Group #20246304 — subgroups

Subgroups — Q₈×C₄⋊Dic₄₇ #20246304

Order 6016.

568 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 333 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×Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	#15689022	C₂	1	6	yes	maximal
3	#15688993	C₂	1	3	yes	maximal
4	C₄×C₄⋊Dic₄₇	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₄₇	—	2	8	no	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	6	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	6	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₂²	1	6	yes	—
12	C₄₇⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₄₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
18	C₂×Q₈×C₄₇	D₄	1	4	yes	—
19	C₂×Q₈×C₄₇	Q₈	1	4	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₇	C₂³	1	6	yes	—
24	C₂²×Dic₄₇	C₂³	1	2	yes	—
25	C₄⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
26	C₄²×C₄₇	C₂×C₄	1	12	yes	—
27	C₄⋊Dic₄₇	—	2	12	no	—
28	C₄⋊Dic₄₇	—	2	6	no	—
29	C₄⋊Dic₄₇	—	4	6	no	—
30	C₄⋊Dic₄₇	Q₈	1	4	yes	direct product factor
31	C₄×Dic₄₇	—	2	12	no	—
32	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
33	Q₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
34	C₂×C₄×C₄₇	—	2	3	no	—
35	C₂×C₄×C₄₇	—	2	3	no	—
36	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
40	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
41	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₄₇	#41	1	2	yes	—
43	C₂×Dic₄₇	—	2	2	no	—
44	C₂×Dic₄₇	—	2	2	no	—
45	C₂×Dic₄₇	—	2	6	no	—
46	C₂×Dic₄₇	—	2	6	no	—
47	C₂×Dic₄₇	—	2	6	no	—
48	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
54	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
56	C₄×C₄₇	—	2	6	no	—
57	C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
58	C₄×C₄₇	C₄×Q₈	1	4	yes	complemented
59	Dic₄₇	—	2	4	no	—
60	Dic₄₇	—	4	6	no	—
61	Q₈×C₄⋊C₄	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₄₇	#520	1	1	yes	characteristic
65	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
66	C₂×C₄₇	#550	1	1	yes	characteristic
67	C₂×C₄₇	#557	1	1	yes	characteristic
68	C₂×C₄₇	Q₈²	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	47	2	no	—
70	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
71	#389	—	47	6	no	—
72	#384	—	47	3	no	—
73	C₄×C₄⋊C₄	—	47	3	no	—
74	C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
76	C₄×Q₈	—	94	8	no	—
77	C₄×Q₈	Dic₄₇	1	8	yes	complemented
78	C₂×C₄⋊C₄	—	47	1	no	—
79	C₂×C₄⋊C₄	—	47	6	no	—
80	C₂×C₄⋊C₄	—	47	3	no	—
81	C₂×C₄⋊C₄	—	47	6	no	—
82	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
83	C₂×C₄²	—	47	6	no	—
84	C₂×C₄²	C₂×D₄₇	1	3	yes	—
85	#95	—	47	6	no	—
86	C₂×Q₈	Dic₉₄	1	4	yes	—
87	C₂×Q₈	D₁₈₈	1	4	yes	—
88	C₂×Q₈	C₂×Dic₄₇	1	4	yes	—
89	C₂²×C₄	—	47	2	no	—
90	C₂²×C₄	—	47	6	no	—
91	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
92	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
93	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
94	C₄⋊C₄	—	47	4	no	—
95	C₄⋊C₄	—	94	12	no	—
96	C₄⋊C₄	—	94	6	no	—
97	C₄⋊C₄	—	188	6	no	—
98	C₄⋊C₄	C₂×Dic₄₇	1	12	yes	—
99	C₄²	—	94	12	no	—
100	C₄²	C₂×Dic₄₇	1	12	yes	—
101	C₂³	C₂³×D₄₇	1	1	yes	characteristic
102	Q₈	C₄⋊Dic₄₇	1	16	yes	direct product factor
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	47	4	no	—
106	C₂×C₄	—	94	2	no	—
107	C₂×C₄	—	94	2	no	—
108	C₂×C₄	—	94	6	no	—
109	C₂×C₄	—	94	6	no	—
110	C₂×C₄	—	94	6	no	—
111	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
112	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
113	C₂×C₄	Q₈×D₄₇	1	2	yes	—
114	C₂×C₄	#10625354	1	2	yes	—
115	C₂×C₄	C₂²×Dic₄₇	1	6	yes	—
116	C₂×C₄	C₂²×Dic₄₇	1	6	yes	—
117	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
118	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
119	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
120	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
121	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
122	C₂²	C₂×#10625354	1	1	yes	characteristic
123	C₂²	#13094070	1	1	yes	characteristic
124	C₂²	#13094071	1	1	yes	characteristic
125	C₄	—	2	6	no	—
126	C₄	—	94	4	no	—
127	C₄	—	188	6	no	—
128	C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
129	C₄	Q₈×Dic₄₇	1	4	yes	complemented
130	C₂	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
131	C₂	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
132	C₂	#15689606	1	1	yes	characteristic
133	C₂	Q₈×D₁₈₈	1	1	yes	characteristic
134	C₂	#15689795	1	1	yes	characteristic
135	C₂	#15689852	1	1	yes	characteristic
136	C₂	Q₈×Dic₉₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₇

