Group #30678639 — subgroups

Subgroups — #30678639

Order 16768.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22554041	C₂	1	1	yes	maximal, characteristic
2	#22553664	C₂	1	3	yes	maximal
3	#22553604	C₂	1	3	yes	maximal
4	#22553522	C₂	1	3	yes	maximal
5	Q₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#14523503	C₂	1	3	yes	maximal
7	Q₈²×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523503	C₂²	1	3	yes	—
9	#16862246	C₂²	1	3	yes	—
10	#16862237	—	2	3	no	—
11	Q₈×Dic₁₃₁	—	2	1	no	—
12	C₄×Dic₂₆₂	C₂²	1	3	yes	—
13	#16862179	C₂²	1	3	yes	—
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	3	yes	—
15	#16862170	C₂²	1	1	yes	characteristic
16	#16862169	C₂²	1	6	yes	—
17	#16862169	C₂²	1	3	yes	—
18	C₄×C₁₃₁⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₃₁	—	2	3	no	—
20	C₄⋊Q₈×C₁₃₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₃₁	—	2	1	no	—
22	C₄×Q₈×C₁₃₁	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₆₂	C₂³	1	3	yes	—
25	#14523503	—	2	6	no	—
26	C₂×C₁₃₁⋊C₈	C₂³	1	3	yes	—
27	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₁₃₁	—	2	3	no	—
29	C₄⋊Dic₁₃₁	—	2	3	no	—
30	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
31	Dic₁₃₁⋊C₄	—	2	3	no	—
32	C₄×Dic₁₃₁	—	2	3	no	—
33	C₂×Q₈×C₁₃₁	—	2	3	no	—
34	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₃₁	—	2	3	no	—
37	C₄⋊C₄×C₁₃₁	—	2	3	no	—
38	C₄⋊C₄×C₁₃₁	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₃₁	C₂³	1	3	yes	—
40	C₄²×C₁₃₁	—	2	3	no	—
41	C₄²×C₁₃₁	C₂³	1	3	yes	—
42	Dic₂₆₂	—	2	3	no	—
43	Dic₂₆₂	—	2	3	no	—
44	C₂×Dic₁₃₁	—	2	1	no	—
45	C₂×Dic₁₃₁	—	2	3	no	—
46	C₁₃₁⋊C₈	—	2	3	no	—
47	C₁₃₁⋊C₈	—	2	1	no	—
48	Q₈×C₁₃₁	—	2	1	no	—
49	Q₈×C₁₃₁	—	8	3	no	—
50	Q₈×C₁₃₁	Q₁₆	1	4	yes	complemented
51	Q₈×C₁₃₁	#41	1	2	yes	—
52	C₂×C₄×C₁₃₁	—	2	1	no	—
53	C₂×C₄×C₁₃₁	—	2	3	no	—
54	C₂×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₃₁	—	2	2	no	—
59	C₄×C₁₃₁	—	2	3	no	—
60	C₄×C₁₃₁	—	4	3	no	—
61	C₄×C₁₃₁	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₃₁	—	4	3	no	—
65	Dic₁₃₁	—	4	1	no	—
66	C₂×C₁₃₁	#548	1	1	yes	characteristic
67	C₂×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₃₁	#577	1	1	yes	characteristic
69	C₁₃₁	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	131	1	no	maximal, Sylow
71	Q₈²	D₁₃₁	1	1	yes	characteristic
72	#557	—	131	1	no	—
73	#487	—	131	3	no	—
74	#477	—	131	3	no	—
75	#462	—	131	3	no	—
76	C₈×Q₈	—	131	1	no	—
77	C₄×Q₁₆	—	131	3	no	—
78	C₂×Q₁₆	—	131	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	131	3	no	—
81	C₄⋊Q₈	C₂×D₁₃₁	1	3	yes	—
82	#125	—	262	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	262	1	no	—
85	C₄×Q₈	—	131	3	no	—
86	C₄×Q₈	C₂×D₁₃₁	1	3	yes	—
87	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
88	#107	—	131	3	no	—
89	C₄⋊C₈	—	131	3	no	—
90	#103	—	131	1	no	—
91	#103	—	131	3	no	—
92	#103	—	131	6	no	—
93	C₄×C₈	—	131	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	131	3	no	—
96	C₂×Q₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
98	Q₁₆	—	262	6	no	—
99	C₂×C₈	—	131	1	no	—
100	C₂×C₈	—	131	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	262	3	no	—
104	C₄⋊C₄	—	262	3	no	—
105	C₄⋊C₄	—	262	3	no	—
106	C₄⋊C₄	—	131	1	no	—
107	C₄⋊C₄	C₂²⋊D₁₃₁	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₃₁	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	262	3	no	—
111	C₄²	C₂²×D₁₃₁	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	262	3	no	—
114	Q₈	—	262	3	no	—
115	Q₈	—	8	3	no	—
116	Q₈	#14523503	1	4	yes	complemented
117	Q₈	#14523514	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	262	1	no	—
121	C₂×C₄	—	262	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	3	yes	—
124	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
125	C₈	—	262	3	no	—
126	C₈	—	262	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	524	1	no	—
132	C₄	—	524	3	no	—
133	C₄	C₂×#14523503	1	6	yes	—
134	C₄	C₂×#14523514	1	1	yes	characteristic
135	C₄	#16862294	1	1	yes	characteristic
136	C₂	#22553973	1	1	yes	characteristic
137	C₂	C₂²×#14523503	1	1	yes	characteristic
138	C₂	#22554123	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554041

