Group #34011798 — subgroups

Subgroups — #34011798

Order 18688.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22625491	C₂	1	2	yes	maximal
2	#22624957	C₂	1	2	yes	maximal, complemented
3	#22624956	C₂	1	1	yes	maximal, characteristic, complemented
4	#22624955	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989608	—	2	2	no	—
7	#17989451	—	2	2	no	—
8	#17989448	C₂²	1	2	yes	—
9	#17989448	C₂²	1	2	yes	—
10	C₂×#14556531	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988362	C₂²	1	1	yes	characteristic
13	#17988362	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
15	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	C₂²⋊#10596833	—	4	2	no	—
20	#14556646	—	4	2	no	—
21	C₂²×#10596831	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
23	#14556531	—	2	2	no	—
24	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
25	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#10596833	—	4	2	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
40	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	SD₁₆	1	2	yes	—
42	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
43	#11924003	—	2	1	no	—
44	#11924003	M₁₆	1	2	yes	—
45	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	2	1	no	—
51	C₁₆×C₇₃	M₁₆	1	2	yes	—
52	C₂³×C₇₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₃	—	4	2	no	—
58	C₂×Dic₇₃	#106	1	2	yes	—
59	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
60	#10596833	—	8	2	no	—
61	#10596833	—	8	2	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
65	#10596831	—	2	1	no	—
66	#10596831	#108	1	2	yes	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	2	no	—
69	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	#423	1	1	yes	characteristic
72	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
73	Dic₇₃	—	2	2	no	—
74	Dic₇₃	—	2	2	no	—
75	#12021	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	4	1	no	—
77	C₂×C₇₃	#1566	1	1	yes	characteristic
78	C₂×C₇₃	#2155	1	2	yes	—
79	C₄×M₃₂	—	73	1	no	—
80	#1568	—	73	2	no	—
81	#1368	—	73	1	no	—
82	#1368	—	73	1	no	—
83	#1368	—	73	2	no	—
84	C₇₃	#12021	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	73	1	no	—
86	C₂×M₃₂	#64408	1	1	yes	characteristic
87	#432	—	146	2	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	#346	—	73	1	no	—
90	#346	—	73	1	no	—
91	C₄×C₁₆	—	73	1	no	—
92	C₄×C₁₆	—	73	1	no	—
93	#335	—	146	2	no	—
94	#334	—	73	2	no	—
95	#334	—	73	2	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
98	C₂×C₄²	—	73	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	146	2	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	73	1	no	—
103	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
104	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
105	C₄⋊C₈	—	146	2	no	—
106	C₄⋊C₈	—	146	2	no	—
107	C₂²⋊C₈	—	292	2	no	—
108	#97	—	292	2	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₂²×C₄	—	73	2	no	—
114	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	292	2	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
123	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
124	C₄²	—	73	2	no	—
125	C₄²	—	73	1	no	—
126	C₄²	—	73	1	no	—
127	C₁₆	—	2	1	no	—
128	C₁₆	—	73	2	no	—
129	C₁₆	—	146	1	no	—
130	C₁₆	#11924037	1	2	yes	—
131	C₂³	#14556659	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	292	2	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	#14556659	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	584	2	no	—
141	C₈	—	584	2	no	—
142	C₈	—	73	2	no	—
143	C₈	—	146	1	no	—
144	C₈	C₂×#11924037	1	1	yes	characteristic
145	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#14556659	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	146	2	no	—
151	C₄	—	146	2	no	—
152	C₄	#17989590	1	1	yes	characteristic
153	C₄	D₇₃⋊#108	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#22625485	1	1	yes	characteristic
156	C₂	#22626279	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625491

C₂

yes

maximal

#22624957

C₂

yes

maximal, complemented

#22624956

C₂

yes

maximal, characteristic, complemented

#22624955

C₂

yes

maximal, characteristic, complemented

M₃₂×Dic₇₃

C₂

yes

maximal, characteristic

#17989608

—

#17989451

—

#17989448

C₂²

yes

—

#17989448

C₂²

yes

—

C₂×#14556531

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988362

C₂²

yes

characteristic

#17988362

C₂²

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

characteristic

C₂×M₃₂×C₇₃

C₄

yes

—

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

#14556531

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂×#10596833

—

C₂²×Dic₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

—

#11924003

M₁₆

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#106

yes

—

C₂×Dic₇₃

C₂×SD₁₆

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#12021

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1566

yes

characteristic

C₂×C₇₃

#2155

yes

—

C₄×M₃₂

—

—

—

—

—

C₇₃

#12021

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

#432

—

146

—

C₂×C₄×C₈

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

146

—

#334

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

146

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂×#64408

yes

characteristic

104

C₂×C₁₆

C₂×#64408

yes

characteristic

105

C₄⋊C₈

—

146

—

106

C₄⋊C₈

—

146

—

107

C₂²⋊C₈

—

292

—

108

#97

—

292

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄⋊#64408

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

C₄⋊#64408

yes

characteristic

123

C₂×C₈

C₂²×#64408

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

146

—

130

C₁₆

#11924037

yes

—

131

C₂³

#14556659

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14556659

yes

characteristic

138

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

139

C₈

—

140

C₈

—

584

—

141

C₈

—

584

—

142

C₈

—

143

C₈

—

146

—

144

C₈

C₂×#11924037

yes

characteristic

145

C₈

D₇₃⋊M₁₆

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#14556659

yes

characteristic

149

C₄

—

150

C₄

—

146

—

151

C₄

—

146

—

152

C₄

#17989590

yes

characteristic

153

C₄

D₇₃⋊#108

yes

characteristic

154

C₂

—

155

C₂

#22625485

yes

characteristic

156

C₂

#22626279

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556659
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	#106
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#12021
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×#14556659

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₈×Dic₇₃

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. #22624957	C₂	semidirect
3. #22624956	C₂	semidirect
4. #22624955	C₂	semidirect
84. C₇₃	#12021	semidirect

H ≅ G/N

type

2. #22624957

C₂

semidirect

3. #22624956

C₂

semidirect

4. #22624955

C₂

semidirect

84. C₇₃

#12021

semidirect

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

subgroup	index
#22625491	2
#22624957	2
#22624956	2
#22624955	2
M₃₂×Dic₇₃	2
#12021	73

subgroup

index

M₃₂×Dic₇₃

Subgroups — #34011798

Conjugacy classes of subgroups

Complements

Maximal subgroups