Group #16848453 — subgroups

Subgroups — #16848453

Order 3968.

444 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 237 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	#14520948	C₂	1	4	yes	maximal
2	#14520948	C₂	1	2	yes	maximal
3	#14520935	C₂	1	4	yes	maximal
4	#14520919	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
6	#14520895	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#125	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₃₁⋊#95	C₂²	1	2	yes	—
19	C₃₁⋊#95	C₂²	1	2	yes	—
20	C₃₁⋊#95	C₂²	1	4	yes	—
21	C₃₁⋊#95	C₂²	1	4	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃₁	C₂³	1	4	yes	—
28	C₂²×Dic₃₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
30	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
31	C₄²×C₃₁	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₃₁	—	2	4	no	—
33	C₄⋊Dic₃₁	—	4	4	no	—
34	C₄⋊Dic₃₁	—	4	4	no	—
35	C₄×Dic₃₁	—	2	2	no	—
36	C₄×Dic₃₁	—	4	4	no	—
37	C₄×Dic₃₁	Q₈	1	4	yes	—
38	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₃₁	#41	1	2	yes	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	2	2	no	—
50	C₂×Dic₃₁	—	2	1	no	—
51	C₂×Dic₃₁	—	2	1	no	—
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	—	2	4	no	—
54	C₂×Dic₃₁	—	2	2	no	—
55	C₂×Dic₃₁	—	2	4	no	—
56	C₂×Dic₃₁	—	2	2	no	—
57	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
58	#2330	—	31	1	no	maximal, Sylow
59	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
62	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
63	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
64	C₄×C₃₁	—	2	8	no	—
65	C₄×C₃₁	—	2	4	no	—
66	C₄×C₃₁	C₄×Q₈	1	4	yes	—
67	Dic₃₁	—	2	4	no	—
68	Dic₃₁	—	4	1	no	—
69	Dic₃₁	—	4	4	no	—
70	Dic₃₁	—	4	1	no	—
71	C₂×#125	D₃₁	1	1	yes	characteristic
72	#389	—	31	4	no	—
73	#389	—	31	2	no	—
74	#387	—	31	4	no	—
75	#384	—	31	1	no	—
76	C₄×C₄⋊C₄	—	31	2	no	—
77	#376	—	31	1	no	—
78	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
79	C₂×C₃₁	#520	1	2	yes	—
80	C₂×C₃₁	#540	1	1	yes	characteristic
81	C₂×C₃₁	#541	1	1	yes	characteristic
82	C₂×C₃₁	#544	1	2	yes	—
83	#125	Dic₃₁	1	8	yes	complemented
84	C₂×C₄⋊C₄	—	31	2	no	—
85	C₂×C₄⋊C₄	—	31	4	no	—
86	C₂×C₄⋊C₄	—	31	4	no	—
87	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
88	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
89	C₂×C₄²	—	31	1	no	—
90	C₂×C₄²	—	31	1	no	—
91	C₂×C₄²	—	31	4	no	—
92	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
93	#95	—	31	2	no	—
94	#95	—	31	2	no	—
95	#95	—	31	4	no	—
96	#95	—	31	4	no	—
97	C₃₁	#2330	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄	—	31	2	no	—
99	C₂²×C₄	—	31	1	no	—
100	C₂²×C₄	—	31	1	no	—
101	C₂²×C₄	—	31	4	no	—
102	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
103	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
104	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
105	C₄⋊C₄	—	62	4	no	—
106	C₄⋊C₄	—	124	4	no	—
107	C₄⋊C₄	—	124	4	no	—
108	C₄⋊C₄	C₂×Dic₃₁	1	16	yes	—
109	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
110	C₄²	—	31	4	no	—
111	C₄²	—	62	2	no	—
112	C₄²	—	124	4	no	—
113	C₄²	C₂×Dic₃₁	1	4	yes	—
114	C₂³	C₂³×D₃₁	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	31	4	no	—
120	C₂×C₄	—	62	2	no	—
121	C₂×C₄	—	62	2	no	—
122	C₂×C₄	—	62	1	no	—
123	C₂×C₄	—	62	1	no	—
124	C₂×C₄	—	62	4	no	—
125	C₂×C₄	—	62	4	no	—
126	C₂×C₄	—	62	2	no	—
127	C₂×C₄	—	62	4	no	—
128	C₂×C₄	—	62	2	no	—
129	C₂×C₄	Q₈×D₃₁	1	2	yes	—
130	C₂×C₄	#92499	1	2	yes	—
131	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
132	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
133	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
134	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
135	C₂²	C₂×#92499	1	1	yes	characteristic
136	C₂²	#11758566	1	2	yes	—
137	C₂²	#11758567	1	2	yes	—
138	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
139	C₄	—	2	8	no	—
140	C₄	—	2	4	no	—
141	C₄	—	62	4	no	—
142	C₄	—	124	1	no	—
143	C₄	—	124	4	no	—
144	C₄	—	124	1	no	—
145	C₄	Q₈×Dic₃₁	1	4	yes	—
146	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
147	C₂	#14521532	1	2	yes	—
148	C₂	#14521655	1	1	yes	characteristic
149	C₂	#14521662	1	1	yes	characteristic
150	C₂	#14521679	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520948

