Group #20248417 — subgroups

Subgroups — #20248417

Order 6016.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13094023	C₂	1	4	yes	maximal
2	#15689043	C₂	1	4	yes	maximal
3	#15689019	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
5	C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	4	yes	—
7	#13094023	—	2	16	no	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
11	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
12	C₄₇⋊#95	C₂²	1	4	yes	—
13	C₄₇⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
20	C₂²×Dic₄₇	C₂³	1	4	yes	—
21	C₂²×Dic₄₇	C₂³	1	4	yes	—
22	C₂×Dic₉₄	—	2	16	no	—
23	C₂×Dic₉₄	—	4	4	no	—
24	C₄⋊C₄×C₄₇	—	2	8	no	—
25	C₄⋊C₄×C₄₇	—	4	2	no	—
26	C₄²×C₄₇	Q₈	1	4	yes	complemented
27	C₄⋊Dic₄₇	D₄	1	8	yes	—
28	C₄⋊Dic₄₇	Q₈	1	8	yes	complemented
29	Dic₄₇⋊C₄	—	2	16	no	—
30	C₄×Dic₄₇	—	2	8	no	—
31	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₄₇	—	2	1	no	—
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	—	2	8	no	—
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
40	C₂×Dic₄₇	—	2	4	no	—
41	C₂×Dic₄₇	—	2	4	no	—
42	C₂×Dic₄₇	—	2	4	no	—
43	C₂×Dic₄₇	—	2	4	no	—
44	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
45	C₂×Dic₄₇	C₂×Q₈	1	8	yes	—
46	Dic₉₄	—	4	16	no	—
47	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
48	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
49	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	—	4	4	no	—
54	C₄×C₄₇	C₄⋊Q₈	1	8	yes	complemented
55	Dic₄₇	—	2	8	no	—
56	Dic₄₇	—	2	8	no	—
57	#2551	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
59	C₂×C₄₇	#544	1	1	yes	characteristic
60	C₂×C₄₇	D₄²	1	1	yes	characteristic
61	C₂×C₄₇	D₄×Q₈	1	2	yes	—
62	C₂×C₄₇	Q₈²	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	—	47	4	no	—
64	#395	—	47	4	no	—
65	#389	—	47	4	no	—
66	#384	D₄₇	1	1	yes	characteristic
67	C₄×C₄⋊C₄	—	47	2	no	—
68	C₄₇	#2551	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	47	4	no	—
70	C₄⋊Q₈	—	94	16	no	—
71	C₂×C₄⋊C₄	—	47	2	no	—
72	C₂×C₄⋊C₄	—	47	2	no	—
73	C₂×C₄⋊C₄	—	47	8	no	—
74	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
76	C₂×C₄²	—	47	4	no	—
77	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
78	#95	—	47	4	no	—
79	#95	—	47	4	no	—
80	C₂×Q₈	—	94	16	no	—
81	C₂×Q₈	—	188	4	no	—
82	C₂²×C₄	—	47	4	no	—
83	C₂²×C₄	—	47	4	no	—
84	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
85	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	47	8	no	—
90	C₄⋊C₄	—	47	8	no	—
91	C₄⋊C₄	—	94	16	no	—
92	C₄²	—	94	8	no	—
93	C₄²	Dic₉₄	1	4	yes	complemented
94	C₂³	C₂³×D₄₇	1	1	yes	characteristic
95	Q₈	—	188	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	47	8	no	—
101	C₂×C₄	—	47	8	no	—
102	C₂×C₄	—	94	4	no	—
103	C₂×C₄	—	94	4	no	—
104	C₂×C₄	—	94	4	no	—
105	C₂×C₄	—	94	4	no	—
106	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
107	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
108	C₂×C₄	D₄×D₄₇	1	4	yes	—
109	C₂×C₄	Q₈×D₄₇	1	4	yes	—
110	C₂²	C₂×D₄×D₄₇	1	2	yes	—
111	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
112	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
113	C₂²	#13094069	1	1	yes	characteristic
114	C₂²	#13094072	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	94	8	no	—
118	C₄	—	94	8	no	—
119	C₄	#13094023	1	8	yes	complemented
120	C₂	C₂×#13094023	1	2	yes	—
121	C₂	#15689754	1	1	yes	characteristic
122	C₂	#15689758	1	1	yes	characteristic
123	C₂	#15689788	1	2	yes	—
124	C₂	#15689853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094023

C₂

yes

maximal

#15689043

C₂

yes

maximal

#15689019

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal

C₄₇×#384

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094023

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

Q₈

yes

complemented

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

Q₈

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

—

#2551

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊Q₈

yes

—

C₂×C₄₇

#544

yes

characteristic

C₂×C₄₇

D₄²

yes

characteristic

C₂×C₄₇

D₄×Q₈

yes

—

C₂×C₄₇

Q₈²

yes

characteristic

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₄₇

yes

characteristic

C₄×C₄⋊C₄

—

C₄₇

#2551

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₉₄

yes

complemented

C₂³

C₂³×D₄₇

yes

characteristic

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₉₄

yes

—

107

C₂×C₄

C₂×Dic₉₄

yes

—

108

C₂×C₄

D₄×D₄₇

yes

—

109

C₂×C₄

Q₈×D₄₇

yes

—

110

C₂²

C₂×D₄×D₄₇

yes

—

111

C₂²

C₂×Q₈×D₄₇

yes

—

112

C₂²

C₂²×Dic₉₄

yes

characteristic

113

C₂²

#13094069

yes

characteristic

114

C₂²

#13094072

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#13094023

yes

complemented

120

C₂

C₂×#13094023

yes

—

121

C₂

#15689754

yes

characteristic

122

C₂

#15689758

yes

characteristic

123

C₂

#15689788

yes

—

124

C₂

#15689853

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#384	C₂
hypercenter	#384	D₄₇
hypocenter	C₄₇	#2551
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#384

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₄₇	Q₈	semidirect
28. C₄⋊Dic₄₇	Q₈	semidirect
54. C₄×C₄₇	C₄⋊Q₈	semidirect
68. C₄₇	#2551	semidirect
93. C₄²	Dic₉₄	semidirect
119. C₄	#13094023	semidirect

H ≅ G/N

type

26. C₄²×C₄₇

Q₈

semidirect

28. C₄⋊Dic₄₇

Q₈

semidirect

54. C₄×C₄₇

C₄⋊Q₈

semidirect

68. C₄₇

#2551

semidirect

93. C₄²

Dic₉₄

semidirect

119. C₄

#13094023

semidirect

Subgroups — #20248417

Conjugacy classes of subgroups

Complements