Group #32973307 — subgroups

Subgroups — #32973307

Order 18688.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22608265	C₂	1	2	yes	maximal
2	C₇₃×#1818	C₂	1	1	yes	maximal, characteristic
3	#17988644	—	2	2	no	—
4	#17988313	C₄	1	4	yes	—
5	#17988311	—	2	2	no	—
6	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
8	#336×C₇₃	—	2	2	no	—
9	C₄⋊#10596831	—	4	2	no	—
10	C₂²⋊#10596831	—	2	4	no	—
11	#14556502	—	4	2	no	—
12	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₇₃	—	2	2	no	—
14	#117×C₇₃	—	4	2	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
16	C₂×C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₇₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	D₄	1	2	yes	—
19	C₂²⋊C₈×C₇₃	—	4	2	no	—
20	C₂×#10596831	—	4	4	no	—
21	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	—	2	2	no	—
23	C₂²×C₄×C₇₃	—	4	2	no	—
24	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₇₃	—	4	4	no	—
29	C₂×C₈×C₇₃	—	4	2	no	—
30	C₄⋊C₄×C₇₃	—	4	2	no	—
31	C₂²⋊C₄×C₇₃	—	4	1	no	—
32	C₂²⋊C₄×C₇₃	—	4	1	no	—
33	C₄²×C₇₃	—	2	2	no	—
34	C₄²×C₇₃	—	2	2	no	—
35	C₂³×C₇₃	—	2	1	no	—
36	C₂³×C₇₃	—	2	2	no	—
37	C₂³×C₇₃	—	4	2	no	—
38	C₂³×C₇₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	4	4	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	#99	1	2	yes	—
48	C₈×C₇₃	—	8	2	no	—
49	#10596831	—	8	4	no	—
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	—	4	1	no	—
52	C₂²×C₇₃	—	4	2	no	—
53	C₂²×C₇₃	—	8	2	no	—
54	C₂²×C₇₃	#342	1	1	yes	characteristic
55	C₂²×C₇₃	#344	1	2	yes	—
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	—	4	1	no	—
58	C₄×C₇₃	—	4	2	no	—
59	C₄×C₇₃	—	4	2	no	—
60	#7327	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	—	2	2	no	—
62	C₂×C₇₃	—	8	1	no	—
63	C₂×C₇₃	#1306	1	1	yes	characteristic
64	C₂×C₇₃	#1395	1	2	yes	—
65	#1818	D₇₃	1	1	yes	characteristic
66	#1464	—	73	2	no	—
67	C₇₃	#7327	1	1	yes	characteristic, complemented, Sylow
68	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
69	#432	—	146	2	no	—
70	C₂×C₂²⋊C₈	Dic₇₃	1	2	yes	—
71	#336	—	2	2	no	—
72	#324	—	73	4	no	—
73	#323	—	146	2	no	—
74	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
75	C₂²×C₈	—	2	2	no	—
76	#117	—	4	2	no	—
77	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
78	C₂×C₂²⋊C₄	Dic₁₄₆	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	D₂₉₂	1	1	yes	characteristic
80	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
81	C₄⋊C₈	—	292	2	no	—
82	C₂²⋊C₈	—	4	2	no	—
83	C₂²⋊C₈	—	146	4	no	—
84	#97	—	292	2	no	—
85	C₂⁴	C₄⋊Dic₇₃	1	1	yes	characteristic
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
89	C₂²×C₄	#11924005	1	2	yes	—
90	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
91	C₂²×C₄	#11924007	1	2	yes	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	292	4	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₂²⋊C₄	—	4	1	no	—
97	C₂²⋊C₄	—	4	1	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	#14556498	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#14556506	1	2	yes	—
113	C₈	—	8	2	no	—
114	C₈	—	584	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	8	2	no	—
119	C₂²	#17988354	1	1	yes	characteristic
120	C₂²	#17988357	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	1	no	—
125	C₂	—	2	2	no	—
126	C₂	—	8	1	no	—
127	C₂	#22607839	1	1	yes	characteristic
128	C₂	#22608060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608265

C₂

yes

maximal

C₇₃×#1818

C₂

yes

maximal, characteristic

#17988644

—

#17988313

C₄

yes

—

#17988311

—

C₂×#117×C₇₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₇₃

C₄

yes

—

#336×C₇₃

—

C₄⋊#10596831

—

C₂²⋊#10596831

—

#14556502

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

—

#117×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₇₃

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂²⋊C₈×C₇₃

—

C₂×#10596831

—

C₂⁴×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#99

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#342

yes

characteristic

C₂²×C₇₃

#344

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#7327

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1306

yes

characteristic

C₂×C₇₃

#1395

yes

—

#1818

D₇₃

yes

characteristic

#1464

—

C₇₃

#7327

yes

characteristic, complemented, Sylow

C₂×#117

C₂×D₇₃

yes

characteristic

#432

—

146

—

C₂×C₂²⋊C₈

Dic₇₃

yes

—

#336

—

#324

—

#323

—

146

—

C₂³×C₄

C₂×Dic₇₃

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

C₄×D₇₃

yes

—

C₂×C₂²⋊C₄

Dic₁₄₆

yes

characteristic

C₂×C₂²⋊C₄

D₂₉₂

yes

characteristic

C₂×C₄²

C₂²⋊D₇₃

yes

—

C₄⋊C₈

—

292

—

C₂²⋊C₈

—

C₂²⋊C₈

—

146

—

#97

—

292

—

C₂⁴

C₄⋊Dic₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×Dic₇₃

yes

characteristic

C₂²×C₄

#11924005

yes

—

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

C₂²×C₄

#11924007

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

#14556498

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#14556506

yes

—

113

C₈

—

114

C₈

—

584

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#17988354

yes

characteristic

120

C₂²

#17988357

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₂

—

126

C₂

—

127

C₂

#22607839

yes

characteristic

128

C₂

#22608060

yes

—

	subgroup	G/H
center	C₂²	#17988354
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂×#117	C₂×D₇₃
Fitting	C₇₃×#1818	C₂
socle	C₂²×C₇₃	#342
prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₂×#117×C₇₃	C₂²
hypercenter	#1818	D₇₃
hypocenter	C₇₃	#7327
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1818

subgroup

G/H

center

C₂²

#17988354

commutator

C₂²×C₄×C₇₃

C₄²

Frattini

C₂×#117

C₂×D₇₃

Fitting

C₇₃×#1818

C₂

socle

C₂²×C₇₃

#342

prime radical

C₂⁴×C₇₃

C₄⋊C₄

cosocle

C₂×#117×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1818

N	H ≅ G/N	type
67. C₇₃	#7327	semidirect

H ≅ G/N

type

67. C₇₃

#7327

semidirect

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

subgroup	index
#22608265	2
C₇₃×#1818	2
#7327	73

subgroup

index

#22608265

C₇₃×#1818

#7327

Subgroups — #32973307

Conjugacy classes of subgroups

Complements

Maximal subgroups