Group #34034075 — subgroups

Subgroups — #34034075

Order 18688.

390 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22627087	C₂	1	2	yes	maximal
2	#22627002	C₂	1	4	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₂²	1	4	yes	—
5	#17989681	C₄	1	8	yes	complemented
6	C₄×#11924036	C₂²	1	1	yes	characteristic
7	C₂×C₈×#64408	C₄	1	2	yes	—
8	C₂×C₈×#64408	C₂²	1	2	yes	—
9	C₄²×#64408	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
11	#14556686	—	2	8	no	—
12	C₂×#11924040	C₂×C₄	1	8	yes	—
13	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
14	C₂×#11924036	C₂³	1	1	yes	characteristic
15	C₂×C₄×#64408	C₈	1	4	yes	—
16	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
17	C₄×#10596833	C₂×C₄	1	4	yes	—
18	C₈×#64408	—	2	4	no	—
19	C₈×#64408	—	2	4	no	—
20	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
21	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₈	1	1	yes	characteristic
23	C₈×Dic₇₃	C₈	1	2	yes	—
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	#11924040	—	2	8	no	—
26	C₂²×#64408	C₂×C₈	1	4	yes	—
27	#11924036	M₁₆	1	4	yes	—
28	#11924036	#41	1	4	yes	—
29	C₂×#10596833	C₄²	1	2	yes	—
30	C₂×#10596833	C₂²×C₄	1	2	yes	—
31	C₄×#64408	—	2	8	no	—
32	C₄×#64408	—	2	2	no	—
33	C₄×#64408	—	2	2	no	—
34	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
35	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×#10596831	C₂×C₈	1	2	yes	—
37	C₈×D₇₃	—	2	2	no	—
38	C₈×D₇₃	—	2	2	no	—
39	C₄×Dic₇₃	C₄²	1	2	yes	—
40	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
42	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
44	C₂×#64408	—	2	4	no	—
45	C₂×#64408	—	2	4	no	—
46	C₂×#64408	M₃₂	1	8	yes	—
47	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
50	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
51	C₄×D₇₃	—	2	4	no	—
52	C₄×D₇₃	#117	1	2	yes	—
53	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
54	#10596833	#97	1	4	yes	—
55	#10596833	#117	1	4	yes	—
56	C₈×C₇₃	—	2	2	no	—
57	#10596831	—	2	2	no	—
58	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×D₇₃	#431	1	2	yes	—
60	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
61	#64408	—	2	8	no	—
62	C₄×C₇₃	—	2	2	no	—
63	C₄×C₇₃	#431	1	2	yes	—
64	Dic₇₃	—	2	2	no	—
65	Dic₇₃	#376	1	1	yes	characteristic
66	Dic₇₃	C₂×#97	1	1	yes	characteristic
67	#12591	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	#1744	1	1	yes	characteristic
69	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
70	D₇₃	#2163	1	4	yes	—
71	C₂×#346	—	73	2	no	—
72	C₄²×C₈	—	73	1	no	—
73	#1374	—	73	4	no	—
74	C₇₃	#12591	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₁₆	—	73	4	no	—
76	C₂×C₄×C₈	—	73	1	no	—
77	C₂×C₄×C₈	—	73	1	no	—
78	C₂×C₄×C₈	—	73	2	no	—
79	C₂×C₄×C₈	—	73	2	no	—
80	C₄³	—	73	1	no	—
81	#346	—	73	8	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂²×C₈	—	73	2	no	—
85	C₂×C₄²	—	73	1	no	—
86	C₂×C₄²	—	73	2	no	—
87	C₂×C₄²	—	73	4	no	—
88	C₂×C₁₆	—	73	8	no	—
89	C₂×C₁₆	—	146	8	no	—
90	C₄×C₈	—	73	4	no	—
91	C₄×C₈	—	73	2	no	—
92	C₄×C₈	—	73	1	no	—
93	C₄×C₈	—	146	4	no	—
94	C₄×C₈	—	146	4	no	—
95	C₄×C₈	#10596842	1	1	yes	characteristic
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	73	4	no	—
98	C₂²×C₄	—	73	2	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	73	4	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₂×C₈	—	146	2	no	—
106	C₂×C₈	C₂×#10596842	1	2	yes	—
107	C₄²	—	73	2	no	—
108	C₄²	—	73	1	no	—
109	C₄²	—	146	2	no	—
110	C₄²	—	146	2	no	—
111	C₄²	—	146	8	no	—
112	C₄²	C₂×#10596842	1	1	yes	characteristic
113	C₁₆	—	146	8	no	—
114	C₂³	—	73	1	no	—
115	C₂×C₄	—	73	1	no	—
116	C₂×C₄	—	73	2	no	—
117	C₂×C₄	—	73	2	no	—
118	C₂×C₄	—	73	8	no	—
119	C₂×C₄	—	73	2	no	—
120	C₂×C₄	—	146	4	no	—
121	C₂×C₄	—	146	4	no	—
122	C₂×C₄	—	146	4	no	—
123	C₂×C₄	C₄×#10596842	1	2	yes	—
124	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	73	4	no	—
127	C₈	—	73	4	no	—
128	C₈	—	146	2	no	—
129	C₂²	—	73	4	no	—
130	C₂²	—	73	2	no	—
131	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	73	1	no	—
134	C₄	—	73	1	no	—
135	C₄	—	146	8	no	—
136	C₄	—	146	2	no	—
137	C₄	#17989700	1	2	yes	—
138	C₂	—	73	4	no	—
139	C₂	#22627065	1	1	yes	characteristic
140	C₂	#64408⋊M₃₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22627087

