Group #14521768 — subgroups

Subgroups — #14521768

Order 1984.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758521	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758513	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
6	#11758456	C₂	1	2	yes	maximal, complemented
7	#11758456	C₂	1	2	yes	maximal, complemented
8	#11758455	C₂	1	2	yes	maximal, complemented
9	#11758454	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₃₁	C₂²	1	2	yes	—
15	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
23	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₃₁	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	Dic₃₁⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₃₁	C₂²	1	2	yes	—
32	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₁	C₂²	1	2	yes	—
35	C₂³×C₃₁	C₂³	1	2	yes	—
36	Q₈×C₃₁	—	2	2	no	—
37	D₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₁	—	2	4	no	—
43	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₃₁	C₂³	1	2	yes	—
45	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₁	C₂³	1	2	yes	—
47	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
49	Dic₆₂	—	2	2	no	—
50	Dic₆₂	—	2	2	no	—
51	C₂²×C₃₁	—	2	2	no	—
52	C₂²×C₃₁	—	2	2	no	—
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₄×C₃₁	—	2	1	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	—	2	1	no	—
58	C₄×C₃₁	#41	1	2	yes	—
59	Dic₃₁	—	2	2	no	—
60	Dic₃₁	—	2	1	no	—
61	Dic₃₁	—	2	2	no	—
62	Dic₃₁	—	2	1	no	—
63	Dic₃₁	—	2	1	no	—
64	Dic₃₁	#41	1	2	yes	—
65	#556	—	31	1	no	maximal, Sylow
66	C₂×C₃₁	—	4	2	no	—
67	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
68	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
70	C₄⋊Q₈	—	31	1	no	—
71	#126	—	31	2	no	—
72	#126	—	31	2	no	—
73	#124	—	31	1	no	—
74	#124	D₃₁	1	1	yes	characteristic
75	C₂²⋊Q₈	—	31	2	no	—
76	C₄×Q₈	—	31	1	no	—
77	C₄×Q₈	—	31	1	no	—
78	C₄×Q₈	—	31	1	no	—
79	C₄×D₄	—	31	1	no	—
80	#117	—	31	2	no	—
81	C₃₁	#556	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	31	1	no	—
83	C₂×Q₈	—	31	1	no	—
84	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
85	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
86	C₂²×C₄	—	31	2	no	—
87	C₄⋊C₄	—	31	1	no	—
88	C₄⋊C₄	—	31	2	no	—
89	C₄⋊C₄	—	31	2	no	—
90	C₄⋊C₄	—	31	1	no	—
91	C₄⋊C₄	—	31	2	no	—
92	C₄⋊C₄	—	31	1	no	—
93	C₄⋊C₄	—	31	2	no	—
94	C₄⋊C₄	—	31	1	no	—
95	C₂²⋊C₄	—	31	2	no	—
96	C₂²⋊C₄	—	31	2	no	—
97	C₂²⋊C₄	—	31	2	no	—
98	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
99	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
100	C₄²	—	31	2	no	—
101	C₄²	—	31	2	no	—
102	C₄²	—	31	1	no	—
103	C₄²	—	31	1	no	—
104	C₄²	C₂×D₃₁	1	1	yes	characteristic
105	C₂³	C₂²×D₃₁	1	2	yes	—
106	Q₈	—	2	2	no	—
107	Q₈	—	62	2	no	—
108	Q₈	—	62	2	no	—
109	D₄	—	2	2	no	—
110	C₂×C₄	—	31	1	no	—
111	C₂×C₄	—	31	1	no	—
112	C₂×C₄	—	31	2	no	—
113	C₂×C₄	—	31	1	no	—
114	C₂×C₄	—	31	2	no	—
115	C₂×C₄	—	31	1	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₃₁	1	2	yes	—
119	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₃₁	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	31	2	no	—
129	C₄	—	62	2	no	—
130	C₄	—	62	2	no	—
131	C₄	—	62	1	no	—
132	C₄	—	62	1	no	—
133	C₄	—	62	1	no	—
134	C₄	#92497	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	C₂×#92497	1	1	yes	characteristic
137	C₂	#41×D₃₁	1	1	yes	characteristic
138	C₂	#11758567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#124

C₂

yes

maximal, characteristic

Q₈×Dic₃₁

C₂

yes

maximal, characteristic, complemented

#11758521

C₂

yes

maximal, characteristic, complemented

#11758513

C₂

yes

maximal, characteristic

D₄×Dic₃₁

C₂

yes

maximal, characteristic

#11758456

C₂

yes

maximal, complemented

#11758456

C₂

yes

maximal, complemented

#11758455

C₂

yes

maximal, complemented

#11758454

C₂

yes

maximal, complemented

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂²×Dic₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

—

C₂³×C₃₁

C₂³

yes

—

Q₈×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#41

yes

—

#556

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

—

—

—

D₃₁

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₃₁

#556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

C₂×D₃₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₃₁

yes

characteristic

105

C₂³

C₂²×D₃₁

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×D₃₁

yes

characteristic

118

C₂×C₄

C₂²×D₃₁

yes

—

119

C₂×C₄

C₂²×D₃₁

yes

characteristic

120

C₂×C₄

C₂²×D₃₁

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₃₁

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#92497

yes

—

135

C₂

—

136

C₂

C₂×#92497

yes

characteristic

137

C₂

#41×D₃₁

yes

characteristic

138

C₂

#11758567

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#124	C₂
prime radical	C₂×D₄×C₃₁	C₂²
hypercenter	#124	D₃₁
hypocenter	C₃₁	#556
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#124

C₂

prime radical

C₂×D₄×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. Q₈×Dic₃₁	C₂	semidirect
3. #11758521	C₂	semidirect
6. #11758456	C₂	semidirect
7. #11758456	C₂	semidirect
8. #11758455	C₂	semidirect
9. #11758454	C₂	semidirect
10. C₄×Dic₆₂	C₂	semidirect
11. C₄×Dic₆₂	C₂	semidirect
81. C₃₁	#556	semidirect

H ≅ G/N

type

2. Q₈×Dic₃₁

C₂

semidirect

3. #11758521

C₂

semidirect

6. #11758456

C₂

semidirect

7. #11758456

C₂

semidirect

8. #11758455

C₂

semidirect

9. #11758454

C₂

semidirect

10. C₄×Dic₆₂

C₂

semidirect

11. C₄×Dic₆₂

C₂

semidirect

81. C₃₁

#556

semidirect

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

subgroup	index
C₃₁×#124	2
Q₈×Dic₃₁	2
#11758521	2
#11758513	2
D₄×Dic₃₁	2
#11758456	2
#11758455	2
#11758454	2
C₄×Dic₆₂	2
#556	31

subgroup

index

C₃₁×#124

Q₈×Dic₃₁

D₄×Dic₃₁

C₄×Dic₆₂

Subgroups — #14521768

Conjugacy classes of subgroups

Complements

Maximal subgroups