Group #26033113 — subgroups

Subgroups — Q₈×#13083563 #26033113

Order 11392.

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×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151996	C₂	1	3	yes	maximal, complemented
4	#19151889	C₂	1	3	yes	maximal
5	#19151884	C₂	1	3	yes	maximal, complemented
6	C₄×#13083563	C₂	1	3	yes	maximal
7	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₈₉	—	2	1	no	—
9	C₂×#13083563	C₂²	1	1	yes	characteristic
10	#15668012	C₂²	1	3	yes	—
11	#15668012	C₂²	1	3	yes	—
12	#15668011	—	2	3	no	—
13	#15667992	—	2	3	no	—
14	Q₈×Dic₈₉	—	2	1	no	—
15	C₄×Dic₁₇₈	C₂²	1	3	yes	—
16	C₄×D₃₅₆	C₂²	1	3	yes	—
17	#15667944	C₂²	1	6	yes	—
18	#15667944	C₂²	1	3	yes	—
19	#15667937	C₂²	1	3	yes	—
20	#15667934	C₂²	1	3	yes	—
21	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
23	C₄×C₈×C₈₉	C₂²	1	3	yes	—
24	C₂×Dic₁₇₈	—	2	3	no	—
25	C₂×Dic₁₇₈	—	2	3	no	—
26	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
27	Q₈×D₈₉	—	4	4	no	—
28	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₈₉	—	2	3	no	—
30	#13083563	Q₈	1	4	yes	direct product factor
31	C₄⋊Dic₈₉	—	2	3	no	—
32	C₄⋊Dic₈₉	—	2	3	no	—
33	C₄⋊Dic₈₉	C₂³	1	3	yes	—
34	C₄⋊Dic₈₉	C₂³	1	3	yes	—
35	#13083553	—	2	3	no	—
36	#13083551	—	2	3	no	—
37	C₄×Dic₈₉	—	2	3	no	—
38	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂³	1	3	yes	—
40	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	D₄	1	3	yes	—
42	C₄²×C₈₉	C₂³	1	3	yes	—
43	C₂²×D₈₉	—	2	1	no	—
44	Q₈×C₈₉	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₈₉	—	2	1	no	—
49	C₂×Dic₈₉	—	2	3	no	—
50	C₂×Dic₈₉	—	2	3	no	—
51	D₃₅₆	—	2	1	no	—
52	D₃₅₆	C₂×Q₈	1	2	yes	—
53	C₄×D₈₉	—	4	6	no	—
54	Dic₁₇₈	—	2	1	no	—
55	Dic₁₇₈	—	4	6	no	—
56	Dic₁₇₈	—	8	3	no	—
57	Dic₁₇₈	C₂×Q₈	1	2	yes	complemented
58	C₈×C₈₉	—	2	3	no	—
59	C₈×C₈₉	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₈₉	—	2	2	no	—
62	C₂×D₈₉	—	4	1	no	—
63	C₂×D₈₉	—	4	1	no	—
64	C₄×C₈₉	—	2	3	no	—
65	C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
66	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₈₉	—	2	2	no	—
69	Dic₈₉	—	4	3	no	—
70	Dic₈₉	—	4	3	no	—
71	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
72	C₂×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₈₉	#577	1	1	yes	characteristic
