Group #14521556 — subgroups

Subgroups — #14521556

Order 1984.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758523	C₂	1	2	yes	maximal, complemented
4	#11758515	C₂	1	2	yes	maximal, complemented
5	#11758514	C₂	1	2	yes	maximal, complemented
6	#11758512	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₃₁	C₂	1	4	yes	maximal, complemented
8	#11758503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₃₁	C₂²	1	1	yes	characteristic
10	#41×C₃₁	—	2	4	no	—
11	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₁	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊D₃₁	C₂²	1	4	yes	—
17	C₂²×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₃₁	—	2	4	no	—
19	C₂×C₄×D₃₁	C₂²	1	2	yes	—
20	C₂×C₄×D₃₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₁	C₂²	1	4	yes	complemented
23	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
24	C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
25	Dic₃₁⋊C₄	C₂²	1	4	yes	complemented
26	C₄×Dic₃₁	C₂²	1	2	yes	complemented
27	C₂³×C₃₁	C₂³	1	2	yes	—
28	C₂³×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×D₃₁	—	2	2	no	—
30	C₂²×D₃₁	—	2	4	no	—
31	C₂²×D₃₁	C₂³	1	2	yes	—
32	Q₈×C₃₁	—	2	2	no	—
33	D₄×C₃₁	—	2	4	no	—
34	D₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	D₄	1	4	yes	complemented
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²⋊D₃₁	—	2	8	no	—
41	C₂×Dic₃₁	—	2	2	no	—
42	C₂×Dic₃₁	C₂³	1	4	yes	—
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	C₄×D₃₁	—	2	4	no	—
45	C₄×D₃₁	—	2	4	no	—
46	C₂²×C₃₁	—	2	2	no	—
47	C₂²×C₃₁	—	2	2	no	—
48	C₂²×C₃₁	—	2	4	no	—
49	C₂²×C₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₃₁	—	2	8	no	—
52	C₂×D₃₁	—	2	2	no	—
53	C₂×D₃₁	—	2	2	no	—
54	C₂×D₃₁	#41	1	4	yes	complemented
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₄×C₃₁	C₂×D₄	1	2	yes	—
58	Dic₃₁	—	2	2	no	—
59	Dic₃₁	—	2	4	no	—
60	#525	—	31	1	no	maximal, Sylow
61	C₂×C₃₁	—	2	2	no	—
62	C₂×C₃₁	—	4	2	no	—
63	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₃₁	C₂×#41	1	2	yes	—
65	D₃₁	—	2	4	no	—
66	C₂×#41	D₃₁	1	1	yes	characteristic, complemented
67	C₂³×C₄	—	31	1	no	—
68	#123	—	31	2	no	—
69	C₂²⋊Q₈	—	31	2	no	—
70	#121	—	31	2	no	—
71	#120	—	31	2	no	—
72	C₄×D₄	—	31	4	no	—
73	#117	—	31	1	no	—
74	C₃₁	#525	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	—	31	1	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
78	C₂×D₄	—	31	4	no	—
79	C₂×D₄	C₂×D₃₁	1	2	yes	—
80	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic, complemented
81	C₂²×C₄	—	31	1	no	—
82	C₂²×C₄	—	31	2	no	—
83	C₂²×C₄	—	31	2	no	—
84	C₂²×C₄	—	62	4	no	—
85	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₃₁	1	2	yes	complemented
87	C₄⋊C₄	—	31	4	no	—
88	C₄⋊C₄	—	31	2	no	—
89	C₂²⋊C₄	—	31	2	no	—
90	C₂²⋊C₄	—	31	4	no	—
91	C₂²⋊C₄	—	31	4	no	—
92	C₄²	—	31	2	no	—
93	C₂³	—	31	2	no	—
94	C₂³	—	62	4	no	—
95	C₂³	—	62	2	no	—
96	C₂³	C₂²×D₃₁	1	2	yes	—
97	C₂³	C₂²×D₃₁	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	62	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	31	4	no	—
104	C₂×C₄	—	31	2	no	—
105	C₂×C₄	—	62	2	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	—	62	4	no	—
108	C₂×C₄	C₂²⋊D₃₁	1	4	yes	complemented
109	C₂×C₄	C₂²×D₃₁	1	2	yes	—
110	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	31	4	no	—
116	C₂²	—	62	8	no	—
117	C₂²	—	62	2	no	—
118	C₂²	—	62	2	no	—
119	C₂²	C₂×C₂²⋊D₃₁	1	2	yes	—
120	C₂²	C₂³×D₃₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	62	4	no	—
123	C₄	—	62	2	no	—
124	C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
125	C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	—	62	4	no	—
129	C₂	#41×D₃₁	1	2	yes	—
130	C₂	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#41×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₁

