Group #20256066 — subgroups

Subgroups — #20256066

Order 6016.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#15689852	C₂	1	1	yes	maximal, characteristic
2	#15689475	C₂	1	3	yes	maximal
3	#15689415	C₂	1	3	yes	maximal
4	#15689333	C₂	1	3	yes	maximal
5	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#10625337	C₂	1	3	yes	maximal
7	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	C₂²	1	3	yes	—
9	#13094024	C₂²	1	3	yes	—
10	#13094015	—	2	3	no	—
11	Q₈×Dic₄₇	—	2	1	no	—
12	C₄×Dic₉₄	C₂²	1	3	yes	—
13	#13093957	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
15	#13093948	C₂²	1	1	yes	characteristic
16	#13093947	C₂²	1	3	yes	—
17	#13093947	C₂²	1	6	yes	—
18	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₄₇	—	2	3	no	—
20	C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×Q₈×C₄₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	—	2	3	no	—
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₉₄	C₂³	1	3	yes	—
28	C₄⋊C₄×C₄₇	—	2	3	no	—
29	C₄⋊C₄×C₄₇	—	2	3	no	—
30	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
31	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
32	C₄²×C₄₇	—	2	3	no	—
33	C₄²×C₄₇	C₂³	1	3	yes	—
34	#10625337	—	2	6	no	—
35	C₄⋊Dic₄₇	—	2	3	no	—
36	C₄⋊Dic₄₇	—	2	3	no	—
37	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
38	Dic₄₇⋊C₄	—	2	3	no	—
39	C₄×Dic₄₇	—	2	3	no	—
40	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
42	Q₈×C₄₇	—	2	1	no	—
43	Q₈×C₄₇	—	8	3	no	—
44	Q₈×C₄₇	Q₁₆	1	4	yes	complemented
45	Q₈×C₄₇	#41	1	2	yes	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	3	no	—
48	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇	—	2	3	no	—
53	Dic₉₄	—	2	3	no	—
54	Dic₉₄	—	2	3	no	—
55	C₄₇⋊C₈	—	2	3	no	—
56	C₄₇⋊C₈	—	2	1	no	—
57	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₄₇	—	2	2	no	—
59	C₄×C₄₇	—	2	3	no	—
60	C₄×C₄₇	—	4	3	no	—
61	C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₄₇	—	4	3	no	—
65	Dic₄₇	—	4	1	no	—
66	#3363	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	#548	1	1	yes	characteristic
68	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₄₇	#577	1	1	yes	characteristic
70	Q₈²	D₄₇	1	1	yes	characteristic
71	#557	—	47	1	no	—
72	#487	—	47	3	no	—
73	#477	—	47	3	no	—
74	#462	—	47	3	no	—
75	C₈×Q₈	—	47	1	no	—
76	C₄×Q₁₆	—	47	3	no	—
77	C₄₇	#3363	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	47	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	47	3	no	—
81	C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
82	#125	—	94	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	47	3	no	—
85	C₄×Q₈	—	94	1	no	—
86	C₄×Q₈	C₂×D₄₇	1	3	yes	—
87	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
88	#107	—	47	3	no	—
89	C₄⋊C₈	—	47	3	no	—
90	#103	—	47	1	no	—
91	#103	—	47	3	no	—
92	#103	—	47	6	no	—
93	C₄×C₈	—	47	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	47	3	no	—
96	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
98	Q₁₆	—	94	6	no	—
99	C₂×C₈	—	47	1	no	—
100	C₂×C₈	—	47	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	47	1	no	—
104	C₄⋊C₄	—	94	3	no	—
105	C₄⋊C₄	—	94	3	no	—
106	C₄⋊C₄	—	94	3	no	—
107	C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	94	3	no	—
111	C₄²	C₂²×D₄₇	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	94	3	no	—
115	Q₈	—	94	3	no	—
116	Q₈	#10625337	1	4	yes	complemented
117	Q₈	#10625354	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	94	1	no	—
121	C₂×C₄	—	94	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
124	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
125	C₈	—	94	3	no	—
126	C₈	—	94	1	no	—
127	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	188	3	no	—
132	C₄	—	188	1	no	—
133	C₄	C₂×#10625337	1	6	yes	—
134	C₄	C₂×#10625354	1	1	yes	characteristic
135	C₄	#13094072	1	1	yes	characteristic
136	C₂	#15689784	1	1	yes	characteristic
137	C₂	C₂²×#10625337	1	1	yes	characteristic
138	C₂	#15689934	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689852

C₂

yes

maximal, characteristic

#15689475

C₂

yes

maximal

#15689415

C₂

yes

maximal

#15689333

C₂

yes

maximal

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄×#10625337

C₂

yes

maximal

Q₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625337

C₂²

yes

—

#13094024

C₂²

yes

—

#13094015

—

Q₈×Dic₄₇

—

C₄×Dic₉₄

C₂²

yes

—

#13093957

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

C₂²

yes

characteristic

#13093947

C₂²

yes

—

#13093947

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

complemented

Q₈×C₄₇

#41

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄₇

#577

yes

characteristic

Q₈²

D₄₇

yes

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄₇

#3363

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

C₂×Q₈

C₂²×D₄₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²⋊D₄₇

yes

—

108

C₄⋊C₄

C₂²×D₄₇

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₄₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

#10625337

yes

complemented

117

Q₈

#10625354

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

124

C₂×C₄

C₂³×D₄₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

188

—

132

C₄

—

188

—

133

C₄

C₂×#10625337

yes

—

134

C₄

C₂×#10625354

yes

characteristic

135

C₄

#13094072

yes

characteristic

136

C₂

#15689784

yes

characteristic

137

C₂

C₂²×#10625337

yes

characteristic

138

C₂

#15689934

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#3363
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

Q₈²×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

Q₈²

D₄₇

hypocenter

C₄₇

#3363

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
44. Q₈×C₄₇	Q₁₆	semidirect
77. C₄₇	#3363	semidirect
116. Q₈	#10625337	semidirect

H ≅ G/N

type

44. Q₈×C₄₇

Q₁₆

semidirect

77. C₄₇

#3363

semidirect

116. Q₈

#10625337

semidirect

Subgroups — #20256066

Conjugacy classes of subgroups

Complements