Group #14521563 — subgroups

Subgroups — #124×D₃₁ #14521563

Order 1984.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758524	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758513	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758462	C₂	1	4	yes	maximal, complemented
7	C₂²⋊C₄×D₃₁	C₂	1	4	yes	maximal, complemented
8	#11758452	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×D₃₁	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	Q₈×D₃₁	—	2	4	no	—
15	D₄×D₃₁	—	2	4	no	—
16	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₃₁	C₂²	1	2	yes	—
19	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₆₂	C₂²	1	1	yes	characteristic, complemented
21	C₂×Dic₆₂	C₂²	1	2	yes	complemented
22	C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
23	C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₂²⋊Dic₃₁	C₂²	1	4	yes	complemented
25	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	complemented
26	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	complemented
27	C₄×Dic₃₁	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	—	2	8	no	—
32	C₂²×D₃₁	C₂³	1	2	yes	—
33	C₂²×D₃₁	C₂³	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	2	no	—
35	D₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²⋊D₃₁	—	2	4	no	—
40	C₂×Dic₃₁	C₂³	1	2	yes	—
41	C₂×Dic₃₁	C₂³	1	2	yes	—
42	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
43	D₁₂₄	—	2	2	no	—
44	C₄×D₃₁	—	2	4	no	—
45	C₄×D₃₁	—	2	4	no	—
46	C₄×D₃₁	D₄	1	4	yes	—
47	Dic₆₂	—	2	2	no	—
48	Dic₆₂	—	2	4	no	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	2	4	no	—
51	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₁	—	2	2	no	—
53	C₂×D₃₁	—	2	4	no	—
54	C₂×D₃₁	—	4	8	no	—
55	C₂×D₃₁	C₂×D₄	1	2	yes	—
56	C₂×D₃₁	#41	1	4	yes	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	C₂×D₄	1	2	yes	—
60	Dic₃₁	—	2	2	no	—
61	Dic₃₁	—	2	2	no	—
62	Dic₃₁	C₂×D₄	1	2	yes	—
63	C₂×#124	—	31	1	no	maximal, Sylow
64	C₂×C₃₁	—	4	2	no	—
65	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×C₃₁	C₂×#41	1	2	yes	—
67	D₃₁	—	4	2	no	—
68	D₃₁	#124	1	4	yes	direct product factor
69	C₂²×Q₈	—	31	1	no	—
70	C₂²×D₄	—	31	1	no	—
71	#124	—	31	1	no	—
72	#124	—	31	1	no	—
73	#124	—	31	4	no	—
74	#124	—	31	1	no	—
75	#124	D₃₁	1	1	yes	characteristic, direct product factor
76	C₂×C₂²⋊C₄	—	31	4	no	—
77	C₂×C₄²	—	31	1	no	—
78	C₃₁	C₂×#124	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	—	31	2	no	—
80	C₂×Q₈	—	31	2	no	—
81	C₂×Q₈	—	31	1	no	—
82	C₂×Q₈	—	62	4	no	—
83	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
84	C₂×D₄	—	31	2	no	—
85	C₂×D₄	—	31	1	no	—
86	C₂×D₄	—	62	4	no	—
87	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
88	C₂²×C₄	—	31	1	no	—
89	C₂²×C₄	—	31	2	no	—
90	C₂²×C₄	—	31	2	no	—
91	C₂²⋊C₄	—	31	4	no	—
92	C₂²⋊C₄	—	31	4	no	—
93	C₂²⋊C₄	—	31	4	no	—
94	C₂²⋊C₄	C₂×D₃₁	1	4	yes	complemented
95	C₄²	—	31	2	no	—
96	C₄²	—	31	1	no	—
97	C₄²	C₂×D₃₁	1	1	yes	characteristic, complemented
98	C₂³	—	31	1	no	—
99	C₂³	—	31	2	no	—
100	C₂³	—	62	4	no	—
101	C₂³	—	62	8	no	—
102	C₂³	C₂²×D₃₁	1	2	yes	—
103	Q₈	—	2	2	no	—
104	Q₈	—	62	4	no	—
105	Q₈	—	62	2	no	—
106	D₄	—	2	2	no	—
107	D₄	—	62	4	no	—
108	D₄	—	62	2	no	—
109	C₂×C₄	—	31	1	no	—
110	C₂×C₄	—	31	4	no	—
111	C₂×C₄	—	31	2	no	—
112	C₂×C₄	—	31	2	no	—
113	C₂×C₄	—	62	4	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	C₂²×D₃₁	1	2	yes	—
116	C₂×C₄	C₂²×D₃₁	1	2	yes	—
117	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	31	2	no	—
121	C₂²	—	31	4	no	—
122	C₂²	—	62	2	no	—
123	C₂²	—	62	4	no	—
124	C₂²	—	124	8	no	—
125	C₂²	C₂³×D₃₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	31	2	no	—
129	C₄	—	62	2	no	—
130	C₄	—	62	2	no	—
131	C₄	D₄×D₃₁	1	2	yes	—
132	C₂	—	4	2	no	—
133	C₂	—	31	4	no	—
134	C₂	—	124	2	no	—
135	C₂	C₂×D₄×D₃₁	1	1	yes	characteristic
136	C₂	#41×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₃₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₃₁