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, complemented

#11758515

C₂

yes

maximal, complemented

#11758514

C₂

yes

maximal, complemented

#11758512

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, complemented

#11758503

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₁

C₂²

yes

characteristic

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₁

C₂²

yes

—

C₂×D₄×C₃₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

complemented

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂²×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

complemented

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

complemented

Dic₃₁⋊C₄

C₂²

yes

complemented

C₄×Dic₃₁

C₂²

yes

complemented

C₂³×C₃₁

C₂³

yes

—

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₂³

yes

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

D₄

yes

complemented

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×D₄

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#41

yes

complemented

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

C₄×C₃₁

C₂×D₄

yes

—

Dic₃₁

—

Dic₃₁

—

#525

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

C₂×#41

yes

—

D₃₁

—

C₂×#41

D₃₁

yes

characteristic, complemented

C₂³×C₄

—

#123

—

C₂²⋊Q₈

—

#121

—

#120

—

C₄×D₄

—

#117

—

C₃₁

#525

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₃₁

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

—

C₂×D₄

C₂×D₃₁

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

C₂×D₃₁

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₃₁

yes

—

C₂³

C₂²×D₃₁

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²⋊D₃₁

yes

complemented

109

C₂×C₄

C₂²×D₃₁

yes

—

110

C₂×C₄

C₂²×D₃₁

yes

characteristic

111

C₂×C₄

C₂²×D₃₁

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×C₂²⋊D₃₁

yes

—

120

C₂²

C₂³×D₃₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×C₂²⋊D₃₁

yes

—

125

C₄

C₂×C₂²⋊D₃₁

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

#41×D₃₁

yes

—

130

C₂

C₂²×C₂²⋊D₃₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Frattini	C₂²	C₂³×D₃₁
Fitting	C₂×#41×C₃₁	C₂
hypercenter	C₂×#41	D₃₁
hypocenter	C₃₁	#525
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂×C₄

C₂²×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Frattini

C₂²

C₂³×D₃₁

Fitting

C₂×#41×C₃₁

C₂

hypercenter

C₂×#41

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#41×C₃₁	C₂	semidirect
2. C₂²×C₄×D₃₁	C₂	semidirect
3. #11758523	C₂	semidirect
4. #11758515	C₂	semidirect
5. #11758514	C₂	semidirect
6. #11758512	C₂	semidirect
7. C₄×C₂²⋊D₃₁	C₂	semidirect
8. #11758503	C₂	semidirect
11. C₂×Q₈×C₃₁	C₂²	semidirect
13. C₂×D₄×C₃₁	C₂²	semidirect
15. C₂²×C₄×C₃₁	C₂²	semidirect
22. C₂²⋊Dic₃₁	C₂²	semidirect
24. C₄⋊Dic₃₁	C₂²	semidirect
25. Dic₃₁⋊C₄	C₂²	semidirect
26. C₄×Dic₃₁	C₂²	semidirect
36. C₂×C₄×C₃₁	D₄	semidirect
54. C₂×D₃₁	#41	semidirect
66. C₂×#41	D₃₁	semidirect
74. C₃₁	#525	semidirect
77. C₂×Q₈	C₂×D₃₁	semidirect
80. C₂×D₄	C₂×D₃₁	semidirect
86. C₂²×C₄	C₂×D₃₁	semidirect
108. C₂×C₄	C₂²⋊D₃₁	semidirect

H ≅ G/N

type

1. C₂×#41×C₃₁

C₂

semidirect

2. C₂²×C₄×D₃₁

C₂

semidirect

3. #11758523

C₂

semidirect

4. #11758515

C₂

semidirect

5. #11758514

C₂

semidirect

6. #11758512

C₂

semidirect

7. C₄×C₂²⋊D₃₁

C₂

semidirect

8. #11758503

C₂

semidirect

11. C₂×Q₈×C₃₁

C₂²

semidirect

13. C₂×D₄×C₃₁

C₂²

semidirect

15. C₂²×C₄×C₃₁

C₂²

semidirect

22. C₂²⋊Dic₃₁

C₂²

semidirect

24. C₄⋊Dic₃₁

C₂²

semidirect

25. Dic₃₁⋊C₄

C₂²

semidirect

26. C₄×Dic₃₁

C₂²

semidirect

36. C₂×C₄×C₃₁

D₄

semidirect

54. C₂×D₃₁

#41

semidirect

66. C₂×#41

D₃₁

semidirect

74. C₃₁

#525

semidirect

77. C₂×Q₈

C₂×D₃₁

semidirect

80. C₂×D₄

C₂×D₃₁

semidirect

86. C₂²×C₄

C₂×D₃₁

semidirect

108. C₂×C₄

C₂²⋊D₃₁

semidirect

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

subgroup	index
C₂×#41×C₃₁	2
C₂²×C₄×D₃₁	2
#11758523	2
#11758515	2
#11758514	2
#11758512	2
C₄×C₂²⋊D₃₁	2
#11758503	2
#525	31

subgroup

index

C₂×#41×C₃₁

C₂²×C₄×D₃₁

C₄×C₂²⋊D₃₁

Subgroups — #14521556

Conjugacy classes of subgroups

Complements

Maximal subgroups