Group #34013298 — subgroups

Subgroups — #34013298

Order 18688.

282 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 104 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	#22625871	C₂	1	2	yes	maximal
2	#22624991	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	C₂²	1	4	yes	—
5	C₂×#14556661	—	2	2	no	—
6	#17989592	—	2	2	no	—
7	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₄×#11924003	C₄	1	1	yes	characteristic
10	C₁₆×Dic₇₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	2	4	no	—
13	#14556662	—	2	8	no	—
14	#14556661	—	4	4	no	—
15	#14556658	—	4	4	no	—
16	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
17	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
18	C₄²×D₇₃	D₄	1	1	yes	characteristic
19	C₂×#11924003	C₂×C₄	1	2	yes	—
20	C₁₆×D₇₃	—	2	4	no	—
21	C₄×#10596831	Q₈	1	1	yes	characteristic
22	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	#11924038	—	4	4	no	—
27	#11924038	—	4	4	no	—
28	C₂×#10596833	—	4	4	no	—
29	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
30	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
31	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
32	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
33	C₈×D₇₃	SD₁₆	1	4	yes	—
34	C₈×D₇₃	#41	1	4	yes	—
35	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₇₃	D₈	1	2	yes	—
37	#11924003	—	2	2	no	—
38	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	2	no	—
42	C₂²×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#107	1	2	yes	—
44	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₇₃	#107	1	2	yes	—
46	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
47	C₄×D₇₃	—	2	4	no	—
48	C₄×D₇₃	#124	1	2	yes	—
49	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
50	#10596833	—	8	4	no	—
51	C₈×C₇₃	#106	1	2	yes	—
52	C₈×C₇₃	#117	1	2	yes	—
53	#10596831	#106	1	2	yes	—
54	#10596831	#125	1	2	yes	—
55	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	#487	1	2	yes	—
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	#382	1	1	yes	characteristic
60	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
61	Dic₇₃	#368	1	4	yes	—
62	Dic₇₃	#486	1	2	yes	—
63	#12891	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	#1834	1	1	yes	characteristic
65	C₂×C₇₃	C₂×#368	1	2	yes	—
66	D₇₃	—	2	2	no	—
67	C₂×C₄×C₁₆	—	73	1	no	—
68	#1848	—	73	2	no	—
69	#1381	—	73	4	no	—
70	C₇₃	#12891	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	—	73	2	no	—
72	C₂×#108	—	73	4	no	—
73	C₂×#108	—	146	2	no	—
74	#423	—	146	2	no	—
75	C₂×C₄×C₈	—	73	1	no	—
76	C₄×C₁₆	—	73	2	no	—
77	C₄×C₁₆	—	73	1	no	—
78	C₄×C₁₆	#64408	1	1	yes	characteristic
79	C₂×M₁₆	—	146	4	no	—
80	C₂²×C₈	—	73	1	no	—
81	C₂²×C₈	—	73	1	no	—
82	C₂×C₄²	—	73	1	no	—
83	C₂×C₁₆	—	73	2	no	—
84	C₂×C₁₆	—	146	4	no	—
85	C₂×C₁₆	C₂×#64408	1	2	yes	—
86	#108	—	292	4	no	—
87	#108	—	146	8	no	—
88	C₄⋊C₈	—	292	4	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	—	73	1	no	—
91	C₄×C₈	—	73	1	no	—
92	C₄×C₈	C₂×#64408	1	1	yes	characteristic
93	C₂²×C₄	—	73	1	no	—
94	C₂²×C₄	—	73	2	no	—
95	M₁₆	—	292	4	no	—
96	M₁₆	—	292	4	no	—
97	C₂×C₈	—	292	4	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	4	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
103	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
104	C₄²	—	73	2	no	—
105	C₄²	—	73	1	no	—
106	C₄²	C₄⋊#64408	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	146	2	no	—
109	C₂³	—	73	1	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	73	2	no	—
114	C₂×C₄	—	146	4	no	—
115	C₂×C₄	#14556660	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
117	C₈	—	584	4	no	—
118	C₈	—	73	2	no	—
119	C₈	—	73	2	no	—
120	C₈	#14556659	1	2	yes	—
121	C₈	#14556670	1	2	yes	—
122	C₂²	—	73	2	no	—
123	C₂²	—	146	2	no	—
124	C₂²	C₂×#14556660	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	73	2	no	—
127	C₄	—	73	4	no	—
128	C₄	#17989531	1	1	yes	characteristic
129	C₄	C₂×#14556659	1	1	yes	characteristic
130	C₂	—	146	2	no	—
131	C₂	#22625852	1	1	yes	characteristic
132	C₂	D₇₃⋊#368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625871

C₂

yes

maximal

#22624991

C₂

yes

maximal

C₄×C₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

C₂²

yes

—

C₂×#14556661

—

#17989592

—

C₂×C₁₆×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×#11924003

C₄

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂³

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₄×D₇₃

Q₁₆

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

SD₁₆

yes

—

C₈×D₇₃

#41

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

—

#11924003

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#107

yes

—

C₂×Dic₇₃

C₂×D₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#124

yes

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

#117

yes

—

#10596831

#106

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#487

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

—

Dic₇₃

#486

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×C₄×C₁₆

—

#1848

—

#1381

—

C₇₃

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#64408

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

—

#108

—

292

—

#108

—

146

—

C₄⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

292

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₄⋊#64408

yes

characteristic

103

C₂×C₈

C₂²×#64408

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

C₄⋊#64408

yes

characteristic

107

C₁₆

—

108

C₁₆

—

146

—

109

C₂³

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

146

—

115

C₂×C₄

#14556660

yes

—

116

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

117

C₈

—

584

—

118

C₈

—

119

C₈

—

120

C₈

#14556659

yes

—

121

C₈

#14556670

yes

—

122

C₂²

—

123

C₂²

—

146

—

124

C₂²

C₂×#14556660

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#17989531

yes

characteristic

129

C₄

C₂×#14556659

yes

characteristic

130

C₂

—

146

—

131

C₂

#22625852

yes

characteristic

132

C₂

D₇₃⋊#368

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556660
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂²×D₇₃	C₂×Q₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#12891
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14556660

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₄×C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂²×D₇₃

C₂×Q₁₆

cosocle

C₂×C₈×D₇₃

C₂³

C₄×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
70. C₇₃	#12891	semidirect

H ≅ G/N

type

70. C₇₃

#12891

semidirect

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

subgroup	index
#22625871	2
#22624991	2
C₄×C₁₆×D₇₃	2
#12891	73

subgroup

index

#22625871

#22624991

C₄×C₁₆×D₇₃

#12891

Subgroups — #34013298

Conjugacy classes of subgroups

Complements

Maximal subgroups