C₄²	Dic₃₁₄	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₅₇	1	1	yes	characteristic
80	C₂×C₄	#14578931	1	2	yes	—
81	C₂×C₄	#14578933	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
83	C₂×C₄	#14578939	1	2	yes	—
84	C₂×C₄	#14578941	1	2	yes	—
85	C₂×C₄	#14578942	1	2	yes	—
86	C₂×C₄	#14578944	1	2	yes	—
87	C₈	—	628	4	no	—
88	C₂²	#19097678	1	1	yes	characteristic
89	C₂²	#19097688	1	1	yes	characteristic
90	C₂²	#19097690	1	1	yes	characteristic
91	C₂²	#19097692	1	2	yes	—
92	C₂²	#19097695	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#19097691	1	2	yes	—
97	C₄	#19097694	1	2	yes	—
98	C₄	#19097702	1	2	yes	—
99	C₄	#19097704	1	2	yes	—
100	C₂	#23757017	1	1	yes	characteristic
101	C₂	#23757019	1	2	yes	—
102	C₂	#23757022	1	1	yes	characteristic
103	C₂	#23757035	1	2	yes	—
104	C₂	#23757042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11929434

C₂

yes

maximal

#384×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

—

C₄⋊#11929434

—

C₄⋊#11929434

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₄

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

—

—

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

Q₈

yes

characteristic

#11929434

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄²

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

D₈

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₂²×C₁₅₇

#103

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#102

yes

—

C₄×C₁₅₇

#103

yes

—

C₄×C₁₅₇

#106

yes

—

C₄×C₁₅₇

#107

yes

—

C₂×C₁₅₇

#331

yes

characteristic

C₂×C₁₅₇

#332

yes

—

C₂×C₁₅₇

#333

yes

characteristic

C₂×C₁₅₇

#340

yes

—

C₂×C₁₅₇

#343

yes

characteristic

C₁₅₇

#1296

yes

characteristic, complemented, Sylow

#1296

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

#384

D₁₅₇

yes

characteristic

C₂²×C₈

—

314

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂×C₈

—

314

—

C₂×C₈

—

628

—

C₂×C₈

—

628

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

D₆₂₈

yes

characteristic

C₄²

Dic₃₁₄

yes

characteristic

C₂³

C₂²⋊Dic₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₅₇

yes

characteristic

C₂×C₄

#14578931

yes

—

C₂×C₄

#14578933

yes

—

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

#14578939

yes

—

C₂×C₄

#14578941

yes

—

C₂×C₄

#14578942

yes

—

C₂×C₄

#14578944

yes

—

C₈

—

628

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#19097695

yes

—

C₄

—

C₄

—

C₄

—

C₄

#19097691

yes

—

C₄

#19097694

yes

—

C₄

#19097702

yes

—

C₄

#19097704

yes

—

100

C₂

#23757017

yes

characteristic

101

C₂

#23757019

yes

—

102

C₂

#23757022

yes

characteristic

103

C₂

#23757035

yes

—

104

C₂

#23757042

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₅₇
commutator	C₂×C₄×C₁₅₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	#384×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	#384	D₁₅₇
hypocenter	C₁₅₇	#1296
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₅₇

commutator

C₂×C₄×C₁₅₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₅₇

Fitting

#384×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₅₇

C₂²

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#11929434	C₄	semidirect
52. C₁₅₇	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#11929434

C₄

semidirect

52. C₁₅₇

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#11929434	2
#384×C₁₅₇	2
#1296	157

subgroup

index

C₂×C₄⋊#11929434

#384×C₁₅₇

#1296

157

Subgroups — #34098019

Conjugacy classes of subgroups

Complements

Maximal subgroups