Group #20252014 — subgroups

Subgroups — C₂×#15689184 #20252014

Order 6016.

412 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 229 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	#15689184	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
10	C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
11	#13093935	C₂²	1	4	yes	complemented
12	C₄×C₄₇⋊C₈	C₂²	1	4	yes	complemented
13	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	#117×C₄₇	C₄	1	4	yes	—
15	#117×C₄₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	—	2	4	no	—
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₄²×C₄₇	C₂³	1	2	yes	—
33	C₄²×C₄₇	C₂³	1	2	yes	—
34	C₂×C₄₇⋊C₈	—	2	4	no	—
35	C₂×C₄₇⋊C₈	—	2	4	no	—
36	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
37	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
38	C₂³×C₄₇	—	2	2	no	—
39	C₂³×C₄₇	—	2	4	no	—
40	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	4	yes	—
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₄₇⋊C₈	—	2	4	no	—
54	C₄₇⋊C₈	—	2	4	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	2	4	no	—
57	C₂²×C₄₇	—	4	4	no	—
58	C₂²×C₄₇	#117	1	2	yes	—
59	C₂²×C₄₇	#131	1	4	yes	—
60	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	2	4	no	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	#117	1	2	yes	—
65	C₄×C₄₇	C₂×#41	1	2	yes	—
66	C₂×#433	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	—	4	2	no	—
68	C₂×C₄₇	#433	1	4	yes	complemented
69	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
70	C₂×C₄₇	C₂×#131	1	2	yes	—
71	C₂×#117	D₄₇	1	1	yes	characteristic
72	#433	—	47	8	no	—
73	C₂×C₄⋊C₈	—	47	2	no	—
74	C₂×C₂²⋊C₈	—	47	2	no	—
75	C₂×#97	—	47	1	no	—
76	C₂×C₄×C₈	—	47	1	no	—
77	C₄₇	C₂×#433	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
79	C₂²×C₈	—	47	2	no	—
80	C₂²×C₈	—	47	2	no	—
81	#117	Dic₄₇	1	4	yes	—
82	#117	C₂×D₄₇	1	4	yes	—
83	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
84	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
85	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	47	8	no	—
88	C₂²⋊C₈	—	47	8	no	—
89	#97	—	47	4	no	—
90	C₄×C₈	—	47	4	no	—
91	C₂⁴	C₂×Dic₄₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
94	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
95	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₈	—	47	4	no	—
100	C₂×C₈	—	47	4	no	—
101	C₂×C₈	—	94	4	no	—
102	C₂×C₈	—	94	4	no	—
103	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
104	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
105	C₄²	C₂×Dic₄₇	1	2	yes	—
106	C₄²	C₂×Dic₄₇	1	2	yes	—
107	C₄²	C₂²×D₄₇	1	2	yes	—
108	C₄²	C₂²×D₄₇	1	2	yes	—
109	C₂³	—	2	2	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂²×Dic₄₇	1	2	yes	—
112	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#10625350	1	4	yes	—
119	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
120	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
121	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
122	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
124	C₈	—	94	4	no	—
125	C₈	—	94	4	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	#13093983	1	2	yes	—
130	C₂²	#13094031	1	4	yes	—
131	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	4	no	—
134	C₄	—	2	4	no	—
135	C₄	#13093983	1	2	yes	—
136	C₄	C₂×#10625350	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	#15689184	1	4	yes	direct product factor
139	C₂	C₂×#13093983	1	1	yes	characteristic
140	C₂	C₂×#13094031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689184

C₂

yes

maximal, direct product factor

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal

C₂×#13093935

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093935

C₂²

yes

complemented

C₄×C₄₇⋊C₈

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₄

yes

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

#131

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

C₂×#41

yes

—

C₂×#433

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#117

D₄₇

yes

characteristic

#433

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

C₄₇

C₂×#433

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

Dic₄₇

yes

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₂²⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

C₂×Dic₄₇

yes

—

104

C₂²⋊C₄

C₂×Dic₄₇

yes

—

105

C₄²

C₂×Dic₄₇

yes

—

106

C₄²

C₂×Dic₄₇

yes

—

107

C₄²

C₂²×D₄₇

yes

—

108

C₄²

C₂²×D₄₇

yes

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂²×Dic₄₇

yes

—

112

C₂³

C₂²×Dic₄₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#10625350

yes

—

119

C₂×C₄

C₂²×Dic₄₇

yes

—

120

C₂×C₄

C₂²×Dic₄₇

yes

—

121

C₂×C₄

C₂²×Dic₄₇

yes

—

122

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

123

C₂×C₄

C₂³×D₄₇

yes

characteristic

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#13093983

yes

—

130

C₂²

#13094031

yes

—

131

C₂²

C₂³×Dic₄₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#13093983

yes

—

136

C₄

C₂×#10625350

yes

—

137

C₂

—

138

C₂

#15689184

yes

direct product factor

139

C₂

C₂×#13093983

yes

characteristic

140

C₂

C₂×#13094031

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #15689184	C₂	direct (N × H)
2. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. C₂×#13093935	C₂	semidirect
5. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
9. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
11. #13093935	C₂²	semidirect
12. C₄×C₄₇⋊C₈	C₂²	semidirect
68. C₂×C₄₇	#433	semidirect
77. C₄₇	C₂×#433	semidirect
138. C₂	#15689184	direct (N × H)

H ≅ G/N

type

1. #15689184

C₂

direct (N × H)

2. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

4. C₂×#13093935

C₂

semidirect

5. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

9. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

11. #13093935

C₂²

semidirect

12. C₄×C₄₇⋊C₈

C₂²

semidirect

68. C₂×C₄₇

#433

semidirect

77. C₄₇

C₂×#433

semidirect

138. C₂

#15689184

direct (N × H)

Subgroups — C₂×#15689184 #20252014

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689184 #20252014

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689184 #20252014