Group #30665597 — subgroups

Subgroups — #30665597

Order 16768.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22553688	C₂	1	1	yes	maximal, characteristic
2	#22553684	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553662	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553660	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553659	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553033	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523501	—	2	2	no	—
9	C₂×#14523498	—	2	2	no	—
10	#16862246	C₂²	1	1	yes	characteristic
11	#16862243	C₂²	1	1	yes	characteristic
12	#16862170	—	2	2	no	—
13	#16862168	—	2	2	no	—
14	#16862166	—	2	2	no	—
15	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	#127×C₁₃₁	C₂²	1	1	yes	characteristic
19	#102×C₁₃₁	—	2	2	no	—
20	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₆₂	—	2	1	no	—
22	C₂×D₅₂₄	—	2	1	no	—
23	C₂×D₅₂₄	—	2	1	no	—
24	C₂×D₅₂₄	—	2	1	no	—
25	#14523501	—	4	4	no	—
26	#14523498	—	4	4	no	—
27	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₁₃₁	—	2	1	no	—
30	C₄⋊Dic₁₃₁	—	2	1	no	—
31	C₂×Q₈×C₁₃₁	—	2	1	no	—
32	C₂×D₄×C₁₃₁	—	2	1	no	—
33	C₂×D₄×C₁₃₁	—	2	1	no	—
34	C₂×D₄×C₁₃₁	—	2	1	no	—
35	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₃₁	—	2	1	no	—
37	C₄⋊C₄×C₁₃₁	—	2	1	no	—
38	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂²×D₁₃₁	—	4	1	no	—
40	Dic₂₆₂	—	4	2	no	—
41	D₅₂₄	—	4	2	no	—
42	D₅₂₄	—	4	2	no	—
43	D₅₂₄	—	4	2	no	—
44	C₂×Dic₁₃₁	—	4	1	no	—
45	C₁₃₁⋊C₈	—	2	2	no	—
46	C₁₃₁⋊C₈	—	2	2	no	—
47	C₂³×C₁₃₁	—	4	1	no	—
48	Q₈×C₁₃₁	—	4	2	no	—
49	D₄×C₁₃₁	—	4	2	no	—
50	D₄×C₁₃₁	—	4	2	no	—
51	D₄×C₁₃₁	—	4	2	no	—
52	C₂×C₄×C₁₃₁	—	4	1	no	—
53	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₃₁	—	2	2	no	—
57	C₂²×C₁₃₁	—	8	1	no	—
58	C₂²×C₁₃₁	—	8	1	no	—
59	C₂²×C₁₃₁	—	8	1	no	—
60	C₂²×C₁₃₁	#127	1	1	yes	characteristic
61	C₂×D₁₃₁	—	8	1	no	—
62	C₂×D₁₃₁	—	8	1	no	—
63	C₂×D₁₃₁	—	8	1	no	—
64	C₄×C₁₃₁	—	8	1	no	—
65	C₄×C₁₃₁	#135	1	2	yes	—
66	C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₁₃₁	—	8	1	no	—
69	C₂×C₁₃₁	—	16	1	no	—
70	C₂×C₁₃₁	#495	1	1	yes	characteristic
71	C₂×C₁₃₁	#496	1	1	yes	characteristic
72	C₂×C₁₃₁	#496	1	1	yes	characteristic
73	D₁₃₁	—	16	1	no	—
74	C₁₃₁	#1717	1	1	yes	characteristic, complemented, Sylow
75	#1717	—	131	1	no	maximal, Sylow
76	#493	—	131	1	no	—
77	#492	—	131	1	no	—
78	#487	—	131	1	no	—
79	#486	—	131	1	no	—
80	#486	—	131	1	no	—
81	#486	D₁₃₁	1	1	yes	characteristic, complemented
82	#322	—	131	1	no	—
83	C₂×SD₁₆	—	262	2	no	—
84	C₂×D₈	—	262	2	no	—
85	C₄⋊Q₈	—	131	1	no	—
86	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
87	#127	—	131	1	no	—
88	#127	C₂×D₁₃₁	1	1	yes	characteristic
89	#103	—	262	2	no	—
90	#102	—	2	2	no	—
91	#102	—	262	2	no	—
92	#102	—	262	2	no	—
93	C₄×C₈	—	131	1	no	—
94	C₄×C₈	—	131	1	no	—
95	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	262	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	262	1	no	—
102	C₂×D₄	—	262	1	no	—
103	C₂×D₄	—	262	1	no	—
104	SD₁₆	—	524	4	no	—
105	D₈	—	524	4	no	—
106	C₂×C₈	—	131	2	no	—
107	C₂×C₈	—	131	2	no	—
108	C₂×C₈	C₂²⋊D₁₃₁	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	262	1	no	—
112	C₄⋊C₄	—	262	1	no	—
113	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	524	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	524	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	524	2	no	—
122	D₄	—	524	2	no	—
123	D₄	—	524	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	524	1	no	—
126	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	262	2	no	—
131	C₈	—	262	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	1048	1	no	—
136	C₂²	—	1048	1	no	—
137	C₂²	—	1048	1	no	—
138	C₂²	#16862242	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	1048	1	no	—
141	C₄	#16862266	1	2	yes	—
142	C₄	#16862270	1	2	yes	—
143	C₄	#16862273	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	2096	1	no	—
146	C₂	#22553694	1	1	yes	characteristic
147	C₂	#22553698	1	1	yes	characteristic
148	C₂	#22553700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553688

