Group #22626114 — subgroups

Subgroups — #22626114

Order 9344.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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	#17989546	C₂	1	1	yes	maximal, characteristic
2	#17989533	C₂	1	1	yes	maximal, characteristic
3	#17989460	C₂	1	4	yes	maximal
4	#125×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	C₂²	1	2	yes	—
6	C₂×C₄⋊#64408	—	2	2	no	—
7	C₂×C₄⋊#64408	C₂²	1	2	yes	—
8	#14556646	—	2	2	no	—
9	#14556583	C₄	1	1	yes	characteristic, complemented
10	#14556582	C₄	1	2	yes	complemented
11	#14556581	—	2	2	no	—
12	C₄⋊C₄×D₇₃	—	2	2	no	—
13	C₄⋊C₄×D₇₃	C₂²	1	2	yes	—
14	C₄²×D₇₃	C₂²	1	1	yes	characteristic
15	#125×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#64408	—	2	2	no	—
17	#11924036	—	2	2	no	—
18	#11924036	Q₈	1	4	yes	—
19	C₂×#10596833	—	2	2	no	—
20	C₄⋊#64408	—	2	4	no	—
21	C₄⋊#64408	—	4	4	no	—
22	C₂×C₄×D₇₃	—	2	2	no	—
23	C₂×C₄×D₇₃	D₄	1	2	yes	—
24	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₇₃	—	2	2	no	—
26	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
27	#11924007	—	2	2	no	—
28	#11924007	—	2	2	no	—
29	#11924007	—	2	2	no	—
30	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₇₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
34	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×#64408	—	4	2	no	—
37	C₂×#64408	—	4	4	no	—
38	C₂×C₄×C₇₃	—	2	2	no	—
39	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₇₃	—	2	2	no	—
42	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₇₃	—	2	2	no	—
45	C₄×D₇₃	—	2	2	no	—
46	C₄×D₇₃	—	4	2	no	—
47	C₄×D₇₃	—	4	2	no	—
48	C₄×D₇₃	C₂×Q₈	1	2	yes	—
49	C₄×D₇₃	#41	1	2	yes	—
50	#10596833	—	2	4	no	—
51	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₇₃	#124	1	1	yes	characteristic
53	C₂×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×D₇₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₂×D₇₃	#137	1	2	yes	—
56	#64408	—	8	2	no	—
57	C₄×C₇₃	—	2	2	no	—
58	C₄×C₇₃	—	4	2	no	—
59	C₄×C₇₃	C₄×Q₈	1	2	yes	—
60	Dic₇₃	—	2	2	no	—
61	Dic₇₃	—	4	2	no	—
62	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₇₃	#391	1	1	yes	characteristic
64	C₂×C₇₃	#417	1	2	yes	—
65	D₇₃	#490	1	1	yes	characteristic
66	D₇₃	#491	1	1	yes	characteristic
67	D₇₃	#501	1	2	yes	—
68	#1992	—	73	1	no	maximal, Sylow
69	C₇₃	#1992	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	—	73	1	no	—
71	C₂×#97	—	73	1	no	—
72	#384	—	73	1	no	—
73	#340	—	73	4	no	—
74	C₂²×C₈	—	73	2	no	—
75	#125	—	73	2	no	—
76	#125	—	73	1	no	—
77	#125	—	146	2	no	—
78	#125	#64408	1	1	yes	characteristic, complemented
79	C₂×C₄⋊C₄	—	73	2	no	—
80	C₂×C₄⋊C₄	—	73	2	no	—
81	C₂×C₄⋊C₄	—	146	2	no	—
82	C₂×C₄⋊C₄	—	146	2	no	—
83	C₂×C₄²	—	73	1	no	—
84	#97	—	146	2	no	—
85	C₂²×C₄	—	73	1	no	—
86	C₂²×C₄	—	73	2	no	—
87	C₂²×C₄	—	146	2	no	—
88	C₂²×C₄	—	146	2	no	—
89	C₂×C₈	—	73	4	no	—
90	C₂×C₈	—	146	2	no	—
91	C₂×C₈	—	146	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	292	4	no	—
94	C₄⋊C₄	—	73	2	no	—
95	C₄⋊C₄	—	146	2	no	—
96	C₄⋊C₄	—	146	2	no	—
97	C₄⋊C₄	—	146	2	no	—
98	C₄⋊C₄	—	146	4	no	—
99	C₄⋊C₄	—	146	2	no	—
100	C₄⋊C₄	C₂×#64408	1	2	yes	—
101	C₄²	—	73	2	no	—
102	C₄²	—	73	1	no	—
103	C₄²	C₂×#64408	1	1	yes	characteristic
104	C₂³	—	73	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	292	2	no	—
107	C₂×C₄	—	292	2	no	—
108	C₂×C₄	—	292	4	no	—
109	C₂×C₄	—	292	2	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	73	2	no	—
114	C₂×C₄	—	146	2	no	—
115	C₂×C₄	—	146	2	no	—
116	C₂×C₄	—	146	2	no	—
117	C₂×C₄	C₂²⋊#64408	1	2	yes	—
118	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
119	C₈	—	146	4	no	—
120	C₂²	—	73	1	no	—
121	C₂²	—	73	1	no	—
122	C₂²	—	73	2	no	—
123	C₂²	—	73	2	no	—
124	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	292	2	no	—
128	C₄	—	584	2	no	—
129	C₄	—	73	2	no	—
130	C₄	—	146	2	no	—
131	C₄	Q₈×#64408	1	2	yes	—
132	C₂	—	73	1	no	—
133	C₂	—	73	1	no	—
134	C₂	—	73	2	no	—
135	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
136	C₂	#17989578	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989546

