Group #36712292 — subgroups

Subgroups — Q₈×D₁₄₂₄ #36712292

Order 22784.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D₇₁₂	C₂	1	2	yes	maximal, complemented
2	#26022527	C₂	1	3	yes	maximal, complemented
3	#26022390	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₄₂₄	C₂	1	3	yes	maximal
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×D₃₅₆	—	2	2	no	—
7	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
8	#19152003	C₂²	1	6	yes	—
9	#19151879	—	2	6	no	—
10	C₄×D₇₁₂	C₂²	1	6	yes	—
11	#19151424	C₂²	1	3	yes	—
12	#19151424	C₂²	1	6	yes	—
13	#19151400	C₂²	1	6	yes	—
14	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₉	C₂²	1	3	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	3	yes	—
17	C₂×Q₈×D₈₉	—	4	2	no	—
18	C₂×D₇₁₂	C₂³	1	2	yes	—
19	#15668012	—	2	6	no	—
20	#15667992	—	4	6	no	—
21	C₄×D₃₅₆	—	2	6	no	—
22	D₁₄₂₄	Q₈	1	4	yes	direct product factor
23	#15667946	—	2	6	no	—
24	#15667946	C₂³	1	6	yes	—
25	#15667934	—	2	6	no	—
26	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂³	1	3	yes	—
28	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	D₄	1	3	yes	—
30	C₄×C₈×C₈₉	C₂³	1	3	yes	—
31	C₂×Dic₁₇₈	—	4	6	no	—
32	Q₈×D₈₉	—	8	8	no	—
33	C₂×D₃₅₆	—	2	2	no	—
34	C₂×C₄×D₈₉	—	4	6	no	—
35	D₇₁₂	—	2	2	no	—
36	D₇₁₂	C₂×Q₈	1	4	yes	complemented
37	C₄⋊Dic₈₉	—	2	6	no	—
38	C₄⋊Dic₈₉	—	4	6	no	—
39	#13083551	—	4	6	no	—
40	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
42	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
43	C₄⋊C₄×C₈₉	D₈	1	3	yes	—
44	C₄²×C₈₉	C₂×D₄	1	3	yes	—
45	C₁₆×C₈₉	—	2	3	no	—
46	C₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
47	C₂²×D₈₉	—	4	2	no	—
48	Q₈×C₈₉	D₁₆	1	4	yes	complemented
49	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
50	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
51	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
52	C₂×Dic₈₉	—	4	6	no	—
53	D₃₅₆	—	2	4	no	—
54	D₃₅₆	—	4	2	no	—
55	C₄×D₈₉	—	8	12	no	—
56	Dic₁₇₈	—	8	12	no	—
57	C₈×C₈₉	—	2	3	no	—
58	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
59	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
60	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
61	C₂×D₈₉	—	4	4	no	—
62	C₂×D₈₉	—	8	2	no	—
63	C₂×D₈₉	—	8	2	no	—
64	C₄×C₈₉	—	2	3	no	—
65	C₄×C₈₉	C₂×D₁₆	1	6	yes	—
66	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
67	C₄×C₈₉	#578	1	1	yes	characteristic
68	Dic₈₉	—	8	6	no	—
69	Q₈×D₁₆	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
