Group #32972713 — subgroups

Subgroups — #32972713

Order 18688.

136 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22609528	C₂	1	1	yes	maximal, characteristic
2	#22608001	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1367	C₂	1	1	yes	maximal, characteristic
4	#17988992	C₄	1	1	yes	characteristic
5	#17988985	—	2	1	no	—
6	#17988967	C₄	1	1	yes	characteristic
7	#17988938	—	2	1	no	—
8	C₄⋊#11924003	—	2	1	no	—
9	C₄⋊C₁₆×C₇₃	—	2	1	no	—
10	C₈²×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₉₂	—	4	1	no	—
12	#14556591	—	2	1	no	—
13	#14556591	—	2	1	no	—
14	C₂×#11924003	—	4	1	no	—
15	#14556525	—	4	1	no	—
16	#14556523	—	4	1	no	—
17	#14556523	—	4	1	no	—
18	#14556516	—	4	1	no	—
19	#14556516	—	4	1	no	—
20	C₂×C₁₆×C₇₃	—	4	1	no	—
21	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₄₆	—	4	1	no	—
25	C₂×Dic₁₄₆	—	4	1	no	—
26	Dic₂₉₂	—	8	2	no	—
27	C₄⋊Dic₇₃	—	4	1	no	—
28	C₄⋊Dic₇₃	—	4	1	no	—
29	C₄⋊Dic₇₃	—	4	1	no	—
30	C₄⋊Dic₇₃	—	4	1	no	—
31	#11924003	—	8	1	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
37	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₁₆×C₇₃	—	8	1	no	—
39	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
40	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
41	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
42	C₂×Dic₇₃	—	8	1	no	—
43	C₂×Dic₇₃	—	8	1	no	—
44	Dic₁₄₆	—	8	2	no	—
45	Dic₁₄₆	—	8	1	no	—
46	Dic₁₄₆	—	8	1	no	—
47	C₈×C₇₃	—	2	2	no	—
48	C₈×C₇₃	—	2	1	no	—
49	C₈×C₇₃	—	2	1	no	—
50	C₈×C₇₃	—	2	1	no	—
51	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
52	C₈×C₇₃	Q₃₂	1	1	yes	characteristic
53	C₂²×C₇₃	#331	1	1	yes	characteristic
54	C₄×C₇₃	—	2	1	no	—
55	C₄×C₇₃	#358	1	1	yes	characteristic
56	C₄×C₇₃	#359	1	1	yes	characteristic
57	C₄×C₇₃	#360	1	1	yes	characteristic
58	C₄×C₇₃	#362	1	1	yes	characteristic
59	Dic₇₃	—	16	1	no	—
60	Dic₇₃	—	16	1	no	—
61	#7095	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	#1359	1	1	yes	characteristic
63	C₂×C₇₃	#1360	1	1	yes	characteristic
64	C₂×C₇₃	#1362	1	1	yes	characteristic
65	#1707	—	73	1	no	—
66	#1370	—	73	1	no	—
67	#1367	D₇₃	1	1	yes	characteristic
68	C₇₃	#7095	1	1	yes	characteristic, complemented, Sylow
69	#500	—	73	1	no	—
70	#498	—	146	1	no	—
71	#494	—	73	1	no	—
72	#487	—	146	1	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	146	1	no	—
75	C₈²	C₂×D₇₃	1	1	yes	characteristic
76	C₂×Q₁₆	—	292	1	no	—
77	C₄⋊Q₈	—	146	1	no	—
78	C₄⋊Q₈	—	146	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	292	1	no	—
81	#107	—	292	1	no	—
82	#106	—	292	1	no	—
83	#106	—	292	1	no	—
84	#103	—	292	1	no	—
85	#103	—	292	1	no	—
86	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
87	C₄×C₈	D₂₉₂	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
89	C₂×Q₈	—	292	1	no	—
90	C₂×Q₈	—	292	1	no	—
91	Q₁₆	—	584	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	#11924013	1	1	yes	characteristic
94	C₂×C₈	D₅₈₄	1	1	yes	characteristic
95	C₂×C₈	#11924015	1	1	yes	characteristic
96	C₂×C₈	#11924017	1	1	yes	characteristic
97	C₄⋊C₄	—	292	1	no	—
98	C₄⋊C₄	—	292	1	no	—
99	C₄⋊C₄	—	292	1	no	—
100	C₄⋊C₄	—	292	1	no	—
101	C₄²	#11924005	1	1	yes	characteristic
102	C₁₆	—	8	1	no	—
103	C₁₆	—	584	1	no	—
104	Q₈	—	584	2	no	—
105	Q₈	—	584	1	no	—
106	Q₈	—	584	1	no	—
107	C₂×C₄	—	584	1	no	—
108	C₂×C₄	—	584	1	no	—
109	C₂×C₄	#14556507	1	1	yes	characteristic
110	C₂×C₄	#14556511	1	1	yes	characteristic
111	C₂×C₄	#14556513	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	#14556536	1	1	yes	characteristic
117	C₈	Dic₅₈₄	1	1	yes	characteristic
118	C₂²	#17988330	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	1168	1	no	—
121	C₄	—	1168	1	no	—
122	C₄	#17988392	1	1	yes	characteristic
123	C₄	#17988393	1	1	yes	characteristic
124	C₄	#17988396	1	1	yes	characteristic
125	C₄	#17988404	1	1	yes	characteristic
126	C₂	#22607971	1	1	yes	characteristic
127	C₂	#22607973	1	1	yes	characteristic
128	C₂	#22607981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609528