C₂

yes

maximal, characteristic

#17989533

C₂

yes

maximal, characteristic

#17989460

C₂

yes

maximal

#125×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

—

C₂×C₄⋊#64408

C₂²

yes

—

#14556646

—

#14556583

C₄

yes

characteristic, complemented

#14556582

C₄

yes

complemented

#14556581

—

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

#125×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924036

—

#11924036

Q₈

yes

—

C₂×#10596833

—

C₄⋊#64408

—

C₄⋊#64408

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

#124

yes

characteristic

C₂×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×D₇₃

AGL(1,ℤ/8)

yes

—

C₂×D₇₃

#137

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²⋊Q₈

yes

—

C₂×C₇₃

#391

yes

characteristic

C₂×C₇₃

#417

yes

—

D₇₃

yes

D₇₃

yes

D₇₃

yes

—

#1992

—

maximal, Sylow

C₇₃

#1992

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#97

—

#384

—

#340

—

C₂²×C₈

—

#125

—

#125

—

#125

—

146

—

#125

#64408

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

#97

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₄⋊C₄

C₂×#64408

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×#64408

yes

characteristic

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

292

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

146

—

115

C₂×C₄

—

146

—

116

C₂×C₄

—

146

—

117

C₂×C₄

C₂²⋊#64408

yes

—

118

C₂×C₄

C₂²×#64408

yes

characteristic

119

C₈

—

146

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊#64408

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

292

—

128

C₄

—

584

—

129

C₄

—

130

C₄

—

146

—

131

C₄

Q₈×#64408

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

136

C₂

#17989578

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#125×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#125	#64408
hypocenter	C₇₃	#1992
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#125×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #14556583	C₄	semidirect
10. #14556582	C₄	semidirect
15. #125×C₇₃	C₄	semidirect
69. C₇₃	#1992	semidirect
78. #125	#64408	semidirect

H ≅ G/N

type

9. #14556583

C₄

semidirect

10. #14556582

C₄

semidirect

15. #125×C₇₃

C₄

semidirect

69. C₇₃

#1992

semidirect

78. #125

#64408

semidirect

Subgroups — #22626114

Conjugacy classes of subgroups

Complements