Group #26019608 — subgroups

Subgroups — #26019608

Order 11392.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151975	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151794	C₂	1	2	yes	maximal, complemented
3	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151341	C₂	1	2	yes	maximal, complemented
5	#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₁₂	—	2	2	no	—
7	C₂×D₇₁₂	—	2	1	no	—
8	C₂×D₇₁₂	—	2	1	no	—
9	C₂×C₈×D₈₉	—	2	1	no	—
10	#15668009	C₂²	1	2	yes	—
11	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	#15667934	—	2	2	no	—
14	#15667924	—	2	1	no	—
15	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	D₄	1	1	yes	characteristic
21	C₂×C₄×D₈₉	—	2	1	no	—
22	D₇₁₂	—	4	2	no	—
23	D₇₁₂	—	4	4	no	—
24	D₇₁₂	—	4	2	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₈×D₈₉	—	4	2	no	—
27	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
28	#13083551	—	2	1	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×D₈₉	—	2	1	no	—
34	C₂²×D₈₉	—	4	2	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
38	C₂×Dic₈₉	—	2	1	no	—
39	D₃₅₆	—	2	1	no	—
40	D₃₅₆	—	4	2	no	—
41	D₃₅₆	—	4	2	no	—
42	D₃₅₆	—	4	2	no	—
43	D₃₅₆	—	4	2	no	—
44	D₃₅₆	—	8	2	no	—
45	D₃₅₆	D₈	1	2	yes	complemented
46	C₄×D₈₉	—	4	1	no	—
47	C₄×D₈₉	—	4	1	no	—
48	C₈×C₈₉	—	2	1	no	—
49	C₈×C₈₉	—	4	2	no	—
50	C₈×C₈₉	D₈	1	2	yes	complemented
51	#10623494	—	4	1	no	—
52	C₂²×C₈₉	#121	1	1	yes	characteristic
53	C₂×D₈₉	—	2	2	no	—
54	C₂×D₈₉	—	4	1	no	—
55	C₂×D₈₉	—	4	1	no	—
56	C₂×D₈₉	—	8	2	no	—
57	C₂×D₈₉	—	8	2	no	—
58	C₂×D₈₉	—	8	2	no	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
61	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
62	C₄×C₈₉	#135	1	1	yes	characteristic
63	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₈₉	—	4	1	no	—
65	C₂×C₈₉	#459	1	1	yes	characteristic
66	C₂×C₈₉	#466	1	1	yes	characteristic
67	C₂×C₈₉	#472	1	1	yes	characteristic
68	D₈₉	—	4	2	no	—
69	D₈₉	—	16	2	no	—
70	#1667	—	89	1	no	maximal, Sylow
71	C₈₉	#1667	1	1	yes	characteristic, complemented, Sylow
72	#493	—	89	1	no	—
73	#459	—	89	2	no	—
74	C₈×D₄	—	89	1	no	—
75	#335	D₈₉	1	1	yes	characteristic, complemented
76	#331	—	89	2	no	—
77	C₂×D₈	—	178	2	no	—
78	C₂×D₈	—	178	1	no	—
79	C₂×D₈	—	178	1	no	—
80	C₂²×C₈	—	178	1	no	—
81	#127	—	89	2	no	—
82	C₄×D₄	—	89	1	no	—
83	C₄⋊C₈	—	89	1	no	—
84	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
85	#102	—	178	2	no	—
86	C₂²⋊C₈	—	178	1	no	—
87	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂×D₄	—	89	1	no	—
89	C₂×D₄	—	178	2	no	—
90	C₂×D₄	—	178	2	no	—
91	C₂×D₄	—	178	2	no	—
92	C₂²×C₄	—	178	1	no	—
93	D₈	—	356	2	no	—
94	D₈	—	356	4	no	—
95	D₈	—	356	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	356	2	no	—
98	C₂×C₈	—	178	1	no	—
99	C₂×C₈	D₃₅₆	1	1	yes	characteristic
100	C₂×C₈	D₃₅₆	1	1	yes	characteristic
101	C₄⋊C₄	—	89	1	no	—
102	C₂²⋊C₄	—	178	1	no	—
103	C₄²	C₂²×D₈₉	1	1	yes	characteristic
104	C₂³	—	356	2	no	—
105	C₂³	—	178	1	no	—
106	D₄	—	89	2	no	—
107	D₄	—	356	2	no	—
108	D₄	—	356	2	no	—
109	D₄	—	356	2	no	—
110	D₄	—	356	2	no	—
111	D₄	—	178	1	no	—
112	D₄	—	712	2	no	—
113	C₂×C₄	—	356	1	no	—
114	C₂×C₄	—	356	1	no	—
115	C₂×C₄	—	178	1	no	—
116	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
118	C₂×C₄	#13083575	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	356	1	no	—
122	C₈	D₇₁₂	1	2	yes	complemented
123	C₂²	—	356	1	no	—
124	C₂²	—	356	1	no	—
125	C₂²	—	178	2	no	—
126	C₂²	—	712	2	no	—
127	C₂²	—	712	2	no	—
128	C₂²	—	712	2	no	—
129	C₂²	#15667987	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	356	1	no	—
132	C₄	D₈×D₈₉	1	1	yes	characteristic
133	C₄	C₂×D₇₁₂	1	1	yes	characteristic
134	C₄	#15668035	1	1	yes	characteristic
135	C₄	#15668040	1	1	yes	characteristic
136	C₂	—	356	2	no	—
137	C₂	—	1424	2	no	—
138	C₂	#19151794	1	1	yes	characteristic
139	C₂	#19151835	1	1	yes	characteristic
140	C₂	#19151869	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151975