C₂

yes

maximal, characteristic

#22553664

C₂

yes

maximal

#22553604

C₂

yes

maximal

#22553522

C₂

yes

maximal

Q₈×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic

C₄×#14523503

C₂

yes

maximal

Q₈²×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#14523503

C₂²

yes

—

#16862246

C₂²

yes

—

#16862237

—

Q₈×Dic₁₃₁

—

C₄×Dic₂₆₂

C₂²

yes

—

#16862179

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

#16862170

C₂²

yes

characteristic

#16862169

C₂²

yes

—

#16862169

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

C₂²

yes

—

C₄×Q₈×C₁₃₁

—

C₄×Q₈×C₁₃₁

C₂²

yes

—

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

C₂³

yes

—

#14523503

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂³

yes

characteristic

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

C₂×Q₈×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₄

yes

—

C₄⋊C₄×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

Q₁₆

yes

complemented

Q₈×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×Q₁₆

yes

—

C₄×C₁₃₁

C₂×#41

yes

characteristic

C₄×C₁₃₁

2₋¹⁺⁴

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

#548

yes

characteristic

C₂×C₁₃₁

C₂²×Q₁₆

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#3363

—

131

maximal, Sylow

Q₈²

D₁₃₁

yes

characteristic

#557

—

131

—

#487

—

131

—

#477

—

131

—

#462

—

131

—

C₈×Q₈

—

131

—

C₄×Q₁₆

—

131

—

C₂×Q₁₆

—

131

—

C₄⋊Q₈

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

—

#125

—

262

—

C₄×Q₈

—

C₄×Q₈

—

262

—

C₄×Q₈

—

131

—

C₄×Q₈

C₂×D₁₃₁

yes

—

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

#107

—

131

—

C₄⋊C₈

—

131

—

#103

—

131

—

#103

—

131

—

#103

—

131

—

C₄×C₈

—

131

—

C₂×Q₈

—

C₂×Q₈

—

131

—

C₂×Q₈

C₂²⋊D₁₃₁

yes

characteristic

C₂×Q₈

C₂²×D₁₃₁

yes

characteristic

Q₁₆

—

262

—

C₂×C₈

—

131

—

100

C₂×C₈

—

131

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

262

—

104

C₄⋊C₄

—

262

—

105

C₄⋊C₄

—

262

—

106

C₄⋊C₄

—

131

—

107

C₄⋊C₄

C₂²⋊D₁₃₁

yes

—

108

C₄⋊C₄

C₂²×D₁₃₁

yes

—

109

C₄²

—

110

C₄²

—

262

—

111

C₄²

C₂²×D₁₃₁

yes

—

112

Q₈

—

113

Q₈

—

262

—

114

Q₈

—

262

—

115

Q₈

—

116

Q₈

#14523503

yes

complemented

117

Q₈

#14523514

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

262

—

121

C₂×C₄

—

262

—

122

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

124

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

125

C₈

—

262

—

126

C₈

—

262

—

127

C₂²

C₂²×C₂²⋊D₁₃₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

524

—

132

C₄

—

524

—

133

C₄

C₂×#14523503

yes

—

134

C₄

C₂×#14523514

yes

characteristic

135

C₄

#16862294

yes

characteristic

136

C₂

#22553973

yes

characteristic

137

C₂

C₂²×#14523503

yes

characteristic

138

C₂

#22554123

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	Q₈²×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	C₂²×D₄
hypercenter	Q₈²	D₁₃₁
hypocenter	C₁₃₁	#3363
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₁

commutator, cosocle

C₂×C₄×C₁₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₁

Fitting

Q₈²×C₁₃₁

C₂

socle, prime radical

C₂²×C₁₃₁

C₂²×D₄

hypercenter

Q₈²

D₁₃₁

hypocenter

C₁₃₁

#3363

LCS

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
50. Q₈×C₁₃₁	Q₁₆	semidirect
69. C₁₃₁	#3363	semidirect
116. Q₈	#14523503	semidirect

H ≅ G/N

type

50. Q₈×C₁₃₁

Q₁₆

semidirect

69. C₁₃₁

#3363

semidirect

116. Q₈

#14523503

semidirect

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

subgroup	index
#22554041	2
#22553664	2
#22553604	2
#22553522	2
Q₈×C₁₃₁⋊C₈	2
C₄×#14523503	2
Q₈²×C₁₃₁	2
#3363	131

subgroup

index

Q₈×C₁₃₁⋊C₈

Q₈²×C₁₃₁

131

Subgroups — #30678639

Conjugacy classes of subgroups

Complements

Maximal subgroups