Group #34070082 — subgroups

Subgroups — #34070082

Order 19328.

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

normal subgroups only

This group has 276 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×#18012739	C₂	1	1	yes	maximal, characteristic
2	C₂×#18012650	C₂	1	1	yes	maximal, characteristic
3	#23751974	C₂	1	4	yes	maximal
4	#382×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₀₂	—	2	1	no	—
6	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
7	#18012739	—	2	2	no	—
8	#18012739	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₅₁	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
11	#18012650	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
14	#95×C₁₅₁	—	2	1	no	—
15	#95×C₁₅₁	—	2	1	no	—
16	C₂×Dic₃₀₂	—	2	4	no	—
17	C₂×Dic₃₀₂	—	4	4	no	—
18	C₂²×Dic₁₅₁	—	2	2	no	—
19	C₂×C₁₅₁⋊C₈	—	2	2	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₂×C₁₅₁⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₅₁	—	2	4	no	—
23	C₄⋊Dic₁₅₁	—	2	2	no	—
24	C₄⋊Dic₁₅₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₅₁	—	2	4	no	—
30	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
31	C₄²×C₁₅₁	D₄	1	2	yes	—
32	Dic₃₀₂	—	4	8	no	—
33	C₂×Dic₁₅₁	—	2	4	no	—
34	C₂×Dic₁₅₁	—	4	4	no	—
35	C₁₅₁⋊C₈	—	2	4	no	—
36	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	—	4	2	no	—
40	C₂×C₄×C₁₅₁	—	4	2	no	—
41	C₂×C₄×C₁₅₁	—	4	2	no	—
42	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₅₁	#41	1	2	yes	—
48	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₅₁	#124	1	1	yes	characteristic
50	C₂²×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₅₁	#137	1	2	yes	—
53	C₄×C₁₅₁	—	2	2	no	—
54	C₄×C₁₅₁	—	2	2	no	—
55	C₄×C₁₅₁	—	8	2	no	—
56	C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
57	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
58	Dic₁₅₁	—	4	4	no	—
59	C₂×C₁₅₁	#391	1	1	yes	characteristic
60	C₂×C₁₅₁	#418	1	1	yes	characteristic
61	C₂×C₁₅₁	#419	1	1	yes	characteristic
62	C₂×C₁₅₁	#489	1	2	yes	—
63	C₂×C₁₅₁	#501	1	2	yes	—
64	C₁₅₁	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	151	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	—	151	1	no	—
67	C₂×#97	—	151	1	no	—
68	#382	D₁₅₁	1	1	yes	characteristic
69	#340	—	151	4	no	—
70	C₂²×Q₈	—	302	1	no	—
71	C₂²×C₈	—	151	2	no	—
72	C₄⋊Q₈	—	302	2	no	—
73	C₄⋊Q₈	—	151	4	no	—
74	C₂×C₄⋊C₄	—	302	1	no	—
75	C₂×C₄⋊C₄	—	151	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
77	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
78	#97	—	302	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	302	4	no	—
82	C₂×Q₈	—	604	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	302	2	no	—
85	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
88	C₂×C₈	—	302	2	no	—
89	C₂×C₈	—	302	2	no	—
90	C₂×C₈	—	151	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	302	4	no	—
93	C₄⋊C₄	—	302	2	no	—
94	C₄⋊C₄	—	151	4	no	—
95	C₄²	C₄×D₁₅₁	1	2	yes	—
96	C₄²	D₆₀₄	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
98	Q₈	—	604	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	302	4	no	—
105	C₂×C₄	—	604	4	no	—
106	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
111	C₂×C₄	#14559003	1	2	yes	—
112	C₈	—	302	4	no	—
113	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
114	C₂²	#18012726	1	1	yes	characteristic
115	C₂²	#18012737	1	1	yes	characteristic
116	C₂²	#18012764	1	2	yes	—
117	C₂²	#18012769	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	604	4	no	—
122	C₄	#18012704	1	2	yes	—
123	C₄	#18012719	1	2	yes	—
124	C₂	#23752117	1	1	yes	characteristic
125	C₂	#23752216	1	1	yes	characteristic
126	C₂	#23752220	1	1	yes	characteristic
127	C₂	#23752571	1	2	yes	—
128	C₂	#23752622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic

C₂×#18012650

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#382×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

—

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

#18012739

—

#18012739

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

#18012650

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

#95×C₁₅₁

—

#95×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

#124

yes

characteristic

C₂²×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₁

#137

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#501

yes

—

C₁₅₁

#1993

yes

characteristic, complemented, Sylow

#1993

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

C₂×#97

—

151

—

#382

D₁₅₁

yes

characteristic

#340

—

151

—

C₂²×Q₈

—

302

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

151

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#97

—

302

—

#95

—

#95

—

C₂×Q₈

—

302

—

C₂×Q₈

—

604

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

D₆₀₄

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

Q₈

—

604

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

302

—

105

C₂×C₄

—

604

—

106

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

109

C₂×C₄

C₂×D₆₀₄

yes

characteristic

110

C₂×C₄

Q₈×D₁₅₁

yes

—

111

C₂×C₄

#14559003

yes

—

112

C₈

—

302

—

113

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

114

C₂²

#18012726

yes

characteristic

115

C₂²

#18012737

yes

characteristic

116

C₂²

#18012764

yes

—

117

C₂²

#18012769

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

604

—

122

C₄

#18012704

yes

—

123

C₄

#18012719

yes

—

124

C₂

#23752117

yes

characteristic

125

C₂

#23752216

yes

characteristic

126

C₂

#23752220

yes

characteristic

127

C₂

#23752571

yes

—

128

C₂

#23752622

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#382×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#382	D₁₅₁
hypocenter	C₁₅₁	#1993
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#382×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #18012739	C₄	semidirect
21. C₂×C₁₅₁⋊C₈	Q₈	semidirect
64. C₁₅₁	#1993	semidirect

H ≅ G/N

type

8. #18012739

C₄

semidirect

21. C₂×C₁₅₁⋊C₈

Q₈

semidirect

64. C₁₅₁

#1993

semidirect

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

subgroup	index
C₂×#18012739	2
C₂×#18012650	2
#23751974	2
#382×C₁₅₁	2
#1993	151

subgroup

index

#382×C₁₅₁

151

Subgroups — #34070082

Conjugacy classes of subgroups

Complements

Maximal subgroups