Group #20241665 — subgroups

Subgroups — C₈×#10625324 #20241665

Order 6016.

224 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 125 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	#15689174	C₂	1	1	yes	maximal, characteristic
2	C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#15688845	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625324	C₄	1	2	yes	—
9	C₂×C₈×D₄₇	C₄	1	2	yes	—
10	C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	Dic₄₇⋊C₈	C₄	1	2	yes	—
12	(C₂×D₄₇)⋊C₈	C₄	1	2	yes	—
13	#13093934	C₄	1	2	yes	—
14	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₈×Dic₄₇	C₄	1	2	yes	—
18	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
34	#10625324	C₈	1	8	yes	direct product factor
35	C₈×D₄₇	—	2	4	no	—
36	C₂²×D₄₇	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₄₇	C₄²	1	1	yes	characteristic
41	C₂×Dic₄₇	C₄²	1	1	yes	characteristic
42	C₂×Dic₄₇	C₄²	1	1	yes	characteristic
43	C₄×D₄₇	—	2	2	no	—
44	C₄×D₄₇	—	2	2	no	—
45	C₄×D₄₇	C₂×C₈	1	4	yes	—
46	C₈×C₄₇	—	2	2	no	—
47	C₈×C₄₇	C₂×C₈	1	4	yes	complemented
48	C₈×C₄₇	M₁₆	1	4	yes	complemented
49	C₄₇⋊C₈	—	2	2	no	—
50	C₄₇⋊C₈	—	2	2	no	—
51	C₄₇⋊C₈	C₂×C₈	1	4	yes	complemented
52	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₄₇	—	2	1	no	—
54	C₂×D₄₇	—	2	1	no	—
55	C₂×D₄₇	C₄×C₈	1	2	yes	—
56	C₄×C₄₇	C₂²×C₈	1	2	yes	—
57	C₄×C₄₇	C₂×M₁₆	1	2	yes	—
58	C₄×C₄₇	#131	1	2	yes	—
59	Dic₄₇	—	2	1	no	—
60	Dic₄₇	—	2	1	no	—
61	Dic₄₇	C₄×C₈	1	2	yes	—
62	C₈×M₁₆	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
65	C₂×C₄₇	#405	1	1	yes	characteristic
66	D₄₇	—	2	2	no	—
67	#431	—	47	1	no	—
68	C₄×M₁₆	—	47	1	no	—
69	C₂×C₄×C₈	—	47	1	no	—
70	#322	—	47	1	no	—
71	#322	—	47	1	no	—
72	C₈²	—	47	1	no	—
73	C₈²	D₄₇	1	1	yes	characteristic, complemented
74	C₄₇	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	47	2	no	—
76	C₂²×C₈	—	47	2	no	—
77	C₂×C₄²	—	47	1	no	—
78	C₄⋊C₈	—	47	2	no	—
79	C₂²⋊C₈	—	47	2	no	—
80	#97	—	47	2	no	—
81	C₄×C₈	—	47	2	no	—
82	C₄×C₈	—	47	1	no	—
83	C₄×C₈	—	47	1	no	—
84	C₄×C₈	—	47	1	no	—
85	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
86	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₂²×C₄	—	47	1	no	—
89	C₂²×C₄	—	47	1	no	—
90	C₂²×C₄	—	47	1	no	—
91	M₁₆	—	47	8	no	—
92	C₂×C₈	—	47	2	no	—
93	C₂×C₈	—	47	2	no	—
94	C₂×C₈	—	47	2	no	—
95	C₂×C₈	—	94	4	no	—
96	C₂×C₈	C₄×D₄₇	1	2	yes	—
97	C₂×C₈	C₄×D₄₇	1	2	yes	—
98	C₂×C₈	C₄×D₄₇	1	2	yes	—
99	C₄²	—	47	1	no	—
100	C₄²	—	47	1	no	—
101	C₄²	—	47	1	no	—
102	C₄²	C₂²×D₄₇	1	1	yes	characteristic
103	C₂³	—	47	1	no	—
104	C₂×C₄	—	47	1	no	—
105	C₂×C₄	—	47	4	no	—
106	C₂×C₄	—	47	1	no	—
107	C₂×C₄	—	47	1	no	—
108	C₂×C₄	—	94	2	no	—
109	C₂×C₄	—	94	2	no	—
110	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	47	4	no	—
115	C₈	—	94	2	no	—
116	C₈	—	94	2	no	—
117	C₈	C₈×D₄₇	1	4	yes	complemented
118	C₈	#10625324	1	4	yes	direct product factor
119	C₂²	—	47	2	no	—
120	C₂²	—	94	1	no	—
121	C₂²	—	94	1	no	—
122	C₂²	C₄²×D₄₇	1	1	yes	characteristic
123	C₄	—	47	2	no	—
124	C₄	—	94	1	no	—
125	C₄	—	94	1	no	—
126	C₄	C₂×C₈×D₄₇	1	2	yes	—
127	C₄	C₂×#10625324	1	2	yes	—
128	C₄	#13094030	1	2	yes	—
129	C₂	—	94	2	no	—
130	C₂	C₄×C₈×D₄₇	1	1	yes	characteristic
131	C₂	C₄×#10625324	1	1	yes	characteristic
132	C₂	#15689077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689174

