Group #34000460 — subgroups

Subgroups — #34000460

Order 18688.

244 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22625815	C₂	1	2	yes	maximal
2	#22611546	C₂	1	1	yes	maximal, characteristic
3	D₇₃⋊#108	—	2	2	no	—
4	#17989573	—	2	2	no	—
5	#17989456	C₄	1	4	yes	—
6	C₂×#14556597	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₄	1	1	yes	characteristic
8	#17988366	—	2	2	no	—
9	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
10	D₇₃⋊M₁₆	—	2	4	no	—
11	#14556662	—	4	2	no	—
12	#14556661	—	4	2	no	—
13	#14556652	—	4	2	no	—
14	#14556651	—	4	2	no	—
15	C₂×#11924028	D₄	1	1	yes	characteristic
16	#14556597	—	4	1	no	—
17	#14556597	—	4	1	no	—
18	C₂×#11924012	Q₈	1	1	yes	characteristic
19	C₂×#11924011	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₇₃	D₄	1	2	yes	—
21	C₂×#11924003	—	2	1	no	—
22	C₂×#11924003	—	4	1	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₇₃	—	4	1	no	—
25	C₂×C₁₆×C₇₃	—	2	1	no	—
26	#11924038	—	4	4	no	—
27	#11924037	—	4	4	no	—
28	#11924036	—	4	4	no	—
29	#11924028	—	4	1	no	—
30	#11924028	—	4	1	no	—
31	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
32	C₂×D₂₉₂	Q₁₆	1	1	yes	characteristic
33	C₂×C₂²⋊D₇₃	SD₁₆	1	2	yes	—
34	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
35	#11924012	—	4	1	no	—
36	#11924011	—	4	2	no	—
37	C₂×#10596831	C₄⋊C₄	1	2	yes	—
38	C₈×D₇₃	—	2	2	no	—
39	C₈×D₇₃	—	2	2	no	—
40	#11924003	—	4	1	no	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	#106	1	1	yes	characteristic
48	C₂²×D₇₃	#103	1	2	yes	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
53	C₂²⋊D₇₃	—	4	2	no	—
54	C₂×Dic₇₃	#102	1	2	yes	—
55	D₂₉₂	—	4	1	no	—
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	—	2	2	no	—
58	Dic₁₄₆	—	4	1	no	—
59	#10596833	—	8	4	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	#99	1	1	yes	characteristic
62	C₈×C₇₃	#99	1	1	yes	characteristic
63	#10596831	—	2	2	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	4	1	no	—
66	C₂²×C₇₃	#340	1	1	yes	characteristic
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	4	2	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
71	C₄×C₇₃	#342	1	1	yes	characteristic
72	Dic₇₃	—	2	2	no	—
73	#7229	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	4	1	no	—
76	C₂×C₇₃	#1289	1	1	yes	characteristic
77	D₇₃	—	4	2	no	—
78	#2113	—	73	1	no	—
79	#1813	—	73	2	no	—
80	C₇₃	#7229	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	73	1	no	—
82	C₂×M₃₂	#64408	1	1	yes	characteristic
83	C₂²×C₁₆	—	73	1	no	—
84	C₂×#108	—	146	2	no	—
85	#413	—	146	2	no	—
86	C₂²⋊C₁₆	—	146	2	no	—
87	#338	—	73	4	no	—
88	C₂×#41	—	73	1	no	—
89	#131	—	292	1	no	—
90	#131	—	292	1	no	—
91	C₂×M₁₆	—	73	1	no	—
92	C₂×M₁₆	—	73	2	no	—
93	C₂×M₁₆	—	146	4	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	292	1	no	—
99	C₂×C₁₆	—	146	1	no	—
100	#108	—	292	2	no	—
101	#108	—	292	2	no	—
102	#101	—	292	2	no	—
103	#100	—	292	2	no	—
104	#41	—	292	1	no	—
105	#41	—	292	1	no	—
106	C₂×Q₈	—	73	1	no	—
107	C₂×D₄	—	73	2	no	—
108	C₂×D₄	—	73	1	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
111	M₁₆	—	292	2	no	—
112	M₁₆	—	292	4	no	—
113	M₁₆	—	292	4	no	—
114	M₁₆	—	292	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	292	4	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	C₄×#64408	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	292	1	no	—
125	C₂³	—	73	2	no	—
126	C₂³	#14556659	1	1	yes	characteristic
127	Q₈	—	292	1	no	—
128	D₄	—	292	1	no	—
129	D₄	—	292	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14556660	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	584	4	no	—
139	C₈	—	146	2	no	—
140	C₈	#14556649	1	1	yes	characteristic
141	C₈	#14556650	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	292	2	no	—
145	C₂²	—	146	2	no	—
146	C₂²	#17989461	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	146	2	no	—
149	C₄	#17989455	1	1	yes	characteristic
150	C₄	#17989465	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	292	2	no	—
154	C₂	#22624794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625815