C₂

yes

maximal, characteristic

C₃₁×#124

C₂

yes

maximal, characteristic, complemented

#11758524

C₂

yes

maximal, characteristic, complemented

#11758513

C₂

yes

maximal, characteristic, complemented

#11758462

C₂

yes

maximal, complemented

C₂²⋊C₄×D₃₁

C₂

yes

maximal, complemented

#11758452

C₂

yes

maximal, characteristic, complemented

C₄²×D₃₁

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

—

Q₈×D₃₁

—

D₄×D₃₁

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

characteristic, complemented

C₂×Dic₆₂

C₂²

yes

complemented

C₂²⋊C₄×C₃₁

C₂²

yes

complemented

C₄²×C₃₁

C₂²

yes

characteristic, complemented

C₂²⋊Dic₃₁

C₂²

yes

complemented

(C₂×D₃₁)⋊C₄

C₂²

yes

complemented

(C₂×D₃₁)⋊C₄

C₂²

yes

complemented

C₄×Dic₃₁

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

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

D₄

yes

—

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×D₄

yes

—

C₂×D₃₁

#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×D₄

yes

—

C₂×#124

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

C₂×#41

yes

—

D₃₁

—

D₃₁

#124

yes

direct product factor

C₂²×Q₈

—

C₂²×D₄

—

—

—

—

—

D₃₁

yes

characteristic, direct product factor

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₃₁

C₂×#124

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂×D₃₁

yes

characteristic, complemented

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×D₃₁

yes

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²×D₃₁

yes

—

116

C₂×C₄

C₂²×D₃₁

yes

—

117

C₂×C₄

C₂²×D₃₁

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

124

—

125

C₂²

C₂³×D₃₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₄×D₃₁

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

124

—

135

C₂

C₂×D₄×D₃₁

yes

characteristic

136

C₂

#41×D₃₁

yes

—

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

C₂

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×Q₈×D₃₁	C₂	semidirect
3. C₃₁×#124	C₂	semidirect
4. #11758524	C₂	semidirect
5. #11758513	C₂	semidirect
6. #11758462	C₂	semidirect
7. C₂²⋊C₄×D₃₁	C₂	semidirect
8. #11758452	C₂	semidirect
9. C₄²×D₃₁	C₂	semidirect
11. C₂×Q₈×C₃₁	C₂²	semidirect
20. C₂×Dic₆₂	C₂²	semidirect
21. C₂×Dic₆₂	C₂²	semidirect
22. C₂²⋊C₄×C₃₁	C₂²	semidirect
23. C₄²×C₃₁	C₂²	semidirect
24. C₂²⋊Dic₃₁	C₂²	semidirect
25. (C₂×D₃₁)⋊C₄	C₂²	semidirect
26. (C₂×D₃₁)⋊C₄	C₂²	semidirect
27. C₄×Dic₃₁	C₂²	semidirect
28. C₄×Dic₃₁	C₂²	semidirect
68. D₃₁	#124	direct (N × H)
75. #124	D₃₁	direct (N × H)
78. C₃₁	C₂×#124	semidirect
83. C₂×Q₈	C₂×D₃₁	semidirect
94. C₂²⋊C₄	C₂×D₃₁	semidirect
97. C₄²	C₂×D₃₁	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₃₁

C₂

semidirect

3. C₃₁×#124

C₂

semidirect

4. #11758524

C₂

semidirect

5. #11758513

C₂

semidirect

6. #11758462

C₂

semidirect

7. C₂²⋊C₄×D₃₁

C₂

semidirect

8. #11758452

C₂

semidirect

9. C₄²×D₃₁

C₂

semidirect

11. C₂×Q₈×C₃₁

C₂²

semidirect

20. C₂×Dic₆₂

C₂²

semidirect

21. C₂×Dic₆₂

C₂²

semidirect

22. C₂²⋊C₄×C₃₁

C₂²

semidirect

23. C₄²×C₃₁

C₂²

semidirect

24. C₂²⋊Dic₃₁

C₂²

semidirect

25. (C₂×D₃₁)⋊C₄

C₂²

semidirect

26. (C₂×D₃₁)⋊C₄

C₂²

semidirect

27. C₄×Dic₃₁

C₂²

semidirect

28. C₄×Dic₃₁

C₂²

semidirect

68. D₃₁

#124

direct (N × H)

75. #124

D₃₁

direct (N × H)

78. C₃₁

C₂×#124

semidirect

83. C₂×Q₈

C₂×D₃₁

semidirect

94. C₂²⋊C₄

C₂×D₃₁

semidirect

97. C₄²

C₂×D₃₁

semidirect

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

subgroup	index
C₂×Q₈×D₃₁	2
C₂×D₄×D₃₁	2
C₃₁×#124	2
#11758524	2
#11758513	2
#11758462	2
C₂²⋊C₄×D₃₁	2
#11758452	2
C₄²×D₃₁	2
C₂×#124	31

subgroup

index

C₂×Q₈×D₃₁

C₂×D₄×D₃₁

C₃₁×#124

C₂²⋊C₄×D₃₁

C₄²×D₃₁

Subgroups — #124×D₃₁ #14521563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #124×D31 #14521563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #124×D₃₁ #14521563