Group #34012317 — subgroups

Subgroups — #34012317

Order 18688.

390 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 176 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	C₂×#17989475	C₂	1	2	yes	maximal
2	#22624970	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	#17989475	C₄	1	8	yes	complemented
6	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
9	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	8	no	—
12	C₂×#11924031	C₂×C₄	1	8	yes	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₈	1	4	yes	—
16	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
17	C₄×#10596831	C₂×C₄	1	4	yes	—
18	C₈×Dic₇₃	—	2	4	no	—
19	C₈×Dic₇₃	—	2	4	no	—
20	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₈	1	4	yes	—
23	#11924031	—	2	8	no	—
24	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
25	C₂×#10596831	C₄²	1	2	yes	—
26	C₂×#10596831	M₁₆	1	4	yes	—
27	C₂×#10596831	C₂²×C₄	1	2	yes	—
28	C₂×#10596831	#41	1	4	yes	—
29	C₄×Dic₇₃	—	2	2	no	—
30	C₄×Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	2	8	no	—
32	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
37	C₄²×C₇₃	C₄²	1	2	yes	—
38	C₄²×C₇₃	C₂×C₈	1	2	yes	—
39	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	4	no	—
41	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#117	1	2	yes	—
44	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
46	C₂×Dic₇₃	—	2	4	no	—
47	C₂×Dic₇₃	—	2	4	no	—
48	C₂×Dic₇₃	M₃₂	1	8	yes	—
49	C₈×C₇₃	—	2	4	no	—
50	#10596831	#97	1	4	yes	—
51	#10596831	#117	1	4	yes	—
52	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂²×C₇₃	#431	1	2	yes	—
54	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
55	C₄×C₇₃	—	2	4	no	—
56	C₄×C₇₃	#376	1	1	yes	characteristic
57	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
58	C₄×C₇₃	#431	1	2	yes	—
59	Dic₇₃	—	2	8	no	—
60	#12591	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	#1744	1	1	yes	characteristic
62	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
63	C₂×C₇₃	#2163	1	4	yes	—
64	C₂×#346	—	73	2	no	—
65	C₄²×C₈	—	73	1	no	—
66	#1374	—	73	4	no	—
67	C₇₃	#12591	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	—	73	4	no	—
69	C₂×C₄×C₈	—	73	1	no	—
70	C₂×C₄×C₈	—	73	2	no	—
71	C₂×C₄×C₈	—	73	2	no	—
72	C₂×C₄×C₈	#64408	1	1	yes	characteristic
73	C₄³	—	73	1	no	—
74	#346	—	73	8	no	—
75	C₂²×C₈	—	73	1	no	—
76	C₂²×C₈	—	73	1	no	—
77	C₂²×C₈	C₂×#64408	1	2	yes	—
78	C₂×C₄²	—	73	4	no	—
79	C₂×C₄²	—	73	2	no	—
80	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
81	C₂×C₁₆	—	73	8	no	—
82	C₂×C₁₆	—	146	8	no	—
83	C₄×C₈	—	73	4	no	—
84	C₄×C₈	—	146	4	no	—
85	C₄×C₈	—	146	4	no	—
86	C₄×C₈	#10596833	1	4	yes	—
87	C₂²×C₄	—	73	4	no	—
88	C₂²×C₄	#11924036	1	2	yes	—
89	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	73	4	no	—
93	C₂×C₈	—	73	2	no	—
94	C₂×C₈	—	73	2	no	—
95	C₂×C₈	—	73	4	no	—
96	C₂×C₈	C₂×#10596833	1	4	yes	—
97	C₄²	—	146	8	no	—
98	C₄²	—	146	2	no	—
99	C₄²	—	146	2	no	—
100	C₄²	C₄×#64408	1	2	yes	—
101	C₄²	C₂×#10596833	1	2	yes	—
102	C₁₆	—	146	8	no	—
103	C₂³	C₂×#11924036	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	73	8	no	—
106	C₂×C₄	—	146	4	no	—
107	C₂×C₄	—	146	4	no	—
108	C₂×C₄	C₄×#10596833	1	4	yes	—
109	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
110	C₂×C₄	#14556670	1	2	yes	—
111	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
112	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	73	4	no	—
115	C₈	—	73	4	no	—
116	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
117	C₂²	#17989605	1	2	yes	—
118	C₂²	#17989643	1	4	yes	—
119	C₄	—	2	4	no	—
120	C₄	—	146	8	no	—
121	C₄	#17989523	1	1	yes	characteristic
122	C₄	#17989546	1	1	yes	characteristic
123	C₄	#17989603	1	2	yes	—
124	C₂	#22625645	1	1	yes	characteristic
125	C₂	#22626137	1	2	yes	—
126	C₂	#22626294	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989475

C₂

yes

maximal

#22624970

C₂

yes

maximal

C₄×C₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

#17989475

C₄

yes

complemented

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₈

yes

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

M₃₂

yes

—

C₈×C₇₃

—

#10596831

#97

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

—

C₄×C₇₃

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

#431

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×M₃₂

yes

—

C₂×C₇₃

#2163

yes

—

C₂×#346

—

C₄²×C₈

—

#1374

—

C₇₃

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

#346

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

#10596833

yes

—

C₂²×C₄

—

C₂²×C₄

#11924036

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#10596833

yes

—

C₄²

—

146

—

C₄²

—

146

—

C₄²

—

146

—

100

C₄²

C₄×#64408

yes

—

101

C₄²

C₂×#10596833

yes

—

102

C₁₆

—

146

—

103

C₂³

C₂×#11924036

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

C₄×#10596833

yes

—

109

C₂×C₄

C₂×C₄×#64408

yes

characteristic

110

C₂×C₄

#14556670

yes

—

111

C₂×C₄

C₂²×#10596833

yes

characteristic

112

C₂×C₄

D₇₃⋊M₁₆

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

C₂×C₄×#10596833

yes

characteristic

117

C₂²

#17989605

yes

—

118

C₂²

#17989643

yes

—

119

C₄

—

120

C₄

—

146

—

121

C₄

#17989523

yes

characteristic

122

C₄

#17989546

yes

characteristic

123

C₄

#17989603

yes

—

124

C₂

#22625645

yes

characteristic

125

C₂

#22626137

yes

—

126

C₂

#22626294

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12591
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator

C₂²×C₇₃

C₂×C₄×C₈

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. #17989475	C₄	semidirect
67. C₇₃	#12591	semidirect

H ≅ G/N

type

5. #17989475

C₄

semidirect

67. C₇₃

#12591

semidirect

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

subgroup	index
C₂×#17989475	2
#22624970	2
C₄×C₈×Dic₇₃	2
#12591	73

subgroup

index

C₂×#17989475

#22624970

C₄×C₈×Dic₇₃

#12591

Subgroups — #34012317

Conjugacy classes of subgroups

Complements

Maximal subgroups