Group #32998004 — subgroups

Subgroups — #32998004

Order 18688.

198 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22608840	C₂	1	2	yes	maximal
2	#22607997	C₂	1	1	yes	maximal, characteristic
3	#22607901	C₂	1	2	yes	maximal
4	#22607851	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2176	C₂	1	1	yes	maximal, characteristic
6	#17988718	—	2	2	no	—
7	#17988363	C₂²	1	1	yes	characteristic
8	#17988338	C₂²	1	2	yes	—
9	Q₈⋊#10596831	—	2	2	no	—
10	#17988308	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
13	#367×C₇₃	C₄	1	1	yes	characteristic
14	#358×C₇₃	C₄	1	2	yes	—
15	#346×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924003	—	2	1	no	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	4	2	no	—
19	#14556502	—	4	2	no	—
20	C₄×#10596831	D₄	1	2	yes	—
21	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₇₃	—	2	2	no	—
23	Q₃₂×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	#107×C₇₃	C₂×C₄	1	2	yes	—
27	#103×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×#10596831	—	2	2	no	—
30	C₂×#10596831	—	4	2	no	—
31	#11924003	—	4	1	no	—
32	C₂×Q₈×C₇₃	—	2	2	no	—
33	Q₁₆×C₇₃	—	2	2	no	—
34	Q₁₆×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	2	2	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄²×C₇₃	—	4	2	no	—
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	1	no	—
42	C₁₆×C₇₃	M₁₆	1	2	yes	—
43	Q₈×C₇₃	—	4	2	no	—
44	Q₈×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₇₃	—	2	1	no	—
50	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
51	C₈×C₇₃	#131	1	1	yes	characteristic
52	#10596831	—	4	2	no	—
53	#10596831	—	8	2	no	—
54	C₂²×C₇₃	#442	1	1	yes	characteristic
55	C₄×C₇₃	—	2	2	no	—
56	C₄×C₇₃	—	8	2	no	—
57	C₄×C₇₃	—	8	2	no	—
58	C₄×C₇₃	#436	1	1	yes	characteristic
59	C₄×C₇₃	#444	1	1	yes	characteristic
60	#12088	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	#1594	1	1	yes	characteristic
62	C₂×C₇₃	#2180	1	2	yes	—
63	#2176	D₇₃	1	1	yes	characteristic
64	#1593	—	73	2	no	—
65	#1368	—	73	1	no	—
66	#1334	—	73	2	no	—
67	#1313	—	73	1	no	—
68	C₇₃	#12088	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	Dic₇₃	1	1	yes	characteristic
70	#446	—	146	2	no	—
71	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
72	#367	Dic₇₃	1	1	yes	characteristic
73	#358	Dic₇₃	1	2	yes	—
74	#346	—	73	1	no	—
75	#346	C₂×D₇₃	1	1	yes	characteristic
76	#334	—	73	2	no	—
77	Q₈⋊C₈	—	146	2	no	—
78	#322	—	73	1	no	—
79	C₂×Q₁₆	C₂×Dic₇₃	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	—	2	2	no	—
82	C₂×C₁₆	—	146	1	no	—
83	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
85	#107	C₂×Dic₇₃	1	2	yes	—
86	C₄⋊C₈	—	292	2	no	—
87	C₄⋊C₈	—	146	2	no	—
88	#103	—	2	2	no	—
89	#97	—	292	2	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	2	2	no	—
95	C₂×C₈	—	292	2	no	—
96	C₂×C₈	—	146	2	no	—
97	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
98	C₂×C₈	#11924027	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₇₃	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	292	1	no	—
105	C₁₆	#11924012	1	2	yes	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
111	C₂×C₄	#14556615	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	292	2	no	—
114	C₈	—	584	2	no	—
115	C₈	C₂×#11924012	1	1	yes	characteristic
116	C₈	#14556598	1	1	yes	characteristic
117	C₂²	#17988699	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	8	2	no	—
120	C₄	—	8	2	no	—
121	C₄	#17988665	1	1	yes	characteristic
122	C₄	#17988709	1	1	yes	characteristic
123	C₂	#22608845	1	1	yes	characteristic
124	C₂	#22611823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608840

C₂

yes

maximal

#22607997

C₂

yes

maximal, characteristic

#22607901

C₂

yes

maximal

#22607851

C₂

yes

maximal, characteristic

C₇₃×#2176

C₂

yes

maximal, characteristic

#17988718

—

#17988363

C₂²

yes

characteristic

#17988338

C₂²

yes

—

Q₈⋊#10596831

—

#17988308

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₄

yes

characteristic

C₄×Q₁₆×C₇₃

C₂²

yes

—

#367×C₇₃

C₄

yes

characteristic

#358×C₇₃

C₄

yes

—

#346×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₇₃

C₂×C₄

yes

—

C₄×Q₈×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₂×C₄

yes

—

#103×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

—

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

#442

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#2176

D₇₃

yes

—

—

—

—

C₇₃

#12088

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₃

yes

characteristic

#446

—

146

—

C₄×Q₁₆

C₂×D₇₃

yes

—

#367

Dic₇₃

yes

characteristic

#358

Dic₇₃

yes

—

#346

—

#346

C₂×D₇₃

yes

characteristic

#334

—

Q₈⋊C₈

—

146

—

#322

—

C₂×Q₁₆

C₂×Dic₇₃

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

#107

C₂×Dic₇₃

yes

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

#97

—

292

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

292

—

C₂×C₈

—

146

—

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

C₂×C₈

#11924027

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₇₃

yes

characteristic

103

C₁₆

—

104

C₁₆

—

292

—

105

C₁₆

#11924012

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₇₃

yes

characteristic

111

C₂×C₄

#14556615

yes

—

112

C₈

—

113

C₈

—

292

—

114

C₈

—

584

—

115

C₈

C₂×#11924012

yes

characteristic

116

C₈

#14556598

yes

characteristic

117

C₂²

#17988699

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#17988665

yes

characteristic

122

C₄

#17988709

yes

characteristic

123

C₂

#22608845

yes

characteristic

124

C₂

#22611823

yes

—

	subgroup	G/H
center	C₂²	#17988699
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2176	C₂
socle, prime radical	C₂²×C₇₃	#442
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#2176	D₇₃
hypocenter	C₇₃	#12088
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#17988699

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2176

C₂

socle, prime radical

C₂²×C₇₃

#442

cosocle

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
68. C₇₃	#12088	semidirect

H ≅ G/N

type

68. C₇₃

#12088

semidirect

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

subgroup	index
#22608840	2
#22607997	2
#22607901	2
#22607851	2
C₇₃×#2176	2
#12088	73

subgroup

index

C₇₃×#2176

Subgroups — #32998004

Conjugacy classes of subgroups

Complements

Maximal subgroups