C₂

yes

maximal, characteristic

#22553684

C₂

yes

maximal, characteristic, complemented

#22553662

C₂

yes

maximal, characteristic, complemented

#22553660

C₂

yes

maximal, characteristic, complemented

#22553659

C₂

yes

maximal, characteristic, complemented

#22553033

C₂

yes

maximal, characteristic, complemented

#486×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#14523501

—

C₂×#14523498

—

#16862246

C₂²

yes

characteristic

#16862243

C₂²

yes

—

—

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

#127×C₁₃₁

C₂²

yes

characteristic

#102×C₁₃₁

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×D₅₂₄

—

C₂×D₅₂₄

—

C₂×D₅₂₄

—

#14523501

—

#14523498

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×C₈×C₁₃₁

D₄

yes

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

C₂²×D₁₃₁

—

Dic₂₆₂

—

D₅₂₄

—

D₅₂₄

—

D₅₂₄

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

—

Q₈×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#127

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#135

yes

—

C₄×C₁₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₁

AGL(1,ℤ/8)

yes

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

D₁₃₁

—

C₁₃₁

#1717

yes

characteristic, complemented, Sylow

#1717

—

131

maximal, Sylow

#493

—

131

—

#492

—

131

—

#487

—

131

—

#486

—

131

—

#486

—

131

—

#486

D₁₃₁

yes

characteristic, complemented

#322

—

131

—

C₂×SD₁₆

—

262

—

C₂×D₈

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

#127

—

131

—

#127

C₂×D₁₃₁

yes

characteristic

#103

—

262

—

#102

—

#102

—

262

—

#102

—

262

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

262

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

262

—

102

C₂×D₄

—

262

—

103

C₂×D₄

—

262

—

104

SD₁₆

—

524

—

105

D₈

—

524

—

106

C₂×C₈

—

131

—

107

C₂×C₈

—

131

—

108

C₂×C₈

C₂²⋊D₁₃₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

262

—

112

C₄⋊C₄

—

262

—

113

C₄²

C₂²×D₁₃₁

yes

characteristic

114

C₂³

—

115

C₂³

—

524

—

116

Q₈

—

117

Q₈

—

524

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

524

—

122

D₄

—

524

—

123

D₄

—

524

—

124

C₂×C₄

—

125

C₂×C₄

—

524

—

126

C₂×C₄

D₄×D₁₃₁

yes

characteristic

127

C₂×C₄

D₄×D₁₃₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

129

C₈

—

130

C₈

—

262

—

131

C₈

—

262

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

1048

—

136

C₂²

—

1048

—

137

C₂²

—

1048

—

138

C₂²

#16862242

yes

characteristic

139

C₄

—

140

C₄

—

1048

—

141

C₄

#16862266

yes

—

142

C₄

#16862270

yes

—

143

C₄

#16862273

yes

—

144

C₂

—

145

C₂

—

2096

—

146

C₂

#22553694

yes

characteristic

147

C₂

#22553698

yes

characteristic

148

C₂

#22553700

yes

characteristic

	subgroup	G/H
center	C₂²	#16862242
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#486×C₁₃₁	C₂
socle	C₂²×C₁₃₁	#127
prime radical	#22553688	C₂
hypercenter	#486	D₁₃₁
hypocenter	C₁₃₁	#1717
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#16862242

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#486×C₁₃₁

C₂

C₂²×C₁₃₁

C₂

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. #22553684	C₂	semidirect
3. #22553662	C₂	semidirect
4. #22553660	C₂	semidirect
5. #22553659	C₂	semidirect
6. #22553033	C₂	semidirect
7. #486×C₁₃₁	C₂	semidirect
74. C₁₃₁	#1717	semidirect
81. #486	D₁₃₁	semidirect

H ≅ G/N

type

2. #22553684

C₂

semidirect

3. #22553662

C₂

semidirect

4. #22553660

C₂

semidirect

5. #22553659

C₂

semidirect

6. #22553033

C₂

semidirect

7. #486×C₁₃₁

C₂

semidirect

74. C₁₃₁

#1717

semidirect

81. #486

D₁₃₁

semidirect

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

subgroup	index
#22553688	2
#22553684	2
#22553662	2
#22553660	2
#22553659	2
#22553033	2
#486×C₁₃₁	2
#1717	131

subgroup

index

#486×C₁₃₁

131

Subgroups — #30665597

Conjugacy classes of subgroups

Complements

Maximal subgroups