C₂

yes

maximal

#15689022

C₂

yes

maximal

#15688993

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

Q₈

yes

direct product factor

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

C₄⋊C₄

yes

complemented

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

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₄×Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

—

Q₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₇

#520

yes

characteristic

C₂×C₄₇

D₄×Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

Q₈²

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₇

yes

characteristic

#389

—

#384

—

C₄×C₄⋊C₄

—

C₄₇

Q₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

#95

—

C₂×Q₈

Dic₉₄

yes

—

C₂×Q₈

D₁₈₈

yes

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₄²

—

100

C₄²

C₂×Dic₄₇

yes

—

101

C₂³

C₂³×D₄₇

yes

characteristic

102

Q₈

C₄⋊Dic₄₇

yes

direct product factor

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×Dic₉₄

yes

—

112

C₂×C₄

C₂×D₁₈₈

yes

—

113

C₂×C₄

Q₈×D₄₇

yes

—

114

C₂×C₄

#10625354

yes

—

115

C₂×C₄

C₂²×Dic₄₇

yes

—

116

C₂×C₄

C₂²×Dic₄₇

yes

—

117

C₂×C₄

C₂²×Dic₄₇

yes

—

118

C₂²

C₂³×Dic₄₇

yes

characteristic

119

C₂²

C₂²×D₁₈₈

yes

characteristic

120

C₂²

C₂×Q₈×D₄₇

yes

characteristic

121

C₂²

C₂²×Dic₉₄

yes

characteristic

122

C₂²

C₂×#10625354

yes

characteristic

123

C₂²

#13094070

yes

characteristic

124

C₂²

#13094071

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

188

—

128

C₄

C₂×C₄⋊Dic₄₇

yes

—

129

C₄

Q₈×Dic₄₇

yes

complemented

130

C₂

C₂²×C₄⋊Dic₄₇

yes

characteristic

131

C₂

C₂×Q₈×Dic₄₇

yes

characteristic

132

C₂

#15689606

yes

characteristic

133

C₂

Q₈×D₁₈₈

yes

characteristic

134

C₂

#15689795

yes

characteristic

135

C₂

#15689852

yes

characteristic

136

C₂

Q₈×Dic₉₄

yes

characteristic

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

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₇

hypocenter

C₄₇

Q₈×C₄⋊C₄

LCS

G ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₄₇	C₄	semidirect
30. C₄⋊Dic₄₇	Q₈	direct (N × H)
33. Q₈×C₄₇	C₄⋊C₄	semidirect
58. C₄×C₄₇	C₄×Q₈	semidirect
74. C₄₇	Q₈×C₄⋊C₄	semidirect
77. C₄×Q₈	Dic₄₇	semidirect
102. Q₈	C₄⋊Dic₄₇	direct (N × H)
129. C₄	Q₈×Dic₄₇	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₄₇

C₄

semidirect

30. C₄⋊Dic₄₇

Q₈

direct (N × H)

33. Q₈×C₄₇

C₄⋊C₄

semidirect

58. C₄×C₄₇

C₄×Q₈

semidirect

74. C₄₇

Q₈×C₄⋊C₄

semidirect

77. C₄×Q₈

Dic₄₇

semidirect

102. Q₈

C₄⋊Dic₄₇

direct (N × H)

129. C₄

Q₈×Dic₄₇

semidirect

Subgroups — Q₈×C₄⋊Dic₄₇ #20246304

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C4⋊Dic47 #20246304

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₄⋊Dic₄₇ #20246304