71	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
72	C₂×C₈₉	#3417	1	1	yes	characteristic
73	D₈₉	—	8	4	no	—
74	Q₈×D₈	—	89	2	no	—
75	#2252	—	89	3	no	—
76	#2224	—	89	6	no	—
77	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
78	C₄×D₁₆	—	89	3	no	—
79	C₈₉	Q₈×D₁₆	1	1	yes	characteristic, complemented, Sylow
80	D₄×Q₈	—	178	2	no	—
81	C₂×D₁₆	—	89	1	no	—
82	#500	—	89	6	no	—
83	#474	—	178	6	no	—
84	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
85	C₄×D₈	—	89	6	no	—
86	#366	—	89	3	no	—
87	#366	—	89	6	no	—
88	C₄⋊C₁₆	C₂×D₈₉	1	3	yes	—
89	#357	—	89	6	no	—
90	C₄×C₁₆	C₂×D₈₉	1	3	yes	—
91	C₂²×Q₈	—	356	2	no	—
92	C₂×D₈	—	89	2	no	—
93	C₄⋊Q₈	—	178	6	no	—
94	C₂²⋊Q₈	—	356	6	no	—
95	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
96	C₄×D₄	—	178	6	no	—
97	D₁₆	—	89	4	no	—
98	C₂×C₁₆	C₂²×D₈₉	1	3	yes	—
99	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
100	#107	—	89	6	no	—
101	#107	—	178	6	no	—
102	C₄⋊C₈	D₃₅₆	1	3	yes	—
103	#102	—	178	6	no	—
104	C₄×C₈	C₂²×D₈₉	1	3	yes	—
105	C₂×Q₈	—	356	6	no	—
106	C₂×Q₈	—	712	8	no	—
107	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
108	C₂×D₄	—	178	2	no	—
109	C₂²×C₄	—	356	6	no	—
110	D₈	—	89	4	no	—
111	D₈	—	178	2	no	—
112	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
113	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
114	C₄⋊C₄	—	356	6	no	—
115	C₄⋊C₄	—	178	6	no	—
116	C₄⋊C₄	D₇₁₂	1	3	yes	—
117	C₂²⋊C₄	—	356	6	no	—
118	C₄²	C₂×D₃₅₆	1	3	yes	—
119	C₁₆	—	2	3	no	—
120	C₁₆	Q₈×D₈₉	1	2	yes	complemented
121	C₂³	—	356	2	no	—
122	Q₈	—	712	12	no	—
123	Q₈	D₁₄₂₄	1	4	yes	direct product factor
124	D₄	—	356	2	no	—
125	D₄	—	178	4	no	—
126	C₂×C₄	—	356	6	no	—
127	C₂×C₄	—	712	12	no	—
128	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
129	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
130	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
131	C₈	—	2	3	no	—
132	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
133	C₈	#15668059	1	1	yes	characteristic
134	C₂²	—	356	4	no	—
135	C₂²	—	712	2	no	—
136	C₂²	—	712	2	no	—
137	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	712	6	no	—
140	C₄	C₂×D₁₄₂₄	1	6	yes	—
141	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
142	C₄	#19152415	1	1	yes	characteristic
143	C₂	—	712	4	no	—
144	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
145	C₂	C₂²×D₁₄₂₄	1	1	yes	characteristic
146	C₂	#26033405	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₇₁₂

