Group #22626117 — subgroups

Subgroups — #22626117

Order 9344.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#17989592	C₂	1	1	yes	maximal, characteristic
2	C₄²×#64408	C₂	1	1	yes	maximal, characteristic
3	#17989457	C₂	1	4	yes	maximal
4	#17989143	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₂²	1	2	yes	—
6	C₂×C₄×#64408	—	2	2	no	—
7	C₂×C₄×#64408	C₂²	1	2	yes	—
8	#14556658	—	2	2	no	—
9	#14556590	C₄	1	2	yes	complemented
10	#14556583	C₄	1	1	yes	characteristic, complemented
11	#14556569	—	2	2	no	—
12	#14556556	—	2	2	no	—
13	#14556549	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	#11924038	—	2	4	no	—
18	C₂×#10596833	—	2	2	no	—
19	C₄×#64408	—	2	8	no	—
20	C₄×#64408	—	2	2	no	—
21	C₄×#64408	Q₈	1	4	yes	complemented
22	C₂×Dic₁₄₆	—	2	2	no	—
23	C₂×C₄×D₇₃	D₄	1	2	yes	—
24	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
26	#11924007	—	2	2	no	—
27	#11924007	—	2	2	no	—
28	#11924005	—	2	2	no	—
29	C₄×Dic₇₃	—	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	—	2	8	no	—
37	C₂×#64408	—	2	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	4	no	—
45	C₄×D₇₃	C₂×Q₈	1	2	yes	—
46	C₄×D₇₃	#41	1	2	yes	—
47	Dic₁₄₆	—	4	4	no	—
48	#10596833	—	4	2	no	—
49	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₇₃	—	2	2	no	—
51	C₂×D₇₃	C₂²⋊Q₈	1	2	yes	—
52	#64408	—	2	8	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	4	2	no	—
55	C₄×C₇₃	C₄×Q₈	1	2	yes	—
56	Dic₇₃	—	4	2	no	—
57	Dic₇₃	#104	1	4	yes	—
58	Dic₇₃	#124	1	1	yes	characteristic
59	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₇₃	#391	1	1	yes	characteristic
61	C₂×C₇₃	C₂×#104	1	2	yes	—
62	D₇₃	—	2	2	no	—
63	#1994	—	73	1	no	maximal, Sylow
64	C₇₃	#1994	1	1	yes	characteristic, complemented, Sylow
65	#533	—	73	1	no	—
66	#423	—	73	1	no	—
67	C₄³	—	73	1	no	—
68	#339	—	73	4	no	—
69	C₂×M₁₆	—	73	2	no	—
70	C₄⋊Q₈	—	73	2	no	—
71	#125	—	73	1	no	—
72	#125	#64408	1	1	yes	characteristic, complemented
73	C₂²⋊Q₈	—	146	2	no	—
74	C₄×Q₈	—	146	2	no	—
75	#117	—	73	2	no	—
76	C₂×C₄²	—	73	1	no	—
77	C₂×C₄²	—	73	2	no	—
78	C₂×C₄²	—	146	2	no	—
79	C₄⋊C₈	—	146	2	no	—
80	C₂×Q₈	—	146	2	no	—
81	C₂²×C₄	—	73	1	no	—
82	C₂²×C₄	—	73	2	no	—
83	C₂²×C₄	—	146	2	no	—
84	M₁₆	—	146	4	no	—
85	C₂×C₈	—	146	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	73	2	no	—
88	C₄⋊C₄	—	146	2	no	—
89	C₄⋊C₄	—	146	2	no	—
90	C₄⋊C₄	C₂×#64408	1	2	yes	—
91	C₂²⋊C₄	—	146	2	no	—
92	C₄²	—	73	2	no	—
93	C₄²	—	73	1	no	—
94	C₄²	—	73	4	no	—
95	C₄²	—	146	2	no	—
96	C₄²	—	146	2	no	—
97	C₄²	—	146	8	no	—
98	C₄²	C₂×#64408	1	1	yes	characteristic
99	C₂³	—	73	1	no	—
100	Q₈	—	292	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	2	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	146	4	no	—
107	C₂×C₄	—	146	2	no	—
108	C₂×C₄	—	146	4	no	—
109	C₂×C₄	—	146	8	no	—
110	C₂×C₄	C₂²⋊#64408	1	2	yes	—
111	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
112	C₈	—	292	2	no	—
113	C₂²	—	73	2	no	—
114	C₂²	—	146	2	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	—	73	1	no	—
120	C₄	—	73	1	no	—
121	C₄	—	73	4	no	—
122	C₄	—	146	8	no	—
123	C₄	Q₈×#64408	1	2	yes	—
124	C₂	—	146	2	no	—
125	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
126	C₂	D₇₃⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989592

C₂

yes

maximal, characteristic

C₄²×#64408

C₂

yes

maximal, characteristic

#17989457

C₂

yes

maximal

#17989143

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

C₂×C₄×#64408

—

C₂×C₄×#64408

C₂²

yes

—

#14556658

—

#14556590

C₄

yes

complemented

#14556583

C₄

yes

characteristic, complemented

#14556569

—

#14556556

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

#125×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

—

Q₈

yes

complemented

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

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

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊Q₈

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

#104

yes

—

Dic₇₃

#124

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#1994

—

maximal, Sylow

C₇₃

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₄³

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#64408

yes

characteristic, complemented

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₄⋊C₈

—

146

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

M₁₆

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

C₂×#64408

yes

—

C₂²⋊C₄

—

146

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

146

—

C₄²

—

146

—

C₄²

—

146

—

C₄²

C₂×#64408

yes

characteristic

C₂³

—

100

Q₈

—

292

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

—

146

—

110

C₂×C₄

C₂²⋊#64408

yes

—

111

C₂×C₄

C₂²×#64408

yes

characteristic

112

C₈

—

292

—

113

C₂²

—

114

C₂²

—

146

—

115

C₂²

C₂×C₂²⋊#64408

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

292

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

146

—

123

C₄

Q₈×#64408

yes

—

124

C₂

—

146

—

125

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

126

C₂

D₇₃⋊#104

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	#1994
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. #14556590	C₄	semidirect
10. #14556583	C₄	semidirect
15. #125×C₇₃	C₄	semidirect
21. C₄×#64408	Q₈	semidirect
64. C₇₃	#1994	semidirect
72. #125	#64408	semidirect

H ≅ G/N

type

9. #14556590

C₄

semidirect

10. #14556583

C₄

semidirect

15. #125×C₇₃

C₄

semidirect

21. C₄×#64408

Q₈

semidirect

64. C₇₃

#1994

semidirect

72. #125

#64408

semidirect

Subgroups — #22626117

Conjugacy classes of subgroups

Complements