Group #33000243 — subgroups

Subgroups — #33000243

Order 18688.

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

normal subgroups only

This group has 216 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	#22609298	C₂	1	2	yes	maximal
3	#22608003	C₂	1	1	yes	maximal, characteristic
4	#22607980	C₂	1	2	yes	maximal
5	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#17988869	—	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	#17988333	—	2	2	no	—
13	C₂×Q₃₂×C₇₃	—	2	1	no	—
14	C₂×Q₃₂×C₇₃	—	2	1	no	—
15	#494×C₇₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
18	C₂×#11924003	—	2	1	no	—
19	#14556522	—	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	C₂×Q₁₆×C₇₃	—	2	2	no	—
26	C₂×Q₁₆×C₇₃	—	2	2	no	—
27	C₄⋊Q₈×C₇₃	—	2	2	no	—
28	Q₃₂×C₇₃	—	4	2	no	—
29	Q₃₂×C₇₃	—	4	2	no	—
30	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
37	#11924007	—	2	1	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	#11924003	—	4	1	no	—
40	C₂×Q₈×C₇₃	—	4	2	no	—
41	Q₁₆×C₇₃	—	4	2	no	—
42	Q₁₆×C₇₃	—	4	2	no	—
43	Q₁₆×C₇₃	—	8	2	no	—
44	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	4	2	no	—
47	C₄⋊C₄×C₇₃	—	4	2	no	—
48	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₃	—	2	1	no	—
50	C₁₆×C₇₃	SD₁₆	1	2	yes	—
51	Q₈×C₇₃	—	8	2	no	—
52	Q₈×C₇₃	—	8	2	no	—
53	C₂×C₄×C₇₃	—	8	2	no	—
54	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	Dic₁₄₆	—	2	1	no	—
60	Dic₁₄₆	Q₃₂	1	2	yes	complemented
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	#10596831	—	4	1	no	—
65	#10596831	—	8	2	no	—
66	C₂²×C₇₃	#466	1	1	yes	characteristic
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	16	2	no	—
69	C₄×C₇₃	#460	1	1	yes	characteristic
70	C₄×C₇₃	#472	1	1	yes	characteristic
71	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₇₃	#508	1	1	yes	characteristic
73	Dic₇₃	—	2	2	no	—
74	Dic₇₃	—	4	1	no	—
75	#12440	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	#1668	1	1	yes	characteristic
77	C₂×C₇₃	#2216	1	1	yes	characteristic
78	C₂×C₇₃	#2222	1	1	yes	characteristic
79	#2247	D₇₃	1	1	yes	characteristic
80	Q₈×C₁₆	—	73	1	no	—
81	#1670	—	73	2	no	—
82	#1371	—	73	1	no	—
83	#1362	—	73	2	no	—
84	C₇₃	#12440	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₃₂	—	2	1	no	—
86	C₂×Q₃₂	—	2	1	no	—
87	#494	C₂×D₇₃	1	2	yes	—
88	#475	—	146	2	no	—
89	C₈×Q₈	—	73	1	no	—
90	C₄⋊C₁₆	—	73	1	no	—
91	C₄⋊C₁₆	—	146	1	no	—
92	C₄×C₁₆	—	146	1	no	—
93	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
94	#335	—	73	2	no	—
95	#332	—	146	2	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	73	1	no	—
100	Q₃₂	—	4	2	no	—
101	Q₃₂	—	4	2	no	—
102	C₂×C₁₆	—	146	1	no	—
103	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
105	#106	—	292	2	no	—
106	C₄⋊C₈	—	73	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	C₄⋊C₈	—	146	2	no	—
109	#103	—	292	2	no	—
110	C₄×C₈	—	146	1	no	—
111	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	73	1	no	—
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	292	2	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
120	C₂×C₈	#11924027	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	73	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄²	—	146	1	no	—
126	C₄²	D₄×D₇₃	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	292	1	no	—
129	C₁₆	#11924016	1	2	yes	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	73	2	no	—
133	Q₈	—	146	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	—	146	1	no	—
137	C₂×C₄	#14556562	1	1	yes	characteristic
138	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
139	C₂×C₄	#14556610	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	292	1	no	—
142	C₈	—	584	2	no	—
143	C₈	C₂×#11924016	1	1	yes	characteristic
144	C₈	#14556617	1	1	yes	characteristic
145	C₂²	#17988820	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	16	2	no	—
148	C₄	—	292	1	no	—
149	C₄	—	146	2	no	—
150	C₄	#17988784	1	1	yes	characteristic
151	C₄	#17988854	1	1	yes	characteristic
152	C₄	Q₃₂×D₇₃	1	1	yes	characteristic
153	C₄	#17989020	1	1	yes	characteristic
154	C₂	#22609288	1	1	yes	characteristic
155	C₂	#22612019	1	1	yes	characteristic
156	C₂	#22612053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₆

