Group #34098860 — subgroups

Subgroups — #34098860

Order 20096.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#23757329	C₂	1	2	yes	maximal
3	#23757022	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757021	C₂	1	2	yes	maximal, complemented
5	#333×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₁₅₇	C₄	1	2	yes	—
7	#19097771	C₂²	1	1	yes	characteristic
8	#19097770	—	2	2	no	—
9	#19097769	C₄	1	2	yes	—
10	C₄⋊C₄×D₁₅₇	—	2	2	no	—
11	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊#11929434	C₂²	1	2	yes	—
13	#19097683	—	2	2	no	—
14	#19097681	—	2	2	no	—
15	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
17	C₂×Dic₃₁₄	—	2	2	no	—
18	C₂×Dic₃₁₄	C₂×C₄	1	2	yes	—
19	Q₈×D₁₅₇	—	4	4	no	—
20	C₂×C₄×D₁₅₇	—	2	2	no	—
21	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₅₇	D₄	1	2	yes	—
23	C₂×#11929434	—	2	2	no	—
24	C₄⋊Dic₁₅₇	—	2	2	no	—
25	#14578933	—	2	4	no	—
26	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₅₇	D₄	1	2	yes	—
28	C₂×Q₈×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₄⋊C₄×C₁₅₇	—	2	2	no	—
31	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₃₁₄	—	4	4	no	—
34	Dic₃₁₄	—	4	2	no	—
35	C₂×Dic₁₅₇	—	2	2	no	—
36	C₂×Dic₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₅₇	C₂²⋊C₄	1	2	yes	—
38	C₄×D₁₅₇	—	2	2	no	—
39	C₄×D₁₅₇	—	2	2	no	—
40	C₄×D₁₅₇	—	2	2	no	—
41	C₄×D₁₅₇	—	4	4	no	—
42	#11929434	—	4	2	no	—
43	Q₈×C₁₅₇	—	4	2	no	—
44	C₂×C₄×C₁₅₇	—	2	2	no	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
47	C₈×C₁₅₇	—	4	2	no	—
48	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₅₇	—	2	2	no	—
50	C₂×D₁₅₇	#100	1	2	yes	—
51	C₄×C₁₅₇	—	2	1	no	—
52	C₄×C₁₅₇	—	4	2	no	—
53	C₄×C₁₅₇	#137	1	2	yes	—
54	C₄×C₁₅₇	#137	1	2	yes	—
55	Dic₁₅₇	—	2	1	no	—
56	Dic₁₅₇	—	2	2	no	—
57	Dic₁₅₇	—	4	2	no	—
58	C₂×C₁₅₇	C₂×#100	1	1	yes	characteristic
59	C₂×C₁₅₇	#419	1	2	yes	—
60	D₁₅₇	—	2	2	no	—
61	C₁₅₇	#1558	1	1	yes	characteristic, complemented, Sylow
62	#1558	—	157	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	157	1	no	—
64	#432	—	157	2	no	—
65	#333	—	157	1	no	—
66	#333	—	157	2	no	—
67	#333	D₁₅₇	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	157	2	no	—
69	C₄⋊Q₈	—	314	2	no	—
70	C₄⋊Q₈	—	157	2	no	—
71	C₄⋊Q₈	—	157	1	no	—
72	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	314	2	no	—
74	C₂×C₄²	—	157	1	no	—
75	C₄⋊C₈	—	157	2	no	—
76	C₄⋊C₈	C₂×D₁₅₇	1	2	yes	—
77	C₂²⋊C₈	—	314	2	no	—
78	#97	—	314	2	no	—
79	C₂×Q₈	—	314	2	no	—
80	C₂×Q₈	—	628	4	no	—
81	C₂×Q₈	—	157	2	no	—
82	C₂×Q₈	C₄×D₁₅₇	1	2	yes	—
83	C₂²×C₄	—	314	2	no	—
84	C₂²×C₄	—	157	1	no	—
85	C₂²×C₄	—	157	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	314	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	314	4	no	—
90	C₄⋊C₄	—	314	2	no	—
91	C₄²	—	157	1	no	—
92	C₄²	—	157	2	no	—
93	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
94	C₂³	—	157	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	628	2	no	—
97	Q₈	—	628	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	314	2	no	—
100	C₂×C₄	—	314	2	no	—
101	C₂×C₄	—	314	2	no	—
102	C₂×C₄	—	314	2	no	—
103	C₂×C₄	—	628	4	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	—	157	1	no	—
106	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	628	2	no	—
110	C₂²	—	314	2	no	—
111	C₂²	—	157	2	no	—
112	C₂²	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	314	1	no	—
116	C₄	—	314	2	no	—
117	C₄	—	628	2	no	—
118	C₄	#19097800	1	2	yes	—
119	C₄	#19097803	1	2	yes	—
120	C₂	—	314	2	no	—
121	C₂	#100×D₁₅₇	1	1	yes	characteristic
122	C₂	#23757280	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₁₅₇

