Group #34070078 — subgroups

Subgroups — #34070078

Order 19328.

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

normal subgroups only

This group has 260 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×#18012731	C₂	1	1	yes	maximal, characteristic
2	C₂×#18012650	C₂	1	1	yes	maximal, characteristic
3	#23751974	C₂	1	4	yes	maximal
4	#384×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	#18012731	—	2	2	no	—
7	#18012731	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
9	C₂×Dic₁₅₁⋊C₄	—	2	2	no	—
10	#18012650	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₁	—	2	2	no	—
15	C₂×C₁₅₁⋊C₈	—	2	2	no	—
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₁₅₁	—	2	2	no	—
19	C₄⋊Dic₁₅₁	C₂×C₄	1	4	yes	—
20	Dic₁₅₁⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₁₅₁	—	2	2	no	—
22	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₅₁	—	2	4	no	—
25	C₄⋊C₄×C₁₅₁	—	4	4	no	—
26	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
27	C₄²×C₁₅₁	D₄	1	2	yes	—
28	C₂×Dic₁₅₁	—	2	4	no	—
29	C₂×Dic₁₅₁	—	4	2	no	—
30	C₂×Dic₁₅₁	—	4	2	no	—
31	C₁₅₁⋊C₈	—	2	4	no	—
32	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₁	—	2	2	no	—
34	C₂×C₄×C₁₅₁	—	2	2	no	—
35	C₂×C₄×C₁₅₁	—	4	2	no	—
36	C₂×C₄×C₁₅₁	—	4	4	no	—
37	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₅₁	#41	1	2	yes	—
42	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₅₁	#124	1	1	yes	characteristic
44	C₂²×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₅₁	#137	1	2	yes	—
47	C₄×C₁₅₁	—	2	4	no	—
48	C₄×C₁₅₁	—	8	2	no	—
49	C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
50	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₅₁	—	4	4	no	—
52	C₂×C₁₅₁	#391	1	1	yes	characteristic
53	C₂×C₁₅₁	#417	1	2	yes	—
54	C₂×C₁₅₁	#490	1	1	yes	characteristic
55	C₂×C₁₅₁	#491	1	1	yes	characteristic
56	C₂×C₁₅₁	#501	1	2	yes	—
57	C₁₅₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	151	1	no	maximal, Sylow
59	C₂×#125	—	151	1	no	—
60	C₂×#97	—	151	1	no	—
61	#384	D₁₅₁	1	1	yes	characteristic
62	#340	—	151	4	no	—
63	C₂²×C₈	—	151	2	no	—
64	#125	—	302	2	no	—
65	#125	—	151	4	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	302	2	no	—
68	C₂×C₄⋊C₄	—	151	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
70	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
71	#97	—	302	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	302	2	no	—
74	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
76	C₂×C₈	—	302	2	no	—
77	C₂×C₈	—	302	2	no	—
78	C₂×C₈	—	151	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	302	8	no	—
82	C₄⋊C₄	—	302	2	no	—
83	C₄⋊C₄	—	151	4	no	—
84	C₄²	C₄×D₁₅₁	1	2	yes	—
85	C₄²	D₆₀₄	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	302	4	no	—
92	C₂×C₄	—	604	2	no	—
93	C₂×C₄	—	604	2	no	—
94	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
96	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
98	C₂×C₄	#14559003	1	2	yes	—
99	C₈	—	302	4	no	—
100	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
101	C₂²	#18012726	1	1	yes	characteristic
102	C₂²	#18012737	1	1	yes	characteristic
103	C₂²	#18012765	1	2	yes	—
104	C₂²	#18012770	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	604	4	no	—
108	C₄	#18012704	1	2	yes	—
109	C₄	#18012719	1	2	yes	—
110	C₂	#23752117	1	1	yes	characteristic
111	C₂	#23752211	1	2	yes	—
112	C₂	#23752576	1	1	yes	characteristic
113	C₂	#23752580	1	1	yes	characteristic
114	C₂	#23752618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012731

C₂

yes

maximal, characteristic

C₂×#18012650

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#384×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

#18012731

—

#18012731

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×Dic₁₅₁⋊C₄

—

#18012650

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

Dic₁₅₁⋊C₄

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

—

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

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×Q₈

yes

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#391

yes

characteristic

C₂×C₁₅₁

#417

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₁₅₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

151

maximal, Sylow

C₂×#125

—

151

—

C₂×#97

—

151

—

#384

D₁₅₁

yes

characteristic

#340

—

151

—

C₂²×C₈

—

151

—

#125

—

302

—

#125

—

151

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#97

—

302

—

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

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

D₆₀₄

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

—

604

—

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂×D₆₀₄

yes

characteristic

C₂×C₄

Q₈×D₁₅₁

yes

—

C₂×C₄

#14559003

yes

—

C₈

—

302

—

100

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

101

C₂²

#18012726

yes

characteristic

102

C₂²

#18012737

yes

characteristic

103

C₂²

#18012765

yes

—

104

C₂²

#18012770

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

604

—

108

C₄

#18012704

yes

—

109

C₄

#18012719

yes

—

110

C₂

#23752117

yes

characteristic

111

C₂

#23752211

yes

—

112

C₂

#23752576

yes

characteristic

113

C₂

#23752580

yes

characteristic

114

C₂

#23752618

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#384×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#384	D₁₅₁
hypocenter	C₁₅₁	#1992
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#384×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 ◁ #384

N	H ≅ G/N	type
7. #18012731	C₄	semidirect
57. C₁₅₁	#1992	semidirect

H ≅ G/N

type

7. #18012731

C₄

semidirect

57. C₁₅₁

#1992

semidirect

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

subgroup	index
C₂×#18012731	2
C₂×#18012650	2
#23751974	2
#384×C₁₅₁	2
#1992	151

subgroup

index

#384×C₁₅₁

151

Subgroups — #34070078

Conjugacy classes of subgroups

Complements

Maximal subgroups