Finite Group Databasebeta

← #37532709

Subgroups — #37532709

Order 23168.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041841C211yesmaximal, characteristic
2#26041593C212yesmaximal, complemented
3#26041204C211yesmaximal, characteristic, complemented
4#26041014C212yesmaximal, complemented
5#323×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181—21no—
7#19153800—22no—
8#19153790—21no—
9#19153790C2212yes—
10#19153786—22no—
11C2×C4⋊Dic181C2211yescharacteristic, complemented
12C2×C22⋊Dic181—21no—
13#19153750—22no—
14#19153738—22no—
15#19153723—21no—
16#19153723C2211yescharacteristic, complemented
17C2×C22⋊C4×C181C2211yescharacteristic
18C22⋊C8×C181C2212yescomplemented
19C23×D181—21no—
20C2×D724—21no—
21C2×D724D412yes—
22C2×C22⋊D181—22no—
23C2×C22⋊D181—22no—
24C2×C22⋊D181—42no—
25C2×C22⋊D181—42no—
26C22×Dic181—21no—
27C22×Dic181—21no—
28C22×Dic181—21no—
29C4⋊Dic181—21no—
30C4⋊Dic181—22no—
31C4⋊Dic181D412yes—
32C22⋊Dic181—42no—
33#15668426—22no—
34#15668426—22no—
35C24×C181D411yescharacteristic
36C22×C4×C181D411yescharacteristic
37C22×C4×C181C2311yescharacteristic
38C2×C8×C181—22no—
39C22⋊C4×C181—22no—
40C22×D181—22no—
41C22×D181—42no—
42C22×D181—81no—
43C22×D181—81no—
44D724—42no—
45D724—81no—
46C22⋊D181—44no—
47C22⋊D181—42no—
48C22⋊D181—42no—
49C2×Dic181—22no—
50C2×Dic181—42no—
51C2×Dic181—41no—
52C2×Dic181—41no—
53C2×Dic181—41no—
54C2×Dic181—41no—
55C2×Dic181—41no—
56C2×Dic181—41no—
57C23×C181—21no—
58C23×C181—21no—
59C23×C181—41no—
60C23×C181—41no—
61C23×C181D812yes—
62C23×C181C2×D411yescharacteristic
63C2×C4×C181—21no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181C2×D412yes—
67C8×C181—42no—
68C22×C181—21no—
69C22×C181—21no—
70C22×C181—22no—
71C22×C181—42no—
72C22×C181—41no—
73C22×C181—41no—
74C22×C181—81no—
75C22×C181#12011yescharacteristic
76C22×C181C2×D811yescharacteristic
77C22×C181AGL(1,ℤ/8)11yescharacteristic
78C2×D181—42no—
79C2×D181—42no—
80C2×D181—42no—
81C2×D181—84no—
82C4×C181—22no—
83C4×C181—41no—
84Dic181—42no—
85Dic181—81no—
86Dic181—81no—
87C2×C181—21no—
88C2×C181—21no—
89C2×C181—42no—
90C2×C181#44711yescharacteristic
91C2×C181#45511yescharacteristic
92C2×C181#45811yescharacteristic
93D181—82no—
94C181#160711yescharacteristic, complemented, Sylow
95#1607—1811nomaximal, Sylow
96#534—1811no—
97#480—1812no—
98#397—1811no—
99#327—1812no—
100#323D18111yescharacteristic, complemented
101C22×D4—3621no—
102#123—3622no—
103#121—3621no—
104#121—1812no—
105#120—3622no—
106C2×C4⋊C4—1811no—
107C2×C22⋊C4—3621no—
108C2×C22⋊C4C2×D18111yescharacteristic
109#107—3622no—
110#102—3622no—
111C22⋊C8C2×D18112yescomplemented
112#95—3621no—
113#95—1811no—
114C24—3621no—
115C24D72411yescharacteristic
116C2×D4—3621no—
117C2×D4—3622no—
118C2×D4—3622no—
119C2×D4—1812no—
120C2×D4—7242no—
121C2×D4—7242no—
122C22×C4—3621no—
123C22×C4—3621no—
124C22×C4—3621no—
125C22×C4D72411yescharacteristic
126C22×C4C22×D18111yescharacteristic
127C2×C8—22no—
128C4⋊C4—3621no—
129C4⋊C4—3622no—
130C4⋊C4—1812no—
131C22⋊C4—22no—
132C22⋊C4—3622no—
133C22⋊C4—3622no—
134C22⋊C4—7242no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—3622no—
140C23—14481no—
141C23—14481no—
142C23—7242no—
143C23D144812yes—
144C23C2×D72411yescharacteristic
145D4—14481no—
146D4—7242no—
147D4—7244no—
148D4—7242no—
149D4—7242no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3622no—
154C2×C4—7241no—
155C2×C4—7241no—
156C2×C4—7241no—
157C2×C4—7241no—
158C2×C4—7241no—
159C2×C4—7241no—
160C2×C4—7242no—
161C2×C4D4×D18112yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—14484no—
171C22—7242no—
172C22—7242no—
173C22—7242no—
174C22#1915378611yescharacteristic
175C22C2×D144811yescharacteristic
176C22#1915384411yescharacteristic
177C4—22no—
178C4—41no—
179C4—14481no—
180C4—14481no—
181C4—7242no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—14482no—
186C2#2604141511yescharacteristic
187C2#2604145911yescharacteristic
188C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#323×C181C2
socleC22×C181#120
prime radical#26041841C2
hypercenter#323D181
hypocenterC181#1607
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26041593C2semidirect
3. #26041204C2semidirect
4. #26041014C2semidirect
5. #323×C181C2semidirect
11. C2×C4⋊Dic181C22semidirect
16. #19153723C22semidirect
18. C22⋊C8×C181C22semidirect
94. C181#1607semidirect
100. #323D181semidirect
111. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418412
#260415932
#260412042
#260410142
#323×C1812
#1607181