Group #22626113 — subgroups

Subgroups — #22626113

Order 9344.

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

normal subgroups only

This group has 252 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	C₄×#11924036	C₂	1	1	yes	maximal, characteristic
2	#17989531	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	C₂²	1	2	yes	—
7	C₄×#10596833	—	2	2	no	—
8	(C₂×#64408)⋊C₄	—	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	C₂×C₄×D₇₃	—	2	2	no	—
22	C₂×C₄×D₇₃	D₄	1	2	yes	—
23	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₇₃	—	2	2	no	—
25	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
26	#11924007	—	2	2	no	—
27	#11924007	—	2	2	no	—
28	#11924007	—	2	2	no	—
29	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₇₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
33	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×#64408	—	4	2	no	—
36	C₂×#64408	—	4	4	no	—
37	C₂×C₄×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₇₃	—	2	2	no	—
41	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
43	C₄×D₇₃	—	2	4	no	—
44	C₄×D₇₃	—	4	4	no	—
45	C₄×D₇₃	C₂×Q₈	1	2	yes	—
46	C₄×D₇₃	#41	1	2	yes	—
47	#10596833	—	2	4	no	—
48	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₇₃	#124	1	1	yes	characteristic
50	C₂×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₇₃	#135	1	4	yes	—
52	#64408	—	8	2	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	4	2	no	—
55	C₄×C₇₃	C₄×Q₈	1	2	yes	—
56	Dic₇₃	—	2	2	no	—
57	Dic₇₃	—	4	2	no	—
58	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
59	C₂×C₇₃	#391	1	1	yes	characteristic
60	C₂×C₇₃	#416	1	2	yes	—
61	D₇₃	#488	1	2	yes	—
62	D₇₃	#499	1	2	yes	—
63	#1991	—	73	1	no	maximal, Sylow
64	C₇₃	#1991	1	1	yes	characteristic, complemented, Sylow
65	C₂×#125	—	73	1	no	—
66	C₂×C₄×C₈	—	73	1	no	—
67	#382	—	73	1	no	—
68	#340	—	73	4	no	—
69	C₂²×C₈	—	73	2	no	—
70	#125	—	73	2	no	—
71	#125	—	73	1	no	—
72	#125	—	146	2	no	—
73	#125	#64408	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	73	2	no	—
75	C₂×C₄⋊C₄	—	73	2	no	—
76	C₂×C₄⋊C₄	—	146	2	no	—
77	C₂×C₄²	—	73	1	no	—
78	C₄×C₈	—	146	2	no	—
79	#95	—	146	2	no	—
80	C₂²×C₄	—	73	1	no	—
81	C₂²×C₄	—	73	2	no	—
82	C₂²×C₄	—	146	2	no	—
83	C₂²×C₄	—	146	2	no	—
84	C₂×C₈	—	73	4	no	—
85	C₂×C₈	—	146	2	no	—
86	C₂×C₈	—	146	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	73	2	no	—
89	C₄⋊C₄	—	146	2	no	—
90	C₄⋊C₄	—	146	2	no	—
91	C₄⋊C₄	—	146	2	no	—
92	C₄⋊C₄	—	146	4	no	—
93	C₄⋊C₄	—	146	2	no	—
94	C₄⋊C₄	C₂×#64408	1	2	yes	—
95	C₄²	—	73	2	no	—
96	C₄²	—	73	1	no	—
97	C₄²	C₂×#64408	1	1	yes	characteristic
98	C₂³	—	73	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	292	4	no	—
101	C₂×C₄	—	292	2	no	—
102	C₂×C₄	—	292	4	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	73	2	no	—
107	C₂×C₄	—	146	4	no	—
108	C₂×C₄	—	146	2	no	—
109	C₂×C₄	C₂²⋊#64408	1	2	yes	—
110	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
111	C₈	—	146	4	no	—
112	C₂²	—	73	1	no	—
113	C₂²	—	73	1	no	—
114	C₂²	—	73	4	no	—
115	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	292	2	no	—
119	C₄	—	584	2	no	—
120	C₄	—	73	2	no	—
121	C₄	—	146	2	no	—
122	C₄	Q₈×#64408	1	2	yes	—
123	C₂	—	73	2	no	—
124	C₂	—	73	2	no	—
125	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
126	C₂	#17989577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11924036

C₂

yes

maximal, characteristic

#17989531

C₂

yes

maximal, characteristic

#17989460

C₂

yes

maximal

#125×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

C₄×#10596833

—

(C₂×#64408)⋊C₄

—

#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

—

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

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

#135

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²⋊Q₈

yes

—

C₂×C₇₃

#391

yes

characteristic

C₂×C₇₃

#416

yes

—

D₇₃

#488

yes

—

D₇₃

#499

yes

—

#1991

—

maximal, Sylow

C₇₃

#1991

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×C₈

—

#382

—

#340

—

C₂²×C₈

—

#125

—

#125

—

#125

—

146

—

#125

#64408

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

C₄×C₈

—

146

—

#95

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

C₂×#64408

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#64408

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

292

—

101

C₂×C₄

—

292

—

102

C₂×C₄

—

292

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

C₂²⋊#64408

yes

—

110

C₂×C₄

C₂²×#64408

yes

characteristic

111

C₈

—

146

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊#64408

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

292

—

119

C₄

—

584

—

120

C₄

—

121

C₄

—

146

—

122

C₄

Q₈×#64408

yes

—

123

C₂

—

124

C₂

—

125

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

126

C₂

#17989577

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	#1991
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
64. C₇₃	#1991	semidirect
73. #125	#64408	semidirect

H ≅ G/N

type

9. #14556583

C₄

semidirect

10. #14556582

C₄

semidirect

15. #125×C₇₃

C₄

semidirect

64. C₇₃

#1991

semidirect

73. #125

#64408

semidirect

Subgroups — #22626113

Conjugacy classes of subgroups

Complements