Group #30665705 — subgroups

Subgroups — #30665705

Order 16768.

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

normal subgroups only

This group has 296 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×C₈×Dic₁₃₁	C₂	1	2	yes	maximal
2	C₄²×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#22553067	C₂	1	1	yes	maximal, characteristic
4	#22553066	C₂	1	2	yes	maximal
5	#336×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₃₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₃₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₃₁	C₄	1	8	yes	complemented
12	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₃₁⋊C₈	—	2	2	no	—
18	C₂×C₁₃₁⋊C₈	—	2	2	no	—
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₃₁	—	2	4	no	—
21	C₄×Dic₁₃₁	—	2	1	no	—
22	C₄×Dic₁₃₁	—	2	1	no	—
23	C₄×Dic₁₃₁	C₈	1	8	yes	complemented
24	C₄×Dic₁₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₁	—	2	2	no	—
28	C₂×C₈×C₁₃₁	—	2	2	no	—
29	C₂×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₁₃₁	—	2	2	no	—
31	C₂×Dic₁₃₁	—	2	2	no	—
32	C₂×Dic₁₃₁	—	2	2	no	—
33	C₂×Dic₁₃₁	C₄²	1	2	yes	—
34	C₂×Dic₁₃₁	C₄²	1	2	yes	—
35	C₂×Dic₁₃₁	C₂×C₈	1	4	yes	—
36	C₂×Dic₁₃₁	C₂×C₈	1	4	yes	—
37	C₂×Dic₁₃₁	M₁₆	1	4	yes	—
38	C₁₃₁⋊C₈	—	2	4	no	—
39	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₃₁	#41	1	2	yes	—
45	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
47	C₈×C₁₃₁	—	2	4	no	—
48	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₁₃₁	#117	1	1	yes	characteristic
50	C₂²×C₁₃₁	#117	1	1	yes	characteristic
51	C₂²×C₁₃₁	C₂²×C₈	1	2	yes	—
52	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₃₁	—	2	4	no	—
54	C₄×C₁₃₁	#117	1	2	yes	—
55	C₄×C₁₃₁	#117	1	2	yes	—
56	Dic₁₃₁	—	2	4	no	—
57	Dic₁₃₁	C₄×C₈	1	4	yes	—
58	Dic₁₃₁	#97	1	4	yes	—
59	C₂×C₁₃₁	#376	1	1	yes	characteristic
60	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₁₃₁	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₃₁	#431	1	2	yes	—
63	C₂×C₁₃₁	#431	1	2	yes	—
64	C₁₃₁	#1744	1	1	yes	characteristic, complemented, Sylow
65	#1744	—	131	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	131	2	no	—
67	C₄³	—	131	1	no	—
68	#336	—	131	2	no	—
69	#336	—	131	1	no	—
70	#336	D₁₃₁	1	1	yes	characteristic
71	C₂²×C₈	—	131	2	no	—
72	C₂²×C₈	C₂×D₁₃₁	1	2	yes	—
73	C₂×C₄²	—	131	2	no	—
74	C₂×C₄²	—	131	1	no	—
75	C₂×C₄²	—	131	2	no	—
76	C₂×C₄²	—	131	1	no	—
77	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
78	C₄×C₈	—	131	8	no	—
79	C₂²×C₄	—	131	2	no	—
80	C₂²×C₄	—	131	1	no	—
81	C₂²×C₄	—	131	1	no	—
82	C₂²×C₄	C₄×D₁₃₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	262	2	no	—
87	C₂×C₈	—	262	2	no	—
88	C₂×C₈	—	131	4	no	—
89	C₂×C₈	C₄×D₁₃₁	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	262	4	no	—
92	C₄²	—	262	1	no	—
93	C₄²	—	262	1	no	—
94	C₄²	—	131	8	no	—
95	C₄²	—	131	4	no	—
96	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	262	2	no	—
100	C₂×C₄	—	262	2	no	—
101	C₂×C₄	—	131	4	no	—
102	C₂×C₄	—	131	4	no	—
103	C₂×C₄	—	131	4	no	—
104	C₂×C₄	—	131	2	no	—
105	C₂×C₄	—	131	2	no	—
106	C₂×C₄	C₈×D₁₃₁	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
108	C₂×C₄	#14523512	1	1	yes	characteristic
109	C₂×C₄	#14523512	1	2	yes	—
110	C₂×C₄	#14523514	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	262	4	no	—
113	C₂²	C₄²×D₁₃₁	1	1	yes	characteristic
114	C₂²	#16862202	1	1	yes	characteristic
115	C₂²	#16862204	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₁₃₁	1	2	yes	—
117	C₂²	M₁₆×D₁₃₁	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	262	4	no	—
120	C₄	—	131	4	no	—
121	C₄	—	131	4	no	—
122	C₄	#16862202	1	2	yes	—
123	C₄	#16862204	1	2	yes	—
124	C₂	#22553157	1	1	yes	characteristic
125	C₂	C₄×C₈×D₁₃₁	1	1	yes	characteristic
126	C₂	#97×D₁₃₁	1	1	yes	characteristic
127	C₂	#22553362	1	2	yes	—
128	C₂	#22553364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₃₁