74	D₈₉	—	4	2	no	—
75	Q₈×SD₁₆	—	89	1	no	maximal, Sylow
76	C₈₉	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	89	1	no	—
78	D₄×Q₈	—	89	1	no	—
79	#498	—	89	3	no	—
80	#476	—	89	3	no	—
81	#475	—	89	3	no	—
82	C₈×Q₈	D₈₉	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	—	89	3	no	—
84	C₂²×Q₈	—	178	1	no	—
85	C₂×SD₁₆	—	89	1	no	—
86	C₄⋊Q₈	—	89	3	no	—
87	C₄⋊Q₈	—	89	3	no	—
88	C₄⋊Q₈	—	178	3	no	—
89	C₂²⋊Q₈	—	178	3	no	—
90	C₄×Q₈	—	89	3	no	—
91	C₄×Q₈	—	178	1	no	—
92	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
93	C₄×D₄	—	89	3	no	—
94	#106	—	89	3	no	—
95	#106	—	89	6	no	—
96	C₄⋊C₈	C₂×D₈₉	1	3	yes	—
97	#103	—	89	3	no	—
98	#102	—	89	3	no	—
99	C₄×C₈	C₂×D₈₉	1	3	yes	—
100	C₂×Q₈	—	89	1	no	—
101	C₂×Q₈	—	356	4	no	—
102	C₂×Q₈	—	178	3	no	—
103	C₂×Q₈	—	178	3	no	—
104	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
105	C₂×D₄	—	89	1	no	—
106	C₂²×C₄	—	178	3	no	—
107	SD₁₆	—	89	4	no	—
108	C₂×C₈	C₂²×D₈₉	1	3	yes	—
109	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
110	C₄⋊C₄	—	89	3	no	—
111	C₄⋊C₄	—	89	3	no	—
112	C₄⋊C₄	—	178	3	no	—
113	C₄⋊C₄	—	178	3	no	—
114	C₄⋊C₄	—	178	3	no	—
115	C₄⋊C₄	D₃₅₆	1	3	yes	—
116	C₂²⋊C₄	—	178	3	no	—
117	C₄²	—	178	3	no	—
118	C₄²	C₂²×D₈₉	1	3	yes	—
119	C₂³	—	178	1	no	—
120	Q₈	—	89	2	no	—
121	Q₈	—	356	6	no	—
122	Q₈	—	178	1	no	—
123	Q₈	—	712	3	no	—
124	Q₈	#13083563	1	4	yes	direct product factor
125	D₄	—	89	2	no	—
126	D₄	—	178	1	no	—
127	C₂×C₄	—	356	6	no	—
128	C₂×C₄	—	178	3	no	—
129	C₂×C₄	—	178	3	no	—
130	C₂×C₄	—	178	1	no	—
131	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
132	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
133	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	Q₈×D₈₉	1	2	yes	complemented
136	C₂²	—	356	1	no	—
137	C₂²	—	356	1	no	—
138	C₂²	—	178	2	no	—
139	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	356	3	no	—
142	C₄	—	356	3	no	—
143	C₄	—	178	2	no	—
144	C₄	C₂×#13083563	1	6	yes	—
145	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
146	C₄	#15668059	1	1	yes	characteristic
147	C₂	—	356	2	no	—
148	C₂	Q₈×D₃₅₆	1	1	yes	characteristic
149	C₂	C₂²×#13083563	1	1	yes	characteristic
150	C₂	#19152410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₅₆

