Group #34068253 — subgroups

Subgroups — #34068253

Order 19328.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#23752928	C₂	1	1	yes	maximal, characteristic
2	#23752499	C₂	1	2	yes	maximal, complemented
3	#23751961	C₂	1	1	yes	maximal, characteristic, complemented
4	#23751941	C₂	1	2	yes	maximal, complemented
5	#494×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012728	—	2	2	no	—
7	#18012714	—	2	2	no	—
8	C₄×D₆₀₄	C₂²	1	2	yes	—
9	#18012697	—	2	2	no	—
10	#18012670	—	2	2	no	—
11	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
12	#18012659	—	2	2	no	—
13	C₂×Q₁₆×C₁₅₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×D₆₀₄	—	2	2	no	—
18	C₂×D₆₀₄	D₄	1	2	yes	—
19	C₂×C₄×D₁₅₁	—	2	2	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₅₁	D₄	1	2	yes	—
22	(C₂×D₁₅₁)⋊C₄	—	2	4	no	—
23	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
24	C₄×Dic₁₅₁	—	2	2	no	—
25	C₂×Q₈×C₁₅₁	—	2	1	no	—
26	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
27	Q₁₆×C₁₅₁	—	4	4	no	—
28	C₂×C₈×C₁₅₁	—	2	1	no	—
29	C₄⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄⋊C₄×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂²×D₁₅₁	—	2	2	no	—
33	D₆₀₄	—	2	2	no	—
34	D₆₀₄	—	2	2	no	—
35	D₆₀₄	—	8	2	no	—
36	C₂×Dic₁₅₁	—	2	2	no	—
37	C₄×D₁₅₁	—	4	2	no	—
38	C₄×D₁₅₁	—	4	2	no	—
39	C₁₅₁⋊C₈	—	4	2	no	—
40	Q₈×C₁₅₁	—	4	2	no	—
41	Q₈×C₁₅₁	—	4	2	no	—
42	C₂×C₄×C₁₅₁	—	2	2	no	—
43	C₂×C₄×C₁₅₁	—	4	1	no	—
44	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₅₁	—	2	2	no	—
47	C₂²×C₁₅₁	#120	1	1	yes	characteristic
48	C₂×D₁₅₁	—	4	2	no	—
49	C₂×D₁₅₁	—	4	2	no	—
50	C₂×D₁₅₁	—	4	2	no	—
51	C₄×C₁₅₁	—	2	1	no	—
52	C₄×C₁₅₁	—	4	2	no	—
53	C₄×C₁₅₁	—	8	1	no	—
54	C₄×C₁₅₁	#135	1	2	yes	—
55	C₄×C₁₅₁	#137	1	2	yes	—
56	Dic₁₅₁	—	4	2	no	—
57	C₂×C₁₅₁	#452	1	2	yes	—
58	C₂×C₁₅₁	#453	1	1	yes	characteristic
59	D₁₅₁	—	8	2	no	—
60	C₁₅₁	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	151	1	no	maximal, Sylow
62	#555	—	151	1	no	—
63	#494	D₁₅₁	1	1	yes	characteristic, complemented
64	#476	—	151	2	no	—
65	#333	—	151	1	no	—
66	D₄⋊C₈	—	151	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
70	#124	—	302	2	no	—
71	#121	—	302	2	no	—
72	C₄×D₄	—	151	2	no	—
73	#117	—	302	2	no	—
74	#106	—	302	2	no	—
75	C₄⋊C₈	—	151	2	no	—
76	#102	—	302	2	no	—
77	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
80	C₂×D₄	—	302	2	no	—
81	C₂×D₄	—	151	2	no	—
82	C₂²×C₄	—	302	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	302	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	151	2	no	—
89	C₂²⋊C₄	—	302	4	no	—
90	C₂²⋊C₄	—	302	2	no	—
91	C₄²	—	302	2	no	—
92	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
93	C₂³	—	302	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	302	2	no	—
97	D₄	—	302	2	no	—
98	D₄	—	1208	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	302	2	no	—
102	C₂×C₄	—	604	2	no	—
103	C₂×C₄	—	604	2	no	—
104	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	604	2	no	—
108	C₂²	—	604	2	no	—
109	C₂²	—	604	2	no	—
110	C₂²	—	604	2	no	—
111	C₂²	#18012707	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	604	2	no	—
116	C₄	#18012762	1	2	yes	—
117	C₄	#18012771	1	2	yes	—
118	C₂	—	1208	2	no	—
119	C₂	#23752373	1	2	yes	—
120	C₂	#23752379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752928

C₂

yes

maximal, characteristic

#23752499

C₂

yes

maximal, complemented

#23751961

C₂

yes

maximal, characteristic, complemented

#23751941

C₂

yes

maximal, complemented

#494×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#18012728

—

#18012714

—

C₄×D₆₀₄

C₂²

yes

—

#18012697

—

#18012670

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012659

—

C₂×Q₁₆×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₆₀₄

—

C₂×D₆₀₄

D₄

yes

—

C₂×C₄×D₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

D₄

yes

—

(C₂×D₁₅₁)⋊C₄

—

(C₂×D₁₅₁)⋊C₄

—

C₄×Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

—

Q₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₂×Dic₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

#120

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#135

yes

—

C₄×C₁₅₁

#137

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#452

yes

—

C₂×C₁₅₁

#453

yes

characteristic

D₁₅₁

—

C₁₅₁

#1634

yes

characteristic, complemented, Sylow

#1634

—

151

maximal, Sylow

#555

—

151

—

#494

D₁₅₁

yes

characteristic, complemented

#476

—

151

—

#333

—

151

—

D₄⋊C₈

—

151

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#124

—

302

—

#121

—

302

—

C₄×D₄

—

151

—

#117

—

302

—

#106

—

302

—

C₄⋊C₈

—

151

—

#102

—

302

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

C₂×D₄

—

302

—

C₂×D₄

—

151

—

C₂²×C₄

—

302

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₄²

—

302

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

302

—

Q₈

—

Q₈

—

D₄

—

302

—

D₄

—

302

—

D₄

—

1208

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

604

—

103

C₂×C₄

—

604

—

104

C₂×C₄

D₄×D₁₅₁

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

106

C₈

—

107

C₈

—

604

—

108

C₂²

—

604

—

109

C₂²

—

604

—

110

C₂²

—

604

—

111

C₂²

#18012707

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

604

—

116

C₄

#18012762

yes

—

117

C₄

#18012771

yes

—

118

C₂

—

1208

—

119

C₂

#23752373

yes

—

120

C₂

#23752379

yes

characteristic

	subgroup	G/H
center	C₂²	#18012707
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#494×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#120
prime radical	#23752928	C₂
hypercenter	#494	D₁₅₁
hypocenter	C₁₅₁	#1634
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#18012707

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#494×C₁₅₁

C₂

C₂²×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #23752499	C₂	semidirect
3. #23751961	C₂	semidirect
4. #23751941	C₂	semidirect
5. #494×C₁₅₁	C₂	semidirect
60. C₁₅₁	#1634	semidirect
63. #494	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752499

C₂

semidirect

3. #23751961

C₂

semidirect

4. #23751941

C₂

semidirect

5. #494×C₁₅₁

C₂

semidirect

60. C₁₅₁

#1634

semidirect

63. #494

D₁₅₁

semidirect

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

subgroup	index
#23752928	2
#23752499	2
#23751961	2
#23751941	2
#494×C₁₅₁	2
#1634	151

subgroup

index

#494×C₁₅₁

151

Subgroups — #34068253

Conjugacy classes of subgroups

Complements

Maximal subgroups