C₂

yes

maximal

#14520948

C₂

yes

maximal

#14520935

C₂

yes

maximal

#14520919

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₃₁

C₂

yes

maximal

#14520895

C₂

yes

maximal, characteristic

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₃₁×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

Q₈

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×Q₈

yes

—

#2330

—

maximal, Sylow

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#125

D₃₁

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

—

C₂×C₃₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₁

#520

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

#125

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#2330

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₃₁

yes

—

103

C₂²×C₄

C₂²×D₃₁

yes

—

104

C₂²×C₄

C₂²×D₃₁

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

124

—

107

C₄⋊C₄

—

124

—

108

C₄⋊C₄

C₂×Dic₃₁

yes

—

109

C₄⋊C₄

C₂×Dic₃₁

yes

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

124

—

113

C₄²

C₂×Dic₃₁

yes

—

114

C₂³

C₂³×D₃₁

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

Q₈×D₃₁

yes

—

130

C₂×C₄

#92499

yes

—

131

C₂×C₄

C₂²×Dic₃₁

yes

—

132

C₂×C₄

C₂²×Dic₃₁

yes

—

133

C₂×C₄

C₂²×Dic₃₁

yes

—

134

C₂²

C₂×Q₈×D₃₁

yes

characteristic

135

C₂²

C₂×#92499

yes

characteristic

136

C₂²

#11758566

yes

—

137

C₂²

#11758567

yes

—

138

C₂²

C₂³×Dic₃₁

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

124

—

143

C₄

—

124

—

144

C₄

—

124

—

145

C₄

Q₈×Dic₃₁

yes

—

146

C₂

C₂×Q₈×Dic₃₁

yes

characteristic

147

C₂

#14521532

yes

—

148

C₂

#14521655

yes

characteristic

149

C₂

#14521662

yes

characteristic

150

C₂

#14521679

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#2330
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
8. C₃₁×#125	C₄	semidirect
83. #125	Dic₃₁	semidirect
97. C₃₁	#2330	semidirect

H ≅ G/N

type

8. C₃₁×#125

C₄

semidirect

83. #125

Dic₃₁

semidirect

97. C₃₁

#2330

semidirect

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

subgroup	index
#14520948	2
#14520935	2
#14520919	2
C₄⋊C₄×Dic₃₁	2
#14520895	2
C₂×C₃₁×#125	2
#2330	31

subgroup

index

C₄⋊C₄×Dic₃₁

C₂×C₃₁×#125

Subgroups — #16848453

Conjugacy classes of subgroups

Complements

Maximal subgroups