C₂

yes

maximal, characteristic

#23757329

C₂

yes

maximal

#23757022

C₂

yes

maximal, characteristic, complemented

#23757021

C₂

yes

maximal, complemented

#333×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₅₇

C₄

yes

—

#19097771

C₂²

yes

characteristic

#19097770

—

#19097769

C₄

yes

—

C₄⋊C₄×D₁₅₇

—

C₄²×D₁₅₇

C₂²

yes

characteristic

C₄⋊#11929434

C₂²

yes

—

#19097683

—

#19097681

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂×C₄

yes

—

Q₈×D₁₅₇

—

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₅₇

D₄

yes

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

#14578933

—

C₄×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×Dic₁₅₇

D₄

yes

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

C₂²⋊C₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅₇

C₂²⋊C₄

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

#100

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#137

yes

—

C₄×C₁₅₇

#137

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

C₂×#100

yes

characteristic

C₂×C₁₅₇

#419

yes

—

D₁₅₇

—

C₁₅₇

#1558

yes

characteristic, complemented, Sylow

#1558

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

#432

—

157

—

#333

—

157

—

#333

—

157

—

#333

D₁₅₇

yes

characteristic, complemented

C₂²×Q₈

—

157

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

C₂×C₄⋊C₄

—

314

—

C₂×C₄²

—

157

—

C₄⋊C₈

—

157

—

C₄⋊C₈

C₂×D₁₅₇

yes

—

C₂²⋊C₈

—

314

—

#97

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

628

—

C₂×Q₈

—

157

—

C₂×Q₈

C₄×D₁₅₇

yes

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂×C₈

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

—

157

—

Q₈

—

Q₈

—

628

—

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

628

—

104

C₂×C₄

—

157

—

105

C₂×C₄

—

157

—

106

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

107

C₂×C₄

D₄×D₁₅₇

yes

—

108

C₈

—

109

C₈

—

628

—

110

C₂²

—

314

—

111

C₂²

—

157

—

112

C₂²

C₂²⋊C₄×D₁₅₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

314

—

116

C₄

—

314

—

117

C₄

—

628

—

118

C₄

#19097800

yes

—

119

C₄

#19097803

yes

—

120

C₂

—

314

—

121

C₂

#100×D₁₅₇

yes

characteristic

122

C₂

#23757280

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#333×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₅₇	C₂²⋊C₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	#333	D₁₅₇
hypocenter	C₁₅₇	#1558
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#333×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₅₇

C₂²⋊C₄

cosocle

C₄²×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #23757022	C₂	semidirect
4. #23757021	C₂	semidirect
5. #333×C₁₅₇	C₂	semidirect
61. C₁₅₇	#1558	semidirect
67. #333	D₁₅₇	semidirect

H ≅ G/N

type

3. #23757022

C₂

semidirect

4. #23757021

C₂

semidirect

5. #333×C₁₅₇

C₂

semidirect

61. C₁₅₇

#1558

semidirect

67. #333

D₁₅₇

semidirect

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

subgroup	index
C₄⋊Q₈×D₁₅₇	2
#23757329	2
#23757022	2
#23757021	2
#333×C₁₅₇	2
#1558	157

subgroup

index

C₄⋊Q₈×D₁₅₇

#333×C₁₅₇

157

Subgroups — #34098860

Conjugacy classes of subgroups

Complements

Maximal subgroups