Group #24920324 — subgroups

Subgroups — Q₈×#13050579 #24920324

Order 10624.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#19128099	C₂	1	3	yes	maximal, complemented
4	#19127992	C₂	1	3	yes	maximal
5	#19127987	C₂	1	3	yes	maximal, complemented
6	C₄×#13050579	C₂	1	3	yes	maximal
7	C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₈₃	—	2	1	no	—
9	C₂×#13050579	C₂²	1	1	yes	characteristic
10	#14581792	C₂²	1	3	yes	—
11	#14581792	C₂²	1	3	yes	—
12	#14581791	—	2	3	no	—
13	#14581772	—	2	3	no	—
14	Q₈×Dic₈₃	—	2	1	no	—
15	C₄×Dic₁₆₆	C₂²	1	3	yes	—
16	C₄×D₃₃₂	C₂²	1	3	yes	—
17	#14581724	C₂²	1	3	yes	—
18	#14581724	C₂²	1	6	yes	—
19	#14581717	C₂²	1	3	yes	—
20	#14581714	C₂²	1	3	yes	—
21	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	3	yes	—
23	C₄×C₈×C₈₃	C₂²	1	3	yes	—
24	C₂×Dic₁₆₆	—	2	3	no	—
25	C₂×Dic₁₆₆	—	2	3	no	—
26	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
27	Q₈×D₈₃	—	4	4	no	—
28	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₈₃	—	2	3	no	—
30	#13050579	Q₈	1	4	yes	direct product factor
31	C₄⋊Dic₈₃	—	2	3	no	—
32	C₄⋊Dic₈₃	—	2	3	no	—
33	C₄⋊Dic₈₃	C₂³	1	3	yes	—
34	C₄⋊Dic₈₃	C₂³	1	3	yes	—
35	Dic₈₃⋊C₄	—	2	3	no	—
36	(C₂×D₈₃)⋊C₄	—	2	3	no	—
37	C₄×Dic₈₃	—	2	3	no	—
38	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	C₂³	1	3	yes	—
40	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₃	D₄	1	3	yes	—
42	C₄²×C₈₃	C₂³	1	3	yes	—
43	C₂²×D₈₃	—	2	1	no	—
44	Q₈×C₈₃	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₈₃	—	2	1	no	—
49	C₂×Dic₈₃	—	2	3	no	—
50	C₂×Dic₈₃	—	2	3	no	—
51	D₃₃₂	—	2	1	no	—
52	D₃₃₂	C₂×Q₈	1	2	yes	—
53	C₄×D₈₃	—	4	6	no	—
54	Dic₁₆₆	—	2	1	no	—
55	Dic₁₆₆	—	4	6	no	—
56	Dic₁₆₆	—	8	3	no	—
57	Dic₁₆₆	C₂×Q₈	1	2	yes	complemented
58	C₈×C₈₃	—	2	3	no	—
59	C₈×C₈₃	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₈₃	—	2	2	no	—
62	C₂×D₈₃	—	4	1	no	—
63	C₂×D₈₃	—	4	1	no	—
64	C₄×C₈₃	—	2	3	no	—
65	C₄×C₈₃	C₂×SD₁₆	1	6	yes	—
66	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₈₃	—	2	2	no	—
69	Dic₈₃	—	4	3	no	—
70	Dic₈₃	—	4	3	no	—
71	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
72	C₂×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₈₃	#577	1	1	yes	characteristic
