Group #26033108 — subgroups

Subgroups — Q₈×#13083561 #26033108

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×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#19151994	C₂	1	3	yes	maximal, complemented
3	#19151885	C₂	1	3	yes	maximal
4	#19151883	C₂	1	3	yes	maximal, complemented
5	Q₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13083561	C₂	1	3	yes	maximal
7	Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₈₉	—	2	1	no	—
9	C₂×#13083561	C₂²	1	1	yes	characteristic
10	#15668012	C₂²	1	3	yes	—
11	#15667992	—	2	3	no	—
12	C₄×D₃₅₆	C₂²	1	3	yes	—
13	#15667943	C₂²	1	6	yes	—
14	#15667943	C₂²	1	3	yes	—
15	C₄⋊#10623494	C₂²	1	3	yes	—
16	#15667936	C₂²	1	3	yes	—
17	#15667932	C₂²	1	3	yes	—
18	C₄×#10623494	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₈₉	—	2	3	no	—
20	C₄⋊Q₈×C₈₉	C₂²	1	3	yes	—
21	C₄×Q₈×C₈₉	—	2	1	no	—
22	C₄×Q₈×C₈₉	C₂²	1	3	yes	—
23	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₇₈	—	2	3	no	—
25	Q₈×D₈₉	—	4	4	no	—
26	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₈₉	—	2	3	no	—
28	#13083561	Q₈	1	4	yes	direct product factor
29	C₂×#10623494	C₂³	1	3	yes	—
30	C₂×#10623494	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₈₉	—	2	3	no	—
32	C₄⋊Dic₈₉	C₂³	1	3	yes	—
33	#13083551	—	2	3	no	—
34	C₂×Q₈×C₈₉	—	2	3	no	—
35	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₈₉	—	2	3	no	—
38	C₄⋊C₄×C₈₉	—	2	3	no	—
39	C₄⋊C₄×C₈₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
41	C₄²×C₈₉	—	2	3	no	—
42	C₄²×C₈₉	C₂³	1	3	yes	—
43	C₂²×D₈₉	—	2	1	no	—
44	Q₈×C₈₉	—	2	1	no	—
45	Q₈×C₈₉	—	8	3	no	—
46	Q₈×C₈₉	SD₁₆	1	4	yes	complemented
47	Q₈×C₈₉	C₂×Q₈	1	2	yes	complemented
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	3	no	—
50	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₈₉	—	2	3	no	—
54	D₃₅₆	—	2	1	no	—
55	D₃₅₆	C₂×Q₈	1	2	yes	—
56	C₄×D₈₉	—	4	6	no	—
57	Dic₁₇₈	—	4	6	no	—
58	#10623494	—	2	3	no	—
59	#10623494	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	2	no	—
65	C₄×C₈₉	—	2	3	no	—
66	C₄×C₈₉	—	4	3	no	—
67	C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
68	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
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	D₈₉	1	1	yes	characteristic, complemented
78	D₄×Q₈	—	89	1	no	—
79	#498	—	89	3	no	—
80	#476	—	89	3	no	—
81	#475	—	89	3	no	—
82	C₈×Q₈	—	89	1	no	—
83	C₄×SD₁₆	—	89	3	no	—
84	C₂²×Q₈	—	178	1	no	—
85	C₂×SD₁₆	—	89	1	no	—
86	C₄⋊Q₈	—	2	3	no	—
87	C₄⋊Q₈	—	89	3	no	—
88	C₄⋊Q₈	C₂×D₈₉	1	3	yes	—
89	C₂²⋊Q₈	—	178	3	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂×D₈₉	1	3	yes	—
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	—	89	3	no	—
97	#103	—	89	3	no	—
98	#102	—	89	3	no	—
99	C₄×C₈	—	89	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	356	4	no	—
102	C₂×Q₈	—	178	3	no	—
103	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
105	C₂×D₄	—	89	1	no	—
106	C₂²×C₄	—	178	3	no	—
107	SD₁₆	—	89	4	no	—
108	C₂×C₈	—	89	1	no	—
109	C₂×C₈	—	89	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	89	3	no	—
113	C₄⋊C₄	—	178	3	no	—
114	C₄⋊C₄	C₂²⋊D₈₉	1	3	yes	—
115	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
116	C₂²⋊C₄	—	178	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂²×D₈₉	1	3	yes	—
119	C₂³	—	178	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	8	3	no	—
122	Q₈	—	356	6	no	—
123	Q₈	#13083561	1	4	yes	direct product factor
124	Q₈	Q₈×D₈₉	1	2	yes	complemented
125	D₄	—	89	2	no	—
126	D₄	—	178	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	356	6	no	—
130	C₂×C₄	—	178	3	no	—
131	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
133	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
134	C₈	—	89	2	no	—
135	C₈	—	178	3	no	—
136	C₂²	—	356	1	no	—
137	C₂²	—	356	1	no	—
138	C₂²	—	178	2	no	—
139	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	3	no	—
143	C₄	—	356	3	no	—
144	C₄	C₂×#13083561	1	6	yes	—
145	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
146	C₄	#15668060	1	1	yes	characteristic
147	C₂	—	356	2	no	—
148	C₂	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂	C₂²×#13083561	1	1	yes	characteristic
150	C₂	#19152409	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₃₅₆

