Finite Group Databasebeta

← #37532653

Subgroups — #37532653

Order 23168.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#26041878C211yesmaximal, characteristic, complemented
2#26041429C211yesmaximal, characteristic, complemented
3#26041415C211yesmaximal, characteristic
4#26041197C211yesmaximal, characteristic, complemented
5#26041012C211yesmaximal, characteristic, complemented
6#26041004C211yesmaximal, characteristic, complemented
7#328×C181C211yesmaximal, characteristic, complemented
8C22×D724C2211yescharacteristic
9C2×#15668436—21no—
10C2×D1448—21no—
11#19153811—21no—
12#19153807—21no—
13#19153801—21no—
14#19153800—21no—
15#19153792—21no—
16#19153790C2211yescharacteristic
17#19153776—21no—
18C2×#15668428C2211yescharacteristic
19C2×#15668426—21no—
20C2×#15668426—21no—
21#19153738—21no—
22#19153736—21no—
23C22⋊#13089678C2211yescharacteristic
24C22⋊Q8×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27C23×D181—21no—
28C2×D724—21no—
29C2×D724—21no—
30C2×D724—42no—
31C2×D724D411yescharacteristic
32C2×C22⋊D181—21no—
33C22×Dic181D411yescharacteristic
34C22×Dic181D411yescharacteristic
35C2×C4×D181—21no—
36#15668436—42no—
37D1448—42no—
38C2×#13089678—21no—
39C4⋊Dic181—21no—
40C4⋊Dic181—21no—
41C4⋊Dic181D411yescharacteristic
42#15668428—21no—
43#15668428—21no—
44#15668428—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49#15668426—42no—
50#15668426—42no—
51C4×Dic181—21no—
52C2×Q8×C181D411yescharacteristic
53C22×C4×C181—21no—
54C22×C4×C181C2311yescharacteristic
55C2×C8×C181—21no—
56C4⋊C4×C181—21no—
57C4⋊C4×C181D411yescharacteristic
58C22⋊C4×C181—21no—
59C22×D181—22no—
60C22×D181—21no—
61C22×D181—41no—
62C22×D181—41no—
63C22×D181—41no—
64C22×D181—41no—
65C22×D181—41no—
66C22×D181—41no—
67D724—22no—
68D724—22no—
69D724—41no—
70D724—41no—
71D724—41no—
72C22⋊D181—41no—
73C22⋊D181—41no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—21no—
78C2×Dic181—21no—
79C2×Dic181—41no—
80C4×D181—81no—
81#13089678—41no—
82C23×C181C2×D411yescharacteristic
83Q8×C181—41no—
84C2×C4×C181—21no—
85C2×C4×C181—21no—
86C2×C4×C181—41no—
87C2×C4×C181—41no—
88C2×C4×C181—41no—
89C2×C4×C181—41no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C8×C181—41no—
93C22×C181—21no—
94C22×C181—21no—
95C22×C181#12011yescharacteristic
96C22×C181#13511yescharacteristic
97C22×C181AGL(1,ℤ/8)11yescharacteristic
98C2×D181—42no—
99C2×D181—42no—
100C2×D181—41no—
101C2×D181—41no—
102C2×D181—41no—
103C2×D181—42no—
104C2×D181—81no—
105C2×D181—81no—
106C2×D181—81no—
107C2×D181—81no—
108C4×C181—21no—
109C4×C181—21no—
110C4×C181—41no—
111C4×C181—41no—
112C4×C181—81no—
113Dic181—41no—
114Dic181—41no—
115Dic181—41no—
116C2×C181—21no—
117C2×C181—21no—
118C2×C181#44911yescharacteristic
119C2×C181#45411yescharacteristic
120C2×C181B(4,𝔽2)11yescharacteristic
121D181—81no—
122D181—82no—
123C181#159711yescharacteristic, complemented, Sylow
124#1597—1811nomaximal, Sylow
125#539—1811no—
126#450—1811no—
127#447—1811no—
128#394—1811no—
129#328D18111yescharacteristic, complemented
130#327—1811no—
131#324—1811no—
132C22×D4—1811no—
133C2×SD16—3621no—
134C2×D8—3621no—
135#126—3621no—
136#125—3621no—
137#123—3621no—
138#123—3621no—
139C22⋊Q8—3621no—
140C22⋊Q8C2×D18111yescharacteristic
141#121—1811no—
142C4×D4—3621no—
143C2×C4⋊C4—1811no—
144C2×C22⋊C4—3621no—
145C2×C22⋊C4—3621no—
146#102—3621no—
147#102—3621no—
148C22⋊C8—1811no—
149C22⋊C8C2×D18111yescharacteristic
150#95C2×D18111yescharacteristic
151C24—3621no—
152C2×Q8D72411yescharacteristic
153C2×D4—3621no—
154C2×D4—3621no—
155C2×D4—3621no—
156C2×D4—1811no—
157C2×D4—7242no—
158C22×C4—21no—
159C22×C4—3621no—
160C22×C4—1811no—
161C22×C4—1811no—
162C22×C4C22×D18111yescharacteristic
163SD16—7242no—
164D8—7242no—
165C2×C8—21no—
166C2×C8—3621no—
167C4⋊C4—21no—
168C4⋊C4—3621no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—3621no—
172C4⋊C4—3621no—
173C4⋊C4—1811no—
174C4⋊C4D72411yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—7242no—
181C22⋊C4—7242no—
182C42—3621no—
183C23—3622no—
184C23—3621no—
185C23—7241no—
186C23—7241no—
187C23—7241no—
188C23—7241no—
189C23—7241no—
190C23—7241no—
191C23D4×D18111yescharacteristic
192Q8—41no—
193D4—3622no—
194D4—3622no—
195D4—7241no—
196D4—7241no—
197D4—7241no—
198D4—7241no—
199D4—7241no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—3621no—
207C2×C4—3621no—
208C2×C4—3621no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—14481no—
212C2×C4—7241no—
213C2×C4D4×D18111yescharacteristic
214C2×C4C2×D72411yescharacteristic
215C8—41no—
216C8—7241no—
217C22—21no—
218C22—21no—
219C22—14481no—
220C22—14481no—
221C22—14481no—
222C22—14481no—
223C22—7242no—
224C22—7242no—
225C22—7241no—
226C22—7241no—
227C22—7241no—
228C22—7242no—
229C22#1915378611yescharacteristic
230C22#1915383911yescharacteristic
231C22#1915384311yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—7241no—
238C4—7241no—
239C4—7241no—
240C2—21no—
241C2—21no—
242C2—14482no—
243C2—14481no—
244C2#2604142711yescharacteristic
245C2#2604145311yescharacteristic
246C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#328×C181C2
socleC22×C181#120
prime radical#26041415C2
hypercenter#328D181
hypocenterC181#1597
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #26041878C2semidirect
2. #26041429C2semidirect
4. #26041197C2semidirect
5. #26041012C2semidirect
6. #26041004C2semidirect
7. #328×C181C2semidirect
123. C181#1597semidirect
129. #328D181semidirect

Maximal subgroups

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

subgroupindex
#260418782
#260414292
#260414152
#260411972
#260410122
#260410042
#328×C1812
#1597181