74	D₈₃	—	4	2	no	—
75	Q₈×SD₁₆	—	83	1	no	maximal, Sylow
76	C₈₃	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	83	1	no	—
78	D₄×Q₈	—	83	1	no	—
79	#498	—	83	3	no	—
80	#476	—	83	3	no	—
81	#475	—	83	3	no	—
82	C₈×Q₈	D₈₃	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	—	83	3	no	—
84	C₂²×Q₈	—	166	1	no	—
85	C₂×SD₁₆	—	83	1	no	—
86	C₄⋊Q₈	—	83	3	no	—
87	C₄⋊Q₈	—	83	3	no	—
88	C₄⋊Q₈	—	166	3	no	—
89	C₂²⋊Q₈	—	166	3	no	—
90	C₄×Q₈	—	83	3	no	—
91	C₄×Q₈	—	166	1	no	—
92	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
93	C₄×D₄	—	83	3	no	—
94	#106	—	83	3	no	—
95	#106	—	83	6	no	—
96	C₄⋊C₈	C₂×D₈₃	1	3	yes	—
97	#103	—	83	3	no	—
98	#102	—	83	3	no	—
99	C₄×C₈	C₂×D₈₃	1	3	yes	—
100	C₂×Q₈	—	332	4	no	—
101	C₂×Q₈	—	83	1	no	—
102	C₂×Q₈	—	166	3	no	—
103	C₂×Q₈	—	166	3	no	—
104	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
105	C₂×D₄	—	83	1	no	—
106	C₂²×C₄	—	166	3	no	—
107	SD₁₆	—	83	4	no	—
108	C₂×C₈	C₂²×D₈₃	1	3	yes	—
109	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
110	C₄⋊C₄	—	83	3	no	—
111	C₄⋊C₄	—	83	3	no	—
112	C₄⋊C₄	—	166	3	no	—
113	C₄⋊C₄	—	166	3	no	—
114	C₄⋊C₄	—	166	3	no	—
115	C₄⋊C₄	D₃₃₂	1	3	yes	—
116	C₂²⋊C₄	—	166	3	no	—
117	C₄²	—	166	3	no	—
118	C₄²	C₂²×D₈₃	1	3	yes	—
119	C₂³	—	166	1	no	—
120	Q₈	—	332	6	no	—
121	Q₈	—	83	2	no	—
122	Q₈	—	664	3	no	—
123	Q₈	—	166	1	no	—
124	Q₈	#13050579	1	4	yes	direct product factor
125	D₄	—	83	2	no	—
126	D₄	—	166	1	no	—
127	C₂×C₄	—	332	6	no	—
128	C₂×C₄	—	166	3	no	—
129	C₂×C₄	—	166	3	no	—
130	C₂×C₄	—	166	1	no	—
131	C₂×C₄	C₂×D₃₃₂	1	3	yes	—
132	C₂×C₄	C₂×D₃₃₂	1	3	yes	—
133	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	Q₈×D₈₃	1	2	yes	complemented
136	C₂²	—	332	1	no	—
137	C₂²	—	332	1	no	—
138	C₂²	—	166	2	no	—
139	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	332	3	no	—
142	C₄	—	332	3	no	—
143	C₄	—	166	2	no	—
144	C₄	C₂×#13050579	1	6	yes	—
145	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
146	C₄	#14581839	1	1	yes	characteristic
147	C₂	—	332	2	no	—
148	C₂	Q₈×D₃₃₂	1	1	yes	characteristic
149	C₂	C₂²×#13050579	1	1	yes	characteristic
150	C₂	#19128513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₆₆

