Group #34009662 — subgroups

Subgroups — #34009662

Order 18688.

298 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 148 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×#17989473	C₂	1	1	yes	maximal, characteristic
2	#22625134	C₂	1	1	yes	maximal, characteristic
3	#22624845	C₂	1	4	yes	maximal, complemented
4	M₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₄	1	2	yes	—
6	#17989603	C₄	1	2	yes	—
7	#17989473	C₄	1	2	yes	—
8	#17989473	C₂²	1	2	yes	—
9	#17989428	C₄	1	2	yes	—
10	C₈×#10596833	C₂²	1	2	yes	—
11	C₂×#14556531	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	#17988362	C₄	1	2	yes	—
14	C₁₆×Dic₇₃	C₂²	1	2	yes	—
15	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	—	2	4	no	—
17	C₂×#11924031	C₂×C₄	1	4	yes	—
18	C₄⋊#10596833	C₈	1	4	yes	—
19	C₂²⋊#10596833	C₈	1	4	yes	—
20	C₄×#10596833	C₂×C₄	1	4	yes	—
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	#14556531	—	2	2	no	—
24	C₂×#11924003	C₂×C₄	1	2	yes	—
25	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₇₃	—	2	2	no	—
31	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×#10596833	C₂×C₈	1	8	yes	—
33	#11924031	—	2	4	no	—
34	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
35	C₂×#10596831	C₄²	1	1	yes	characteristic
36	C₂×#10596831	M₁₆	1	4	yes	—
37	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
39	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
40	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
41	#11924003	—	2	2	no	—
42	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	2	no	—
48	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#97	1	2	yes	—
51	C₂×C₄×C₇₃	#97	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	4	no	—
54	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
55	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
56	#10596833	—	2	8	no	—
57	C₈×C₇₃	#97	1	2	yes	—
58	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	#10596831	#97	1	2	yes	—
61	#10596831	C₂×M₁₆	1	2	yes	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	#322	1	2	yes	—
64	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₇₃	#322	1	2	yes	—
66	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
67	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	#504	1	4	yes	—
70	#11401	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	2	2	no	—
72	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
73	C₂×C₇₃	#2108	1	2	yes	—
74	C₄×M₃₂	—	73	1	no	—
75	C₂×C₄×C₁₆	—	73	1	no	—
76	#1450	—	73	1	no	—
77	#1312	—	73	4	no	—
78	C₇₃	#11401	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₃₂	—	73	1	no	—
80	C₂×M₃₂	#64408	1	1	yes	characteristic
81	C₂²×C₁₆	—	73	2	no	—
82	#431	—	73	2	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	#346	—	73	2	no	—
85	C₄×C₁₆	—	73	2	no	—
86	C₄×C₁₆	—	73	2	no	—
87	C₄×C₁₆	—	73	2	no	—
88	#322	—	73	2	no	—
89	C₈²	—	73	2	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
92	C₂×C₄²	—	73	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	146	2	no	—
95	C₂×C₁₆	—	73	4	no	—
96	C₂×C₁₆	—	73	2	no	—
97	C₂×C₁₆	—	146	4	no	—
98	C₂×C₁₆	C₂×#64408	1	2	yes	—
99	C₄⋊C₈	—	73	4	no	—
100	C₂²⋊C₈	—	73	4	no	—
101	C₄×C₈	—	73	4	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₂²×C₄	—	73	2	no	—
107	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
108	C₂×C₈	—	73	4	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	8	no	—
112	C₂×C₈	C₄×#64408	1	1	yes	characteristic
113	C₂×C₈	#11924037	1	2	yes	—
114	C₂×C₈	#11924038	1	2	yes	—
115	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
116	C₄²	—	73	2	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	146	4	no	—
121	C₁₆	—	146	2	no	—
122	C₂³	C₈×#64408	1	1	yes	characteristic
123	C₂×C₄	—	73	2	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	146	4	no	—
126	C₂×C₄	C₈×#64408	1	1	yes	characteristic
127	C₂×C₄	#14556645	1	2	yes	—
128	C₂×C₄	#14556646	1	2	yes	—
129	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
130	C₈	—	73	2	no	—
131	C₈	—	73	2	no	—
132	C₈	—	146	8	no	—
133	C₈	#14556646	1	2	yes	—
134	C₈	C₂×#11924037	1	1	yes	characteristic
135	C₈	C₂×#11924038	1	1	yes	characteristic
136	C₂²	—	2	2	no	—
137	C₂²	#17989429	1	2	yes	—
138	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
139	C₄	—	73	4	no	—
140	C₄	—	146	2	no	—
141	C₄	#17989429	1	2	yes	—
142	C₄	C₂×#14556645	1	1	yes	characteristic
143	C₄	C₂×#14556646	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×#17989429	1	1	yes	characteristic
146	C₂	#22626143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989473

C₂

yes

maximal, characteristic

#22625134

C₂

yes

maximal, characteristic

#22624845

C₂

yes

maximal, complemented

M₃₂×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#11924031

C₄

yes

—

#17989603

C₄

yes

—

#17989473

C₄

yes

—

#17989473

C₂²

yes

—

#17989428

C₄

yes

—

C₈×#10596833

C₂²

yes

—

C₂×#14556531

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988362

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂×M₃₂×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×#11924031

C₂×C₄

yes

—

C₄⋊#10596833

C₈

yes

—

C₂²⋊#10596833

C₈

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

#14556531

—

C₂×#11924003

C₂×C₄

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×#10596833

C₂×C₈

yes

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

#11924003

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂²×C₈

yes

—

#10596833

—

C₈×C₇₃

#97

yes

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

#97

yes

—

#10596831

C₂×M₁₆

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#322

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

#322

yes

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

Dic₇₃

—

Dic₇₃

#504

yes

—

#11401

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#322

yes

characteristic

C₂×C₇₃

#2108

yes

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

#1312

—

C₇₃

#11401

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×#64408

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄×#64408

yes

characteristic

113

C₂×C₈

#11924037

yes

—

114

C₂×C₈

#11924038

yes

—

115

C₂×C₈

C₂²×#64408

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

146

—

121

C₁₆

—

146

—

122

C₂³

C₈×#64408

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

146

—

126

C₂×C₄

C₈×#64408

yes

characteristic

127

C₂×C₄

#14556645

yes

—

128

C₂×C₄

#14556646

yes

—

129

C₂×C₄

C₂×C₄×#64408

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

146

—

133

C₈

#14556646

yes

—

134

C₈

C₂×#11924037

yes

characteristic

135

C₈

C₂×#11924038

yes

characteristic

136

C₂²

—

137

C₂²

#17989429

yes

—

138

C₂²

C₂×C₈×#64408

yes

characteristic

139

C₄

—

140

C₄

—

146

—

141

C₄

#17989429

yes

—

142

C₄

C₂×#14556645

yes

characteristic

143

C₄

C₂×#14556646

yes

characteristic

144

C₂

—

145

C₂

C₂×#17989429

yes

characteristic

146

C₂

#22626143

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#11401
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₇₃

C₄

prime radical

C₂³×C₇₃

C₄×C₈

cosocle

C₈×Dic₇₃

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #22624845	C₂	semidirect
78. C₇₃	#11401	semidirect

H ≅ G/N

type

3. #22624845

C₂

semidirect

78. C₇₃

#11401

semidirect

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

subgroup	index
C₂×#17989473	2
#22625134	2
#22624845	2
M₃₂×Dic₇₃	2
#11401	73

subgroup

index

M₃₂×Dic₇₃

Subgroups — #34009662

Conjugacy classes of subgroups

Complements

Maximal subgroups