Group #14521617 — subgroups

Subgroups — #14521617

Order 1984.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#92499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758524	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758517	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758461	C₂	1	4	yes	maximal, complemented
7	#11758459	C₂	1	4	yes	maximal, complemented
8	#11758451	C₂	1	1	yes	maximal, characteristic, complemented
9	#11758449	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₃₁	C₂²	1	2	yes	—
11	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₃₁	C₂²	1	4	yes	complemented
15	#92499	—	2	4	no	—
16	D₄×D₃₁	—	2	4	no	—
17	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
18	C₂×D₁₂₄	C₂²	1	2	yes	—
19	C₂×D₁₂₄	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₁₂₄	C₂²	1	4	yes	complemented
21	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₃₁	C₂²	1	2	yes	complemented
23	C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
24	C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
25	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	complemented
27	Dic₃₁⋊C₄	C₂²	1	2	yes	complemented
28	C₄×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₃₁	C₂³	1	2	yes	—
30	C₂²×D₃₁	—	2	4	no	—
31	C₂²×D₃₁	—	4	4	no	—
32	C₂²×D₃₁	C₂³	1	2	yes	—
33	C₂²×D₃₁	C₂³	1	2	yes	—
34	C₂²×D₃₁	C₂³	1	1	yes	characteristic
35	Q₈×C₃₁	—	2	2	no	—
36	D₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²⋊D₃₁	—	2	4	no	—
41	C₂²⋊D₃₁	—	4	4	no	—
42	C₂×Dic₃₁	C₂³	1	2	yes	—
43	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
44	D₁₂₄	—	2	4	no	—
45	D₁₂₄	—	2	2	no	—
46	D₁₂₄	—	2	2	no	—
47	D₁₂₄	—	4	4	no	—
48	C₄×D₃₁	—	2	4	no	—
49	C₄×D₃₁	D₄	1	4	yes	complemented
50	C₂²×C₃₁	—	2	2	no	—
51	C₂²×C₃₁	—	2	4	no	—
52	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×D₃₁	—	2	2	no	—
54	C₂×D₃₁	—	2	2	no	—
55	C₂×D₃₁	—	2	2	no	—
56	C₂×D₃₁	—	2	4	no	—
57	C₂×D₃₁	—	2	4	no	—
58	C₂×D₃₁	—	4	8	no	—
59	C₂×D₃₁	C₂×D₄	1	2	yes	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	C₂×D₄	1	2	yes	—
63	Dic₃₁	—	2	2	no	—
64	Dic₃₁	C₂×D₄	1	2	yes	—
65	#535	—	31	1	no	maximal, Sylow
66	C₂×C₃₁	—	4	2	no	—
67	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₃₁	H(5,𝔽₂)	1	2	yes	—
69	D₃₁	—	2	2	no	—
70	D₃₁	—	4	2	no	—
71	D₃₁	—	4	2	no	—
72	C₂×#41	—	31	1	no	—
73	C₂²×D₄	—	31	1	no	—
74	#127	—	31	1	no	—
75	#127	—	31	1	no	—
76	#124	—	31	1	no	—
77	#124	D₃₁	1	1	yes	characteristic, complemented
78	#121	—	31	4	no	—
79	#120	—	31	4	no	—
80	#117	—	31	1	no	—
81	C₃₁	#535	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	31	2	no	—
83	#41	—	62	4	no	—
84	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
85	C₂×D₄	—	31	2	no	—
86	C₂×D₄	—	31	4	no	—
87	C₂×D₄	—	31	2	no	—
88	C₂×D₄	—	31	1	no	—
89	C₂×D₄	—	31	1	no	—
90	C₂×D₄	—	31	4	no	—
91	C₂×D₄	—	62	4	no	—
92	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
93	C₂²×C₄	—	31	1	no	—
94	C₂²×C₄	—	31	2	no	—
95	C₄⋊C₄	—	31	2	no	—
96	C₂²⋊C₄	—	31	2	no	—
97	C₂²⋊C₄	—	31	4	no	—
98	C₂²⋊C₄	C₂×D₃₁	1	4	yes	complemented
99	C₄²	—	31	1	no	—
100	C₄²	C₂×D₃₁	1	1	yes	characteristic, complemented
101	C₂³	—	31	1	no	—
102	C₂³	—	31	2	no	—
103	C₂³	—	31	2	no	—
104	C₂³	—	62	4	no	—
105	C₂³	—	124	4	no	—
106	C₂³	C₂²×D₃₁	1	2	yes	—
107	Q₈	—	2	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	62	4	no	—
110	D₄	—	62	4	no	—
111	D₄	—	62	2	no	—
112	D₄	—	62	2	no	—
113	D₄	—	124	4	no	—
114	D₄	—	124	4	no	—
115	C₂×C₄	—	31	1	no	—
116	C₂×C₄	—	31	4	no	—
117	C₂×C₄	—	31	2	no	—
118	C₂×C₄	—	62	4	no	—
119	C₂×C₄	C₂²×D₃₁	1	2	yes	—
120	C₂×C₄	C₂²×D₃₁	1	2	yes	—
121	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	31	2	no	—
125	C₂²	—	62	2	no	—
126	C₂²	—	62	2	no	—
127	C₂²	—	62	2	no	—
128	C₂²	—	62	4	no	—
129	C₂²	—	62	4	no	—
130	C₂²	—	124	8	no	—
131	C₂²	C₂³×D₃₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	31	2	no	—
135	C₄	—	62	2	no	—
136	C₄	D₄×D₃₁	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	62	2	no	—
139	C₂	—	124	2	no	—
140	C₂	—	124	2	no	—
141	C₂	C₂×D₄×D₃₁	1	1	yes	characteristic
142	C₂	#11758566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92499