C₂

yes

maximal

C₄²×Dic₁₃₁

C₂

yes

maximal, characteristic

#22553067

C₂

yes

maximal, characteristic

#22553066

C₂

yes

maximal

#336×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×C₁₃₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₃₁

C₄

yes

—

C₂×C₄×Dic₁₃₁

C₄

yes

—

C₂×C₄×Dic₁₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₁

C₂²

yes

characteristic

C₈×Dic₁₃₁

C₄

yes

complemented

C₂²×C₈×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂²×Dic₁₃₁

C₂×C₄

yes

—

C₂²×Dic₁₃₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₁

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

C₈

yes

complemented

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

C₄²

yes

—

C₂×Dic₁₃₁

C₄²

yes

—

C₂×Dic₁₃₁

C₂×C₈

yes

—

C₂×Dic₁₃₁

C₂×C₈

yes

—

C₂×Dic₁₃₁

M₁₆

yes

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₈×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄²

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

C₂²×C₈

yes

—

C₂²×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#117

yes

—

C₄×C₁₃₁

#117

yes

—

Dic₁₃₁

—

Dic₁₃₁

C₄×C₈

yes

—

Dic₁₃₁

#97

yes

—

C₂×C₁₃₁

#376

yes

characteristic

C₂×C₁₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₁

C₂×#97

yes

characteristic

C₂×C₁₃₁

#431

yes

—

C₂×C₁₃₁

#431

yes

—

C₁₃₁

#1744

yes

characteristic, complemented, Sylow

#1744

—

131

maximal, Sylow

C₂×C₄×C₈

—

131

—

C₄³

—

131

—

#336

—

131

—

#336

—

131

—

#336

D₁₃₁

yes

characteristic

C₂²×C₈

—

131

—

C₂²×C₈

C₂×D₁₃₁

yes

—

C₂×C₄²

—

131

—

C₂×C₄²

—

131

—

C₂×C₄²

—

131

—

C₂×C₄²

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄×C₈

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₄×D₁₃₁

yes

—

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

262

—

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

C₄×D₁₃₁

yes

—

C₄²

—

C₄²

—

262

—

C₄²

—

262

—

C₄²

—

262

—

C₄²

—

131

—

C₄²

—

131

—

C₂³

C₂×C₄×D₁₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

100

C₂×C₄

—

262

—

101

C₂×C₄

—

131

—

102

C₂×C₄

—

131

—

103

C₂×C₄

—

131

—

104

C₂×C₄

—

131

—

105

C₂×C₄

—

131

—

106

C₂×C₄

C₈×D₁₃₁

yes

—

107

C₂×C₄

C₂×C₄×D₁₃₁

yes

—

108

C₂×C₄

#14523512

yes

characteristic

109

C₂×C₄

#14523512

yes

—

110

C₂×C₄

#14523514

yes

characteristic

111

C₈

—

112

C₈

—

262

—

113

C₂²

C₄²×D₁₃₁

yes

characteristic

114

C₂²

#16862202

yes

characteristic

115

C₂²

#16862204

yes

characteristic

116

C₂²

C₂×C₈×D₁₃₁

yes

—

117

C₂²

M₁₆×D₁₃₁

yes

—

118

C₄

—

119

C₄

—

262

—

120

C₄

—

131

—

121

C₄

—

131

—

122

C₄

#16862202

yes

—

123

C₄

#16862204

yes

—

124

C₂

#22553157

yes

characteristic

125

C₂

C₄×C₈×D₁₃₁

yes

characteristic

126

C₂

#97×D₁₃₁

yes

characteristic

127

C₂

#22553362

yes

—

128

C₂

#22553364

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Fitting	#336×C₁₃₁	C₂
socle, prime radical	C₂³×C₁₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	#336	D₁₃₁
hypocenter	C₁₃₁	#1744
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₁

commutator

C₂×C₁₃₁

C₂×C₄×C₈

Fitting

#336×C₁₃₁

C₂

socle, prime radical

C₂³×C₁₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. C₈×Dic₁₃₁	C₄	semidirect
23. C₄×Dic₁₃₁	C₈	semidirect
64. C₁₃₁	#1744	semidirect

H ≅ G/N

type

11. C₈×Dic₁₃₁

C₄

semidirect

23. C₄×Dic₁₃₁

C₈

semidirect

64. C₁₃₁

#1744

semidirect

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

subgroup	index
C₂×C₈×Dic₁₃₁	2
C₄²×Dic₁₃₁	2
#22553067	2
#22553066	2
#336×C₁₃₁	2
#1744	131

subgroup

index

C₂×C₈×Dic₁₃₁

C₄²×Dic₁₃₁

#22553067

#22553066

#336×C₁₃₁

#1744

131

Subgroups — #30665705

Conjugacy classes of subgroups

Complements

Maximal subgroups