Group #16845372 — subgroups

Subgroups — #16845372

Order 3968.

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

normal subgroups only

This group has 186 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	#14521445	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521428	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521400	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521395	C₂	1	1	yes	maximal, characteristic
5	#14521395	C₂	1	1	yes	maximal, characteristic
6	#14520770	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#486	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₂×#92482	—	2	2	no	—
11	#11758446	C₂²	1	1	yes	characteristic
12	#11758446	C₂²	1	1	yes	characteristic
13	C₃₁×#102	—	2	2	no	—
14	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	#11758420	—	2	2	no	—
16	#11758420	—	2	2	no	—
17	#11758404	—	2	2	no	—
18	#11758396	—	2	2	no	—
19	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₁	—	2	1	no	—
22	C₂×D₄×C₃₁	—	2	1	no	—
23	C₂×D₄×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	—	2	1	no	—
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×Dic₆₂	—	2	1	no	—
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	#92482	—	4	4	no	—
32	C₄⋊Dic₃₁	—	2	1	no	—
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
37	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
38	C₂³×C₃₁	—	4	1	no	—
39	Q₈×C₃₁	—	4	2	no	—
40	D₄×C₃₁	—	4	2	no	—
41	D₄×C₃₁	—	4	2	no	—
42	D₄×C₃₁	—	4	2	no	—
43	C₂×C₄×C₃₁	—	4	1	no	—
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₃₁	—	4	1	no	—
48	C₂×Dic₃₁	—	4	1	no	—
49	Dic₆₂	—	4	2	no	—
50	Dic₆₂	—	4	2	no	—
51	C₈×C₃₁	—	2	2	no	—
52	C₃₁⋊C₈	—	2	2	no	—
53	C₃₁⋊C₈	—	2	2	no	—
54	#1720	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	8	1	no	—
56	C₂²×C₃₁	—	8	1	no	—
57	C₂²×C₃₁	—	8	1	no	—
58	C₂²×C₃₁	#127	1	1	yes	characteristic
59	C₄×C₃₁	—	8	1	no	—
60	C₄×C₃₁	#135	1	2	yes	—
61	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₃₁	#137	1	2	yes	—
63	Dic₃₁	—	8	1	no	—
64	Dic₃₁	—	8	1	no	—
65	#498	—	31	1	no	—
66	#494	—	31	1	no	—
67	#487	—	31	1	no	—
68	#486	—	31	1	no	—
69	#486	—	31	1	no	—
70	#486	D₃₁	1	1	yes	characteristic
71	#322	—	31	1	no	—
72	C₂×C₃₁	—	16	1	no	—
73	C₂×C₃₁	#495	1	1	yes	characteristic
74	C₂×C₃₁	#496	1	1	yes	characteristic
75	C₂×C₃₁	#497	1	1	yes	characteristic
76	C₂×Q₁₆	—	62	2	no	—
77	C₄⋊Q₈	—	31	1	no	—
78	C₄⋊Q₈	—	31	1	no	—
79	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
80	#127	C₂×D₃₁	1	1	yes	characteristic
81	#106	—	62	2	no	—
82	#103	—	62	2	no	—
83	#102	—	2	2	no	—
84	#102	—	62	2	no	—
85	#102	—	62	2	no	—
86	C₄×C₈	—	31	1	no	—
87	C₄×C₈	—	31	1	no	—
88	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
89	C₃₁	#1720	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	62	1	no	—
92	C₂×Q₈	—	62	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	Q₁₆	—	124	4	no	—
97	C₂×C₈	—	31	2	no	—
98	C₂×C₈	—	31	2	no	—
99	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	62	1	no	—
103	C₄⋊C₄	—	62	1	no	—
104	C₄⋊C₄	—	62	1	no	—
105	C₄⋊C₄	—	62	1	no	—
106	C₄²	C₂²×D₃₁	1	1	yes	characteristic
107	C₂³	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	124	2	no	—
110	Q₈	—	124	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	124	1	no	—
116	C₂×C₄	—	124	1	no	—
117	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	62	2	no	—
122	C₈	—	62	2	no	—
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	#11758517	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	248	1	no	—
129	C₄	—	248	1	no	—
130	C₄	#11758488	1	2	yes	—
131	C₄	#11758492	1	2	yes	—
132	C₄	#11758527	1	2	yes	—
133	C₂	—	16	1	no	—
134	C₂	#14521431	1	1	yes	characteristic
135	C₂	#14521437	1	1	yes	characteristic
136	C₂	#14521440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521445

C₂

yes

maximal, characteristic, complemented

#14521428

C₂

yes

maximal, characteristic, complemented

#14521400

C₂

yes

maximal, characteristic, complemented

#14521395

C₂

yes

maximal, characteristic

#14521395

C₂

yes

maximal, characteristic

#14520770

C₂

yes

maximal, characteristic, complemented

C₃₁×#486

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

characteristic

C₂×#92482

—

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×#102

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1720

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#127

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic

#322

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

C₂×D₃₁

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1720

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂²×D₃₁

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

124

—

110

Q₈

—

124

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

124

—

116

C₂×C₄

—

124

—

117

C₂×C₄

D₄×D₃₁

yes

characteristic

118

C₂×C₄

D₄×D₃₁

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#11758517

yes

characteristic

127

C₄

—

128

C₄

—

248

—

129

C₄

—

248

—

130

C₄

#11758488

yes

—

131

C₄

#11758492

yes

—

132

C₄

#11758527

yes

—

133

C₂

—

134

C₂

#14521431

yes

characteristic

135

C₂

#14521437

yes

characteristic

136

C₂

#14521440

yes

characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#486	C₂
socle	C₂²×C₃₁	#127
prime radical	C₃₁×#127	C₂²
hypercenter	#486	D₃₁
hypocenter	C₃₁	#1720
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#486

C₂

socle

C₂²×C₃₁

#127

prime radical

C₃₁×#127

C₂²

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #14521445	C₂	semidirect
2. #14521428	C₂	semidirect
3. #14521400	C₂	semidirect
6. #14520770	C₂	semidirect
89. C₃₁	#1720	semidirect

H ≅ G/N

type

1. #14521445

C₂

semidirect

2. #14521428

C₂

semidirect

3. #14521400

C₂

semidirect

6. #14520770

C₂

semidirect

89. C₃₁

#1720

semidirect

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

subgroup	index
#14521445	2
#14521428	2
#14521400	2
#14521395	2
#14520770	2
C₃₁×#486	2
#1720	31

subgroup

index

C₃₁×#486

Subgroups — #16845372

Conjugacy classes of subgroups

Complements

Maximal subgroups