Group #34032964 — subgroups

Subgroups — #34032964

Order 18688.

210 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22627135	C₂	1	2	yes	maximal
2	#22625125	C₂	1	1	yes	maximal, characteristic
3	#17989690	—	2	2	no	—
4	#17989684	C₄	1	4	yes	—
5	#17989681	—	2	2	no	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	#17989428	—	2	2	no	—
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924040	—	2	4	no	—
11	C₂×#11924036	—	2	1	no	—
12	C₂×#11924036	C₂×C₄	1	2	yes	—
13	C₄×#10596833	—	2	1	no	—
14	C₄×#10596833	C₈	1	2	yes	—
15	C₄×#10596833	D₄	1	2	yes	—
16	C₄×#10596833	D₄	1	1	yes	characteristic
17	C₄×#10596833	Q₈	1	1	yes	characteristic
18	C₂×C₈×D₇₃	—	2	1	no	—
19	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	C₈	1	1	yes	characteristic
21	C₈×Dic₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
23	#11924040	—	4	4	no	—
24	#11924036	—	4	2	no	—
25	#11924036	—	4	2	no	—
26	C₂×#10596833	—	2	2	no	—
27	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
28	C₂×#10596833	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
31	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
32	C₂×#10596831	—	2	1	no	—
33	C₈×D₇₃	—	2	4	no	—
34	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₇₃	#97	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
43	C₂×Dic₇₃	#95	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
46	C₄×D₇₃	—	2	2	no	—
47	C₄×D₇₃	—	2	2	no	—
48	C₄×D₇₃	—	2	2	no	—
49	#10596833	—	2	2	no	—
50	#10596833	—	2	2	no	—
51	#10596833	—	4	2	no	—
52	C₈×C₇₃	—	2	2	no	—
53	#10596831	—	2	2	no	—
54	C₂²×C₇₃	#336	1	1	yes	characteristic
55	C₂×D₇₃	—	2	1	no	—
56	C₂×D₇₃	—	2	1	no	—
57	C₂×D₇₃	#347	1	2	yes	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	2	1	no	—
61	Dic₇₃	—	2	1	no	—
62	Dic₇₃	#344	1	2	yes	—
63	Dic₇₃	#364	1	2	yes	—
64	#7196	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1281	1	1	yes	characteristic
66	C₂×C₇₃	#1375	1	1	yes	characteristic
67	C₂×C₇₃	#1376	1	1	yes	characteristic
68	D₇₃	—	2	2	no	—
69	#2163	—	73	2	no	—
70	C₂×#322	—	73	1	no	—
71	C₇₃	#7196	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×C₈	—	73	1	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	C₂×C₄×C₈	—	73	1	no	—
75	C₄⋊C₁₆	—	146	2	no	—
76	C₂²⋊C₁₆	—	73	4	no	—
77	#346	—	146	2	no	—
78	#322	—	146	2	no	—
79	C₂²×C₈	—	73	2	no	—
80	C₂²×C₈	—	146	1	no	—
81	C₂²×C₈	—	146	1	no	—
82	C₂×C₄²	—	73	1	no	—
83	C₂×C₁₆	—	146	4	no	—
84	C₄×C₈	—	73	2	no	—
85	C₄×C₈	—	73	1	no	—
86	C₄×C₈	—	73	1	no	—
87	C₄×C₈	—	73	2	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	—	146	1	no	—
90	C₄×C₈	—	146	1	no	—
91	C₄×C₈	#10596842	1	1	yes	characteristic
92	C₂²×C₄	—	73	1	no	—
93	C₂²×C₄	—	73	1	no	—
94	C₂²×C₄	—	73	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	292	2	no	—
97	C₂×C₈	—	292	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	146	4	no	—
101	C₂×C₈	—	146	2	no	—
102	C₂×C₈	—	146	1	no	—
103	C₄²	—	73	1	no	—
104	C₄²	—	73	1	no	—
105	C₄²	—	73	1	no	—
106	C₄²	C₂×#10596842	1	1	yes	characteristic
107	C₁₆	—	292	4	no	—
108	C₂³	—	73	1	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	73	1	no	—
112	C₂×C₄	—	146	2	no	—
113	C₂×C₄	—	146	2	no	—
114	C₂×C₄	—	146	2	no	—
115	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
117	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	292	2	no	—
120	C₈	—	146	2	no	—
121	C₈	—	146	2	no	—
122	C₈	—	146	2	no	—
123	C₂²	—	73	2	no	—
124	C₂²	—	146	1	no	—
125	C₂²	—	146	1	no	—
126	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	73	2	no	—
131	C₄	—	73	2	no	—
132	C₄	—	146	1	no	—
133	C₂	—	146	2	no	—
134	C₂	#22626941	1	1	yes	characteristic
135	C₂	#22627004	1	1	yes	characteristic
136	C₂	#22627005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627135

C₂

yes

maximal

#22625125

C₂

yes

maximal, characteristic

#17989690

—

#17989684

C₄

yes

—

#17989681

—

C₄×#11924036

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

#17989428

—

C₄×C₈×D₇₃

C₄

yes

—

—

C₂×C₄

yes

—

C₄×#10596833

—

C₄×#10596833

C₈

yes

—

C₄×#10596833

D₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

#95

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#347

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

#344

yes

—

Dic₇₃

#364

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

C₂×#322

—

C₇₃

#7196

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

#346

—

146

—

#322

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

#10596842

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₂×C₈

—

146

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#10596842

yes

characteristic

107

C₁₆

—

292

—

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

146

—

113

C₂×C₄

—

146

—

114

C₂×C₄

—

146

—

115

C₂×C₄

C₄×#10596842

yes

characteristic

116

C₂×C₄

C₂²⋊#10596842

yes

characteristic

117

C₂×C₄

C₄⋊#10596842

yes

characteristic

118

C₈

—

119

C₈

—

292

—

120

C₈

—

146

—

121

C₈

—

146

—

122

C₈

—

146

—

123

C₂²

—

124

C₂²

—

146

—

125

C₂²

—

146

—

126

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

146

—

133

C₂

—

146

—

134

C₂

#22626941

yes

characteristic

135

C₂

#22627004

yes

characteristic

136

C₂

#22627005

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	#97
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7196
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

(C₂×#10596842)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

C₂²×C₇₃

C₂²×D₇₃

C₂²

C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
71. C₇₃	#7196	semidirect

H ≅ G/N

type

71. C₇₃

#7196

semidirect

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

subgroup	index
#22627135	2
#22625125	2
#7196	73

subgroup

index

#22627135

#22625125

#7196

Subgroups — #34032964

Conjugacy classes of subgroups

Complements

Maximal subgroups