Group #26033134 — subgroups

Subgroups — Q₈×#13083564 #26033134

Order 11392.

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

normal subgroups only

This group has 336 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
2	#19152001	C₂	1	3	yes	maximal
3	#19151893	C₂	1	3	yes	maximal
4	#19151890	C₂	1	3	yes	maximal
5	Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
6	C₄×#13083564	C₂	1	3	yes	maximal
7	Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083564	C₂²	1	1	yes	characteristic
9	#15668012	C₂²	1	3	yes	—
10	#15668011	—	2	3	no	—
11	Q₈×Dic₈₉	—	2	1	no	—
12	C₄×Dic₁₇₈	C₂²	1	3	yes	—
13	#15667945	C₂²	1	6	yes	—
14	#15667945	C₂²	1	3	yes	—
15	C₄⋊#10623494	C₂²	1	3	yes	—
16	#15667936	C₂²	1	3	yes	—
17	#15667935	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	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
26	#13083564	Q₈	1	4	yes	direct product factor
27	C₂×#10623494	C₂³	1	3	yes	—
28	C₂×#10623494	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₈₉	—	2	3	no	—
30	C₄⋊Dic₈₉	C₂³	1	3	yes	—
31	#13083553	—	2	3	no	—
32	C₄×Dic₈₉	—	2	3	no	—
33	C₂×Q₈×C₈₉	—	2	3	no	—
34	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	2	3	no	—
37	C₄⋊C₄×C₈₉	—	2	3	no	—
38	C₄⋊C₄×C₈₉	D₄	1	3	yes	—
39	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
40	C₄²×C₈₉	—	2	3	no	—
41	C₄²×C₈₉	C₂³	1	3	yes	—
42	Q₈×C₈₉	—	2	1	no	—
43	Q₈×C₈₉	—	8	3	no	—
44	Q₈×C₈₉	Q₁₆	1	4	yes	complemented
45	Q₈×C₈₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	3	no	—
48	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₈₉	—	2	1	no	—
52	C₂×Dic₈₉	—	2	3	no	—
53	Dic₁₇₈	—	2	1	no	—
54	Dic₁₇₈	—	8	3	no	—
55	Dic₁₇₈	C₂×Q₈	1	2	yes	—
56	#10623494	—	2	3	no	—
57	#10623494	C₂×Q₈	1	2	yes	—
58	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₈₉	—	2	2	no	—
60	C₄×C₈₉	—	2	3	no	—
61	C₄×C₈₉	—	4	3	no	—
62	C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
63	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
65	Dic₈₉	—	2	2	no	—
66	Dic₈₉	—	4	3	no	—
67	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
68	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₈₉	#576	1	1	yes	characteristic
70	Q₈×Q₁₆	—	89	1	no	maximal, Sylow
71	C₈₉	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
72	Q₈²	—	89	1	no	—
73	Q₈²	D₈₉	1	1	yes	characteristic
74	#500	—	89	3	no	—
75	#477	—	89	3	no	—
76	#477	—	89	3	no	—
77	C₈×Q₈	—	89	1	no	—
78	C₄×Q₁₆	—	89	3	no	—
79	C₂×Q₁₆	—	89	1	no	—
80	C₄⋊Q₈	—	2	3	no	—
81	C₄⋊Q₈	—	89	3	no	—
82	C₄⋊Q₈	—	178	3	no	—
83	C₄⋊Q₈	C₂×D₈₉	1	3	yes	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	89	3	no	—
86	C₄×Q₈	—	178	1	no	—
87	C₄×Q₈	C₂×D₈₉	1	3	yes	—
88	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
89	#107	—	89	3	no	—
90	#107	—	89	6	no	—
91	C₄⋊C₈	—	89	3	no	—
92	#103	—	89	3	no	—
93	#103	—	89	3	no	—
94	C₄×C₈	—	89	3	no	—
95	C₂×Q₈	—	2	3	no	—
96	C₂×Q₈	—	89	1	no	—
97	C₂×Q₈	—	178	3	no	—
98	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
99	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
100	Q₁₆	—	89	4	no	—
101	C₂×C₈	—	89	1	no	—
102	C₂×C₈	—	89	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	2	3	no	—
105	C₄⋊C₄	—	89	3	no	—
106	C₄⋊C₄	—	178	3	no	—
107	C₄⋊C₄	—	178	3	no	—
108	C₄⋊C₄	C₂²⋊D₈₉	1	3	yes	—
109	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
110	C₄²	—	2	3	no	—
111	C₄²	—	178	3	no	—
112	C₄²	C₂²×D₈₉	1	3	yes	—
113	Q₈	—	2	1	no	—
114	Q₈	—	8	3	no	—
115	Q₈	—	89	2	no	—
116	Q₈	—	178	1	no	—
117	Q₈	—	712	3	no	—
118	Q₈	#13083564	1	4	yes	direct product factor
119	Q₈	Q₈×D₈₉	1	2	yes	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	178	3	no	—
123	C₂×C₄	—	178	1	no	—
124	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
125	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
126	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
127	C₈	—	89	2	no	—
128	C₈	—	178	3	no	—
129	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₄	—	2	3	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	3	no	—
133	C₄	—	356	3	no	—
134	C₄	—	178	2	no	—
135	C₄	C₂×#13083564	1	6	yes	—
136	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
137	C₄	#15668060	1	1	yes	characteristic
138	C₂	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂	C₂²×#13083564	1	1	yes	characteristic
140	C₂	#19152401	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₇₈

