Group #33000348 — subgroups

Subgroups — #33000348

Order 18688.

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

normal subgroups only

This group has 192 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	#22609335	C₂	1	2	yes	maximal
3	#22608003	C₂	1	1	yes	maximal, characteristic
4	#22607988	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988882	—	2	2	no	—
8	#17988413	—	2	1	no	—
9	#17988413	—	2	1	no	—
10	#17988340	C₂²	1	2	yes	—
11	#17988335	—	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	#14556591	—	2	2	no	—
18	#14556525	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556524	—	4	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	#14556515	—	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	C₂×#10596831	—	4	2	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
41	C₄²×C₇₃	—	2	1	no	—
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	1	no	—
44	C₁₆×C₇₃	—	4	1	no	—
45	C₁₆×C₇₃	Q₁₆	1	2	yes	—
46	Q₈×C₇₃	—	2	1	no	—
47	Q₈×C₇₃	Q₃₂	1	2	yes	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
53	C₂×Dic₇₃	—	8	2	no	—
54	Dic₁₄₆	—	16	2	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	—	4	1	no	—
57	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₇₃	#137	1	1	yes	characteristic
59	#10596831	—	8	2	no	—
60	C₂²×C₇₃	#466	1	1	yes	characteristic
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	4	1	no	—
64	C₄×C₇₃	#462	1	1	yes	characteristic
65	C₄×C₇₃	#473	1	1	yes	characteristic
66	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
67	C₄×C₇₃	#508	1	1	yes	characteristic
68	Dic₇₃	—	16	2	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	—	73	1	no	—
74	Q₈×C₁₆	D₇₃	1	1	yes	characteristic
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	—	73	2	no	—
80	#477	—	146	2	no	—
81	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
82	#366	—	146	1	no	—
83	#366	—	146	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
86	C₄×C₁₆	—	2	1	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#335	—	73	2	no	—
89	#333	—	146	2	no	—
90	C₄⋊Q₈	—	146	2	no	—
91	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
95	#107	—	292	2	no	—
96	#107	—	146	2	no	—
97	#107	—	146	2	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	146	2	no	—
100	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
101	#103	—	292	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×Q₈	—	292	2	no	—
105	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	292	2	no	—
108	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
109	C₂×C₈	#11924028	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	292	2	no	—
113	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
114	C₄²	—	2	1	no	—
115	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	4	1	no	—
118	C₁₆	#11924018	1	2	yes	—
119	Q₈	—	2	1	no	—
120	Q₈	—	1168	2	no	—
121	Q₈	Dic₅₈₄	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	584	2	no	—
125	C₂×C₄	#14556565	1	1	yes	characteristic
126	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
127	C₂×C₄	#14556613	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	584	2	no	—
131	C₈	C₂×#11924018	1	1	yes	characteristic
132	C₈	#14556621	1	1	yes	characteristic
133	C₂²	#17988823	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	1168	2	no	—
138	C₄	#17988797	1	1	yes	characteristic
139	C₄	#17988862	1	1	yes	characteristic
140	C₄	C₂×Dic₅₈₄	1	1	yes	characteristic
141	C₄	#17989019	1	1	yes	characteristic
142	C₂	#22609325	1	1	yes	characteristic
143	C₂	#22612022	1	1	yes	characteristic
144	C₂	#22612051	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22609335

C₂

yes

maximal

#22608003

C₂

yes

maximal, characteristic

#22607988

C₂

yes

maximal

Q₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988335

—

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

—

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

—

Q₈×C₇₃

—

Q₈×C₇₃

Q₃₂

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

—

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×Q₃₂

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

Q₈×C₁₆

D₇₃

yes

—

—

—

C₇₃

#12458

yes

characteristic, complemented, Sylow

#500

—

#477

—

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

—

#333

—

146

—

C₄⋊Q₈

—

146

—

C₄×Q₈

D₂₉₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

#107

—

292

—

#107

—

146

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

100

C₄⋊C₈

D₂₉₂

yes

characteristic

101

#103

—

292

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

C₂×Q₈

—

292

—

105

C₂×Q₈

D₅₈₄

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

292

—

108

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

109

C₂×C₈

#11924028

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

292

—

113

C₄⋊C₄

D₅₈₄

yes

characteristic

114

C₄²

—

115

C₄²

C₂×D₂₉₂

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

#11924018

yes

—

119

Q₈

—

120

Q₈

—

1168

—

121

Q₈

Dic₅₈₄

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

584

—

125

C₂×C₄

#14556565

yes

characteristic

126

C₂×C₄

C₂×D₅₈₄

yes

characteristic

127

C₂×C₄

#14556613

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

584

—

131

C₈

C₂×#11924018

yes

characteristic

132

C₈

#14556621

yes

characteristic

133

C₂²

#17988823

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

1168

—

138

C₄

#17988797

yes

characteristic

139

C₄

#17988862

yes

characteristic

140

C₄

C₂×Dic₅₈₄

yes

characteristic

141

C₄

#17989019

yes

characteristic

142

C₂

#22609325

yes

characteristic

143

C₂

#22612022

yes

characteristic

144

C₂

#22612051

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	#12458
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

#12458

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

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
#22612206	2
#22609335	2
#22608003	2
#22607988	2
Q₈×C₁₆×C₇₃	2
#12458	73

subgroup

index

Q₈×C₁₆×C₇₃

Subgroups — #33000348

Conjugacy classes of subgroups

Complements

Maximal subgroups