C₂

yes

maximal

#22611546

C₂

yes

maximal, characteristic

D₇₃⋊#108

—

#17989573

—

#17989456

C₄

yes

—

C₂×#14556597

C₂²

yes

characteristic

C₂²×#11924003

C₄

yes

characteristic

#17988366

—

C₂×M₃₂×C₇₃

C₄

yes

characteristic

D₇₃⋊M₁₆

—

—

—

—

—

D₄

yes

—

—

Q₈

yes

characteristic

C₂×#11924011

C₂×C₄

yes

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×#11924003

—

C₂×#11924003

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

—

—

—

—

—

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×D₂₉₂

Q₁₆

yes

characteristic

C₂×C₂²⋊D₇₃

SD₁₆

yes

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

—

#11924012

—

#11924011

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂²×D₇₃

#103

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂²⋊D₇₃

—

C₂×Dic₇₃

#102

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7229

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1289

yes

characteristic

D₇₃

—

#2113

—

#1813

—

C₇₃

#7229

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

#64408

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

146

—

#413

—

146

—

C₂²⋊C₁₆

—

146

—

#338

—

C₂×#41

—

#131

—

292

—

#131

—

292

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

100

#108

—

292

—

101

#108

—

292

—

102

#101

—

292

—

103

#100

—

292

—

104

#41

—

292

—

105

#41

—

292

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄⋊#64408

yes

characteristic

111

M₁₆

—

292

—

112

M₁₆

—

292

—

113

M₁₆

—

292

—

114

M₁₆

—

292

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₂×C₈

C₄×#64408

yes

characteristic

122

C₂×C₈

C₂²⋊#64408

yes

characteristic

123

C₁₆

—

124

C₁₆

—

292

—

125

C₂³

—

126

C₂³

#14556659

yes

characteristic

127

Q₈

—

292

—

128

D₄

—

292

—

129

D₄

—

292

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

136

C₂×C₄

#14556660

yes

characteristic

137

C₈

—

138

C₈

—

584

—

139

C₈

—

146

—

140

C₈

#14556649

yes

characteristic

141

C₈

#14556650

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

292

—

145

C₂²

—

146

—

146

C₂²

#17989461

yes

characteristic

147

C₄

—

148

C₄

—

146

—

149

C₄

#17989455

yes

characteristic

150

C₄

#17989465

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

292

—

154

C₂

#22624794

yes

characteristic

	subgroup	G/H
center	C₄	#17989455
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂×C₇₃	#1289
prime radical	C₂×#11924028	D₄
cosocle	C₂×#14556597	C₂²
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#7229
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#17989455

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

C₂×C₇₃

D₄

C₂²

C₂×M₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
80. C₇₃	#7229	semidirect

H ≅ G/N

type

80. C₇₃

#7229

semidirect

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

subgroup	index
#22625815	2
#22611546	2
#7229	73

subgroup

index

#22625815

#22611546

#7229

Subgroups — #34000460

Conjugacy classes of subgroups

Complements

Maximal subgroups