C₂

yes

maximal, characteristic

#22609298

C₂

yes

maximal

#22608003

C₂

yes

maximal, characteristic

#22607980

C₂

yes

maximal

C₇₃×#2247

C₂

yes

maximal, characteristic

#17988869

—

C₈×Dic₁₄₆

C₂²

yes

characteristic

C₄⋊#11924003

C₂²

yes

characteristic

#17988405

—

C₁₆×Dic₇₃

—

#17988340

C₂²

yes

—

#17988333

—

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

—

#494×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×Dic₁₄₆

D₄

yes

—

—

—

D₄

yes

characteristic

#14556520

—

#14556514

—

C₈×Dic₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

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

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×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

SD₁₆

yes

—

Q₈×C₇₃

—

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

complemented

C₈×C₇₃

—

C₈×C₇₃

C₂×SD₁₆

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

—

#1670

—

#1371

—

#1362

—

C₇₃

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₇₃

yes

—

#475

—

146

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

#332

—

146

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

100

Q₃₂

—

101

Q₃₂

—

102

C₂×C₁₆

—

146

—

103

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

105

#106

—

292

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

#103

—

292

—

110

C₄×C₈

—

146

—

111

C₄×C₈

C₂²×D₇₃

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

146

—

119

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

120

C₂×C₈

#11924027

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

146

—

125

C₄²

—

146

—

126

C₄²

D₄×D₇₃

yes

characteristic

127

C₁₆

—

128

C₁₆

—

292

—

129

C₁₆

#11924016

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

146

—

134

C₂×C₄

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

#14556562

yes

characteristic

138

C₂×C₄

D₈×D₇₃

yes

characteristic

139

C₂×C₄

#14556610

yes

characteristic

140

C₈

—

141

C₈

—

292

—

142

C₈

—

584

—

143

C₈

C₂×#11924016

yes

characteristic

144

C₈

#14556617

yes

characteristic

145

C₂²

#17988820

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

292

—

149

C₄

—

146

—

150

C₄

#17988784

yes

characteristic

151

C₄

#17988854

yes

characteristic

152

C₄

Q₃₂×D₇₃

yes

characteristic

153

C₄

#17989020

yes

characteristic

154

C₂

#22609288

yes

characteristic

155

C₂

#22612019

yes

characteristic

156

C₂

#22612053

yes

characteristic

	subgroup	G/H
center	C₂²	#17988820
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2247	C₂
socle, prime radical	C₂²×C₇₃	#466
hypercenter	#2247	D₇₃
hypocenter	C₇₃	#12440
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#17988820

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2247

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
60. Dic₁₄₆	Q₃₂	semidirect
84. C₇₃	#12440	semidirect

H ≅ G/N

type

60. Dic₁₄₆

Q₃₂

semidirect

84. C₇₃

#12440

semidirect

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

subgroup	index
C₁₆×Dic₁₄₆	2
#22609298	2
#22608003	2
#22607980	2
C₇₃×#2247	2
#12440	73

subgroup

index

C₁₆×Dic₁₄₆

C₇₃×#2247

Subgroups — #33000243

Conjugacy classes of subgroups

Complements

Maximal subgroups