Group #33000345 — subgroups

Subgroups — #33000345

Order 18688.

192 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#22609332	C₂	1	2	yes	maximal
3	#22608003	C₂	1	1	yes	maximal, characteristic
4	#22607989	C₂	1	2	yes	maximal
5	C₇₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#17988879	—	2	2	no	—
7	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
8	C₄⋊#11924003	C₂²	1	1	yes	characteristic
9	#17988405	—	2	1	no	—
10	C₁₆×Dic₇₃	—	2	1	no	—
11	#17988340	C₂²	1	2	yes	—
12	#17988336	—	2	2	no	—
13	#500×C₇₃	C₂²	1	2	yes	—
14	#366×C₇₃	—	2	1	no	—
15	#366×C₇₃	—	2	1	no	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
18	C₂×#11924003	—	2	1	no	—
19	#14556524	—	4	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	D₄	1	1	yes	characteristic
22	#14556520	—	2	1	no	—
23	#14556514	—	4	2	no	—
24	C₈×Dic₇₃	—	2	1	no	—
25	C₄⋊Q₈×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
28	#107×C₇₃	—	2	2	no	—
29	#107×C₇₃	—	2	2	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
35	#11924007	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	#11924003	—	4	1	no	—
38	C₂×Q₈×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	4	2	no	—
42	C₄⋊C₄×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	Q₁₆	1	2	yes	—
46	Q₈×C₇₃	—	16	2	no	—
47	C₂×C₄×C₇₃	—	8	2	no	—
48	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	1	no	—
53	Dic₁₄₆	—	2	1	no	—
54	Dic₁₄₆	Q₃₂	1	2	yes	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₇₃	#137	1	1	yes	characteristic
58	#10596831	—	4	1	no	—
59	#10596831	—	8	2	no	—
60	C₂²×C₇₃	#466	1	1	yes	characteristic
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	16	2	no	—
63	C₄×C₇₃	#462	1	1	yes	characteristic
64	C₄×C₇₃	#473	1	1	yes	characteristic
65	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₇₃	#508	1	1	yes	characteristic
67	Dic₇₃	—	2	2	no	—
68	Dic₇₃	—	4	1	no	—
69	#12458	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	#1674	1	1	yes	characteristic
71	C₂×C₇₃	#2216	1	1	yes	characteristic
72	C₂×C₇₃	#2221	1	1	yes	characteristic
73	#2252	D₇₃	1	1	yes	characteristic
74	Q₈×C₁₆	—	73	1	no	—
75	#1676	—	73	2	no	—
76	#1371	—	73	1	no	—
77	#1365	—	73	2	no	—
78	C₇₃	#12458	1	1	yes	characteristic, complemented, Sylow
79	#500	C₂×D₇₃	1	2	yes	—
80	#477	—	146	2	no	—
81	C₈×Q₈	—	73	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	C₄⋊C₁₆	—	73	1	no	—
85	C₄⋊C₁₆	—	146	1	no	—
86	C₄×C₁₆	—	146	1	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#335	—	73	2	no	—
89	#333	—	146	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	73	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	292	2	no	—
98	C₄⋊C₈	—	73	1	no	—
99	C₄⋊C₈	—	146	1	no	—
100	C₄⋊C₈	—	146	2	no	—
101	#103	—	292	2	no	—
102	C₄×C₈	—	146	1	no	—
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	73	1	no	—
106	C₂×C₈	—	292	2	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
109	C₂×C₈	#11924027	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	73	1	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₄²	—	146	1	no	—
115	C₄²	D₄×D₇₃	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	292	1	no	—
118	C₁₆	#11924018	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	73	2	no	—
121	Q₈	—	146	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	146	1	no	—
124	C₂×C₄	—	146	1	no	—
125	C₂×C₄	#14556562	1	1	yes	characteristic
126	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
127	C₂×C₄	#14556610	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	292	1	no	—
130	C₈	—	584	2	no	—
131	C₈	C₂×#11924018	1	1	yes	characteristic
132	C₈	#14556622	1	1	yes	characteristic
133	C₂²	#17988820	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	292	1	no	—
137	C₄	—	146	2	no	—
138	C₄	#17988794	1	1	yes	characteristic
139	C₄	#17988859	1	1	yes	characteristic
140	C₄	Q₃₂×D₇₃	1	1	yes	characteristic
141	C₄	#17989020	1	1	yes	characteristic
142	C₂	#22609322	1	1	yes	characteristic
143	C₂	#22612019	1	1	yes	characteristic
144	C₂	#22612048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₆

C₂

yes

maximal, characteristic

#22609332

C₂

yes

maximal

#22608003

C₂

yes

maximal, characteristic

#22607989

C₂

yes

maximal

C₇₃×#2252

C₂

yes

maximal, characteristic

#17988879

—

C₈×Dic₁₄₆

C₂²

yes

characteristic

C₄⋊#11924003

C₂²

yes

characteristic

#17988405

—

C₁₆×Dic₇₃

—

#17988340

C₂²

yes

—

#17988336

—

#500×C₇₃

C₂²

yes

—

#366×C₇₃

—

#366×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×Dic₁₄₆

D₄

yes

—

—

—

D₄

yes

characteristic

#14556520

—

#14556514

—

C₈×Dic₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

D₈

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

Q₁₆

yes

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

Q₃₂

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₇₃

#12458

yes

characteristic, complemented, Sylow

#500

C₂×D₇₃

yes

—

#477

—

146

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

#333

—

146

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

—

—

—

292

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

100

C₄⋊C₈

—

146

—

101

#103

—

292

—

102

C₄×C₈

—

146

—

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

292

—

107

C₂×C₈

—

146

—

108

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

109

C₂×C₈

#11924027

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

146

—

114

C₄²

—

146

—

115

C₄²

D₄×D₇₃

yes

characteristic

116

C₁₆

—

117

C₁₆

—

292

—

118

C₁₆

#11924018

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

146

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

#14556562

yes

characteristic

126

C₂×C₄

D₈×D₇₃

yes

characteristic

127

C₂×C₄

#14556610

yes

characteristic

128

C₈

—

129

C₈

—

292

—

130

C₈

—

584

—

131

C₈

C₂×#11924018

yes

characteristic

132

C₈

#14556622

yes

characteristic

133

C₂²

#17988820

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

292

—

137

C₄

—

146

—

138

C₄

#17988794

yes

characteristic

139

C₄

#17988859

yes

characteristic

140

C₄

Q₃₂×D₇₃

yes

characteristic

141

C₄

#17989020

yes

characteristic

142

C₂

#22609322

yes

characteristic

143

C₂

#22612019

yes

characteristic

144

C₂

#22612048

yes

characteristic

	subgroup	G/H
center	C₂²	#17988820
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2252	C₂
socle, prime radical	C₂²×C₇₃	#466
hypercenter	#2252	D₇₃
hypocenter	C₇₃	#12458
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#17988820

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2252

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
78. C₇₃	#12458	semidirect

H ≅ G/N

type

78. C₇₃

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₄₆	2
#22609332	2
#22608003	2
#22607989	2
C₇₃×#2252	2
#12458	73

subgroup

index

C₁₆×Dic₁₄₆

C₇₃×#2252

Subgroups — #33000345

Conjugacy classes of subgroups

Complements

Maximal subgroups