Group #20252308 — subgroups

Subgroups — C₂×#15689247 #20252308

Order 6016.

440 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 253 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×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#15689247	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₄₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₉₄	C₄	1	4	yes	—
11	C₄×Dic₉₄	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	Dic₄₇⋊C₈	C₂²	1	8	yes	—
16	C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
17	C₈×Dic₄₇	C₂²	1	8	yes	—
18	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₂×Dic₉₄	C₈	1	8	yes	complemented
27	C₂×Dic₉₄	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₄₇	—	2	4	no	—
29	C₂×C₈×C₄₇	C₂³	1	4	yes	—
30	C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₄²×C₄₇	C₂³	1	2	yes	—
32	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
34	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₄×Dic₄₇	C₂³	1	4	yes	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	Q₈	1	4	yes	complemented
38	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	4	no	—
50	C₂×Dic₄₇	C₂×C₈	1	8	yes	—
51	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
53	Dic₉₄	C₂×C₈	1	16	yes	complemented
54	C₈×C₄₇	—	2	4	no	—
55	C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊C₈	C₂×Q₈	1	4	yes	complemented
57	C₂²×C₄₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
59	C₂²×C₄₇	#131	1	2	yes	—
60	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	C₄×Q₈	1	2	yes	—
63	C₄×C₄₇	C₂²×C₈	1	4	yes	—
64	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
66	Dic₄₇	—	2	4	no	—
67	Dic₄₇	C₂²×C₈	1	8	yes	—
68	C₂×C₈×Q₈	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	C₈×Q₈	1	4	yes	complemented
70	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄₇	C₂³×C₈	1	1	yes	characteristic
72	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
73	C₂×C₄×Q₈	—	47	1	no	—
74	C₈×Q₈	—	47	8	no	—
75	C₂×C₄⋊C₈	—	47	2	no	—
76	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
77	C₂×C₄×C₈	—	47	2	no	—
78	C₂×C₄×C₈	—	47	1	no	—
79	C₄₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	47	1	no	—
81	C₂²×C₈	—	47	1	no	—
82	C₂²×C₈	—	47	1	no	—
83	C₂²×C₈	C₂×D₄₇	1	2	yes	—
84	C₄×Q₈	—	47	4	no	—
85	C₄×Q₈	—	47	4	no	—
86	C₂×C₄⋊C₄	—	47	2	no	—
87	C₂×C₄⋊C₄	—	47	1	no	—
88	C₂×C₄²	—	47	2	no	—
89	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
90	C₄⋊C₈	—	47	8	no	—
91	C₄⋊C₈	C₂×D₄₇	1	4	yes	—
92	C₄×C₈	—	47	8	no	—
93	C₄×C₈	—	47	4	no	—
94	C₂×Q₈	—	47	8	no	—
95	C₂×Q₈	—	47	4	no	—
96	C₂²×C₄	—	47	2	no	—
97	C₂²×C₄	—	47	2	no	—
98	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	47	4	no	—
103	C₂×C₈	—	47	2	no	—
104	C₂×C₈	—	47	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₂×C₈	C₂²×D₄₇	1	4	yes	—
107	C₄⋊C₄	—	47	8	no	—
108	C₄⋊C₄	—	47	4	no	—
109	C₄²	—	47	4	no	—
110	C₄²	—	47	4	no	—
111	C₄²	C₄×D₄₇	1	2	yes	—
112	C₄²	C₂²×D₄₇	1	2	yes	—
113	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
114	Q₈	—	47	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	47	8	no	—
117	C₂×C₄	—	47	4	no	—
118	C₂×C₄	—	47	4	no	—
119	C₂×C₄	—	94	4	no	—
120	C₂×C₄	C₈×D₄₇	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
124	C₂×C₄	#10625352	1	2	yes	—
125	C₂×C₄	Q₈×D₄₇	1	2	yes	—
126	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	47	4	no	—
129	C₈	—	94	2	no	—
130	C₂²	#13093989	1	2	yes	—
131	C₂²	C₂×C₈×D₄₇	1	2	yes	—
132	C₂²	#13094032	1	2	yes	—
133	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	47	8	no	—
136	C₄	—	94	4	no	—
137	C₄	#13093989	1	2	yes	—
138	C₄	C₂×C₈×D₄₇	1	4	yes	—
139	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
140	C₄	C₂×#10625352	1	1	yes	characteristic
141	C₂	#15689247	1	4	yes	direct product factor
142	C₂	C₂×#13093989	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₄₇	1	1	yes	characteristic
144	C₂	C₂×#13094032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic

#15689247

C₂

yes

maximal, direct product factor

C₂×Dic₄₇⋊C₈

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₄₇

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×Dic₉₄

C₄

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

C₈

yes

complemented

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

Q₈

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×C₈

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

Dic₉₄

C₂×C₈

yes

complemented

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₄×Q₈

yes

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

#131

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄×Q₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

C₂²×C₈

yes

—

C₂×C₈×Q₈

—

maximal, Sylow

C₂×C₄₇

C₈×Q₈

yes

complemented

C₂×C₄₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₇

C₂³×C₈

yes

characteristic

C₂×C₄₇

C₂×#131

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄₇

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×D₄₇

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₄₇

yes

—

112

C₄²

C₂²×D₄₇

yes

—

113

C₂³

C₂×C₄×D₄₇

yes

characteristic

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₈×D₄₇

yes

—

121

C₂×C₄

C₂×C₄×D₄₇

yes

—

122

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

123

C₂×C₄

C₂×C₄×D₄₇

yes

—

124

C₂×C₄

#10625352

yes

—

125

C₂×C₄

Q₈×D₄₇

yes

—

126

C₂×C₄

C₂³×D₄₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

#13093989

yes

—

131

C₂²

C₂×C₈×D₄₇

yes

—

132

C₂²

#13094032

yes

—

133

C₂²

C₂²×C₄×D₄₇

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#13093989

yes

—

138

C₄

C₂×C₈×D₄₇

yes

—

139

C₄

C₂×Q₈×D₄₇

yes

characteristic

140

C₄

C₂×#10625352

yes

characteristic

141

C₂

#15689247

yes

direct product factor

142

C₂

C₂×#13093989

yes

characteristic

143

C₂

C₂²×C₈×D₄₇

yes

characteristic

144

C₂

C₂×#13094032

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₇
commutator	C₂×C₄₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×C₈×Q₈
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₄₇

commutator

C₂×C₄₇

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

C₂×C₈×Q₈

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #15689247	C₂	direct (N × H)
26. C₂×Dic₉₄	C₈	semidirect
37. C₂×C₄₇⋊C₈	Q₈	semidirect
53. Dic₉₄	C₂×C₈	semidirect
56. C₄₇⋊C₈	C₂×Q₈	semidirect
69. C₂×C₄₇	C₈×Q₈	semidirect
79. C₄₇	C₂×C₈×Q₈	semidirect
141. C₂	#15689247	direct (N × H)

H ≅ G/N

type

2. #15689247

C₂

direct (N × H)

26. C₂×Dic₉₄

C₈

semidirect

37. C₂×C₄₇⋊C₈

Q₈

semidirect

53. Dic₉₄

C₂×C₈

semidirect

56. C₄₇⋊C₈

C₂×Q₈

semidirect

69. C₂×C₄₇

C₈×Q₈

semidirect

79. C₄₇

C₂×C₈×Q₈

semidirect

141. C₂

#15689247

direct (N × H)

Subgroups — C₂×#15689247 #20252308

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689247 #20252308

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689247 #20252308