Group #14521758 — subgroups

Subgroups — #14521758

Order 1984.

228 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	D₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#11758501	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₃₁⋊C₄	C₂	1	2	yes	maximal, complemented
6	#11758467	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758465	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758455	C₂	1	2	yes	maximal, complemented
9	#11758455	C₂	1	2	yes	maximal, complemented
10	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×Dic₃₁	C₂²	1	2	yes	—
15	C₂²×Dic₃₁	C₂²	1	2	yes	—
16	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₂	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₄²×C₃₁	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
26	C₄⋊Dic₃₁	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
28	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	Dic₃₁⋊C₄	C₂²	1	2	yes	—
31	Dic₃₁⋊C₄	C₂²	1	2	yes	—
32	Dic₃₁⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
35	C₂³×C₃₁	C₂³	1	2	yes	—
36	D₄×C₃₁	Q₈	1	4	yes	complemented
37	C₂×C₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₁	—	2	4	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₁	C₂³	1	2	yes	—
48	C₂×Dic₃₁	C₂³	1	2	yes	—
49	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
50	Dic₆₂	—	2	2	no	—
51	Dic₆₂	—	4	2	no	—
52	C₂²×C₃₁	—	2	2	no	—
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
56	C₄×C₃₁	—	2	1	no	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	1	no	—
59	C₄×C₃₁	C₂×Q₈	1	2	yes	—
60	Dic₃₁	—	2	1	no	—
61	Dic₃₁	—	2	2	no	—
62	Dic₃₁	—	2	1	no	—
63	Dic₃₁	—	2	2	no	—
64	Dic₃₁	—	2	1	no	—
65	Dic₃₁	#41	1	2	yes	—
66	#554	—	31	1	no	maximal, Sylow
67	C₂×C₃₁	—	2	4	no	—
68	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
71	C₄⋊Q₈	—	31	1	no	—
72	#125	—	31	1	no	—
73	#125	—	31	1	no	—
74	C₂²⋊Q₈	—	31	2	no	—
75	C₂²⋊Q₈	—	31	2	no	—
76	C₂²⋊Q₈	—	31	2	no	—
77	C₄×Q₈	—	31	1	no	—
78	C₄×D₄	—	31	1	no	—
79	C₄×D₄	—	31	1	no	—
80	C₄×D₄	D₃₁	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	31	2	no	—
82	C₃₁	#554	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	31	1	no	—
84	C₂×Q₈	—	31	2	no	—
85	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
86	C₂²×C₄	—	31	2	no	—
87	C₂²×C₄	—	31	2	no	—
88	C₂²×C₄	C₂×D₃₁	1	2	yes	—
89	C₄⋊C₄	—	31	1	no	—
90	C₄⋊C₄	—	31	2	no	—
91	C₄⋊C₄	—	31	2	no	—
92	C₄⋊C₄	—	31	2	no	—
93	C₄⋊C₄	—	31	1	no	—
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	—	31	1	no	—
96	C₄⋊C₄	—	31	1	no	—
97	C₄⋊C₄	—	31	1	no	—
98	C₄⋊C₄	—	31	2	no	—
99	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
100	C₂²⋊C₄	—	31	2	no	—
101	C₂²⋊C₄	—	31	2	no	—
102	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
103	C₄²	—	31	1	no	—
104	C₄²	—	31	1	no	—
105	C₄²	C₂×D₃₁	1	1	yes	characteristic
106	C₂³	C₂²×D₃₁	1	2	yes	—
107	Q₈	—	62	2	no	—
108	Q₈	—	124	2	no	—
109	D₄	Dic₆₂	1	4	yes	complemented
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	31	1	no	—
112	C₂×C₄	—	31	1	no	—
113	C₂×C₄	—	31	1	no	—
114	C₂×C₄	—	31	1	no	—
115	C₂×C₄	—	31	2	no	—
116	C₂×C₄	—	31	2	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	124	2	no	—
119	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₃₁	1	2	yes	—
122	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×Dic₆₂	1	4	yes	—
126	C₂²	C₂³×D₃₁	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	31	2	no	—
131	C₄	—	62	1	no	—
132	C₄	—	62	2	no	—
133	C₄	—	62	1	no	—
134	C₄	—	62	2	no	—
135	C₄	—	62	1	no	—
136	C₄	C₂×Dic₆₂	1	2	yes	—
137	C₂	—	2	4	no	—
138	C₂	C₂²×Dic₆₂	1	1	yes	characteristic
139	C₂	#11758561	1	1	yes	characteristic
140	C₂	#41×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄×C₃₁

