Group #16859113 — subgroups

Subgroups — #16859113

Order 3968.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#14521678	C₂	1	12	yes	maximal, complemented
2	#14521675	C₂	1	3	yes	maximal
3	#14521601	C₂	1	12	yes	maximal, complemented
4	C₂×#11758454	C₂	1	3	yes	maximal
5	C₃₁×#543	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	—
8	C₂×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₃₁	C₂²	1	3	yes	—
10	Q₈×Dic₃₁	C₂²	1	4	yes	complemented
11	#11758521	C₂²	1	12	yes	complemented
12	#11758501	C₂²	1	12	yes	—
13	C₂×Dic₃₁⋊C₄	C₂²	1	6	yes	—
14	C₂×C₄×Dic₃₁	C₂²	1	6	yes	—
15	#11758466	C₂²	1	24	yes	complemented
16	#11758465	C₂²	1	12	yes	complemented
17	#11758464	C₂²	1	12	yes	complemented
18	#11758455	C₂²	1	24	yes	—
19	#11758454	C₂²	1	12	yes	—
20	#11758454	C₂²	1	12	yes	—
21	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
24	C₂²×Dic₃₁	Q₈	1	8	yes	complemented
25	C₂²×Dic₃₁	C₂³	1	6	yes	—
26	C₂²×Dic₃₁	C₂³	1	3	yes	—
27	C₂²×Dic₃₁	C₂³	1	3	yes	—
28	C₂×Dic₆₂	C₂³	1	12	yes	—
29	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
30	C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
31	C₂²⋊Dic₃₁	C₂³	1	12	yes	—
32	C₄⋊Dic₃₁	C₂³	1	12	yes	—
33	Dic₃₁⋊C₄	C₂³	1	24	yes	—
34	Dic₃₁⋊C₄	C₂³	1	24	yes	—
35	C₄×Dic₃₁	C₂³	1	12	yes	—
36	C₄×Dic₃₁	C₂³	1	12	yes	—
37	C₂³×C₃₁	—	4	2	no	—
38	C₂³×C₃₁	C₂×Q₈	1	4	yes	—
39	C₂³×C₃₁	C₂⁴	1	3	yes	—
40	Q₈×C₃₁	—	4	4	no	—
41	C₂×C₄×C₃₁	—	4	6	no	—
42	C₂×C₄×C₃₁	C₂⁴	1	12	yes	—
43	C₂×Dic₃₁	—	4	6	no	—
44	C₂×Dic₃₁	C₂×Q₈	1	12	yes	—
45	C₂×Dic₃₁	C₂×Q₈	1	12	yes	—
46	C₂×Dic₃₁	C₂⁴	1	12	yes	—
47	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₁	C₂⁴	1	3	yes	—
49	Dic₆₂	—	4	12	no	—
50	#3458	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	—	2	6	no	—
52	C₂²×C₃₁	—	4	4	no	—
53	C₂²×C₃₁	C₂²×Q₈	1	6	yes	—
54	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
55	C₄×C₃₁	—	2	12	no	—
56	Dic₃₁	—	2	12	no	—
57	Dic₃₁	C₂²×Q₈	1	2	yes	—
58	Dic₃₁	C₂²×Q₈	1	6	yes	—
59	#544	—	31	12	no	—
60	#543	—	31	3	no	—
61	#543	D₃₁	1	1	yes	characteristic
62	#533	—	31	12	no	—
63	C₂×#117	—	31	3	no	—
64	C₂×C₃₁	—	2	6	no	—
65	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
66	C₂×C₃₁	#585	1	2	yes	—
67	C₂³×C₄	—	31	1	no	—
68	C₄⋊Q₈	—	31	12	no	—
69	C₄⋊Q₈	—	31	12	no	—
70	#125	—	31	24	no	—
71	C₂²⋊Q₈	—	31	12	no	—
72	C₂²⋊Q₈	—	31	24	no	—
73	C₂²⋊Q₈	C₂×D₃₁	1	12	yes	—
74	C₄×Q₈	—	31	4	no	—
75	C₄×Q₈	—	31	12	no	—
76	#117	—	31	12	no	—
77	#117	—	31	12	no	—
78	C₂×C₄⋊C₄	—	31	6	no	—
79	C₂×C₂²⋊C₄	—	31	3	no	—
80	C₂×C₂²⋊C₄	C₂×D₃₁	1	3	yes	—
81	C₂×C₄²	—	31	6	no	—
82	C₃₁	#3458	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
84	C₂×Q₈	—	31	12	no	—
85	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
86	C₂²×C₄	—	31	8	no	—
87	C₂²×C₄	—	31	3	no	—
88	C₂²×C₄	—	31	3	no	—
89	C₂²×C₄	—	31	6	no	—
90	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
91	C₄⋊C₄	—	31	24	no	—
92	C₄⋊C₄	—	31	24	no	—
93	C₄⋊C₄	—	31	12	no	—
94	C₄⋊C₄	C₂²×D₃₁	1	12	yes	—
95	C₂²⋊C₄	—	31	12	no	—
96	C₂²⋊C₄	C₂²×D₃₁	1	12	yes	—
97	C₄²	—	31	12	no	—
98	C₄²	—	31	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₃₁	1	4	yes	—
101	C₂³	C₂³×D₃₁	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	124	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	31	12	no	—
106	C₂×C₄	—	31	12	no	—
107	C₂×C₄	—	31	1	no	—
108	C₂×C₄	—	31	3	no	—
109	C₂×C₄	—	31	12	no	—
110	C₂×C₄	—	124	6	no	—
111	C₂×C₄	C₂³×D₃₁	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₃₁	1	6	yes	—
115	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	31	2	no	—
118	C₄	—	31	6	no	—
119	C₄	—	62	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₃₁	1	1	yes	characteristic
122	C₂	#14521884	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521678

