Group #34079451 — subgroups

Subgroups — C₂×#23752582 #34079451

Order 19328.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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	#23752582	C₂	1	8	yes	maximal, direct product factor
3	C₂×#18012662	C₂	1	4	yes	maximal
4	C₂×#18012650	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	2	yes	—
7	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
8	#18012739	—	2	2	no	—
9	#18012739	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₅₁	—	2	2	no	—
11	#18012662	C₂²	1	16	yes	—
12	#18012650	C₂²	1	4	yes	—
13	#125×C₁₅₁	—	2	2	no	—
14	#125×C₁₅₁	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₀₂	—	2	2	no	—
19	C₂×Dic₃₀₂	—	2	8	no	—
20	C₂×Dic₃₀₂	C₂³	1	4	yes	—
21	C₂²×Dic₁₅₁	—	2	2	no	—
22	C₂×C₁₅₁⋊C₈	—	2	4	no	—
23	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
24	C₄⋊Dic₁₅₁	—	2	8	no	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₁	—	2	8	no	—
29	C₄⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
31	C₄²×C₁₅₁	D₄	1	2	yes	—
32	C₄²×C₁₅₁	C₂³	1	2	yes	—
33	Dic₃₀₂	—	2	8	no	—
34	Dic₃₀₂	—	4	4	no	—
35	C₂×Dic₁₅₁	—	2	4	no	—
36	C₂×Dic₁₅₁	—	4	4	no	—
37	C₁₅₁⋊C₈	—	2	4	no	—
38	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	—	2	4	no	—
40	C₂×C₄×C₁₅₁	—	2	4	no	—
41	C₂×C₄×C₁₅₁	—	4	4	no	—
42	C₂×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	4	yes	—
45	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
46	C₂²×C₁₅₁	#124	1	2	yes	—
47	C₂²×C₁₅₁	#137	1	4	yes	—
48	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₅₁	—	2	4	no	—
50	C₄×C₁₅₁	—	4	4	no	—
51	C₄×C₁₅₁	#124	1	2	yes	—
52	C₄×C₁₅₁	C₂×#41	1	2	yes	—
53	Dic₁₅₁	—	4	4	no	—
54	C₂×C₁₅₁	#491	1	4	yes	complemented
55	C₂×C₁₅₁	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₅₁	C₂×#137	1	2	yes	—
57	C₁₅₁	C₂×#491	1	1	yes	characteristic, complemented, Sylow
58	C₂×#491	—	151	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	—	151	1	no	—
60	C₂×#125	D₁₅₁	1	1	yes	characteristic
61	#491	—	151	8	no	—
62	C₂×#103	—	151	4	no	—
63	C₂×#97	—	151	1	no	—
64	C₂²×Q₈	—	151	2	no	—
65	C₂²×C₈	—	151	2	no	—
66	C₄⋊Q₈	—	302	2	no	—
67	C₄⋊Q₈	—	151	4	no	—
68	#125	—	2	2	no	—
69	#125	C₂×D₁₅₁	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	302	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
73	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
74	#103	—	151	16	no	—
75	#97	—	151	4	no	—
76	C₂×Q₈	—	302	2	no	—
77	C₂×Q₈	—	302	8	no	—
78	C₂×Q₈	—	151	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	302	2	no	—
81	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
83	C₂×C₈	—	302	4	no	—
84	C₂×C₈	—	151	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	302	8	no	—
88	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
89	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
90	C₄²	C₂²×D₁₅₁	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
92	Q₈	—	302	8	no	—
93	Q₈	—	604	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	302	4	no	—
98	C₂×C₄	—	604	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
101	C₂×C₄	#14559005	1	4	yes	—
102	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
103	C₈	—	302	4	no	—
104	C₂²	#18012728	1	2	yes	—
105	C₂²	#18012769	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	604	4	no	—
110	C₄	#18012728	1	2	yes	—
111	C₄	C₂×#14559005	1	2	yes	—
112	C₂	#23752582	1	4	yes	direct product factor
113	C₂	C₂×#18012728	1	1	yes	characteristic
114	C₂	C₂×#18012769	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic

#23752582

C₂

yes

maximal, direct product factor

C₂×#18012662

C₂

yes

maximal

C₂×#18012650

C₂

yes

maximal, characteristic

C₂×#125×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

#18012739

—

#18012739

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

—

#18012662

C₂²

yes

—

#18012650

C₂²

yes

—

#125×C₁₅₁

—

#125×C₁₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

C₂³

yes

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

—

Dic₃₀₂

—

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×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₂²×C₁₅₁

#124

yes

—

C₂²×C₁₅₁

#137

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#124

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₁₅₁

C₂×#491

yes

characteristic, complemented, Sylow

C₂×#491

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

C₂×#125

D₁₅₁

yes

characteristic

#491

—

151

—

C₂×#103

—

151

—

C₂×#97

—

151

—

C₂²×Q₈

—

151

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

151

—

#125

—

#125

C₂×D₁₅₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#103

—

151

—

#97

—

151

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₄²

C₂²×D₁₅₁

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

Q₈

—

302

—

Q₈

—

604

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

101

C₂×C₄

#14559005

yes

—

102

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

103

C₈

—

302

—

104

C₂²

#18012728

yes

—

105

C₂²

#18012769

yes

—

106

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

604

—

110

C₄

#18012728

yes

—

111

C₄

C₂×#14559005

yes

—

112

C₂

#23752582

yes

direct product factor

113

C₂

C₂×#18012728

yes

characteristic

114

C₂

C₂×#18012769

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	C₂×#125×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	C₂×#125	D₁₅₁
hypocenter	C₁₅₁	C₂×#491
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

C₂×#125×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #23752582	C₂	direct (N × H)
54. C₂×C₁₅₁	#491	semidirect
57. C₁₅₁	C₂×#491	semidirect
112. C₂	#23752582	direct (N × H)

H ≅ G/N

type

2. #23752582

C₂

direct (N × H)

54. C₂×C₁₅₁

#491

semidirect

57. C₁₅₁

C₂×#491

semidirect

112. C₂

#23752582

direct (N × H)

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

subgroup	index
C₂×#18012739	2
#23752582	2
C₂×#18012662	2
C₂×#18012650	2
C₂×#125×C₁₅₁	2
C₂×#491	151

subgroup

index

C₂×#125×C₁₅₁

151

Subgroups — C₂×#23752582 #34079451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23752582 #34079451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23752582 #34079451