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₃₂

C₂

yes

maximal, characteristic

#19128099

C₂

yes

maximal, complemented

#19127992

C₂

yes

maximal

#19127987

C₂

yes

maximal, complemented

C₄×#13050579

C₂

yes

maximal

C₈×Q₈×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₃

—

C₂×#13050579

C₂²

yes

characteristic

#14581792

C₂²

yes

—

#14581792

C₂²

yes

—

#14581791

—

#14581772

—

Q₈×Dic₈₃

—

C₄×Dic₁₆₆

C₂²

yes

—

C₄×D₃₃₂

C₂²

yes

—

#14581724

C₂²

yes

—

#14581724

C₂²

yes

—

#14581717

C₂²

yes

—

#14581714

C₂²

yes

—

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

characteristic

Q₈×D₈₃

—

C₂×D₃₃₂

C₂³

yes

characteristic

C₂×C₄×D₈₃

—

#13050579

Q₈

yes

direct product factor

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

—

C₂²×D₈₃

—

Q₈×C₈₃

SD₁₆

yes

complemented

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

C₂×Q₈

yes

—

C₄×D₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

C₂×Q₈

yes

complemented

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₈

yes

complemented

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×SD₁₆

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

2₋¹⁺⁴

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

D₄×Q₈

yes

characteristic

C₂×C₈₃

C₂²×SD₁₆

yes

characteristic

C₂×C₈₃

#577

yes

characteristic

D₈₃

—

Q₈×SD₁₆

—

maximal, Sylow

C₈₃

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

D₈₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₂²×Q₈

—

166

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

C₄×Q₈

—

166

—

C₄×Q₈

C₂×D₈₃

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₈₃

yes

—

100

C₂×Q₈

—

332

—

101

C₂×Q₈

—

102

C₂×Q₈

—

166

—

103

C₂×Q₈

—

166

—

104

C₂×Q₈

D₃₃₂

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

166

—

107

SD₁₆

—

108

C₂×C₈

C₂²×D₈₃

yes

—

109

C₂×C₈

C₂²×D₈₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

166

—

113

C₄⋊C₄

—

166

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

D₃₃₂

yes

—

116

C₂²⋊C₄

—

166

—

117

C₄²

—

166

—

118

C₄²

C₂²×D₈₃

yes

—

119

C₂³

—

166

—

120

Q₈

—

332

—

121

Q₈

—

122

Q₈

—

664

—

123

Q₈

—

166

—

124

Q₈

#13050579

yes

direct product factor

125

D₄

—

126

D₄

—

166

—

127

C₂×C₄

—

332

—

128

C₂×C₄

—

166

—

129

C₂×C₄

—

166

—

130

C₂×C₄

—

166

—

131

C₂×C₄

C₂×D₃₃₂

yes

—

132

C₂×C₄

C₂×D₃₃₂

yes

—

133

C₂×C₄

C₂³×D₈₃

yes

characteristic

134

C₈

—

135

C₈

Q₈×D₈₃

yes

complemented

136

C₂²

—

332

—

137

C₂²

—

332

—

138

C₂²

—

166

—

139

C₂²

C₂²×D₃₃₂

yes

characteristic

140

C₄

—

141

C₄

—

332

—

142

C₄

—

332

—

143

C₄

—

166

—

144

C₄

C₂×#13050579

yes

—

145

C₄

C₂×Q₈×D₈₃

yes

characteristic

146

C₄

#14581839

yes

characteristic

147

C₂

—

332

—

148

C₂

Q₈×D₃₃₂

yes

characteristic

149

C₂

C₂²×#13050579

yes

characteristic

150

C₂

#19128513

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₈×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₃₃₂	C₂³
hypercenter	C₈×Q₈	D₈₃
hypocenter	C₈₃	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₃₂

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₈×Q₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂×D₃₃₂

C₂³

hypercenter

C₈×Q₈

D₈₃

hypocenter

C₈₃

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₆₆	C₂	semidirect
3. #19128099	C₂	semidirect
5. #19127987	C₂	semidirect
7. C₈×Q₈×C₈₃	C₂	semidirect
30. #13050579	Q₈	direct (N × H)
44. Q₈×C₈₃	SD₁₆	semidirect
57. Dic₁₆₆	C₂×Q₈	semidirect
59. C₈×C₈₃	C₂×Q₈	semidirect
76. C₈₃	Q₈×SD₁₆	semidirect
82. C₈×Q₈	D₈₃	semidirect
124. Q₈	#13050579	direct (N × H)
135. C₈	Q₈×D₈₃	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₆₆

C₂

semidirect

3. #19128099

C₂

semidirect

5. #19127987

C₂

semidirect

7. C₈×Q₈×C₈₃

C₂

semidirect

30. #13050579

Q₈

direct (N × H)

44. Q₈×C₈₃

SD₁₆

semidirect

57. Dic₁₆₆

C₂×Q₈

semidirect

59. C₈×C₈₃

C₂×Q₈

semidirect

76. C₈₃

Q₈×SD₁₆

semidirect

82. C₈×Q₈

D₈₃

semidirect

124. Q₈

#13050579

direct (N × H)

135. C₈

Q₈×D₈₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Dic₁₆₆	2
Q₈×D₃₃₂	2
#19128099	2
#19127992	2
#19127987	2
C₄×#13050579	2
C₈×Q₈×C₈₃	2
Q₈×SD₁₆	83

subgroup

index

Q₈×Dic₁₆₆

Q₈×D₃₃₂

C₈×Q₈×C₈₃

Q₈×SD₁₆

Subgroups — Q₈×#13050579 #24920324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13050579 #24920324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13050579 #24920324