C₂

yes

maximal, complemented

#26022527

C₂

yes

maximal, complemented

#26022390

C₂

yes

maximal, complemented

C₄×D₁₄₂₄

C₂

yes

maximal

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₅₆

—

C₂×D₁₄₂₄

C₂²

yes

characteristic

#19152003

C₂²

yes

—

#19151879

—

C₄×D₇₁₂

C₂²

yes

—

#19151424

C₂²

yes

—

#19151424

C₂²

yes

—

#19151400

C₂²

yes

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₂×Q₈×D₈₉

—

C₂×D₇₁₂

C₂³

yes

—

#15668012

—

#15667992

—

C₄×D₃₅₆

—

D₁₄₂₄

Q₈

yes

direct product factor

#15667946

—

#15667946

C₂³

yes

—

#15667934

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

C₂×Q₈

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

D₈

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

complemented

C₂²×D₈₉

—

Q₈×C₈₉

D₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×Q₈

yes

characteristic

C₈×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

D₄×Q₈

yes

characteristic

C₄×C₈₉

#578

yes

characteristic

Dic₈₉

—

Q₈×D₁₆

—

maximal, Sylow

C₂×C₈₉

Q₈×D₈

yes

characteristic

C₂×C₈₉

C₂²×D₁₆

yes

characteristic

C₂×C₈₉

#3417

yes

characteristic

D₈₉

—

Q₈×D₈

—

#2252

—

#2224

—

Q₈×C₁₆

D₈₉

yes

characteristic, complemented

C₄×D₁₆

—

C₈₉

Q₈×D₁₆

yes

characteristic, complemented, Sylow

D₄×Q₈

—

178

—

C₂×D₁₆

—

#500

—

#474

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₈

—

#366

—

#366

—

C₄⋊C₁₆

C₂×D₈₉

yes

—

#357

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₂²×Q₈

—

356

—

C₂×D₈

—

C₄⋊Q₈

—

178

—

C₂²⋊Q₈

—

356

—

C₄×Q₈

D₃₅₆

yes

characteristic

C₄×D₄

—

178

—

D₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

100

#107

—

101

#107

—

178

—

102

C₄⋊C₈

D₃₅₆

yes

—

103

#102

—

178

—

104

C₄×C₈

C₂²×D₈₉

yes

—

105

C₂×Q₈

—

356

—

106

C₂×Q₈

—

712

—

107

C₂×Q₈

D₇₁₂

yes

characteristic

108

C₂×D₄

—

178

—

109

C₂²×C₄

—

356

—

110

D₈

—

111

D₈

—

178

—

112

C₂×C₈

C₂×D₃₅₆

yes

—

113

C₂×C₈

C₂³×D₈₉

yes

characteristic

114

C₄⋊C₄

—

356

—

115

C₄⋊C₄

—

178

—

116

C₄⋊C₄

D₇₁₂

yes

—

117

C₂²⋊C₄

—

356

—

118

C₄²

C₂×D₃₅₆

yes

—

119

C₁₆

—

120

C₁₆

Q₈×D₈₉

yes

complemented

121

C₂³

—

356

—

122

Q₈

—

712

—

123

Q₈

D₁₄₂₄

yes

direct product factor

124

D₄

—

356

—

125

D₄

—

178

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

712

—

128

C₂×C₄

C₂×D₇₁₂

yes

—

129

C₂×C₄

C₂×D₇₁₂

yes

—

130

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

131

C₈

—

132

C₈

C₂×Q₈×D₈₉

yes

characteristic

133

C₈

#15668059

yes

characteristic

134

C₂²

—

356

—

135

C₂²

—

712

—

136

C₂²

—

712

—

137

C₂²

C₂²×D₇₁₂

yes

characteristic

138

C₄

—

139

C₄

—

712

—

140

C₄

C₂×D₁₄₂₄

yes

—

141

C₄

Q₈×D₃₅₆

yes

characteristic

142

C₄

#19152415

yes

characteristic

143

C₂

—

712

—

144

C₂

Q₈×D₇₁₂

yes

characteristic

145

C₂

C₂²×D₁₄₂₄

yes

characteristic

146

C₂

#26033405

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	Q₈×D₁₆
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

C₂²×D₇₁₂

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

Q₈×C₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₈

prime radical

C₂×D₁₄₂₄

C₂²

hypercenter

Q₈×C₁₆

D₈₉

hypocenter

C₈₉

Q₈×D₁₆

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
2. #26022527	C₂	semidirect
3. #26022390	C₂	semidirect
5. Q₈×C₁₆×C₈₉	C₂	semidirect
22. D₁₄₂₄	Q₈	direct (N × H)
36. D₇₁₂	C₂×Q₈	semidirect
46. C₁₆×C₈₉	C₂×Q₈	semidirect
48. Q₈×C₈₉	D₁₆	semidirect
77. Q₈×C₁₆	D₈₉	semidirect
79. C₈₉	Q₈×D₁₆	semidirect
120. C₁₆	Q₈×D₈₉	semidirect
123. Q₈	D₁₄₂₄	direct (N × H)

H ≅ G/N

type

1. Q₈×D₇₁₂

C₂

semidirect

2. #26022527

C₂

semidirect

3. #26022390

C₂

semidirect

5. Q₈×C₁₆×C₈₉

C₂

semidirect

22. D₁₄₂₄

Q₈

direct (N × H)

36. D₇₁₂

C₂×Q₈

semidirect

46. C₁₆×C₈₉

C₂×Q₈

semidirect

48. Q₈×C₈₉

D₁₆

semidirect

77. Q₈×C₁₆

D₈₉

semidirect

79. C₈₉

Q₈×D₁₆

semidirect

120. C₁₆

Q₈×D₈₉

semidirect

123. Q₈

D₁₄₂₄

direct (N × H)

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

subgroup	index
Q₈×D₇₁₂	2
#26022527	2
#26022390	2
C₄×D₁₄₂₄	2
Q₈×C₁₆×C₈₉	2
Q₈×D₁₆	89

subgroup

index

Q₈×D₇₁₂

#26022527

#26022390

C₄×D₁₄₂₄

Q₈×C₁₆×C₈₉

Q₈×D₁₆

Subgroups — Q₈×D₁₄₂₄ #36712292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×D1424 #36712292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₄₂₄ #36712292