Group #34000485 — subgroups

Subgroups — #34000485

Order 18688.

148 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22626307	C₂	1	1	yes	maximal, characteristic
2	#22626267	C₂	1	1	yes	maximal, characteristic
3	#22611745	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556666	C₄	1	1	yes	characteristic
5	C₂×#14556664	C₄	1	1	yes	characteristic
6	#17989510	—	2	1	no	—
7	#17989508	—	2	1	no	—
8	C₄⋊#11924031	—	2	1	no	—
9	#17989484	—	2	1	no	—
10	#17989480	C₄	1	2	yes	—
11	#17988544	C₂²	1	1	yes	characteristic
12	#17988407	—	2	1	no	—
13	#17988368	—	2	1	no	—
14	#17988365	C₄	1	1	yes	characteristic
15	#347×C₇₃	C₄	1	1	yes	characteristic
16	#14556666	—	4	1	no	—
17	#14556665	—	2	2	no	—
18	#14556664	—	4	1	no	—
19	C₂×#11924031	—	2	1	no	—
20	C₂×#11924031	—	2	1	no	—
21	C₂×#11924011	D₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
23	#14556548	Q₈	1	1	yes	characteristic
24	#14556531	—	2	1	no	—
25	#14556501	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	M₃₂×C₇₃	—	2	1	no	—
29	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924031	—	4	2	no	—
31	#11924031	—	4	1	no	—
32	#11924031	—	4	1	no	—
33	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
34	#11924011	—	4	1	no	—
35	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
37	C₈×D₇₃	—	2	1	no	—
38	C₈×D₇₃	—	2	1	no	—
39	#11924007	—	2	1	no	—
40	#11924005	—	2	1	no	—
41	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
42	#11924003	—	4	1	no	—
43	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂²×D₇₃	#108	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	#108	1	1	yes	characteristic
52	C₄×D₇₃	—	2	1	no	—
53	C₄×D₇₃	—	2	1	no	—
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	#10596831	—	2	1	no	—
57	#10596831	#100	1	1	yes	characteristic
58	#10596831	#101	1	1	yes	characteristic
59	C₂²×C₇₃	#341	1	1	yes	characteristic
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	—	4	1	no	—
62	C₄×C₇₃	—	4	1	no	—
63	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
64	C₄×C₇₃	#343	1	1	yes	characteristic
65	Dic₇₃	—	2	1	no	—
66	Dic₇₃	—	4	1	no	—
67	#7239	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	—	2	1	no	—
69	C₂×C₇₃	#1290	1	1	yes	characteristic
70	D₇₃	—	4	1	no	—
71	#2166	—	73	1	no	—
72	#2166	—	73	1	no	—
73	#2151	—	73	1	no	—
74	C₇₃	#7239	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₃₂	—	73	1	no	—
76	C₂²×C₁₆	—	73	1	no	—
77	#405	—	73	1	no	—
78	#364	—	146	1	no	—
79	#364	—	146	1	no	—
80	C₄⋊C₁₆	—	146	1	no	—
81	C₄⋊C₁₆	—	146	1	no	—
82	#349	—	146	1	no	—
83	#349	—	146	1	no	—
84	#347	—	73	2	no	—
85	#347	—	73	1	no	—
86	#347	#64408	1	1	yes	characteristic
87	C₂×M₁₆	—	73	1	no	—
88	C₂²×C₈	—	73	1	no	—
89	#117	—	73	1	no	—
90	M₃₂	—	2	1	no	—
91	M₃₂	—	292	1	no	—
92	M₃₂	—	146	2	no	—
93	M₃₂	—	146	1	no	—
94	C₂×C₁₆	—	292	1	no	—
95	C₂×C₁₆	—	146	1	no	—
96	C₂×C₁₆	—	146	1	no	—
97	#97	—	73	1	no	—
98	#97	C₂×#64408	1	1	yes	characteristic
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₂²×C₄	—	73	1	no	—
102	M₁₆	—	292	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	—	146	1	no	—
107	C₂×C₈	C₄×#64408	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
109	C₄⋊C₄	—	146	1	no	—
110	C₂²⋊C₄	—	146	1	no	—
111	C₄²	—	73	1	no	—
112	C₄²	C₄⋊#64408	1	1	yes	characteristic
113	C₁₆	—	4	1	no	—
114	C₁₆	—	292	1	no	—
115	C₁₆	—	292	1	no	—
116	C₁₆	—	292	1	no	—
117	C₁₆	—	292	2	no	—
118	C₂³	—	73	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	146	1	no	—
122	C₂×C₄	—	146	1	no	—
123	C₂×C₄	—	146	1	no	—
124	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	73	1	no	—
128	C₈	—	73	1	no	—
129	C₈	—	146	1	no	—
130	C₂²	—	292	1	no	—
131	C₂²	—	146	1	no	—
132	C₂²	#17989462	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	292	1	no	—
135	C₄	—	146	1	no	—
136	C₄	#17989454	1	1	yes	characteristic
137	C₄	#17989466	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	292	1	no	—
140	C₂	#22624795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626307

C₂

yes

maximal, characteristic

#22626267

C₂

yes

maximal, characteristic

#22611745

C₂

yes

maximal, characteristic

C₂×#14556666

C₄

yes

characteristic

C₂×#14556664

C₄

yes

—

—

—

—

C₄

yes

—

#17988544

C₂²

yes

—

—

C₄

yes

characteristic

#347×C₇₃

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

Q₈

yes

characteristic

#14556531

—

#14556501

C₂×C₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

#97×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

#11924011

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7239

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1290

yes

characteristic

D₇₃

—

#2166

—

#2166

—

#2151

—

C₇₃

#7239

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

146

—

#364

—

146

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

#349

—

146

—

#349

—

146

—

—

—

yes

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

292

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

—

yes

C₄×C₈

—

100

C₄×C₈

—

101

C₂²×C₄

—

102

M₁₆

—

292

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

—

146

—

107

C₂×C₈

C₄×#64408

yes

characteristic

108

C₂×C₈

C₂²⋊#64408

yes

characteristic

109

C₄⋊C₄

—

146

—

110

C₂²⋊C₄

—

146

—

111

C₄²

—

112

C₄²

C₄⋊#64408

yes

characteristic

113

C₁₆

—

114

C₁₆

—

292

—

115

C₁₆

—

292

—

116

C₁₆

—

292

—

117

C₁₆

—

292

—

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

146

—

130

C₂²

—

292

—

131

C₂²

—

146

—

132

C₂²

#17989462

yes

characteristic

133

C₄

—

134

C₄

—

292

—

135

C₄

—

146

—

136

C₄

#17989454

yes

characteristic

137

C₄

#17989466

yes

characteristic

138

C₂

—

139

C₂

—

292

—

140

C₂

#22624795

yes

characteristic

	subgroup	G/H
center	C₄	#17989454
commutator	C₂×C₈×C₇₃	C₄²
Frattini	#97	C₂×#64408
Fitting	#347×C₇₃	C₄
socle	C₂×C₇₃	#1290
prime radical	C₂²×D₇₃	#108
cosocle	#17988544	C₂²
hypercenter	#347	#64408
hypocenter	C₇₃	#7239
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

subgroup

G/H

center

C₄

#17989454

commutator

C₂×C₈×C₇₃

C₄²

#347×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

N	H ≅ G/N	type
74. C₇₃	#7239	semidirect

H ≅ G/N

type

74. C₇₃

#7239

semidirect

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

subgroup	index
#22626307	2
#22626267	2
#22611745	2
#7239	73

subgroup

index

Subgroups — #34000485

Conjugacy classes of subgroups

Complements

Maximal subgroups