Group #34005444 — subgroups

Subgroups — #34005444

Order 18688.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#22625494	C₂	1	2	yes	maximal
2	#22624788	C₂	1	4	yes	maximal
3	#22608216	C₂	1	1	yes	maximal, characteristic
4	#17989609	C₂²	1	4	yes	—
5	#17989451	—	2	2	no	—
6	#17989449	—	2	2	no	—
7	#17988638	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	#17988308	C₄	1	1	yes	characteristic
10	C₈×#10596831	C₄	1	2	yes	—
11	#322×C₇₃	C₄	1	1	yes	characteristic
12	C₄⋊#10596833	—	2	4	no	—
13	#14556648	—	2	4	no	—
14	#14556646	—	2	4	no	—
15	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
16	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
17	C₄²×D₇₃	C₂³	1	1	yes	characteristic
18	#14556520	—	2	2	no	—
19	#14556503	—	2	2	no	—
20	C₄×#10596831	C₂×C₄	1	2	yes	—
21	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₇₃	D₄	1	1	yes	characteristic
23	C₈×Dic₇₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596833	—	4	4	no	—
27	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×D₇₃	#41	1	2	yes	—
29	C₂×#10596831	—	2	2	no	—
30	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
31	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
32	C₈×D₇₃	—	2	2	no	—
33	C₈×D₇₃	—	2	2	no	—
34	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
35	C₄×Dic₇₃	#41	1	2	yes	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×D₇₃	#125	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#117	1	2	yes	—
43	C₂×Dic₇₃	#124	1	2	yes	—
44	C₂×Dic₇₃	#125	1	1	yes	characteristic
45	C₄×D₇₃	—	2	1	no	—
46	C₄×D₇₃	—	2	1	no	—
47	C₄×D₇₃	—	4	2	no	—
48	#10596833	—	8	4	no	—
49	C₈×C₇₃	—	2	1	no	—
50	C₈×C₇₃	—	2	1	no	—
51	C₈×C₇₃	—	4	2	no	—
52	#10596831	—	4	2	no	—
53	#10596831	#108	1	2	yes	—
54	#10596831	#108	1	2	yes	—
55	C₂²×C₇₃	#382	1	1	yes	characteristic
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
60	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
61	Dic₇₃	—	2	2	no	—
62	Dic₇₃	—	2	1	no	—
63	#8796	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	#1833	1	2	yes	—
65	C₂×C₇₃	#1837	1	1	yes	characteristic
66	D₇₃	—	4	1	no	—
67	#1568	—	73	2	no	—
68	#1450	—	73	1	no	—
69	#1287	—	73	4	no	—
70	C₇₃	#8796	1	1	yes	characteristic, complemented, Sylow
71	#432	—	73	4	no	—
72	#431	—	73	2	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	#335	—	146	2	no	—
75	#334	—	146	2	no	—
76	#322	—	73	1	no	—
77	#322	#64408	1	1	yes	characteristic
78	C₈²	—	73	2	no	—
79	C₂²×C₈	—	73	1	no	—
80	C₂²×C₈	—	73	1	no	—
81	C₂×C₄²	—	73	1	no	—
82	C₄⋊C₈	—	146	2	no	—
83	C₄⋊C₈	—	146	4	no	—
84	C₂²⋊C₈	—	146	2	no	—
85	C₂²⋊C₈	—	146	4	no	—
86	#97	—	146	4	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	C₂×#64408	1	2	yes	—
92	C₄×C₈	C₂×#64408	1	1	yes	characteristic
93	C₂²×C₄	—	73	2	no	—
94	C₂²×C₄	—	73	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	292	4	no	—
97	C₂×C₈	—	73	1	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	146	2	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
103	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
104	C₄²	—	73	1	no	—
105	C₄²	—	73	2	no	—
106	C₄²	C₂²×#64408	1	1	yes	characteristic
107	C₂³	—	73	1	no	—
108	C₂×C₄	—	292	2	no	—
109	C₂×C₄	—	73	2	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	146	1	no	—
112	C₂×C₄	—	146	1	no	—
113	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
114	C₂×C₄	#14556670	1	2	yes	—
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	4	2	no	—
118	C₈	—	292	2	no	—
119	C₈	—	584	4	no	—
120	C₈	—	73	2	no	—
121	C₈	—	73	2	no	—
122	C₂²	—	146	2	no	—
123	C₂²	—	146	1	no	—
124	C₂²	#17989531	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	146	2	no	—
127	C₄	—	146	1	no	—
128	C₄	D₇₃⋊#108	1	1	yes	characteristic
129	C₄	D₇₃⋊#108	1	1	yes	characteristic
130	C₂	—	292	1	no	—
131	C₂	#22625851	1	2	yes	—
132	C₂	#22625856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625494

C₂

yes

maximal

#22624788

C₂

yes

maximal

#22608216

C₂

yes

maximal, characteristic

#17989609

C₂²

yes

—

#17989451

—

#17989449

—

#17988638

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988308

C₄

yes

characteristic

C₈×#10596831

C₄

yes

—

#322×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₇₃

#41

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

#125

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

#124

yes

—

C₂×Dic₇₃

#125

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#108

yes

—

#10596831

#108

yes

—

C₂²×C₇₃

#382

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#8796

—

maximal, Sylow

C₂×C₇₃

#1833

yes

—

C₂×C₇₃

#1837

yes

characteristic

D₇₃

—

#1568

—

#1450

—

#1287

—

C₇₃

#8796

yes

characteristic, complemented, Sylow

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

146

—

#334

—

146

—

—

yes

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₂×C₈

C₄⋊#64408

yes

characteristic

103

C₂×C₈

C₄⋊#64408

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²×#64408

yes

characteristic

107

C₂³

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

146

—

112

C₂×C₄

—

146

—

113

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

114

C₂×C₄

#14556670

yes

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

292

—

119

C₈

—

584

—

120

C₈

—

121

C₈

—

122

C₂²

—

146

—

123

C₂²

—

146

—

124

C₂²

#17989531

yes

characteristic

125

C₄

—

126

C₄

—

146

—

127

C₄

—

146

—

128

C₄

D₇₃⋊#108

yes

characteristic

129

C₄

D₇₃⋊#108

yes

characteristic

130

C₂

—

292

—

131

C₂

#22625851

yes

—

132

C₂

#22625856

yes

characteristic

	subgroup	G/H
center	C₂²	#17989531
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	#382
prime radical	C₂²×D₇₃	#125
cosocle	C₄²×D₇₃	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	#8796
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17989531

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

#322×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
70. C₇₃	#8796	semidirect

H ≅ G/N

type

70. C₇₃

#8796

semidirect

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

subgroup	index
#22625494	2
#22624788	2
#22608216	2
#8796	73

subgroup

index

Subgroups — #34005444

Conjugacy classes of subgroups

Complements

Maximal subgroups