C₂

yes

maximal, characteristic

D₄×Dic₃₁

C₂

yes

maximal, characteristic

D₄×Dic₃₁

C₂

yes

maximal, characteristic

#11758501

C₂

yes

maximal, complemented

C₂×Dic₃₁⋊C₄

C₂

yes

maximal, complemented

#11758467

C₂

yes

maximal, characteristic, complemented

#11758465

C₂

yes

maximal, characteristic, complemented

#11758455

C₂

yes

maximal, complemented

#11758455

C₂

yes

maximal, complemented

#11758447

C₂

yes

maximal, characteristic, complemented

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

—

C₂²×Dic₃₁

C₂²

yes

—

C₂²×Dic₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₂³×C₃₁

C₂³

yes

—

D₄×C₃₁

Q₈

yes

complemented

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×Q₈

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#41

yes

—

#554

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×Q₈

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₃₁

#554

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

characteristic

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

C₂×D₃₁

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₃₁

yes

characteristic

106

C₂³

C₂²×D₃₁

yes

—

107

Q₈

—

108

Q₈

—

124

—

109

D₄

Dic₆₂

yes

complemented

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

C₂²×D₃₁

yes

characteristic

120

C₂×C₄

C₂²×D₃₁

yes

characteristic

121

C₂×C₄

C₂²×D₃₁

yes

—

122

C₂×C₄

C₂²×D₃₁

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×Dic₆₂

yes

—

126

C₂²

C₂³×D₃₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×Dic₆₂

yes

—

137

C₂

—

138

C₂

C₂²×Dic₆₂

yes

characteristic

139

C₂

#11758561

yes

characteristic

140

C₂

#41×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₄×D₄×C₃₁	C₂
prime radical	C₂×D₄×C₃₁	C₂²
hypercenter	C₄×D₄	D₃₁
hypocenter	C₃₁	#554
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₄×D₄×C₃₁

C₂

prime radical

C₂×D₄×C₃₁

C₂²

hypercenter

C₄×D₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
4. #11758501	C₂	semidirect
5. C₂×Dic₃₁⋊C₄	C₂	semidirect
6. #11758467	C₂	semidirect
7. #11758465	C₂	semidirect
8. #11758455	C₂	semidirect
9. #11758455	C₂	semidirect
10. #11758447	C₂	semidirect
11. C₄×Dic₆₂	C₂	semidirect
36. D₄×C₃₁	Q₈	semidirect
82. C₃₁	#554	semidirect
109. D₄	Dic₆₂	semidirect

H ≅ G/N

type

4. #11758501

C₂

semidirect

5. C₂×Dic₃₁⋊C₄

C₂

semidirect

6. #11758467

C₂

semidirect

7. #11758465

C₂

semidirect

8. #11758455

C₂

semidirect

9. #11758455

C₂

semidirect

10. #11758447

C₂

semidirect

11. C₄×Dic₆₂

C₂

semidirect

36. D₄×C₃₁

Q₈

semidirect

82. C₃₁

#554

semidirect

109. D₄

Dic₆₂

semidirect

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

subgroup	index
C₄×D₄×C₃₁	2
D₄×Dic₃₁	2
#11758501	2
C₂×Dic₃₁⋊C₄	2
#11758467	2
#11758465	2
#11758455	2
#11758447	2
C₄×Dic₆₂	2
#554	31

subgroup

index

C₄×D₄×C₃₁

D₄×Dic₃₁

C₂×Dic₃₁⋊C₄

C₄×Dic₆₂

Subgroups — #14521758

Conjugacy classes of subgroups

Complements

Maximal subgroups