Group #26021404 — subgroups

Subgroups — #26021404

Order 11392.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#15668004	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667937	C₂	1	2	yes	maximal
3	#19151504	C₂	1	1	yes	maximal, characteristic
4	#19151361	C₂	1	2	yes	maximal
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	#15668004	—	2	4	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×#13083553	—	2	2	no	—
11	#15667937	—	2	4	no	—
12	#15667919	—	2	2	no	—
13	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₈	—	2	2	no	—
16	C₂×Dic₁₇₈	—	2	2	no	—
17	C₂×Dic₁₇₈	—	2	1	no	—
18	C₂×Dic₁₇₈	Q₈	1	2	yes	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂²×Dic₈₉	—	2	1	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	D₄	1	4	yes	—
25	#13083553	—	2	8	no	—
26	C₂²×C₄×C₈₉	D₄	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	—	2	2	no	—
32	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	2	yes	—
40	C₂×Dic₈₉	—	2	4	no	—
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	4	1	no	—
43	C₂×Dic₈₉	—	4	1	no	—
44	C₂×Dic₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	C₂×Dic₈₉	—	4	1	no	—
47	C₂×Dic₈₉	—	4	2	no	—
48	Dic₁₇₈	—	2	4	no	—
49	Dic₁₇₈	—	4	2	no	—
50	C₈×C₈₉	—	4	2	no	—
51	C₂²×C₈₉	#120	1	1	yes	characteristic
52	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₈₉	#124	1	1	yes	characteristic
54	C₂²×C₈₉	#135	1	2	yes	—
55	C₂²×C₈₉	#137	1	2	yes	—
56	C₄×C₈₉	—	4	2	no	—
57	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₈₉	#124	1	2	yes	—
59	Dic₈₉	—	4	2	no	—
60	Dic₈₉	—	4	4	no	—
61	Dic₈₉	—	8	1	no	—
62	C₂×C₈₉	#393	1	1	yes	characteristic
63	C₂×C₈₉	#452	1	2	yes	—
64	C₂×C₈₉	#479	1	2	yes	—
65	C₂×C₈₉	#488	1	1	yes	characteristic
66	C₂×C₈₉	#491	1	1	yes	characteristic
67	#2026	—	89	1	no	maximal, Sylow
68	C₈₉	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	—	89	1	no	—
70	C₂×#103	—	89	2	no	—
71	#391	—	89	1	no	—
72	#340	—	89	2	no	—
73	#336	D₈₉	1	1	yes	characteristic
74	C₂²×Q₈	—	89	1	no	—
75	C₂²×C₈	C₂×D₈₉	1	2	yes	—
76	#125	—	178	4	no	—
77	C₂×C₄⋊C₄	—	89	2	no	—
78	C₂×C₄⋊C₄	—	89	1	no	—
79	C₂×C₄⋊C₄	—	178	2	no	—
80	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
81	#103	—	178	4	no	—
82	#95	—	178	2	no	—
83	C₂×Q₈	—	89	2	no	—
84	C₂×Q₈	—	178	2	no	—
85	C₂×Q₈	—	178	2	no	—
86	C₂×Q₈	—	178	1	no	—
87	C₂²×C₄	—	178	1	no	—
88	C₂²×C₄	—	178	1	no	—
89	C₂²×C₄	—	178	2	no	—
90	C₂²×C₄	D₃₅₆	1	2	yes	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	89	4	no	—
96	C₄⋊C₄	—	178	8	no	—
97	C₄⋊C₄	—	178	2	no	—
98	C₄⋊C₄	—	178	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
101	Q₈	—	356	2	no	—
102	Q₈	—	178	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	356	1	no	—
107	C₂×C₄	—	356	1	no	—
108	C₂×C₄	—	356	2	no	—
109	C₂×C₄	—	356	2	no	—
110	C₂×C₄	—	356	1	no	—
111	C₂×C₄	—	356	2	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	D₄×D₈₉	1	2	yes	—
115	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
116	C₂×C₄	#13083573	1	1	yes	characteristic
117	C₂×C₄	#13083574	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#15667982	1	1	yes	characteristic
120	C₂²	#15667992	1	1	yes	characteristic
121	C₂²	#15668000	1	1	yes	characteristic
122	C₂²	#15668034	1	2	yes	—
123	C₂²	#15668045	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	356	4	no	—
126	C₄	—	356	2	no	—
127	C₄	—	712	1	no	—
128	C₄	#15667990	1	2	yes	—
129	C₄	#15667998	1	2	yes	—
130	C₂	#19151511	1	1	yes	characteristic
131	C₂	#19151762	1	2	yes	—
132	C₂	#19151906	1	2	yes	—
133	C₂	#19151953	1	1	yes	characteristic
134	C₂	#19151967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

#19151504

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

#15667937

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#393

yes

characteristic

C₂×C₈₉

#452

yes

—

C₂×C₈₉

#479

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₉

yes

—

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

100

C₂³

C₂×D₃₅₆

yes

characteristic

101

Q₈

—

356

—

102

Q₈

—

178

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

356

—

110

C₂×C₄

—

356

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

D₄×D₈₉

yes

—

115

C₂×C₄

Q₈×D₈₉

yes

characteristic

116

C₂×C₄

#13083573

yes

characteristic

117

C₂×C₄

#13083574

yes

—

118

C₈

—

119

C₂²

#15667982

yes

characteristic

120

C₂²

#15667992

yes

characteristic

121

C₂²

#15668000

yes

characteristic

122

C₂²

#15668034

yes

—

123

C₂²

#15668045

yes

—

124

C₄

—

125

C₄

—

356

—

126

C₄

—

356

—

127

C₄

—

712

—

128

C₄

#15667990

yes

—

129

C₄

#15667998

yes

—

130

C₂

#19151511

yes

characteristic

131

C₂

#19151762

yes

—

132

C₂

#19151906

yes

—

133

C₂

#19151953

yes

characteristic

134

C₂

#19151967

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#336×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#336	D₈₉
hypocenter	C₈₉	#2026
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#336×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
68. C₈₉	#2026	semidirect

H ≅ G/N

type

68. C₈₉

#2026

semidirect

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

subgroup	index
C₂×#15668004	2
C₂×#15667937	2
#19151504	2
#19151361	2
#336×C₈₉	2
#2026	89

subgroup

index

#336×C₈₉

Subgroups — #26021404

Conjugacy classes of subgroups

Complements

Maximal subgroups