C₂

yes

maximal, characteristic

#19152001

C₂

yes

maximal

#19151893

C₂

yes

maximal

#19151890

C₂

yes

maximal

Q₈×#10623494

C₂

yes

maximal, characteristic

C₄×#13083564

C₂

yes

maximal

Q₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083564

C₂²

yes

characteristic

#15668012

C₂²

yes

—

#15668011

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667945

C₂²

yes

—

#15667945

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667936

C₂²

yes

—

#15667935

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

—

C₂×Dic₁₇₈

C₂³

yes

characteristic

#13083564

Q₈

yes

direct product factor

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₄×Dic₈₉

—

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

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

Q₁₆

yes

complemented

Q₈×C₈₉

C₂×Q₈

yes

—

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

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₂×Q₈

yes

—

#10623494

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

D₄×Q₈

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

characteristic

C₂×C₈₉

#576

yes

characteristic

Q₈×Q₁₆

—

maximal, Sylow

C₈₉

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

Q₈²

D₈₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₄×Q₈

C₂×D₈₉

yes

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

C₂×Q₈

C₂²×D₈₉

yes

characteristic

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

C₂²⋊D₈₉

yes

—

109

C₄⋊C₄

C₂²×D₈₉

yes

—

110

C₄²

—

111

C₄²

—

178

—

112

C₄²

C₂²×D₈₉

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

178

—

117

Q₈

—

712

—

118

Q₈

#13083564

yes

direct product factor

119

Q₈

Q₈×D₈₉

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

125

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

126

C₂×C₄

C₂³×D₈₉

yes

characteristic

127

C₈

—

128

C₈

—

178

—

129

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

356

—

134

C₄

—

178

—

135

C₄

C₂×#13083564

yes

—

136

C₄

C₂×Q₈×D₈₉

yes

characteristic

137

C₄

#15668060

yes

characteristic

138

C₂

Q₈×C₂²⋊D₈₉

yes

characteristic

139

C₂

C₂²×#13083564

yes

characteristic

140

C₂

#19152401

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, prime radical	C₂²×C₈₉	C₂²×D₄
hypercenter	Q₈²	D₈₉
hypocenter	C₈₉	Q₈×Q₁₆
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, prime radical

C₂²×C₈₉

C₂²×D₄

hypercenter

Q₈²

D₈₉

hypocenter

C₈₉

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
26. #13083564	Q₈	direct (N × H)
44. Q₈×C₈₉	Q₁₆	semidirect
71. C₈₉	Q₈×Q₁₆	semidirect
118. Q₈	#13083564	direct (N × H)

H ≅ G/N

type

26. #13083564

Q₈

direct (N × H)

44. Q₈×C₈₉

Q₁₆

semidirect

71. C₈₉

Q₈×Q₁₆

semidirect

118. Q₈

#13083564

direct (N × H)

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

subgroup	index
Q₈×Dic₁₇₈	2
#19152001	2
#19151893	2
#19151890	2
Q₈×#10623494	2
C₄×#13083564	2
Q₈²×C₈₉	2
Q₈×Q₁₆	89

subgroup

index

Q₈×Dic₁₇₈

Q₈²×C₈₉

Q₈×Q₁₆

Subgroups — Q₈×#13083564 #26033134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13083564 #26033134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13083564 #26033134