C₂

yes

maximal

#22627002

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₂×#14556686

C₂²

yes

—

#17989681

C₄

yes

complemented

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₄

yes

—

C₂×C₈×#64408

C₂²

yes

—

C₄²×#64408

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

#14556686

—

C₂×#11924040

C₂×C₄

yes

—

C₂×#11924036

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

C₈

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₈×#64408

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

C₂×C₈

yes

—

#11924036

M₁₆

yes

—

#11924036

#41

yes

—

C₂×#10596833

C₄²

yes

—

C₂×#10596833

C₂²×C₄

yes

—

C₄×#64408

—

C₄×#64408

—

C₄×#64408

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²×C₈

yes

—

—

M₃₂

yes

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂×C₄²

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#117

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

#10596833

#97

yes

—

#10596833

#117

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

#431

yes

—

C₂×D₇₃

C₂×M₃₂

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

—

Dic₇₃

—

Dic₇₃

#376

yes

characteristic

Dic₇₃

C₂×#97

yes

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×M₃₂

yes

—

D₇₃

#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

—

C₄³

—

#346

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

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

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂×#10596842

yes

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

146

—

110

C₄²

—

146

—

111

C₄²

—

146

—

112

C₄²

C₂×#10596842

yes

characteristic

113

C₁₆

—

146

—

114

C₂³

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

C₄×#10596842

yes

—

124

C₂×C₄

C₂²×#10596842

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

146

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₄×#10596842

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

146

—

136

C₄

—

146

—

137

C₄

#17989700

yes

—

138

C₂

—

139

C₂

#22627065

yes

characteristic

140

C₂

#64408⋊M₃₂

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄×C₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#12591
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

C₄×C₈×C₇₃

C₈

prime radical

C₂²×D₇₃

C₂²×C₈

C₂³

C₄×C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
5. #17989681	C₄	semidirect
74. C₇₃	#12591	semidirect

H ≅ G/N

type

5. #17989681

C₄

semidirect

74. C₇₃

#12591

semidirect

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

subgroup	index
#22627087	2
#22627002	2
C₄×C₈×#64408	2
#12591	73

subgroup

index

#22627087

#22627002

C₄×C₈×#64408

#12591

Subgroups — #34034075

Conjugacy classes of subgroups

Complements

Maximal subgroups