Group #17989341 — subgroups

Subgroups — #17989341

Order 4672.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#14556590	C₂	1	2	yes	maximal, complemented
2	#14556586	C₂	1	2	yes	maximal, complemented
3	#14556586	C₂	1	2	yes	maximal, complemented
4	#14556583	C₂	1	1	yes	maximal, characteristic, complemented
5	#14556579	C₂	1	1	yes	maximal, characteristic
6	#14556576	C₂	1	2	yes	maximal
7	#14556571	C₂	1	2	yes	maximal, complemented
8	#14556569	C₂	1	2	yes	maximal, complemented
9	#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₄₆	C₂²	1	2	yes	—
11	C₂×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	C₂×D₂₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₇₃	C₂²	1	2	yes	—
14	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇₃	C₂²	1	2	yes	—
16	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
17	#11924007	C₂²	1	2	yes	—
18	#11924007	C₂²	1	2	yes	—
19	#11924007	C₂²	1	2	yes	—
20	#11924005	C₂²	1	2	yes	—
21	#11924005	C₂²	1	2	yes	—
22	#11924005	C₂²	1	2	yes	—
23	#11924005	C₂²	1	2	yes	—
24	#11924005	C₂²	1	2	yes	—
25	C₄×Dic₇₃	C₂²	1	2	yes	—
26	C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
28	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
30	C₄²×C₇₃	C₂²	1	1	yes	characteristic
31	C₂²×D₇₃	C₂³	1	2	yes	—
32	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂³	1	2	yes	—
36	C₂×C₄×C₇₃	C₂³	1	2	yes	—
37	C₂×Dic₇₃	C₂³	1	2	yes	—
38	C₂×Dic₇₃	C₂³	1	2	yes	—
39	C₂×Dic₇₃	C₂³	1	2	yes	—
40	D₂₉₂	—	4	1	no	—
41	C₄×D₇₃	—	4	2	no	—
42	Dic₁₄₆	—	4	1	no	—
43	Dic₁₄₆	—	4	2	no	—
44	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₇₃	—	2	2	no	—
46	C₂×D₇₃	—	2	2	no	—
47	C₂×D₇₃	—	2	2	no	—
48	C₄×C₇₃	—	2	2	no	—
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	—	2	2	no	—
52	C₄×C₇₃	—	2	1	no	—
53	Dic₇₃	—	2	2	no	—
54	Dic₇₃	—	2	2	no	—
55	Dic₇₃	—	2	2	no	—
56	C₂×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
59	D₇₃	—	4	2	no	—
60	C₇₃	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	73	1	no	maximal, Sylow
62	C₄⋊Q₈	—	73	2	no	—
63	#126	—	73	2	no	—
64	#126	—	73	2	no	—
65	#125	—	73	1	no	—
66	#125	D₇₃	1	1	yes	characteristic, complemented
67	#124	—	73	1	no	—
68	#123	—	73	2	no	—
69	C₂²⋊Q₈	—	73	2	no	—
70	C₂²⋊Q₈	—	73	2	no	—
71	C₂×Q₈	—	73	2	no	—
72	C₂×Q₈	—	73	1	no	—
73	C₂×D₄	—	73	1	no	—
74	C₂²×C₄	—	73	2	no	—
75	C₄⋊C₄	—	73	2	no	—
76	C₄⋊C₄	—	73	2	no	—
77	C₄⋊C₄	—	73	2	no	—
78	C₄⋊C₄	—	73	1	no	—
79	C₄⋊C₄	—	73	2	no	—
80	C₄⋊C₄	—	73	1	no	—
81	C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	73	2	no	—
86	C₂²⋊C₄	—	73	2	no	—
87	C₂²⋊C₄	—	73	2	no	—
88	C₂²⋊C₄	—	73	2	no	—
89	C₂²⋊C₄	—	73	2	no	—
90	C₄²	—	73	2	no	—
91	C₄²	C₂×D₇₃	1	1	yes	characteristic
92	C₂³	—	73	2	no	—
93	Q₈	—	292	2	no	—
94	Q₈	—	292	1	no	—
95	D₄	—	292	1	no	—
96	C₂×C₄	—	292	2	no	—
97	C₂×C₄	—	73	2	no	—
98	C₂×C₄	—	73	2	no	—
99	C₂×C₄	—	73	2	no	—
100	C₂×C₄	C₂²×D₇₃	1	2	yes	—
101	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₇₃	1	2	yes	—
105	C₂²	—	146	2	no	—
106	C₂²	—	146	2	no	—
107	C₂²	—	146	2	no	—
108	C₂²	C₂³×D₇₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	146	2	no	—
115	C₄	—	146	2	no	—
116	C₄	—	146	2	no	—
117	C₂	—	292	2	no	—
118	C₂	#14556637	1	1	yes	characteristic
119	C₂	#14556638	1	1	yes	characteristic
120	C₂	#14556639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556590

