Group #30666010 — subgroups

Subgroups — #30666010

Order 16768.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16862254	C₂	1	1	yes	maximal, characteristic
2	C₂×#16862181	C₂	1	2	yes	maximal, complemented
3	#22553294	C₂	1	2	yes	maximal
4	C₁₃₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523513	C₂²	1	1	yes	characteristic
7	#16862254	—	2	4	no	—
8	C₂×#14523497	C₂²	1	2	yes	—
9	C₂×#14523496	C₄	1	2	yes	—
10	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₃₁	C₄	1	2	yes	—
12	#16862181	—	2	4	no	—
13	#16862164	—	2	2	no	—
14	#16862162	—	2	2	no	—
15	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
17	#14523513	—	2	2	no	—
18	#14523513	—	2	2	no	—
19	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
20	C₂×D₅₂₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₃₁	Q₈	1	2	yes	—
22	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
23	#14523497	—	2	2	no	—
24	#14523497	—	2	2	no	—
25	#14523496	—	2	4	no	—
26	C₂×C₁₃₁⋊C₈	—	2	2	no	—
27	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃₁⋊C₈	D₄	1	4	yes	—
30	C₈×D₁₃₁	—	2	4	no	—
31	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₃₁	—	2	2	no	—
33	M₁₆×C₁₃₁	—	2	2	no	—
34	M₁₆×C₁₃₁	—	2	2	no	—
35	C₂×C₈×C₁₃₁	—	2	2	no	—
36	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
37	C₂²×D₁₃₁	C₄⋊C₄	1	2	yes	—
38	Dic₂₆₂	—	2	2	no	—
39	D₅₂₄	—	2	2	no	—
40	C₂²⋊D₁₃₁	—	2	4	no	—
41	C₂×Dic₁₃₁	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₃₁	—	2	2	no	—
43	C₄×D₁₃₁	—	2	2	no	—
44	C₁₃₁⋊C₈	—	4	2	no	—
45	C₁₃₁⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₁₃₁⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₁	#41	1	2	yes	—
52	C₈×C₁₃₁	—	2	2	no	—
53	C₈×C₁₃₁	—	4	2	no	—
54	C₂²×C₁₃₁	—	4	1	no	—
55	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₃₁	—	2	2	no	—
58	C₂×D₁₃₁	—	4	2	no	—
59	C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₃₁	#117	1	1	yes	characteristic
61	C₄×C₁₃₁	#121	1	2	yes	—
62	Dic₁₃₁	—	2	2	no	—
63	C₂×C₁₃₁	—	2	1	no	—
64	C₂×C₁₃₁	—	2	2	no	—
65	C₂×C₁₃₁	#380	1	1	yes	characteristic
66	D₁₃₁	—	4	2	no	—
67	C₁₃₁	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	131	1	no	maximal, Sylow
69	C₂×#131	—	131	1	no	—
70	C₂×#108	—	131	2	no	—
71	#413	—	131	2	no	—
72	#338	—	131	1	no	—
73	#338	D₁₃₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	131	1	no	—
75	#131	—	262	4	no	—
76	C₂×M₁₆	—	131	2	no	—
77	C₂×M₁₆	—	131	2	no	—
78	C₂×M₁₆	C₂×D₁₃₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
80	C₂²×C₈	—	131	2	no	—
81	C₂²×C₈	—	131	1	no	—
82	#108	—	262	4	no	—
83	#101	—	262	2	no	—
84	#100	—	262	2	no	—
85	#41	—	262	2	no	—
86	#41	—	262	2	no	—
87	C₂×Q₈	—	131	1	no	—
88	C₂×D₄	—	131	2	no	—
89	C₂×D₄	—	131	1	no	—
90	C₂²×C₄	—	131	2	no	—
91	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	262	4	no	—
96	M₁₆	—	262	2	no	—
97	M₁₆	—	262	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	262	4	no	—
100	C₂×C₈	—	262	2	no	—
101	C₂×C₈	—	131	1	no	—
102	C₂×C₈	—	131	1	no	—
103	C₂×C₈	—	131	4	no	—
104	C₂×C₈	C₄×D₁₃₁	1	2	yes	—
105	C₂³	—	131	2	no	—
106	C₂³	Q₈×D₁₃₁	1	1	yes	characteristic
107	Q₈	—	262	2	no	—
108	D₄	—	262	4	no	—
109	D₄	—	262	2	no	—
110	C₂×C₄	—	262	2	no	—
111	C₂×C₄	—	262	2	no	—
112	C₂×C₄	—	131	2	no	—
113	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
115	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
116	C₂×C₄	#14523513	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	524	2	no	—
120	C₈	—	131	2	no	—
121	C₈	—	131	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	262	2	no	—
124	C₂²	—	524	2	no	—
125	C₂²	C₄⋊C₄×D₁₃₁	1	1	yes	characteristic
126	C₂²	#16862224	1	2	yes	—
127	C₄	—	262	2	no	—
128	C₄	C₂²⋊C₄×D₁₃₁	1	1	yes	characteristic
129	C₄	#16862203	1	1	yes	characteristic
130	C₄	#16862219	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	524	2	no	—
134	C₂	#22553171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862254

