Group #34069228 — subgroups

Subgroups — #34069228

Order 19328.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×Dic₁₅₁⋊C₈	C₂	1	2	yes	maximal
2	#23752058	C₂	1	1	yes	maximal, characteristic
3	#23751967	C₂	1	1	yes	maximal, characteristic
4	#23751966	C₂	1	2	yes	maximal
5	#336×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	Dic₁₅₁⋊C₈	—	2	4	no	—
10	C₁₅₁⋊#95	C₄	1	2	yes	—
11	C₁₅₁⋊#95	C₄	1	2	yes	—
12	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₅₁	C₂×C₄	1	2	yes	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₂×C₁₅₁⋊C₈	—	2	2	no	—
20	C₄×Dic₁₅₁	—	2	2	no	—
21	C₄×Dic₁₅₁	D₄	1	2	yes	—
22	C₄×Dic₁₅₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₅₁	—	2	2	no	—
26	C₂×C₈×C₁₅₁	—	2	2	no	—
27	C₂×C₈×C₁₅₁	—	2	2	no	—
28	C₄²×C₁₅₁	—	2	2	no	—
29	C₂×Dic₁₅₁	—	2	2	no	—
30	C₂×Dic₁₅₁	—	2	2	no	—
31	C₂×Dic₁₅₁	—	2	1	no	—
32	C₂×Dic₁₅₁	—	2	1	no	—
33	C₂×Dic₁₅₁	—	2	2	no	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₂×Dic₁₅₁	C₄⋊C₄	1	4	yes	—
36	C₁₅₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	—	2	2	no	—
40	C₂×C₄×C₁₅₁	—	2	2	no	—
41	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	#41	1	2	yes	—
44	C₂×C₄×C₁₅₁	#41	1	2	yes	—
45	C₈×C₁₅₁	—	4	2	no	—
46	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₁	#117	1	1	yes	characteristic
48	C₂²×C₁₅₁	#117	1	1	yes	characteristic
49	C₂²×C₁₅₁	#131	1	2	yes	—
50	C₂²×C₁₅₁	#131	1	2	yes	—
51	C₄×C₁₅₁	—	4	2	no	—
52	C₄×C₁₅₁	#124	1	2	yes	—
53	C₄×C₁₅₁	#125	1	2	yes	—
54	Dic₁₅₁	—	2	4	no	—
55	Dic₁₅₁	—	4	1	no	—
56	Dic₁₅₁	—	4	1	no	—
57	C₂×C₁₅₁	#382	1	1	yes	characteristic
58	C₂×C₁₅₁	#424	1	2	yes	—
59	C₂×C₁₅₁	#433	1	2	yes	—
60	C₂×C₁₅₁	#433	1	1	yes	characteristic
61	C₂×C₁₅₁	#433	1	1	yes	characteristic
62	C₁₅₁	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	151	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	151	2	no	—
65	#376	—	151	1	no	—
66	#336	—	151	2	no	—
67	#336	—	151	1	no	—
68	#336	D₁₅₁	1	1	yes	characteristic
69	C₂²×C₈	—	151	2	no	—
70	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
71	C₂×C₄²	—	151	1	no	—
72	C₂×C₄²	—	151	1	no	—
73	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
74	C₄⋊C₈	—	302	4	no	—
75	#95	—	151	2	no	—
76	#95	—	151	2	no	—
77	C₂²×C₄	—	151	1	no	—
78	C₂²×C₄	—	151	1	no	—
79	C₂²×C₄	—	151	2	no	—
80	C₂²×C₄	C₄×D₁₅₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	302	2	no	—
86	C₂×C₈	—	302	2	no	—
87	C₂×C₈	—	302	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	302	2	no	—
90	C₄²	—	151	2	no	—
91	C₄²	—	151	2	no	—
92	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	302	2	no	—
97	C₂×C₄	—	302	2	no	—
98	C₂×C₄	—	302	1	no	—
99	C₂×C₄	—	302	1	no	—
100	C₂×C₄	—	302	2	no	—
101	C₂×C₄	—	302	2	no	—
102	C₂×C₄	—	151	4	no	—
103	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
105	C₂×C₄	#14559003	1	2	yes	—
106	C₂×C₄	#14559004	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	604	2	no	—
109	C₂²	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
110	C₂²	#18012695	1	1	yes	characteristic
111	C₂²	#18012696	1	1	yes	characteristic
112	C₂²	#18012745	1	2	yes	—
113	C₂²	#18012747	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	302	4	no	—
116	C₄	—	604	1	no	—
117	C₄	—	604	1	no	—
118	C₄	#18012725	1	2	yes	—
119	C₄	#18012729	1	2	yes	—
120	C₂	#23752076	1	1	yes	characteristic
121	C₂	#23752237	1	2	yes	—
122	C₂	#23752270	1	2	yes	—
123	C₂	#23752271	1	1	yes	characteristic
124	C₂	#23752274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₅₁⋊C₈

C₂

yes

maximal

#23752058

C₂

yes

maximal, characteristic

#23751967

C₂

yes

maximal, characteristic

#23751966

C₂

yes

maximal

#336×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

Dic₁₅₁⋊C₈

—

C₁₅₁⋊#95

C₄

yes

—

C₁₅₁⋊#95

C₄

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

Q₈

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₄⋊C₄

yes

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#131

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#124

yes

—

C₄×C₁₅₁

#125

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#382

yes

characteristic

C₂×C₁₅₁

#424

yes

—

C₂×C₁₅₁

#433

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1832

—

151

maximal, Sylow

C₂×C₄⋊C₈

—

151

—

#376

—

151

—

#336

—

151

—

#336

—

151

—

#336

D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

302

—

#95

—

151

—

#95

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄²

—

C₄²

—

302

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

100

C₂×C₄

—

302

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

151

—

103

C₂×C₄

D₄×D₁₅₁

yes

characteristic

104

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

105

C₂×C₄

#14559003

yes

—

106

C₂×C₄

#14559004

yes

—

107

C₈

—

108

C₈

—

604

—

109

C₂²

C₄⋊C₄×D₁₅₁

yes

characteristic

110

C₂²

#18012695

yes

characteristic

111

C₂²

#18012696

yes

characteristic

112

C₂²

#18012745

yes

—

113

C₂²

#18012747

yes

—

114

C₄

—

115

C₄

—

302

—

116

C₄

—

604

—

117

C₄

—

604

—

118

C₄

#18012725

yes

—

119

C₄

#18012729

yes

—

120

C₂

#23752076

yes

characteristic

121

C₂

#23752237

yes

—

122

C₂

#23752270

yes

—

123

C₂

#23752271

yes

characteristic

124

C₂

#23752274

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₁
commutator, socle, prime radical	C₂³×C₁₅₁	C₂²×C₄
Fitting	#336×C₁₅₁	C₂
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#336	D₁₅₁
hypocenter	C₁₅₁	#1832
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₁

commutator, socle, prime radical

C₂³×C₁₅₁

C₂²×C₄

Fitting

#336×C₁₅₁

C₂

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₅₁	#1832	semidirect

H ≅ G/N

type

62. C₁₅₁

#1832

semidirect

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

subgroup	index
C₂×Dic₁₅₁⋊C₈	2
#23752058	2
#23751967	2
#23751966	2
#336×C₁₅₁	2
#1832	151

subgroup

index

C₂×Dic₁₅₁⋊C₈

#336×C₁₅₁

151

Subgroups — #34069228

Conjugacy classes of subgroups

Complements

Maximal subgroups