Group #32979788 — subgroups

Subgroups — #32979788

Order 18688.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22610103	C₂	1	2	yes	maximal
2	#22608690	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556527	—	2	4	no	—
5	#17988611	C₂²	1	4	yes	—
6	#17988340	—	2	8	no	—
7	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
9	C₈²×C₇₃	C₄	1	1	yes	characteristic
10	C₈²×C₇₃	C₄	1	2	yes	—
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924012	—	2	4	no	—
13	#14556527	—	4	8	no	—
14	C₄⋊#10596831	—	2	8	no	—
15	C₂²×C₈×C₇₃	D₄	1	2	yes	—
16	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
17	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₃	D₄	1	4	yes	—
23	C₄×C₈×C₇₃	Q₈	1	2	yes	—
24	#11924012	—	4	8	no	—
25	C₂×#10596831	—	4	4	no	—
26	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
28	C₂×C₈×C₇₃	—	2	8	no	—
29	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₇₃	D₈	1	4	yes	—
34	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
35	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₃	C₂×D₄	1	2	yes	—
37	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	4	no	—
40	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₇₃	—	2	4	no	—
47	C₈×C₇₃	#107	1	4	yes	—
48	C₈×C₇₃	#107	1	4	yes	—
49	C₈×C₇₃	#108	1	8	yes	complemented
50	#10596831	—	8	4	no	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	#384	1	1	yes	characteristic
53	C₂²×C₇₃	#500	1	2	yes	—
54	C₄×C₇₃	—	2	2	no	—
55	C₄×C₇₃	C₂×#107	1	2	yes	—
56	C₄×C₇₃	C₂×#108	1	4	yes	—
57	C₄×C₇₃	#493	1	1	yes	characteristic
58	C₄×C₇₃	#494	1	1	yes	characteristic
59	#8823	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	2	2	no	—
61	C₂×C₇₃	#1845	1	1	yes	characteristic
62	C₂×C₇₃	#1848	1	2	yes	—
63	#1848	—	73	2	no	—
64	#1567	—	73	4	no	—
65	C₂×C₈²	D₇₃	1	1	yes	characteristic
66	C₇₃	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	146	4	no	—
68	#423	—	73	4	no	—
69	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
71	#335	—	146	8	no	—
72	C₈²	Dic₇₃	1	1	yes	characteristic
73	C₈²	Dic₇₃	1	2	yes	—
74	C₈²	Dic₇₃	1	1	yes	characteristic
75	C₂×M₁₆	—	146	4	no	—
76	C₂²×C₈	Dic₁₄₆	1	4	yes	—
77	C₂²×C₈	D₂₉₂	1	2	yes	—
78	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
79	#108	—	292	8	no	—
80	C₄⋊C₈	—	146	8	no	—
81	C₄×C₈	Dic₁₄₆	1	2	yes	—
82	C₄×C₈	D₂₉₂	1	4	yes	—
83	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
84	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
87	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
89	M₁₆	—	292	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	292	4	no	—
92	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
96	C₂×C₈	D₅₈₄	1	4	yes	—
97	C₂×C₈	Dic₂₉₂	1	4	yes	—
98	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
99	C₄²	C₂×D₂₉₂	1	2	yes	—
100	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
101	C₂³	#14556591	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
105	C₂×C₄	#14556588	1	1	yes	characteristic
106	C₂×C₄	#14556591	1	2	yes	—
107	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	584	4	no	—
111	C₈	#14556525	1	4	yes	—
112	C₈	#14556525	1	4	yes	—
113	C₈	#14556527	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#17988457	1	1	yes	characteristic
116	C₂²	#17988992	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#14556525	1	2	yes	—
119	C₄	C₂×#14556527	1	4	yes	—
120	C₄	#17988964	1	1	yes	characteristic
121	C₄	#17988967	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#22610091	1	1	yes	characteristic
124	C₂	#22610103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610103

C₂

yes

maximal

#22608690

C₂

yes

maximal

C₂×C₈²×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556527

—

#17988611

C₂²

yes

—

#17988340

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

—

C₈²×C₇₃

C₄

yes

—

—

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

Q₈

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

#11924012

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

#108

yes

complemented

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#384

yes

characteristic

C₂²×C₇₃

#500

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

C₂×#108

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1845

yes

characteristic

C₂×C₇₃

#1848

yes

—

#1848

—

#1567

—

C₂×C₈²

D₇₃

yes

characteristic

C₇₃

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄×C₈

C₂×D₇₃

yes

—

#335

—

146

—

C₈²

Dic₇₃

yes

characteristic

C₈²

Dic₇₃

yes

—

C₈²

Dic₇₃

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

Dic₁₄₆

yes

—

C₂²×C₈

D₂₉₂

yes

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#108

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂×Dic₇₃

yes

—

C₄×C₈

C₂×Dic₇₃

yes

characteristic

C₄×C₈

C₂×Dic₇₃

yes

characteristic

C₄×C₈

C₂×Dic₇₃

yes

—

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₄₆

yes

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

D₅₈₄

yes

—

C₂×C₈

Dic₂₉₂

yes

—

C₄²

C₂²×Dic₇₃

yes

characteristic

C₄²

C₂×D₂₉₂

yes

—

100

C₄²

C₂×Dic₁₄₆

yes

characteristic

101

C₂³

#14556591

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

—

105

C₂×C₄

#14556588

yes

characteristic

106

C₂×C₄

#14556591

yes

—

107

C₂×C₄

C₂×D₅₈₄

yes

—

108

C₂×C₄

C₂×Dic₂₉₂

yes

—

109

C₈

—

110

C₈

—

584

—

111

C₈

#14556525

yes

—

112

C₈

#14556525

yes

—

113

C₈

#14556527

yes

complemented

114

C₂²

—

115

C₂²

#17988457

yes

characteristic

116

C₂²

#17988992

yes

—

117

C₄

—

118

C₄

C₂×#14556525

yes

—

119

C₄

C₂×#14556527

yes

—

120

C₄

#17988964

yes

characteristic

121

C₄

#17988967

yes

characteristic

122

C₂

—

123

C₂

#22610091

yes

characteristic

124

C₂

#22610103

yes

—

	subgroup	G/H
center	C₂×C₄	#14556588
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
socle	C₂²×C₇₃	#384
prime radical	C₂³×C₇₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#8823
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14556588

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₂×C₈²×C₇₃

C₂

socle

C₂²×C₇₃

#384

prime radical

C₂³×C₇₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₃

C₂³

hypercenter

C₂×C₈²

D₇₃

hypocenter

C₇₃

#8823

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₇₃	#108	semidirect
66. C₇₃	#8823	semidirect
113. C₈	#14556527	semidirect

H ≅ G/N

type

49. C₈×C₇₃

#108

semidirect

66. C₇₃

#8823

semidirect

113. C₈

#14556527

semidirect

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

subgroup	index
#22610103	2
#22608690	2
C₂×C₈²×C₇₃	2
#8823	73

subgroup

index

#22610103

#22608690

C₂×C₈²×C₇₃

#8823

Subgroups — #32979788

Conjugacy classes of subgroups

Complements

Maximal subgroups