Group #34069931 — subgroups

Subgroups — #34069931

Order 19328.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#18012663	C₂	1	4	yes	maximal
2	C₂×#18012650	C₂	1	1	yes	maximal, characteristic
3	#23752078	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
7	#18012663	—	2	8	no	—
8	#18012650	—	2	2	no	—
9	C₁₅₁⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₁₅₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₅₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
14	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₁	—	2	2	no	—
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₅₁	—	2	4	no	—
20	C₂×Q₈×C₁₅₁	—	2	2	no	—
21	C₂×Q₈×C₁₅₁	—	2	4	no	—
22	C₂×Q₈×C₁₅₁	—	2	4	no	—
23	C₂×Q₈×C₁₅₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₅₁	—	2	2	no	—
25	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₅₁	—	2	8	no	—
28	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₅₁	D₄	1	2	yes	—
30	C₂×Dic₁₅₁	—	4	2	no	—
31	C₂×Dic₁₅₁	—	4	4	no	—
32	C₁₅₁⋊C₈	—	2	4	no	—
33	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₅₁	—	2	8	no	—
35	Q₈×C₁₅₁	—	4	4	no	—
36	C₂×C₄×C₁₅₁	—	2	4	no	—
37	C₂×C₄×C₁₅₁	—	2	4	no	—
38	C₂×C₄×C₁₅₁	—	4	2	no	—
39	C₂×C₄×C₁₅₁	—	4	2	no	—
40	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	2	yes	—
45	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₅₁	#124	1	1	yes	characteristic
47	C₂²×C₁₅₁	#127	1	1	yes	characteristic
48	C₂²×C₁₅₁	#137	1	4	yes	—
49	C₄×C₁₅₁	—	2	4	no	—
50	C₄×C₁₅₁	—	4	4	no	—
51	C₄×C₁₅₁	C₄×D₄	1	2	yes	—
52	C₄×C₁₅₁	#121	1	2	yes	—
53	Dic₁₅₁	—	8	2	no	—
54	C₂×C₁₅₁	#390	1	1	yes	characteristic
55	C₂×C₁₅₁	#419	1	2	yes	—
56	C₂×C₁₅₁	#491	1	2	yes	—
57	C₂×C₁₅₁	#497	1	2	yes	—
58	C₁₅₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	151	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₁₅₁	1	1	yes	characteristic
61	C₂×#103	—	151	4	no	—
62	C₂×#97	—	151	1	no	—
63	#382	—	151	1	no	—
64	C₂²×Q₈	C₂×D₁₅₁	1	2	yes	—
65	C₂²×C₈	—	151	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₅₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	151	2	no	—
70	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
71	#103	—	302	8	no	—
72	#97	—	302	2	no	—
73	#95	—	302	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₅₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	302	2	no	—
80	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
82	C₂×C₈	—	302	2	no	—
83	C₂×C₈	—	302	2	no	—
84	C₂×C₈	—	151	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	302	4	no	—
87	C₄²	C₂×Dic₁₅₁	1	2	yes	—
88	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	604	2	no	—
97	C₂×C₄	—	604	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₅₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₅₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
102	C₂×C₄	#14559003	1	2	yes	—
103	C₈	—	302	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₅₁	1	1	yes	characteristic
105	C₂²	#18012726	1	1	yes	characteristic
106	C₂²	#18012735	1	1	yes	characteristic
107	C₂²	#18012770	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1208	2	no	—
111	C₄	D₄×Dic₁₅₁	1	2	yes	—
112	C₄	#18012710	1	2	yes	—
113	C₂	#23752115	1	1	yes	characteristic
114	C₂	#23752221	1	2	yes	—
115	C₂	#23752580	1	2	yes	—
116	C₂	#23752604	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012663

C₂

yes

maximal

C₂×#18012650

C₂

yes

maximal, characteristic

#23752078

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

#18012663

—

#18012650

—

C₁₅₁⋊#95

—

C₂²×Q₈×C₁₅₁

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄×D₄

yes

—

C₄×C₁₅₁

#121

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#390

yes

characteristic

C₂×C₁₅₁

#419

yes

—

C₂×C₁₅₁

#491

yes

—

C₂×C₁₅₁

#497

yes

—

C₁₅₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

151

maximal, Sylow

C₂×C₄⋊Q₈

D₁₅₁

yes

characteristic

C₂×#103

—

151

—

C₂×#97

—

151

—

#382

—

151

—

C₂²×Q₈

C₂×D₁₅₁

yes

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₅₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#103

—

302

—

#97

—

302

—

#95

—

302

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₅₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄²

C₂×Dic₁₅₁

yes

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

604

—

C₂×C₄

—

604

—

C₂×C₄

C₂²⋊Dic₁₅₁

yes

—

C₂×C₄

C₂²×Dic₁₅₁

yes

characteristic

100

C₂×C₄

D₄×D₁₅₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

102

C₂×C₄

#14559003

yes

—

103

C₈

—

302

—

104

C₂²

C₂×C₂²⋊Dic₁₅₁

yes

characteristic

105

C₂²

#18012726

yes

characteristic

106

C₂²

#18012735

yes

characteristic

107

C₂²

#18012770

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1208

—

111

C₄

D₄×Dic₁₅₁

yes

—

112

C₄

#18012710

yes

—

113

C₂

#23752115

yes

characteristic

114

C₂

#23752221

yes

—

115

C₂

#23752580

yes

—

116

C₂

#23752604

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×C₄⋊Q₈×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅₁
hypocenter	C₁₅₁	#1960
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×C₄⋊Q₈×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₅₁

hypocenter

C₁₅₁

#1960

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₅₁	C₄	semidirect
58. C₁₅₁	#1960	semidirect
67. C₄⋊Q₈	Dic₁₅₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₅₁

C₄

semidirect

58. C₁₅₁

#1960

semidirect

67. C₄⋊Q₈

Dic₁₅₁

semidirect

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

subgroup	index
C₂×#18012663	2
C₂×#18012650	2
#23752078	2
C₂×C₄⋊Q₈×C₁₅₁	2
#1960	151

subgroup

index

C₂×C₄⋊Q₈×C₁₅₁

151

Subgroups — #34069931

Conjugacy classes of subgroups

Complements

Maximal subgroups