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₃₁

C₂

yes

maximal, characteristic, complemented

C₃₁×#124

C₂

yes

maximal, characteristic, complemented

#11758524

C₂

yes

maximal, characteristic, complemented

#11758517

C₂

yes

maximal, characteristic, complemented

#11758461

C₂

yes

maximal, complemented

#11758459

C₂

yes

maximal, complemented

#11758451

C₂

yes

maximal, characteristic, complemented

#11758449

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₁

C₂²

yes

—

C₂×Q₈×C₃₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×C₂²⋊D₃₁

C₂²

yes

complemented

#92499

—

D₄×D₃₁

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×D₁₂₄

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

characteristic, complemented

C₂×D₁₂₄

C₂²

yes

complemented

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

complemented

C₂²⋊C₄×C₃₁

C₂²

yes

complemented

C₄²×C₃₁

C₂²

yes

characteristic, complemented

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

complemented

Dic₃₁⋊C₄

C₂²

yes

complemented

C₄×Dic₃₁

C₂²

yes

characteristic, complemented

C₂³×C₃₁

C₂³

yes

—

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

D₄

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×D₄

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

Dic₃₁

—

Dic₃₁

C₂×D₄

yes

—

#535

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

—

D₃₁

—

D₃₁

—

D₃₁

—

C₂×#41

—

C₂²×D₄

—

—

—

—

D₃₁

yes

characteristic, complemented

#121

—

#120

—

#117

—

C₃₁

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₃₁

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

complemented

C₄²

—

100

C₄²

C₂×D₃₁

yes

characteristic, complemented

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

124

—

106

C₂³

C₂²×D₃₁

yes

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

124

—

114

D₄

—

124

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×D₃₁

yes

—

120

C₂×C₄

C₂²×D₃₁

yes

—

121

C₂×C₄

C₂²×D₃₁

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

124

—

131

C₂²

C₂³×D₃₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×D₃₁

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

124

—

140

C₂

—

124

—

141

C₂

C₂×D₄×D₃₁

yes

characteristic

142

C₂

#11758566

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#124	C₂
hypercenter	#124	D₃₁
hypocenter	C₃₁	#535
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

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×#92499	C₂	semidirect
2. C₂×D₄×D₃₁	C₂	semidirect
3. C₃₁×#124	C₂	semidirect
4. #11758524	C₂	semidirect
5. #11758517	C₂	semidirect
6. #11758461	C₂	semidirect
7. #11758459	C₂	semidirect
8. #11758451	C₂	semidirect
9. #11758449	C₂	semidirect
11. C₂×Q₈×C₃₁	C₂²	semidirect
14. C₂×C₂²⋊D₃₁	C₂²	semidirect
19. C₂×D₁₂₄	C₂²	semidirect
20. C₂×D₁₂₄	C₂²	semidirect
22. C₂×C₄×D₃₁	C₂²	semidirect
23. C₂²⋊C₄×C₃₁	C₂²	semidirect
24. C₄²×C₃₁	C₂²	semidirect
26. (C₂×D₃₁)⋊C₄	C₂²	semidirect
27. Dic₃₁⋊C₄	C₂²	semidirect
28. C₄×Dic₃₁	C₂²	semidirect
49. C₄×D₃₁	D₄	semidirect
77. #124	D₃₁	semidirect
81. C₃₁	#535	semidirect
84. C₂×Q₈	C₂×D₃₁	semidirect
98. C₂²⋊C₄	C₂×D₃₁	semidirect
100. C₄²	C₂×D₃₁	semidirect

H ≅ G/N

type

1. C₂×#92499

C₂

semidirect

2. C₂×D₄×D₃₁

C₂

semidirect

3. C₃₁×#124

C₂

semidirect

4. #11758524

C₂

semidirect

5. #11758517

C₂

semidirect

6. #11758461

C₂

semidirect

7. #11758459

C₂

semidirect

8. #11758451

C₂

semidirect

9. #11758449

C₂

semidirect

11. C₂×Q₈×C₃₁

C₂²

semidirect

14. C₂×C₂²⋊D₃₁

C₂²

semidirect

19. C₂×D₁₂₄

C₂²

semidirect

20. C₂×D₁₂₄

C₂²

semidirect

22. C₂×C₄×D₃₁

C₂²

semidirect

23. C₂²⋊C₄×C₃₁

C₂²

semidirect

24. C₄²×C₃₁

C₂²

semidirect

26. (C₂×D₃₁)⋊C₄

C₂²

semidirect

27. Dic₃₁⋊C₄

C₂²

semidirect

28. C₄×Dic₃₁

C₂²

semidirect

49. C₄×D₃₁

D₄

semidirect

77. #124

D₃₁

semidirect

81. C₃₁

#535

semidirect

84. C₂×Q₈

C₂×D₃₁

semidirect

98. C₂²⋊C₄

C₂×D₃₁

semidirect

100. C₄²

C₂×D₃₁

semidirect

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

subgroup	index
C₂×#92499	2
C₂×D₄×D₃₁	2
C₃₁×#124	2
#11758524	2
#11758517	2
#11758461	2
#11758459	2
#11758451	2
#11758449	2
#535	31

subgroup

index

C₂×#92499

C₂×D₄×D₃₁

C₃₁×#124

Subgroups — #14521617

Conjugacy classes of subgroups

Complements

Maximal subgroups