C₂

yes

maximal, characteristic

C₄×#10625324

C₂

yes

maximal, characteristic

C₄×C₈×D₄₇

C₂

yes

maximal, characteristic

#15688845

C₂

yes

maximal, characteristic, complemented

#15688844

C₂

yes

maximal, characteristic, complemented

C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₈²×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625324

C₄

yes

—

C₂×C₈×D₄₇

C₄

yes

—

C₄²×D₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₄

yes

—

(C₂×D₄₇)⋊C₈

C₄

yes

—

#13093934

C₄

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

C₄

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

#10625324

C₈

yes

direct product factor

C₈×D₄₇

—

C₂²×D₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×Dic₄₇

C₄²

yes

characteristic

C₂×Dic₄₇

C₄²

yes

characteristic

C₂×Dic₄₇

C₄²

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

C₂×C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₈×C₄₇

M₁₆

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

complemented

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₄×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

#131

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₄×C₈

yes

—

C₈×M₁₆

—

maximal, Sylow

C₂×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

#405

yes

characteristic

D₄₇

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₈²

—

C₈²

D₄₇

yes

characteristic, complemented

C₄₇

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄₇

yes

—

C₂×C₈

C₄×D₄₇

yes

—

C₂×C₈

C₄×D₄₇

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₄₇

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

111

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

112

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

C₈×D₄₇

yes

complemented

118

C₈

#10625324

yes

direct product factor

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄²×D₄₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₈×D₄₇

yes

—

127

C₄

C₂×#10625324

yes

—

128

C₄

#13094030

yes

—

129

C₂

—

130

C₂

C₄×C₈×D₄₇

yes

characteristic

131

C₂

C₄×#10625324

yes

characteristic

132

C₂

#15689077

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄²
hypercenter	C₈²	D₄₇
hypocenter	C₄₇	C₈×M₁₆
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×D₄₇

commutator

C₂×C₄₇

C₂×C₄×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈²×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₄²

hypercenter

C₈²

D₄₇

hypocenter

C₄₇

C₈×M₁₆

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
4. #15688845	C₂	semidirect
5. #15688844	C₂	semidirect
6. C₈×C₄₇⋊C₈	C₂	semidirect
7. C₈²×C₄₇	C₂	semidirect
34. #10625324	C₈	direct (N × H)
47. C₈×C₄₇	C₂×C₈	semidirect
48. C₈×C₄₇	M₁₆	semidirect
51. C₄₇⋊C₈	C₂×C₈	semidirect
73. C₈²	D₄₇	semidirect
74. C₄₇	C₈×M₁₆	semidirect
117. C₈	C₈×D₄₇	semidirect
118. C₈	#10625324	direct (N × H)

H ≅ G/N

type

4. #15688845

C₂

semidirect

5. #15688844

C₂

semidirect

6. C₈×C₄₇⋊C₈

C₂

semidirect

7. C₈²×C₄₇

C₂

semidirect

34. #10625324

C₈

direct (N × H)

47. C₈×C₄₇

C₂×C₈

semidirect

48. C₈×C₄₇

M₁₆

semidirect

51. C₄₇⋊C₈

C₂×C₈

semidirect

73. C₈²

D₄₇

semidirect

74. C₄₇

C₈×M₁₆

semidirect

117. C₈

C₈×D₄₇

semidirect

118. C₈

#10625324

direct (N × H)

Subgroups — C₈×#10625324 #20241665

Conjugacy classes of subgroups

Complements

Subgroups — C8×#10625324 #20241665

Conjugacy classes of subgroups

Complements

Subgroups — C₈×#10625324 #20241665