C₂

yes

maximal, characteristic

C₂×#16862181

C₂

yes

maximal, complemented

#22553294

C₂

yes

maximal

C₁₃₁⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#14523513

C₂²

yes

characteristic

#16862254

—

C₂×#14523497

C₂²

yes

—

C₂×#14523496

C₄

yes

—

C₂²×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₃₁

C₄

yes

—

#16862181

—

#16862164

—

#16862162

—

C₂×M₁₆×C₁₃₁

C₂²

yes

—

C₂×M₁₆×C₁₃₁

C₂²

yes

characteristic

#14523513

—

#14523513

—

C₂×Dic₂₆₂

D₄

yes

characteristic

C₂×D₅₂₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₃₁

Q₈

yes

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

—

#14523497

—

#14523497

—

#14523496

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₈×D₁₃₁

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

M₁₆×C₁₃₁

—

M₁₆×C₁₃₁

—

M₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂²×D₁₃₁

C₄⋊C₄

yes

—

Dic₂₆₂

—

D₅₂₄

—

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

C₄⋊C₄

yes

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

C₂²⋊C₄

yes

—

C₁₃₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₁

C₂²⋊Q₈

yes

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

C₂×C₂²⋊C₄

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#380

yes

characteristic

D₁₃₁

—

C₁₃₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

131

maximal, Sylow

C₂×#131

—

131

—

C₂×#108

—

131

—

#413

—

131

—

#338

—

131

—

#338

D₁₃₁

yes

characteristic, complemented

C₂×#41

—

131

—

#131

—

262

—

C₂×M₁₆

—

131

—

C₂×M₁₆

—

131

—

C₂×M₁₆

C₂×D₁₃₁

yes

—

C₂×M₁₆

C₂×D₁₃₁

yes

characteristic

C₂²×C₈

—

131

—

C₂²×C₈

—

131

—

#108

—

262

—

#101

—

262

—

#100

—

262

—

#41

—

262

—

#41

—

262

—

C₂×Q₈

—

131

—

C₂×D₄

—

131

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

262

—

M₁₆

—

262

—

M₁₆

—

262

—

C₂×C₈

—

C₂×C₈

—

262

—

100

C₂×C₈

—

262

—

101

C₂×C₈

—

131

—

102

C₂×C₈

—

131

—

103

C₂×C₈

—

131

—

104

C₂×C₈

C₄×D₁₃₁

yes

—

105

C₂³

—

131

—

106

C₂³

Q₈×D₁₃₁

yes

characteristic

107

Q₈

—

262

—

108

D₄

—

262

—

109

D₄

—

262

—

110

C₂×C₄

—

262

—

111

C₂×C₄

—

262

—

112

C₂×C₄

—

131

—

113

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

114

C₂×C₄

D₄×D₁₃₁

yes

characteristic

115

C₂×C₄

D₄×D₁₃₁

yes

—

116

C₂×C₄

#14523513

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

524

—

120

C₈

—

131

—

121

C₈

—

131

—

122

C₂²

—

123

C₂²

—

262

—

124

C₂²

—

524

—

125

C₂²

C₄⋊C₄×D₁₃₁

yes

characteristic

126

C₂²

#16862224

yes

—

127

C₄

—

262

—

128

C₄

C₂²⋊C₄×D₁₃₁

yes

characteristic

129

C₄

#16862203

yes

characteristic

130

C₄

#16862219

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

524

—

134

C₂

#22553171

yes

characteristic

	subgroup	G/H
center	C₄	#16862203
commutator	C₂×C₄×C₁₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₁
Fitting	#338×C₁₃₁	C₂
socle	C₂×C₁₃₁	#380
prime radical	C₂×#14523513	C₂²
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	#338	D₁₃₁
hypocenter	C₁₃₁	#1813
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#16862203

commutator

C₂×C₄×C₁₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₁

Fitting

#338×C₁₃₁

C₂

C₂×C₁₃₁

C₂²

C₂²×C₄×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#16862181	C₂	semidirect
4. C₁₃₁⋊#338	C₂	semidirect
5. #338×C₁₃₁	C₂	semidirect
67. C₁₃₁	#1813	semidirect
73. #338	D₁₃₁	semidirect

H ≅ G/N

type

2. C₂×#16862181

C₂

semidirect

4. C₁₃₁⋊#338

C₂

semidirect

5. #338×C₁₃₁

C₂

semidirect

67. C₁₃₁

#1813

semidirect

73. #338

D₁₃₁

semidirect

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

subgroup	index
C₂×#16862254	2
C₂×#16862181	2
#22553294	2
C₁₃₁⋊#338	2
#338×C₁₃₁	2
#1813	131

subgroup

index

C₂×#16862254

C₂×#16862181

#22553294

C₁₃₁⋊#338

#338×C₁₃₁

#1813

131

Subgroups — #30666010

Conjugacy classes of subgroups

Complements

Maximal subgroups