Group #32997620 — subgroups

Subgroups — #32997620

Order 18688.

232 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22608687	C₂	1	2	yes	maximal
2	#22607997	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988611	—	2	2	no	—
5	#17988338	—	2	2	no	—
6	#17988338	C₂²	1	4	yes	—
7	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
9	#346×C₇₃	C₂²	1	2	yes	—
10	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924012	—	4	2	no	—
13	C₄⋊#10596831	—	2	4	no	—
14	C₄⋊#10596831	—	4	2	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	M₃₂×C₇₃	—	2	4	no	—
19	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	#11924012	—	8	4	no	—
26	C₂×#10596831	—	4	4	no	—
27	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₇₃	—	2	2	no	—
30	C₂×C₈×C₇₃	—	2	2	no	—
31	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₇₃	SD₁₆	1	2	yes	—
37	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₇₃	—	2	2	no	—
39	C₁₆×C₇₃	M₁₆	1	4	yes	complemented
40	C₂³×C₇₃	#106	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	4	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#106	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
48	C₈×C₇₃	—	2	1	no	—
49	C₈×C₇₃	—	2	1	no	—
50	C₈×C₇₃	#108	1	2	yes	—
51	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
52	#10596831	—	8	4	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	—	2	2	no	—
58	C₄×C₇₃	#365	1	4	yes	—
59	C₄×C₇₃	#423	1	1	yes	characteristic
60	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
61	#12020	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	4	1	no	—
63	C₂×C₇₃	#1566	1	1	yes	characteristic
64	C₂×C₇₃	C₂×#365	1	2	yes	—
65	C₄×M₃₂	D₇₃	1	1	yes	characteristic
66	#1566	—	73	2	no	—
67	#1368	—	73	4	no	—
68	C₇₃	#12020	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	Dic₇₃	1	2	yes	—
70	#423	—	146	2	no	—
71	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
72	#346	C₂×D₇₃	1	2	yes	—
73	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
74	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
75	#334	—	73	4	no	—
76	#334	—	146	2	no	—
77	C₂×M₁₆	—	292	2	no	—
78	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
80	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
84	C₄⋊C₈	—	292	2	no	—
85	C₄⋊C₈	—	146	4	no	—
86	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
87	C₄×C₈	D₂₉₂	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
91	C₂²×C₄	#11924017	1	2	yes	—
92	M₁₆	—	584	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	292	4	no	—
96	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
98	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
99	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
101	C₄²	#11924017	1	2	yes	—
102	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11924012	1	4	yes	complemented
105	C₂³	#14556523	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#14556523	1	1	yes	characteristic
110	C₂×C₄	#14556523	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×#11924017	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	584	4	no	—
116	C₈	#14556527	1	2	yes	—
117	C₈	C₂×#11924012	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#14556523	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	#17988410	1	4	yes	—
124	C₄	#17988611	1	1	yes	characteristic
125	C₄	C₂×#14556527	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#22608687	1	1	yes	characteristic
128	C₂	C₂×#17988410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608687

C₂

yes

maximal

#22607997

C₂

yes

maximal, complemented

C₄×M₃₂×C₇₃

C₂

yes

maximal, characteristic

#17988611

—

#17988338

—

#17988338

C₂²

yes

—

C₂×M₃₂×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₄

yes

characteristic

C₄×C₁₆×C₇₃

C₄

yes

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

#11924012

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

complemented

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

#106

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#108

yes

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#365

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1566

yes

characteristic

C₂×C₇₃

C₂×#365

yes

—

C₄×M₃₂

D₇₃

yes

characteristic

#1566

—

#1368

—

C₇₃

#12020

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₃

yes

—

#423

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#346

C₂×D₇₃

yes

—

C₄×C₁₆

Dic₇₃

yes

characteristic

C₄×C₁₆

Dic₇₃

yes

characteristic

#334

—

#334

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

D₂₉₂

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₇₃

yes

—

C₂×C₁₆

C₂×Dic₇₃

yes

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

Dic₁₄₆

yes

characteristic

C₄×C₈

D₂₉₂

yes

characteristic

C₄×C₈

C₂×Dic₇₃

yes

characteristic

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

C₂²×C₄

#11924017

yes

—

M₁₆

—

584

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

100

C₄²

C₄⋊Dic₇₃

yes

characteristic

101

C₄²

#11924017

yes

—

102

C₄²

C₂×D₂₉₂

yes

characteristic

103

C₁₆

—

104

C₁₆

#11924012

yes

complemented

105

C₂³

#14556523

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14556523

yes

characteristic

110

C₂×C₄

#14556523

yes

—

111

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

112

C₂×C₄

C₂×#11924017

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

584

—

116

C₈

#14556527

yes

—

117

C₈

C₂×#11924012

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14556523

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#17988410

yes

—

124

C₄

#17988611

yes

characteristic

125

C₄

C₂×#14556527

yes

characteristic

126

C₂

—

127

C₂

#22608687

yes

characteristic

128

C₂

C₂×#17988410

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556523
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	#106
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#12020
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#14556523

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₄×M₃₂×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₄×C₈×C₇₃

C₂³

hypercenter

C₄×M₃₂

D₇₃

hypocenter

C₇₃

#12020

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #22607997	C₂	semidirect
39. C₁₆×C₇₃	M₁₆	semidirect
68. C₇₃	#12020	semidirect
104. C₁₆	#11924012	semidirect

H ≅ G/N

type

2. #22607997

C₂

semidirect

39. C₁₆×C₇₃

M₁₆

semidirect

68. C₇₃

#12020

semidirect

104. C₁₆

#11924012

semidirect

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

subgroup	index
#22608687	2
#22607997	2
C₄×M₃₂×C₇₃	2
#12020	73

subgroup

index

#22608687

#22607997

C₄×M₃₂×C₇₃

#12020

Subgroups — #32997620

Conjugacy classes of subgroups

Complements

Maximal subgroups