Group #33000343 — subgroups

Subgroups — #33000343

Order 18688.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22609325	C₂	1	2	yes	maximal
3	#22608003	C₂	1	1	yes	maximal, characteristic
4	#22607985	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988797	—	2	2	no	—
8	#17988413	—	2	1	no	—
9	#17988413	—	2	1	no	—
10	#17988340	C₂²	1	2	yes	—
11	#17988332	—	2	2	no	—
12	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₃	—	2	1	no	—
14	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₃	—	2	1	no	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924018	—	4	2	no	—
18	#14556591	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	#14556514	—	4	2	no	—
23	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	—	2	1	no	—
28	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₆	—	4	2	no	—
32	#11924018	—	8	4	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
42	C₄²×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	—	4	1	no	—
46	C₁₆×C₇₃	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₃	—	2	1	no	—
48	Q₈×C₇₃	D₁₆	1	2	yes	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
54	C₂×Dic₇₃	—	8	2	no	—
55	Dic₁₄₆	—	8	2	no	—
56	Dic₁₄₆	—	8	2	no	—
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₇₃	#137	1	1	yes	characteristic
61	#10596831	—	8	2	no	—
62	C₂²×C₇₃	#466	1	1	yes	characteristic
63	C₄×C₇₃	—	2	2	no	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	4	1	no	—
66	C₄×C₇₃	#462	1	1	yes	characteristic
67	C₄×C₇₃	#473	1	1	yes	characteristic
68	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₃	#508	1	1	yes	characteristic
70	Dic₇₃	—	16	2	no	—
71	#12457	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	#1674	1	1	yes	characteristic
73	C₂×C₇₃	#2215	1	1	yes	characteristic
74	C₂×C₇₃	#2222	1	1	yes	characteristic
75	#2252	—	73	1	no	—
76	Q₈×C₁₆	D₇₃	1	1	yes	characteristic
77	#1674	—	73	2	no	—
78	#1371	—	73	1	no	—
79	#1364	—	73	2	no	—
80	C₇₃	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	73	2	no	—
82	#462	—	146	2	no	—
83	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
84	#366	—	146	1	no	—
85	#366	—	146	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
90	#335	—	73	2	no	—
91	#332	—	146	2	no	—
92	C₂×Q₁₆	—	292	2	no	—
93	C₄⋊Q₈	—	146	2	no	—
94	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
98	#107	—	146	2	no	—
99	#107	—	146	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	146	2	no	—
102	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
103	#103	—	292	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
106	C₂×Q₈	—	292	2	no	—
107	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
108	Q₁₆	—	584	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂×C₈	#11924028	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	292	2	no	—
115	C₄⋊C₄	—	292	2	no	—
116	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11924018	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	584	2	no	—
124	Q₈	—	584	2	no	—
125	Q₈	D₁₁₆₈	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	584	2	no	—
129	C₂×C₄	#14556565	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
131	C₂×C₄	#14556613	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	584	2	no	—
135	C₈	C₂×#11924018	1	1	yes	characteristic
136	C₈	#14556621	1	1	yes	characteristic
137	C₂²	#17988823	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1168	2	no	—
142	C₄	#17988797	1	1	yes	characteristic
143	C₄	#17988862	1	1	yes	characteristic
144	C₄	C₂×D₁₁₆₈	1	1	yes	characteristic
145	C₄	#17989019	1	1	yes	characteristic
146	C₂	#22609325	1	1	yes	characteristic
147	C₂	#22612017	1	1	yes	characteristic
148	C₂	#22612056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22609325

C₂

yes

maximal

#22608003

C₂

yes

maximal, characteristic

#22607985

C₂

yes

maximal

Q₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988332

—

C₈×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

—

—

C₄×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

#11924018

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

Q₁₆

yes

complemented

Q₈×C₇₃

—

Q₈×C₇₃

D₁₆

yes

—

C₂×C₄×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

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

Q₈×C₁₆

D₇₃

yes

—

—

—

C₇₃

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

146

—

C₈×Q₈

C₂×D₇₃

yes

characteristic

#366

—

146

—

#366

—

146

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

#332

—

146

—

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

146

—

C₄×Q₈

D₂₉₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

#107

—

146

—

#107

—

146

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

146

—

102

C₄⋊C₈

D₂₉₂

yes

characteristic

103

#103

—

292

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₇₃

yes

characteristic

106

C₂×Q₈

—

292

—

107

C₂×Q₈

D₅₈₄

yes

characteristic

108

Q₁₆

—

584

—

109

C₂×C₈

—

110

C₂×C₈

—

292

—

111

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

112

C₂×C₈

#11924028

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

292

—

115

C₄⋊C₄

—

292

—

116

C₄⋊C₄

D₅₈₄

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₉₂

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11924018

yes

complemented

122

Q₈

—

123

Q₈

—

584

—

124

Q₈

—

584

—

125

Q₈

D₁₁₆₈

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

584

—

129

C₂×C₄

#14556565

yes

characteristic

130

C₂×C₄

C₂×D₅₈₄

yes

characteristic

131

C₂×C₄

#14556613

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

584

—

135

C₈

C₂×#11924018

yes

characteristic

136

C₈

#14556621

yes

characteristic

137

C₂²

#17988823

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1168

—

142

C₄

#17988797

yes

characteristic

143

C₄

#17988862

yes

characteristic

144

C₄

C₂×D₁₁₆₈

yes

characteristic

145

C₄

#17989019

yes

characteristic

146

C₂

#22609325

yes

characteristic

147

C₂

#22612017

yes

characteristic

148

C₂

#22612056

yes

characteristic

	subgroup	G/H
center	C₂²	#17988823
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	Q₈×C₁₆×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	#466
hypercenter	Q₈×C₁₆	D₇₃
hypocenter	C₇₃	#12457
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#17988823

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

Q₈×C₁₆×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

#466

hypercenter

Q₈×C₁₆

D₇₃

hypocenter

C₇₃

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₇₃	Q₁₆	semidirect
80. C₇₃	#12457	semidirect
121. C₁₆	#11924018	semidirect

H ≅ G/N

type

46. C₁₆×C₇₃

Q₁₆

semidirect

80. C₇₃

#12457

semidirect

121. C₁₆

#11924018

semidirect

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

subgroup	index
#22612206	2
#22609325	2
#22608003	2
#22607985	2
Q₈×C₁₆×C₇₃	2
#12457	73

subgroup

index

Q₈×C₁₆×C₇₃

Subgroups — #33000343

Conjugacy classes of subgroups

Complements

Maximal subgroups