C₂

yes

maximal, complemented

#14556586

C₂

yes

maximal, complemented

#14556586

C₂

yes

maximal, complemented

#14556583

C₂

yes

maximal, characteristic, complemented

#14556579

C₂

yes

maximal, characteristic

#14556576

C₂

yes

maximal

#14556571

C₂

yes

maximal, complemented

#14556569

C₂

yes

maximal, complemented

#125×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₆

C₂²

yes

—

C₂×Dic₁₄₆

C₂²

yes

characteristic

C₂×D₂₉₂

C₂²

yes

characteristic

C₂×C₄×D₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

characteristic

C₄⋊Dic₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

characteristic

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

C₄×Dic₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₄²×C₇₃

C₂²

yes

characteristic

C₂²×D₇₃

C₂³

yes

—

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

—

C₂×C₄×C₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₂²×C₇₃

C₂⁴

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂×C₇₃

2₋¹⁺⁴

yes

characteristic

C₂×C₇₃

2₋¹⁺⁴

yes

characteristic

D₇₃

—

C₇₃

#563

yes

characteristic, complemented, Sylow

#563

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

D₇₃

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇₃

yes

characteristic

C₄⋊C₄

C₂×D₇₃

yes

—

C₄⋊C₄

C₂×D₇₃

yes

—

C₄⋊C₄

C₂×D₇₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₇₃

yes

characteristic

C₂³

—

Q₈

—

292

—

Q₈

—

292

—

D₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²×D₇₃

yes

—

101

C₂×C₄

C₂²×D₇₃

yes

characteristic

102

C₂×C₄

C₂²×D₇₃

yes

characteristic

103

C₂×C₄

C₂²×D₇₃

yes

characteristic

104

C₂×C₄

C₂²×D₇₃

yes

—

105

C₂²

—

146

—

106

C₂²

—

146

—

107

C₂²

—

146

—

108

C₂²

C₂³×D₇₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

146

—

115

C₄

—

146

—

116

C₄

—

146

—

117

C₂

—

292

—

118

C₂

#14556637

yes

characteristic

119

C₂

#14556638

yes

characteristic

120

C₂

#14556639

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	#125×C₇₃	C₂
prime radical	C₂×D₂₉₂	C₂²
hypercenter	#125	D₇₃
hypocenter	C₇₃	#563
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₃

commutator, socle, cosocle

C₂²×C₇₃

C₂⁴

Fitting

#125×C₇₃

C₂

prime radical

C₂×D₂₉₂

C₂²

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #14556590	C₂	semidirect
2. #14556586	C₂	semidirect
3. #14556586	C₂	semidirect
4. #14556583	C₂	semidirect
7. #14556571	C₂	semidirect
8. #14556569	C₂	semidirect
9. #125×C₇₃	C₂	semidirect
60. C₇₃	#563	semidirect
66. #125	D₇₃	semidirect

H ≅ G/N

type

1. #14556590

C₂

semidirect

2. #14556586

C₂

semidirect

3. #14556586

C₂

semidirect

4. #14556583

C₂

semidirect

7. #14556571

C₂

semidirect

8. #14556569

C₂

semidirect

9. #125×C₇₃

C₂

semidirect

60. C₇₃

#563

semidirect

66. #125

D₇₃

semidirect

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

subgroup	index
#14556590	2
#14556586	2
#14556583	2
#14556579	2
#14556576	2
#14556571	2
#14556569	2
#125×C₇₃	2
#563	73

subgroup

index

#125×C₇₃

Subgroups — #17989341

Conjugacy classes of subgroups

Complements

Maximal subgroups