C₂

yes

maximal, characteristic

#22608001

C₂

yes

maximal, characteristic

C₇₃×#1367

C₂

yes

maximal, characteristic

#17988992

C₄

yes

characteristic

#17988985

—

#17988967

C₄

yes

characteristic

#17988938

—

C₄⋊#11924003

—

C₄⋊C₁₆×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

characteristic

C₈×C₇₃

Q₃₂

yes

characteristic

C₂²×C₇₃

#331

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₇₃

#7095

yes

characteristic, complemented, Sylow

#500

—

#498

—

146

—

#494

—

#487

—

146

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

292

—

#107

—

292

—

#106

—

292

—

#106

—

292

—

#103

—

292

—

#103

—

292

—

C₄×C₈

C₄×D₇₃

yes

characteristic

C₄×C₈

D₂₉₂

yes

characteristic

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

Q₁₆

—

584

—

C₂×C₈

—

C₂×C₈

#11924013

yes

characteristic

C₂×C₈

D₅₈₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

100

C₄⋊C₄

—

292

—

101

C₄²

#11924005

yes

characteristic

102

C₁₆

—

103

C₁₆

—

584

—

104

Q₈

—

584

—

105

Q₈

—

584

—

106

Q₈

—

584

—

107

C₂×C₄

—

584

—

108

C₂×C₄

—

584

—

109

C₂×C₄

#14556507

yes

characteristic

110

C₂×C₄

#14556511

yes

characteristic

111

C₂×C₄

#14556513

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#14556536

yes

characteristic

117

C₈

Dic₅₈₄

yes

characteristic

118

C₂²

#17988330

yes

characteristic

119

C₄

—

120

C₄

—

1168

—

121

C₄

—

1168

—

122

C₄

#17988392

yes

characteristic

123

C₄

#17988393

yes

characteristic

124

C₄

#17988396

yes

characteristic

125

C₄

#17988404

yes

characteristic

126

C₂

#22607971

yes

characteristic

127

C₂

#22607973

yes

characteristic

128

C₂

#22607981

yes

characteristic

	subgroup	G/H
center	C₂²	#17988330
commutator	C₄×C₈×C₇₃	C₂×C₄
Frattini	C₈²	C₂×D₇₃
Fitting	C₇₃×#1367	C₂
socle, prime radical	C₂²×C₇₃	#331
cosocle	C₈²×C₇₃	C₂²
hypercenter	#1367	D₇₃
hypocenter	C₇₃	#7095
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#17988330

commutator

C₄×C₈×C₇₃

C₂×C₄

Frattini

C₈²

C₂×D₇₃

Fitting

C₇₃×#1367

C₂

socle, prime radical

C₂²×C₇₃

#331

cosocle

C₈²×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
68. C₇₃	#7095	semidirect

H ≅ G/N

type

68. C₇₃

#7095

semidirect

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

subgroup	index
#22609528	2
#22608001	2
C₇₃×#1367	2
#7095	73

subgroup

index

#22609528

#22608001

C₇₃×#1367

#7095

Subgroups — #32972713

Conjugacy classes of subgroups

Complements

Maximal subgroups