C₂

yes

maximal, characteristic

#19151994

C₂

yes

maximal, complemented

#19151885

C₂

yes

maximal

#19151883

C₂

yes

maximal, complemented

Q₈×#10623494

C₂

yes

maximal, characteristic, complemented

C₄×#13083561

C₂

yes

maximal

Q₈²×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

—

C₂×#13083561

C₂²

yes

characteristic

#15668012

C₂²

yes

—

#15667992

—

C₄×D₃₅₆

C₂²

yes

—

#15667943

C₂²

yes

—

#15667943

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667936

C₂²

yes

—

#15667932

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₄×D₈₉

—

#13083561

Q₈

yes

direct product factor

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083551

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×Q₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

SD₁₆

yes

complemented

Q₈×C₈₉

C₂×Q₈

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

C₂×Q₈

yes

—

C₄×D₈₉

—

Dic₁₇₈

—

#10623494

—

#10623494

C₂×Q₈

yes

complemented

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

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

D₈₉

yes

characteristic, complemented

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

178

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₉

yes

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

356

—

102

C₂×Q₈

—

178

—

103

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

104

C₂×Q₈

C₂²×D₈₉

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

178

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

C₂²⋊D₈₉

yes

—

115

C₄⋊C₄

C₂²×D₈₉

yes

—

116

C₂²⋊C₄

—

178

—

117

C₄²

—

118

C₄²

C₂²×D₈₉

yes

—

119

C₂³

—

178

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

356

—

123

Q₈

#13083561

yes

direct product factor

124

Q₈

Q₈×D₈₉

yes

complemented

125

D₄

—

126

D₄

—

178

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

356

—

130

C₂×C₄

—

178

—

131

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

133

C₂×C₄

C₂³×D₈₉

yes

characteristic

134

C₈

—

135

C₈

—

178

—

136

C₂²

—

356

—

137

C₂²

—

356

—

138

C₂²

—

178

—

139

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₄

C₂×#13083561

yes

—

145

C₄

C₂×Q₈×D₈₉

yes

characteristic

146

C₄

#15668060

yes

characteristic

147

C₂

—

356

—

148

C₂

Q₈×C₂²⋊D₈₉

yes

characteristic

149

C₂

C₂²×#13083561

yes

characteristic

150

C₂

#19152409

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	Q₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	C₂³
hypercenter	Q₈²	D₈₉
hypocenter	C₈₉	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

Q₈²×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×D₃₅₆

C₂³

hypercenter

Q₈²

D₈₉

hypocenter

C₈₉

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
2. #19151994	C₂	semidirect
4. #19151883	C₂	semidirect
5. Q₈×#10623494	C₂	semidirect
7. Q₈²×C₈₉	C₂	semidirect
28. #13083561	Q₈	direct (N × H)
46. Q₈×C₈₉	SD₁₆	semidirect
47. Q₈×C₈₉	C₂×Q₈	semidirect
59. #10623494	C₂×Q₈	semidirect
76. C₈₉	Q₈×SD₁₆	semidirect
77. Q₈²	D₈₉	semidirect
123. Q₈	#13083561	direct (N × H)
124. Q₈	Q₈×D₈₉	semidirect

H ≅ G/N

type

2. #19151994

C₂

semidirect

4. #19151883

C₂

semidirect

5. Q₈×#10623494

C₂

semidirect

7. Q₈²×C₈₉

C₂

semidirect

28. #13083561

Q₈

direct (N × H)

46. Q₈×C₈₉

SD₁₆

semidirect

47. Q₈×C₈₉

C₂×Q₈

semidirect

59. #10623494

C₂×Q₈

semidirect

76. C₈₉

Q₈×SD₁₆

semidirect

77. Q₈²

D₈₉

semidirect

123. Q₈

#13083561

direct (N × H)

124. Q₈

Q₈×D₈₉

semidirect

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

subgroup	index
Q₈×D₃₅₆	2
#19151994	2
#19151885	2
#19151883	2
Q₈×#10623494	2
C₄×#13083561	2
Q₈²×C₈₉	2
Q₈×SD₁₆	89

subgroup

index

Q₈×D₃₅₆

Q₈²×C₈₉

Q₈×SD₁₆

Subgroups — Q₈×#13083561 #26033108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13083561 #26033108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13083561 #26033108