Group #16845838 — subgroups

Subgroups — #16845838

Order 3968.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14520919	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal
4	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
6	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	1	no	—
10	C₂×C₄⋊Dic₃₁	—	2	1	no	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	C₄×C₃₁⋊C₈	C₄	1	4	yes	complemented
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₄⋊C₄×C₃₁	—	2	4	no	—
20	C₄⋊C₄×C₃₁	—	4	2	no	—
21	C₄⋊C₄×C₃₁	—	4	2	no	—
22	C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₄²×C₃₁	Q₈	1	2	yes	—
24	C₄⋊Dic₃₁	—	2	4	no	—
25	C₄⋊Dic₃₁	—	4	2	no	—
26	C₄⋊Dic₃₁	—	4	2	no	—
27	C₂×C₃₁⋊C₈	—	2	2	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	—	4	2	no	—
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	—	4	2	no	—
34	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	D₈	1	2	yes	—
37	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
39	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
40	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	2	yes	—
43	C₂×C₄×C₃₁	#41	1	2	yes	—
44	C₂×Dic₃₁	—	4	2	no	—
45	C₂×Dic₃₁	—	4	2	no	—
46	C₂×Dic₃₁	—	4	2	no	—
47	C₃₁⋊C₈	—	2	4	no	—
48	#1834	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₁	#124	1	1	yes	characteristic
51	C₂²×C₃₁	#124	1	1	yes	characteristic
52	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₃₁	—	8	2	no	—
57	C₄×C₃₁	#106	1	4	yes	—
58	C₄×C₃₁	#107	1	4	yes	—
59	C₄×C₃₁	#117	1	2	yes	—
60	C₄×C₃₁	#125	1	2	yes	—
61	Dic₃₁	—	8	2	no	—
62	C₂×C₄×C₈	—	31	1	no	—
63	#384	—	31	1	no	—
64	#384	D₃₁	1	1	yes	characteristic
65	#340	—	31	4	no	—
66	C₂×C₃₁	#382	1	1	yes	characteristic
67	C₂×C₃₁	C₂×#106	1	1	yes	characteristic
68	C₂×C₃₁	C₂×#107	1	1	yes	characteristic
69	C₂×C₃₁	#486	1	1	yes	characteristic
70	C₂×C₃₁	#486	1	1	yes	characteristic
71	C₂×C₃₁	#487	1	1	yes	characteristic
72	C₂×C₃₁	#487	1	1	yes	characteristic
73	C₂²×C₈	—	31	2	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	62	1	no	—
78	C₂×C₄⋊C₄	—	62	1	no	—
79	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
80	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
81	C₄×C₈	—	31	4	no	—
82	C₃₁	#1834	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	62	2	no	—
85	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	C₂×C₈	—	31	4	no	—
89	C₂×C₈	—	62	2	no	—
90	C₂×C₈	—	62	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	62	4	no	—
95	C₄⋊C₄	—	124	2	no	—
96	C₄⋊C₄	—	124	2	no	—
97	C₄²	Dic₆₂	1	2	yes	—
98	C₄²	C₄×D₃₁	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	124	2	no	—
104	C₂×C₄	—	124	2	no	—
105	C₂×C₄	—	124	2	no	—
106	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
108	C₂×C₄	#92479	1	2	yes	—
109	C₂×C₄	#92480	1	2	yes	—
110	C₂×C₄	#92481	1	2	yes	—
111	C₂×C₄	#92482	1	2	yes	—
112	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
114	C₂×C₄	#92497	1	2	yes	—
115	C₂×C₄	#92499	1	2	yes	—
116	C₈	—	62	4	no	—
117	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
118	C₂²	C₂×#92479	1	1	yes	characteristic
119	C₂²	C₂×#92480	1	1	yes	characteristic
120	C₂²	#11758513	1	1	yes	characteristic
121	C₂²	C₂×#92481	1	1	yes	characteristic
122	C₂²	C₂×#92482	1	1	yes	characteristic
123	C₂²	#11758524	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	248	2	no	—
126	C₄	#11758395	1	4	yes	—
127	C₄	#11758396	1	4	yes	—
128	C₄	#11758467	1	2	yes	—
129	C₄	#11758469	1	2	yes	—
130	C₂	#14520914	1	1	yes	characteristic
131	C₂	C₂×#11758396	1	1	yes	characteristic
132	C₂	C₂×#11758395	1	1	yes	characteristic
133	C₂	#14521395	1	1	yes	characteristic
134	C₂	#14521397	1	1	yes	characteristic
135	C₂	#14521399	1	1	yes	characteristic
136	C₂	#14521401	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520919

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

C₃₁×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₄×C₃₁⋊C₈

C₄

yes

complemented

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

D₈

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1834

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#106

yes

—

C₄×C₃₁

#107

yes

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#125

yes

—

Dic₃₁

—

C₂×C₄×C₈

—

#384

—

#384

D₃₁

yes

characteristic

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄×C₈

—

C₃₁

#1834

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

—

C₄²

C₄×D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

124

—

104

C₂×C₄

—

124

—

105

C₂×C₄

—

124

—

106

C₂×C₄

Dic₃₁⋊C₄

yes

—

107

C₂×C₄

Dic₃₁⋊C₄

yes

—

108

C₂×C₄

#92479

yes

—

109

C₂×C₄

#92480

yes

—

110

C₂×C₄

#92481

yes

—

111

C₂×C₄

#92482

yes

—

112

C₂×C₄

C₂×Dic₆₂

yes

characteristic

113

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

114

C₂×C₄

#92497

yes

—

115

C₂×C₄

#92499

yes

—

116

C₈

—

117

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

118

C₂²

C₂×#92479

yes

characteristic

119

C₂²

C₂×#92480

yes

characteristic

120

C₂²

#11758513

yes

characteristic

121

C₂²

C₂×#92481

yes

characteristic

122

C₂²

C₂×#92482

yes

characteristic

123

C₂²

#11758524

yes

characteristic

124

C₄

—

125

C₄

—

248

—

126

C₄

#11758395

yes

—

127

C₄

#11758396

yes

—

128

C₄

#11758467

yes

—

129

C₄

#11758469

yes

—

130

C₂

#14520914

yes

characteristic

131

C₂

C₂×#11758396

yes

characteristic

132

C₂

C₂×#11758395

yes

characteristic

133

C₂

#14521395

yes

characteristic

134

C₂

#14521397

yes

characteristic

135

C₂

#14521399

yes

characteristic

136

C₂

#14521401

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#384	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#384	D₃₁
hypocenter	C₃₁	#1834
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#384

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
13. C₄×C₃₁⋊C₈	C₄	semidirect
82. C₃₁	#1834	semidirect

H ≅ G/N

type

13. C₄×C₃₁⋊C₈

C₄

semidirect

82. C₃₁

#1834

semidirect

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

subgroup	index
C₂×C₄×C₃₁⋊C₈	2
#14520919	2
#14520811	2
C₃₁×#384	2
#1834	31

subgroup

index

C₂×C₄×C₃₁⋊C₈

#14520919

#14520811

C₃₁×#384

#1834

Subgroups — #16845838

Conjugacy classes of subgroups

Complements

Maximal subgroups