Group #16856092 — subgroups

Subgroups — #16856092

Order 3968.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758446	C₂	1	1	yes	maximal, characteristic
3	#14521418	C₂	1	4	yes	maximal, complemented
4	#14521400	C₂	1	4	yes	maximal, complemented
5	#14521059	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₂	C₂²	1	2	yes	—
8	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	#11758503	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₁	—	2	2	no	—
12	#11758501	—	2	2	no	—
13	#11758447	C₂²	1	2	yes	—
14	#11758446	—	2	2	no	—
15	#11758446	C₂²	1	2	yes	—
16	#11758446	C₂²	1	2	yes	—
17	#11758446	C₂²	1	2	yes	—
18	C₄×Dic₆₂	—	2	2	no	—
19	C₃₁×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	#11758404	C₂²	1	16	yes	—
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₂×Dic₆₂	—	2	8	no	—
26	C₂×Dic₆₂	—	2	2	no	—
27	C₂×Dic₆₂	—	2	2	no	—
28	C₂×Dic₆₂	C₂³	1	4	yes	—
29	M₁₆×C₃₁	—	2	4	no	—
30	C₂×C₈×C₃₁	C₂³	1	4	yes	—
31	C₄²×C₃₁	D₄	1	2	yes	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	—	2	2	no	—
36	C₄⋊Dic₃₁	—	2	4	no	—
37	C₄⋊Dic₃₁	—	2	4	no	—
38	C₄⋊Dic₃₁	C₂³	1	4	yes	—
39	Dic₃₁⋊C₄	—	2	4	no	—
40	C₄×Dic₃₁	—	2	2	no	—
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₁	#41	1	4	yes	—
47	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	4	4	no	—
51	Dic₆₂	—	2	8	no	—
52	Dic₆₂	—	4	4	no	—
53	Dic₆₂	—	4	4	no	—
54	C₈×C₃₁	—	2	4	no	—
55	#3139	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	#124	1	2	yes	—
58	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	#124	1	2	yes	—
61	C₄×C₃₁	#137	1	4	yes	—
62	C₄×C₃₁	C₂×#41	1	2	yes	—
63	Dic₃₁	—	4	4	no	—
64	Dic₃₁	—	4	4	no	—
65	#533	—	31	1	no	—
66	C₂×C₄⋊Q₈	—	31	1	no	—
67	#491	—	31	4	no	—
68	#487	—	31	4	no	—
69	#419	—	31	4	no	—
70	C₄×M₁₆	D₃₁	1	1	yes	characteristic
71	C₂×C₃₁	—	2	2	no	—
72	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#137	1	2	yes	—
74	C₂²×Q₈	—	31	2	no	—
75	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
76	C₄⋊Q₈	—	31	2	no	—
77	C₄⋊Q₈	—	31	2	no	—
78	C₄⋊Q₈	—	31	2	no	—
79	C₄⋊Q₈	—	62	2	no	—
80	#125	—	31	2	no	—
81	C₂²⋊Q₈	—	62	2	no	—
82	C₄×Q₈	—	62	2	no	—
83	#117	—	31	2	no	—
84	C₂×C₄⋊C₄	—	62	2	no	—
85	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
86	#103	—	31	16	no	—
87	#97	C₂×D₃₁	1	2	yes	—
88	C₄×C₈	C₂×D₃₁	1	2	yes	—
89	C₃₁	#3139	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	31	4	no	—
91	C₂×Q₈	—	62	2	no	—
92	C₂×Q₈	—	62	2	no	—
93	C₂×Q₈	—	62	8	no	—
94	C₂²×C₄	—	62	2	no	—
95	C₂²×C₄	D₁₂₄	1	2	yes	—
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	M₁₆	—	2	4	no	—
98	C₂×C₈	C₂²×D₃₁	1	4	yes	—
99	C₄⋊C₄	—	31	4	no	—
100	C₄⋊C₄	—	62	4	no	—
101	C₄⋊C₄	—	62	4	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄⋊C₄	—	62	2	no	—
104	C₂²⋊C₄	—	62	2	no	—
105	C₄²	—	62	2	no	—
106	C₄²	D₁₂₄	1	2	yes	—
107	C₄²	C₂²×D₃₁	1	1	yes	characteristic
108	C₄²	C₂²×D₃₁	1	1	yes	characteristic
109	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
110	Q₈	—	62	8	no	—
111	Q₈	—	124	4	no	—
112	Q₈	—	124	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	—	62	4	no	—
116	C₂×C₄	—	124	4	no	—
117	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
118	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
120	C₂×C₄	#92495	1	4	yes	—
121	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#11758452	1	2	yes	—
125	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	124	4	no	—
128	C₄	—	124	4	no	—
129	C₄	#11758452	1	2	yes	—
130	C₄	#11758486	1	4	yes	—
131	C₄	C₂×#92495	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#11758452	1	1	yes	characteristic
134	C₂	C₂×#11758486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

C₂×#11758446

C₂

yes

maximal, characteristic

#14521418

C₂

yes

maximal, complemented

#14521400

C₂

yes

maximal, complemented

#14521059

C₂

yes

maximal

C₄×M₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×C₄⋊Dic₃₁

—

#11758501

—

#11758447

C₂²

yes

—

#11758446

—

#11758446

C₂²

yes

—

#11758446

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

C₃₁×#97

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758404

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂³

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

#3139

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

#137

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

Dic₃₁

—

#533

—

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

characteristic

C₂×C₃₁

C₂×#137

yes

—

C₂²×Q₈

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

#97

C₂×D₃₁

yes

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

#3139

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

D₁₂₄

yes

—

107

C₄²

C₂²×D₃₁

yes

characteristic

108

C₄²

C₂²×D₃₁

yes

characteristic

109

C₂³

C₂×D₁₂₄

yes

characteristic

110

Q₈

—

111

Q₈

—

124

—

112

Q₈

—

124

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

124

—

117

C₂×C₄

C₂×D₁₂₄

yes

—

118

C₂×C₄

C₂×D₁₂₄

yes

characteristic

119

C₂×C₄

C₂×D₁₂₄

yes

—

120

C₂×C₄

#92495

yes

—

121

C₂×C₄

C₂³×D₃₁

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#11758452

yes

—

125

C₂²

C₂²×D₁₂₄

yes

characteristic

126

C₄

—

127

C₄

—

124

—

128

C₄

—

124

—

129

C₄

#11758452

yes

—

130

C₄

#11758486

yes

—

131

C₄

C₂×#92495

yes

—

132

C₂

—

133

C₂

C₂×#11758452

yes

characteristic

134

C₂

C₂×#11758486

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	#3139
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₄×M₁₆×C₃₁

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₄×M₁₆

D₃₁

hypocenter

C₃₁

#3139

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14521418	C₂	semidirect
4. #14521400	C₂	semidirect
89. C₃₁	#3139	semidirect

H ≅ G/N

type

3. #14521418

C₂

semidirect

4. #14521400

C₂

semidirect

89. C₃₁

#3139

semidirect

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

subgroup	index
#14521606	2
C₂×#11758446	2
#14521418	2
#14521400	2
#14521059	2
C₄×M₁₆×C₃₁	2
#3139	31

subgroup

index

C₄×M₁₆×C₃₁

Subgroups — #16856092

Conjugacy classes of subgroups

Complements

Maximal subgroups