C₂

yes

maximal, characteristic

#19151996

C₂

yes

maximal, complemented

#19151889

C₂

yes

maximal

#19151884

C₂

yes

maximal, complemented

C₄×#13083563

C₂

yes

maximal

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

—

C₂×#13083563

C₂²

yes

characteristic

#15668012

C₂²

yes

—

#15668012

C₂²

yes

—

#15668011

—

#15667992

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×D₃₅₆

C₂²

yes

—

#15667944

C₂²

yes

—

#15667944

C₂²

yes

—

#15667937

C₂²

yes

—

#15667934

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

characteristic

Q₈×D₈₉

—

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₄×D₈₉

—

#13083563

Q₈

yes

direct product factor

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂²×D₈₉

—

Q₈×C₈₉

SD₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

C₂×Q₈

yes

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₂×Q₈

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₈

yes

complemented

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

D₄×Q₈

yes

characteristic

C₂×C₈₉

C₂²×SD₁₆

yes

characteristic

C₂×C₈₉

#577

yes

characteristic

D₈₉

—

Q₈×SD₁₆

—

maximal, Sylow

C₈₉

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

D₈₉

yes

characteristic, complemented

C₄×SD₁₆

—

C₂²×Q₈

—

178

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₈₉

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

356

—

102

C₂×Q₈

—

178

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

D₃₅₆

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

178

—

107

SD₁₆

—

108

C₂×C₈

C₂²×D₈₉

yes

—

109

C₂×C₈

C₂²×D₈₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

178

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄⋊C₄

D₃₅₆

yes

—

116

C₂²⋊C₄

—

178

—

117

C₄²

—

178

—

118

C₄²

C₂²×D₈₉

yes

—

119

C₂³

—

178

—

120

Q₈

—

121

Q₈

—

356

—

122

Q₈

—

178

—

123

Q₈

—

712

—

124

Q₈

#13083563

yes

direct product factor

125

D₄

—

126

D₄

—

178

—

127

C₂×C₄

—

356

—

128

C₂×C₄

—

178

—

129

C₂×C₄

—

178

—

130

C₂×C₄

—

178

—

131

C₂×C₄

C₂×D₃₅₆

yes

—

132

C₂×C₄

C₂×D₃₅₆

yes

—

133

C₂×C₄

C₂³×D₈₉

yes

characteristic

134

C₈

—

135

C₈

Q₈×D₈₉

yes

complemented

136

C₂²

—

356

—

137

C₂²

—

356

—

138

C₂²

—

178

—

139

C₂²

C₂²×D₃₅₆

yes

characteristic

140

C₄

—

141

C₄

—

356

—

142

C₄

—

356

—

143

C₄

—

178

—

144

C₄

C₂×#13083563

yes

—

145

C₄

C₂×Q₈×D₈₉

yes

characteristic

146

C₄

#15668059

yes

characteristic

147

C₂

—

356

—

148

C₂

Q₈×D₃₅₆

yes

characteristic

149

C₂

C₂²×#13083563

yes

characteristic

150

C₂

#19152410

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	C₂³
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₈×Q₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×D₃₅₆

C₂³

hypercenter

C₈×Q₈

D₈₉

hypocenter

C₈₉

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₇₈	C₂	semidirect
3. #19151996	C₂	semidirect
5. #19151884	C₂	semidirect
7. C₈×Q₈×C₈₉	C₂	semidirect
30. #13083563	Q₈	direct (N × H)
44. Q₈×C₈₉	SD₁₆	semidirect
57. Dic₁₇₈	C₂×Q₈	semidirect
59. C₈×C₈₉	C₂×Q₈	semidirect
76. C₈₉	Q₈×SD₁₆	semidirect
82. C₈×Q₈	D₈₉	semidirect
124. Q₈	#13083563	direct (N × H)
135. C₈	Q₈×D₈₉	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₇₈

C₂

semidirect

3. #19151996

C₂

semidirect

5. #19151884

C₂

semidirect

7. C₈×Q₈×C₈₉

C₂

semidirect

30. #13083563

Q₈

direct (N × H)

44. Q₈×C₈₉

SD₁₆

semidirect

57. Dic₁₇₈

C₂×Q₈

semidirect

59. C₈×C₈₉

C₂×Q₈

semidirect

76. C₈₉

Q₈×SD₁₆

semidirect

82. C₈×Q₈

D₈₉

semidirect

124. Q₈

#13083563

direct (N × H)

135. C₈

Q₈×D₈₉

semidirect

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

subgroup	index
Q₈×Dic₁₇₈	2
Q₈×D₃₅₆	2
#19151996	2
#19151889	2
#19151884	2
C₄×#13083563	2
C₈×Q₈×C₈₉	2
Q₈×SD₁₆	89

subgroup

index

Q₈×Dic₁₇₈

Q₈×D₃₅₆

C₈×Q₈×C₈₉

Q₈×SD₁₆

Subgroups — Q₈×#13083563 #26033113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13083563 #26033113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13083563 #26033113