Group #16845760 — subgroups

Subgroups — #16845760

Order 3968.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#11758484	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758410	C₂	1	2	yes	maximal, complemented
3	#14521031	C₂	1	2	yes	maximal
4	C₃₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#92495	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92474	C₂²	1	2	yes	—
10	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	#11758484	—	2	4	no	—
12	C₂×#92469	C₄	1	2	yes	—
13	C₂×C₈×D₃₁	C₄	1	2	yes	—
14	#11758412	—	2	2	no	—
15	#11758411	—	2	2	no	—
16	#11758410	—	2	4	no	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×C₂²⋊D₃₁	Q₈	1	2	yes	—
19	#92495	—	2	2	no	—
20	#92495	—	2	2	no	—
21	C₂×D₁₂₄	D₄	1	1	yes	characteristic
22	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
23	C₂×Dic₆₂	D₄	1	1	yes	characteristic
24	M₁₆×C₃₁	—	2	2	no	—
25	M₁₆×C₃₁	—	2	2	no	—
26	M₁₆×C₃₁	—	2	2	no	—
27	C₂×C₈×C₃₁	—	2	2	no	—
28	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	#92474	—	2	2	no	—
30	#92474	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
35	#92469	—	2	4	no	—
36	C₈×D₃₁	—	2	4	no	—
37	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₃₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	2	yes	—
43	C₂²⋊D₃₁	—	2	4	no	—
44	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
45	D₁₂₄	—	2	2	no	—
46	C₄×D₃₁	—	2	2	no	—
47	C₄×D₃₁	—	2	2	no	—
48	Dic₆₂	—	2	2	no	—
49	C₈×C₃₁	—	2	2	no	—
50	C₈×C₃₁	—	4	2	no	—
51	C₃₁⋊C₈	—	4	2	no	—
52	C₃₁⋊C₈	C₂²⋊C₄	1	2	yes	—
53	C₃₁⋊C₈	C₂²⋊C₄	1	2	yes	—
54	#1813	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	4	1	no	—
56	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
58	C₂×D₃₁	—	2	2	no	—
59	C₂×D₃₁	—	4	2	no	—
60	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₃₁	#117	1	1	yes	characteristic
62	C₄×C₃₁	#121	1	2	yes	—
63	Dic₃₁	—	2	2	no	—
64	C₂×#131	—	31	1	no	—
65	C₂×#108	—	31	2	no	—
66	#413	—	31	2	no	—
67	#338	—	31	1	no	—
68	#338	D₃₁	1	1	yes	characteristic, complemented
69	C₂×C₃₁	—	2	1	no	—
70	C₂×C₃₁	—	2	2	no	—
71	C₂×C₃₁	#380	1	1	yes	characteristic
72	D₃₁	—	4	2	no	—
73	C₂×#41	—	31	1	no	—
74	#131	—	62	4	no	—
75	C₂×M₁₆	—	31	2	no	—
76	C₂×M₁₆	—	31	2	no	—
77	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
78	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
79	C₂²×C₈	—	31	1	no	—
80	C₂²×C₈	—	31	2	no	—
81	#108	—	62	4	no	—
82	#101	—	62	2	no	—
83	#100	—	62	2	no	—
84	C₃₁	#1813	1	1	yes	characteristic, complemented, Sylow
85	#41	—	62	2	no	—
86	#41	—	62	2	no	—
87	C₂×Q₈	—	31	1	no	—
88	C₂×D₄	—	31	2	no	—
89	C₂×D₄	—	31	1	no	—
90	C₂²×C₄	—	31	2	no	—
91	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	62	4	no	—
96	M₁₆	—	62	2	no	—
97	M₁₆	—	62	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	31	1	no	—
100	C₂×C₈	—	31	1	no	—
101	C₂×C₈	—	31	4	no	—
102	C₂×C₈	—	62	4	no	—
103	C₂×C₈	—	62	2	no	—
104	C₂×C₈	C₄×D₃₁	1	2	yes	—
105	C₂³	—	31	2	no	—
106	C₂³	Q₈×D₃₁	1	1	yes	characteristic
107	Q₈	—	62	2	no	—
108	D₄	—	62	4	no	—
109	D₄	—	62	2	no	—
110	C₂×C₄	—	31	2	no	—
111	C₂×C₄	—	62	2	no	—
112	C₂×C₄	—	62	2	no	—
113	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
114	C₂×C₄	#92495	1	2	yes	—
115	C₂×C₄	D₄×D₃₁	1	2	yes	—
116	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	31	2	no	—
120	C₈	—	31	2	no	—
121	C₈	—	124	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	62	2	no	—
124	C₂²	—	124	2	no	—
125	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
126	C₂²	#11758473	1	2	yes	—
127	C₄	—	62	2	no	—
128	C₄	#11758449	1	1	yes	characteristic
129	C₄	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
130	C₄	#11758461	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	124	2	no	—
134	C₂	#14520908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758484

C₂

yes

maximal, characteristic

C₂×#11758410

C₂

yes

maximal, complemented

#14521031

C₂

yes

maximal

C₃₁⋊#338

C₂

yes

maximal, characteristic, complemented

C₃₁×#338

C₂

yes

maximal, characteristic, complemented

C₂×#92495

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758484

—

C₂×#92469

C₄

yes

—

C₂×C₈×D₃₁

C₄

yes

—

#11758412

—

#11758411

—

#11758410

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₂²⋊D₃₁

Q₈

yes

—

#92495

—

#92495

—

C₂×D₁₂₄

D₄

yes

characteristic

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×Dic₆₂

D₄

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

—

#92469

—

C₈×D₃₁

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂²×D₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂²⋊C₄

yes

—

C₃₁⋊C₈

C₂²⋊C₄

yes

—

#1813

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

D₃₁

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₃₁

#1813

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×D₃₁

yes

—

105

C₂³

—

106

C₂³

Q₈×D₃₁

yes

characteristic

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

114

C₂×C₄

#92495

yes

—

115

C₂×C₄

D₄×D₃₁

yes

—

116

C₂×C₄

D₄×D₃₁

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

124

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

124

—

125

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

126

C₂²

#11758473

yes

—

127

C₄

—

128

C₄

#11758449

yes

characteristic

129

C₄

C₂²⋊C₄×D₃₁

yes

characteristic

130

C₄

#11758461

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

124

—

134

C₂

#14520908

yes

characteristic

	subgroup	G/H
center	C₄	#11758449
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#338	C₂
socle	C₂×C₃₁	#380
prime radical	C₂×#92495	C₂²
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#338	D₃₁
hypocenter	C₃₁	#1813
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#11758449

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#338

C₂

C₂×C₃₁

C₂²

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#11758410	C₂	semidirect
4. C₃₁⋊#338	C₂	semidirect
5. C₃₁×#338	C₂	semidirect
68. #338	D₃₁	semidirect
84. C₃₁	#1813	semidirect

H ≅ G/N

type

2. C₂×#11758410

C₂

semidirect

4. C₃₁⋊#338

C₂

semidirect

5. C₃₁×#338

C₂

semidirect

68. #338

D₃₁

semidirect

84. C₃₁

#1813

semidirect

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

subgroup	index
C₂×#11758484	2
C₂×#11758410	2
#14521031	2
C₃₁⋊#338	2
C₃₁×#338	2
#1813	31

subgroup

index

C₂×#11758484

C₂×#11758410

#14521031

C₃₁⋊#338

C₃₁×#338

#1813

Subgroups — #16845760

Conjugacy classes of subgroups

Complements

Maximal subgroups