Group #14521624 — subgroups

Subgroups — #14521624

Order 1984.

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

normal subgroups only

This group has 270 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
2	C₂×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758524	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758521	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758473	C₂	1	4	yes	maximal, complemented
7	#11758471	C₂	1	4	yes	maximal, complemented
8	#11758452	C₂	1	1	yes	maximal, characteristic, complemented
9	#11758449	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	#92499	—	2	4	no	—
13	Q₈×D₃₁	—	2	4	no	—
14	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
15	C₂×D₁₂₄	C₂²	1	2	yes	—
16	C₂×C₄×D₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
20	C₂×Dic₆₂	C₂²	1	2	yes	—
21	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
22	C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
25	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	4	yes	—
28	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	C₂²×D₃₁	C₂³	1	2	yes	—
30	C₂²×D₃₁	C₂³	1	1	yes	characteristic
31	Q₈×C₃₁	—	2	2	no	—
32	Q₈×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	C₂³	1	2	yes	—
34	C₂×C₄×C₃₁	C₂³	1	2	yes	—
35	C₂×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×Dic₃₁	C₂³	1	2	yes	—
38	C₂×Dic₃₁	C₂³	1	2	yes	—
39	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
40	D₁₂₄	—	2	4	no	—
41	D₁₂₄	—	2	2	no	—
42	C₄×D₃₁	—	2	4	no	—
43	C₄×D₃₁	—	2	4	no	—
44	C₄×D₃₁	D₄	1	4	yes	—
45	Dic₆₂	—	2	2	no	—
46	Dic₆₂	—	2	4	no	—
47	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×D₃₁	—	2	2	no	—
49	C₂×D₃₁	—	2	2	no	—
50	C₂×D₃₁	—	2	4	no	—
51	C₂×D₃₁	C₂×D₄	1	2	yes	—
52	C₄×C₃₁	—	2	2	no	—
53	C₄×C₃₁	—	2	2	no	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	C₂×D₄	1	2	yes	—
56	Dic₃₁	—	2	2	no	—
57	Dic₃₁	—	2	2	no	—
58	Dic₃₁	C₂×D₄	1	2	yes	—
59	#536	—	31	1	no	maximal, Sylow
60	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
62	D₃₁	—	2	2	no	—
63	D₃₁	—	4	2	no	—
64	C₂×#41	—	31	1	no	—
65	C₂²×Q₈	—	31	1	no	—
66	C₄⋊Q₈	—	31	1	no	—
67	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
68	#124	—	31	1	no	—
69	#124	—	31	1	no	—
70	#123	—	31	4	no	—
71	C₂²⋊Q₈	—	31	4	no	—
72	#117	—	31	1	no	—
73	C₃₁	#536	1	1	yes	characteristic, complemented, Sylow
74	#41	—	62	4	no	—
75	C₂×Q₈	—	31	2	no	—
76	C₂×Q₈	—	31	1	no	—
77	C₂×Q₈	—	62	4	no	—
78	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
80	C₂×D₄	—	31	2	no	—
81	C₂×D₄	—	31	1	no	—
82	C₂²×C₄	—	31	1	no	—
83	C₂²×C₄	—	31	2	no	—
84	C₂²×C₄	—	31	2	no	—
85	C₄⋊C₄	—	31	2	no	—
86	C₄⋊C₄	—	31	4	no	—
87	C₄⋊C₄	C₂×D₃₁	1	4	yes	—
88	C₂²⋊C₄	—	31	2	no	—
89	C₂²⋊C₄	—	31	4	no	—
90	C₂²⋊C₄	—	31	4	no	—
91	C₄²	—	31	1	no	—
92	C₄²	C₂×D₃₁	1	1	yes	characteristic
93	C₂³	—	31	1	no	—
94	C₂³	—	31	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	62	4	no	—
98	Q₈	—	62	2	no	—
99	D₄	—	62	4	no	—
100	D₄	—	62	2	no	—
101	C₂×C₄	—	31	1	no	—
102	C₂×C₄	—	31	4	no	—
103	C₂×C₄	—	31	2	no	—
104	C₂×C₄	—	31	2	no	—
105	C₂×C₄	—	62	4	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	C₂²×D₃₁	1	2	yes	—
108	C₂×C₄	C₂²×D₃₁	1	2	yes	—
109	C₂×C₄	C₂²×D₃₁	1	2	yes	—
110	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
111	C₂²	—	31	2	no	—
112	C₂²	—	62	2	no	—
113	C₂²	—	62	2	no	—
114	C₂²	—	62	4	no	—
115	C₂²	C₂³×D₃₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	31	2	no	—
120	C₄	—	62	2	no	—
121	C₄	—	62	2	no	—
122	C₄	D₄×D₃₁	1	2	yes	—
123	C₂	—	62	2	no	—
124	C₂	—	124	2	no	—
125	C₂	C₂×D₄×D₃₁	1	1	yes	characteristic
126	C₂	#11758564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92499

C₂

yes

maximal, characteristic

C₂×Q₈×D₃₁

C₂

yes

maximal, characteristic, complemented

C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11758524

C₂

yes

maximal, characteristic, complemented

#11758521

C₂

yes

maximal, characteristic, complemented

#11758473

C₂

yes

maximal, complemented

#11758471

C₂

yes

maximal, complemented

#11758452

C₂

yes

maximal, characteristic, complemented

#11758449

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×Q₈×C₃₁

C₂²

yes

characteristic

#92499

—

Q₈×D₃₁

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×D₁₂₄

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×D₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

D₄

yes

—

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×D₄

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×D₄

yes

—

#536

—

maximal, Sylow

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

—

D₃₁

—

D₃₁

—

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₃₁

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

C₃₁

#536

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₃₁

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×D₃₁

yes

—

108

C₂×C₄

C₂²×D₃₁

yes

—

109

C₂×C₄

C₂²×D₃₁

yes

—

110

C₂×C₄

C₂²×D₃₁

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₃₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₃₁

yes

—

123

C₂

—

124

C₂

—

124

—

125

C₂

C₂×D₄×D₃₁

yes

characteristic

126

C₂

#11758564

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
prime radical	C₂×#92499	C₂
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#536
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×C₄⋊Q₈

C₂

prime radical

C₂×#92499

C₂

hypercenter

C₄⋊Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₃₁	C₂	semidirect
3. C₃₁×C₄⋊Q₈	C₂	semidirect
4. #11758524	C₂	semidirect
5. #11758521	C₂	semidirect
6. #11758473	C₂	semidirect
7. #11758471	C₂	semidirect
8. #11758452	C₂	semidirect
9. #11758449	C₂	semidirect
67. C₄⋊Q₈	D₃₁	semidirect
73. C₃₁	#536	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₃₁

C₂

semidirect

3. C₃₁×C₄⋊Q₈

C₂

semidirect

4. #11758524

C₂

semidirect

5. #11758521

C₂

semidirect

6. #11758473

C₂

semidirect

7. #11758471

C₂

semidirect

8. #11758452

C₂

semidirect

9. #11758449

C₂

semidirect

67. C₄⋊Q₈

D₃₁

semidirect

73. C₃₁

#536

semidirect

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

subgroup	index
C₂×#92499	2
C₂×Q₈×D₃₁	2
C₃₁×C₄⋊Q₈	2
#11758524	2
#11758521	2
#11758473	2
#11758471	2
#11758452	2
#11758449	2
#536	31

subgroup

index

C₂×#92499

C₂×Q₈×D₃₁

C₃₁×C₄⋊Q₈

Subgroups — #14521624

Conjugacy classes of subgroups

Complements

Maximal subgroups