Group #34068705 — subgroups

Subgroups — #34068705

Order 19328.

180 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23752615	C₂	1	1	yes	maximal, characteristic
2	#23752610	C₂	1	1	yes	maximal, characteristic
3	#23752591	C₂	1	1	yes	maximal, characteristic
4	#23752564	C₂	1	1	yes	maximal, characteristic
5	#23752562	C₂	1	1	yes	maximal, characteristic
6	C₈×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14558994	—	2	2	no	—
9	#18012739	C₂²	1	1	yes	characteristic
10	#18012739	C₂²	1	1	yes	characteristic
11	#18012672	—	2	2	no	—
12	#18012671	—	2	2	no	—
13	#18012663	—	2	2	no	—
14	#18012662	—	2	2	no	—
15	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	#106×C₁₅₁	—	2	2	no	—
20	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₀₂	—	2	1	no	—
22	C₂×Dic₃₀₂	—	2	1	no	—
23	#14558994	—	4	4	no	—
24	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₅₁	—	2	1	no	—
27	C₄⋊Dic₁₅₁	—	2	1	no	—
28	C₄⋊Dic₁₅₁	—	2	1	no	—
29	C₄⋊Dic₁₅₁	—	2	1	no	—
30	C₂×Q₈×C₁₅₁	—	2	1	no	—
31	C₂×Q₈×C₁₅₁	—	2	1	no	—
32	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₅₁	—	2	1	no	—
34	C₄⋊C₄×C₁₅₁	—	2	1	no	—
35	C₄⋊C₄×C₁₅₁	—	2	1	no	—
36	C₄⋊C₄×C₁₅₁	—	2	1	no	—
37	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
38	Dic₃₀₂	—	4	2	no	—
39	Dic₃₀₂	—	4	2	no	—
40	C₂×Dic₁₅₁	—	4	1	no	—
41	C₂×Dic₁₅₁	—	4	1	no	—
42	C₁₅₁⋊C₈	—	2	2	no	—
43	C₁₅₁⋊C₈	SD₁₆	1	4	yes	—
44	Q₈×C₁₅₁	—	4	2	no	—
45	Q₈×C₁₅₁	—	4	2	no	—
46	C₂×C₄×C₁₅₁	—	4	1	no	—
47	C₂×C₄×C₁₅₁	—	4	1	no	—
48	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₅₁	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₅₁	#127	1	1	yes	characteristic
53	C₄×C₁₅₁	—	8	1	no	—
54	C₄×C₁₅₁	—	8	1	no	—
55	C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₅₁	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₅₁	#135	1	2	yes	—
58	Dic₁₅₁	—	8	1	no	—
59	Dic₁₅₁	—	8	1	no	—
60	C₂×C₁₅₁	#492	1	1	yes	characteristic
61	C₂×C₁₅₁	#494	1	1	yes	characteristic
62	C₂×C₁₅₁	#495	1	1	yes	characteristic
63	C₁₅₁	#1707	1	1	yes	characteristic, complemented, Sylow
64	#1707	—	151	1	no	maximal, Sylow
65	#500	—	151	1	no	—
66	#498	—	151	1	no	—
67	#498	D₁₅₁	1	1	yes	characteristic
68	#494	—	151	1	no	—
69	#487	—	151	1	no	—
70	#487	—	151	1	no	—
71	C₈²	—	151	1	no	—
72	C₂×Q₁₆	—	302	2	no	—
73	C₄⋊Q₈	—	151	1	no	—
74	C₄⋊Q₈	—	151	1	no	—
75	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
77	#107	—	302	2	no	—
78	#106	—	2	2	no	—
79	#106	—	302	2	no	—
80	#103	—	302	2	no	—
81	#103	—	302	2	no	—
82	C₄×C₈	—	151	1	no	—
83	C₄×C₈	—	151	1	no	—
84	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	302	1	no	—
88	C₂×Q₈	—	302	1	no	—
89	Q₁₆	—	604	4	no	—
90	C₂×C₈	—	151	2	no	—
91	C₂×C₈	—	151	2	no	—
92	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	302	1	no	—
98	C₄⋊C₄	—	302	1	no	—
99	C₄⋊C₄	—	302	1	no	—
100	C₄⋊C₄	—	302	1	no	—
101	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	604	2	no	—
105	Q₈	—	604	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	604	1	no	—
109	C₂×C₄	—	604	1	no	—
110	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
113	C₈	—	302	2	no	—
114	C₈	—	151	4	no	—
115	C₈	#14558994	1	4	yes	complemented
116	C₂²	#18012735	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1208	1	no	—
120	C₄	—	1208	1	no	—
121	C₄	SD₁₆×D₁₅₁	1	2	yes	—
122	C₄	C₂×#14558994	1	2	yes	—
123	C₄	#18012760	1	2	yes	—
124	C₂	#23752583	1	1	yes	characteristic
125	C₂	#23752591	1	1	yes	characteristic
126	C₂	#23752593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752615

C₂

yes

maximal, characteristic

#23752610

C₂

yes

maximal, characteristic

#23752591

C₂

yes

maximal, characteristic

#23752564

C₂

yes

maximal, characteristic

#23752562

C₂

yes

maximal, characteristic

C₈×C₁₅₁⋊C₈

C₂

yes

maximal, characteristic

#498×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558994

—

#18012739

C₂²

yes

characteristic

#18012739

C₂²

yes

—

—

—

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#106×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

#14558994

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

Q₁₆

yes

complemented

C₂²×C₁₅₁

#127

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

C₂×Q₁₆

yes

—

C₄×C₁₅₁

#135

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1707

—

151

maximal, Sylow

#500

—

151

—

#498

—

151

—

#498

D₁₅₁

yes

characteristic

#494

—

151

—

#487

—

151

—

#487

—

151

—

C₈²

—

151

—

C₂×Q₁₆

—

302

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#107

—

302

—

#106

—

#106

—

302

—

#103

—

302

—

#103

—

302

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

Q₁₆

—

604

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

100

C₄⋊C₄

—

302

—

101

C₄²

C₂²×D₁₅₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

604

—

105

Q₈

—

604

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

604

—

109

C₂×C₄

—

604

—

110

C₂×C₄

D₄×D₁₅₁

yes

characteristic

111

C₂×C₄

D₄×D₁₅₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

113

C₈

—

302

—

114

C₈

—

151

—

115

C₈

#14558994

yes

complemented

116

C₂²

#18012735

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1208

—

120

C₄

—

1208

—

121

C₄

SD₁₆×D₁₅₁

yes

—

122

C₄

C₂×#14558994

yes

—

123

C₄

#18012760

yes

—

124

C₂

#23752583

yes

characteristic

125

C₂

#23752591

yes

characteristic

126

C₂

#23752593

yes

characteristic

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#498×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	#127
hypercenter	#498	D₁₅₁
hypocenter	C₁₅₁	#1707
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#498×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
51. C₈×C₁₅₁	Q₁₆	semidirect
63. C₁₅₁	#1707	semidirect
115. C₈	#14558994	semidirect

H ≅ G/N

type

51. C₈×C₁₅₁

Q₁₆

semidirect

63. C₁₅₁

#1707

semidirect

115. C₈

#14558994

semidirect

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

subgroup	index
#23752615	2
#23752610	2
#23752591	2
#23752564	2
#23752562	2
C₈×C₁₅₁⋊C₈	2
#498×C₁₅₁	2
#1707	151

subgroup

index

C₈×C₁₅₁⋊C₈

#498×C₁₅₁

#1707

151

Subgroups — #34068705

Conjugacy classes of subgroups

Complements

Maximal subgroups