C₂

yes

maximal, complemented

#14521675

C₂

yes

maximal

#14521601

C₂

yes

maximal, complemented

C₂×#11758454

C₂

yes

maximal

C₃₁×#543

C₂

yes

maximal, characteristic

C₂³×Dic₃₁

C₂²

yes

characteristic

C₃₁×C₂²⋊Q₈

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

Q₈×Dic₃₁

C₂²

yes

complemented

#11758521

C₂²

yes

complemented

#11758501

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758466

C₂²

yes

complemented

#11758465

C₂²

yes

complemented

#11758464

C₂²

yes

complemented

#11758455

C₂²

yes

—

#11758454

C₂²

yes

—

#11758454

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

Q₈

yes

complemented

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

C₂³

yes

—

Dic₃₁⋊C₄

C₂³

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×Q₈

yes

—

C₂³×C₃₁

C₂⁴

yes

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

C₂⁴

yes

—

C₂×Dic₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

C₂⁴

yes

—

Dic₆₂

—

#3458

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²×Q₈

yes

—

C₂²×C₃₁

C₂⁵

yes

characteristic

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

C₂²×Q₈

yes

—

Dic₃₁

C₂²×Q₈

yes

—

#544

—

#543

—

#543

D₃₁

yes

characteristic

#533

—

C₂×#117

—

C₂×C₃₁

—

C₂×C₃₁

C₂³×Q₈

yes

characteristic

C₂×C₃₁

#585

yes

—

C₂³×C₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₃₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₃₁

#3458

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₃₁

yes

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

Q₈×D₃₁

yes

—

101

C₂³

C₂³×D₃₁

yes

—

102

Q₈

—

103

Q₈

—

124

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

124

—

111

C₂×C₄

C₂³×D₃₁

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₃₁

yes

—

115

C₂²

C₂⁴×D₃₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₂

—

121

C₂

C₂²×Q₈×D₃₁

yes

characteristic

122

C₂

#14521884

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁵
Fitting	C₃₁×#543	C₂
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	#543	D₃₁
hypocenter	C₃₁	#3458
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁵

Fitting

C₃₁×#543

C₂

prime radical

C₂⁴×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #14521678	C₂	semidirect
3. #14521601	C₂	semidirect
10. Q₈×Dic₃₁	C₂²	semidirect
11. #11758521	C₂²	semidirect
15. #11758466	C₂²	semidirect
16. #11758465	C₂²	semidirect
17. #11758464	C₂²	semidirect
24. C₂²×Dic₃₁	Q₈	semidirect
82. C₃₁	#3458	semidirect

H ≅ G/N

type

1. #14521678

C₂

semidirect

3. #14521601

C₂

semidirect

10. Q₈×Dic₃₁

C₂²

semidirect

11. #11758521

C₂²

semidirect

15. #11758466

C₂²

semidirect

16. #11758465

C₂²

semidirect

17. #11758464

C₂²

semidirect

24. C₂²×Dic₃₁

Q₈

semidirect

82. C₃₁

#3458

semidirect

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

subgroup	index
#14521678	2
#14521675	2
#14521601	2
C₂×#11758454	2
C₃₁×#543	2
#3458	31

subgroup

index

C₃₁×#543

Subgroups — #16859113

Conjugacy classes of subgroups

Complements

Maximal subgroups