C₂

yes

maximal, characteristic, complemented

#19151794

C₂

yes

maximal, complemented

C₈×D₃₅₆

C₂

yes

maximal, characteristic, complemented

#19151341

C₂

yes

maximal, complemented

#335×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×C₈×D₈₉

—

#15668009

C₂²

yes

—

C₄×D₃₅₆

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

#15667934

—

#15667924

—

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

—

C₈×D₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083551

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

D₈

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

D₈

yes

complemented

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

D₈₉

—

#1667

—

maximal, Sylow

C₈₉

#1667

yes

characteristic, complemented, Sylow

#493

—

#459

—

C₈×D₄

—

#335

D₈₉

yes

characteristic, complemented

#331

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₂²×C₈

—

178

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

178

—

C₂²⋊C₈

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

D₈

—

356

—

D₈

—

356

—

D₈

—

356

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

characteristic

100

C₂×C₈

D₃₅₆

yes

characteristic

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

178

—

103

C₄²

C₂²×D₈₉

yes

characteristic

104

C₂³

—

356

—

105

C₂³

—

178

—

106

D₄

—

107

D₄

—

356

—

108

D₄

—

356

—

109

D₄

—

356

—

110

D₄

—

356

—

111

D₄

—

178

—

112

D₄

—

712

—

113

C₂×C₄

—

356

—

114

C₂×C₄

—

356

—

115

C₂×C₄

—

178

—

116

C₂×C₄

D₄×D₈₉

yes

characteristic

117

C₂×C₄

C₂×D₃₅₆

yes

characteristic

118

C₂×C₄

#13083575

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

356

—

122

C₈

D₇₁₂

yes

complemented

123

C₂²

—

356

—

124

C₂²

—

356

—

125

C₂²

—

178

—

126

C₂²

—

712

—

127

C₂²

—

712

—

128

C₂²

—

712

—

129

C₂²

#15667987

yes

characteristic

130

C₄

—

131

C₄

—

356

—

132

C₄

D₈×D₈₉

yes

characteristic

133

C₄

C₂×D₇₁₂

yes

characteristic

134

C₄

#15668035

yes

characteristic

135

C₄

#15668040

yes

characteristic

136

C₂

—

356

—

137

C₂

—

1424

—

138

C₂

#19151794

yes

characteristic

139

C₂

#19151835

yes

characteristic

140

C₂

#19151869

yes

characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#335×C₈₉	C₂
socle	C₂²×C₈₉	#121
hypercenter	#335	D₈₉
hypocenter	C₈₉	#1667
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#335×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #19151975	C₂	semidirect
2. #19151794	C₂	semidirect
3. C₈×D₃₅₆	C₂	semidirect
4. #19151341	C₂	semidirect
5. #335×C₈₉	C₂	semidirect
45. D₃₅₆	D₈	semidirect
50. C₈×C₈₉	D₈	semidirect
71. C₈₉	#1667	semidirect
75. #335	D₈₉	semidirect
122. C₈	D₇₁₂	semidirect

H ≅ G/N

type

1. #19151975

C₂

semidirect

2. #19151794

C₂

semidirect

3. C₈×D₃₅₆

C₂

semidirect

4. #19151341

C₂

semidirect

5. #335×C₈₉

C₂

semidirect

45. D₃₅₆

D₈

semidirect

50. C₈×C₈₉

D₈

semidirect

71. C₈₉

#1667

semidirect

75. #335

D₈₉

semidirect

122. C₈

D₇₁₂

semidirect

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

subgroup	index
#19151975	2
#19151794	2
C₈×D₃₅₆	2
#19151341	2
#335×C₈₉	2
#1667	89

subgroup

index

C₈×D₃₅₆

#335×C₈₉

Subgroups — #26019608

Conjugacy classes of subgroups

Complements

Maximal subgroups