Group #20246006 — subgroups

Subgroups — #20246006

Order 6016.

600 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 309 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×#13093983	C₂	1	2	yes	maximal
2	#15689022	C₂	1	8	yes	maximal, complemented
3	#15688983	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#13093983	—	2	8	no	—
6	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	8	yes	—
8	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
10	C₄₇⋊#95	C₂²	1	8	yes	—
11	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
12	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₄₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
15	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	—	4	4	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₄₇	D₄	1	4	yes	—
20	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂²×Dic₄₇	C₂³	1	8	yes	—
24	C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
25	C₂²⋊Dic₄₇	—	2	8	no	—
26	C₄⋊Dic₄₇	—	2	8	no	—
27	C₄⋊Dic₄₇	—	4	8	no	—
28	C₄×Dic₄₇	—	2	8	no	—
29	C₂³×C₄₇	—	2	2	no	—
30	C₂³×C₄₇	—	2	2	no	—
31	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
32	C₂³×C₄₇	C₂×D₄	1	2	yes	—
33	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
34	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	8	no	—
36	C₂×C₄×C₄₇	—	4	8	no	—
37	C₂×C₄×C₄₇	C₄⋊C₄	1	16	yes	complemented
38	C₂×C₄×C₄₇	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
42	C₂×Dic₄₇	—	2	8	no	—
43	C₂×Dic₄₇	—	2	8	no	—
44	C₂×Dic₄₇	—	2	8	no	—
45	C₂²×C₄₇	—	2	2	no	—
46	C₂²×C₄₇	—	2	2	no	—
47	C₂²×C₄₇	—	2	4	no	—
48	C₂²×C₄₇	—	2	4	no	—
49	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
50	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
51	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
55	C₄×C₄₇	—	2	8	no	—
56	C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
57	Dic₄₇	—	4	8	no	—
58	#2316	—	47	1	no	maximal, Sylow
59	C₂×C₄₇	—	2	4	no	—
60	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄₇	#518	1	1	yes	characteristic
62	C₂×C₄₇	#519	1	1	yes	characteristic
63	C₂×C₄₇	#537	1	2	yes	—
64	C₂×C₄₇	#544	1	2	yes	—
65	C₂×#117	—	47	2	no	—
66	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
67	#389	—	47	8	no	—
68	#380	—	47	4	no	—
69	C₄₇	#2316	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	C₂×D₄₇	1	2	yes	—
71	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
72	#117	—	94	8	no	—
73	C₂×C₄⋊C₄	—	47	4	no	—
74	C₂×C₄⋊C₄	—	47	8	no	—
75	C₂×C₄⋊C₄	Dic₄₇	1	8	yes	complemented
76	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
77	C₂×C₂²⋊C₄	—	47	4	no	—
78	C₂×C₄²	—	47	4	no	—
79	#95	—	47	8	no	—
80	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	—	47	8	no	—
83	C₂²×C₄	Dic₉₄	1	4	yes	—
84	C₂²×C₄	D₁₈₈	1	4	yes	—
85	C₂²×C₄	C₂×Dic₄₇	1	8	yes	—
86	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
87	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
88	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
89	C₄⋊C₄	—	94	8	no	—
90	C₄⋊C₄	—	188	8	no	—
91	C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
92	C₂²⋊C₄	—	94	8	no	—
93	C₄²	—	94	8	no	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Dic₉₄	1	2	yes	—
97	C₂³	C₂×D₁₈₈	1	2	yes	—
98	C₂³	C₂²×Dic₄₇	1	2	yes	—
99	C₂³	C₂³×D₄₇	1	1	yes	characteristic
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	4	8	no	—
102	C₂×C₄	—	94	8	no	—
103	C₂×C₄	—	94	8	no	—
104	C₂×C₄	—	94	8	no	—
105	C₂×C₄	C₄⋊Dic₄₇	1	16	yes	complemented
106	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
107	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
108	C₂×C₄	C₂²×Dic₄₇	1	8	yes	—
109	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	C₂×C₄⋊Dic₄₇	1	4	yes	—
115	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
116	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
117	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
118	C₂²	#13094069	1	2	yes	—
119	C₂²	#13094072	1	2	yes	—
120	C₄	—	2	8	no	—
121	C₄	—	188	8	no	—
122	C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
123	C₂	—	2	4	no	—
124	C₂	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
125	C₂	#15689598	1	1	yes	characteristic
126	C₂	#15689601	1	1	yes	characteristic
127	C₂	#15689704	1	2	yes	—
128	C₂	#15689754	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal

#15689022

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

#13093983

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

#2316

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#544

yes

—

C₂×#117

—

C₂²×C₄⋊C₄

D₄₇

yes

characteristic

#389

—

#380

—

C₄₇

#2316

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

—

C₂²×C₄

D₁₈₈

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₉₄

yes

—

C₂³

C₂×D₁₈₈

yes

—

C₂³

C₂²×Dic₄₇

yes

—

C₂³

C₂³×D₄₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄⋊Dic₄₇

yes

complemented

106

C₂×C₄

C₂×Dic₉₄

yes

—

107

C₂×C₄

C₂×D₁₈₈

yes

—

108

C₂×C₄

C₂²×Dic₄₇

yes

—

109

C₂×C₄

C₂²×Dic₄₇

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₄⋊Dic₄₇

yes

—

115

C₂²

C₂³×Dic₄₇

yes

characteristic

116

C₂²

C₂²×D₁₈₈

yes

characteristic

117

C₂²

C₂²×Dic₉₄

yes

characteristic

118

C₂²

#13094069

yes

—

119

C₂²

#13094072

yes

—

120

C₄

—

121

C₄

—

188

—

122

C₄

C₂×C₄⋊Dic₄₇

yes

—

123

C₂

—

124

C₂

C₂²×C₄⋊Dic₄₇

yes

characteristic

125

C₂

#15689598

yes

characteristic

126

C₂

#15689601

yes

characteristic

127

C₂

#15689704

yes

—

128

C₂

#15689754

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2316
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #15689022	C₂	semidirect
13. C₂×C₄⋊C₄×C₄₇	C₄	semidirect
37. C₂×C₄×C₄₇	C₄⋊C₄	semidirect
69. C₄₇	#2316	semidirect
75. C₂×C₄⋊C₄	Dic₄₇	semidirect
105. C₂×C₄	C₄⋊Dic₄₇	semidirect

H ≅ G/N

type

2. #15689022

C₂

semidirect

13. C₂×C₄⋊C₄×C₄₇

C₄

semidirect

37. C₂×C₄×C₄₇

C₄⋊C₄

semidirect

69. C₄₇

#2316

semidirect

75. C₂×C₄⋊C₄

Dic₄₇

semidirect

105. C₂×C₄

C₄⋊Dic₄₇

semidirect

Subgroups — #20246006

Conjugacy classes of subgroups

Complements