Finite Group Databasebeta

← #36361856

Subgroups — #36361856

Order 22144.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D173C211yesmaximal, characteristic, complemented
2C2×#19149663C211yesmaximal, characteristic, complemented
3C2×#19149662C211yesmaximal, characteristic, complemented
4C2×#19149649C211yesmaximal, characteristic
5#26011369C214yesmaximal, complemented
6#26011365C214yesmaximal, complemented
7#26011352C211yesmaximal, characteristic, complemented
8#26011346C211yesmaximal, characteristic, complemented
9#381×C173C211yesmaximal, characteristic, complemented
10C24×D173C2211yescharacteristic
11C22×D692C2212yes—
12C22×C22⋊D173C2211yescharacteristic
13C23×Dic173C2211yescharacteristic
14C22×C4×D173—24no—
15C22×C4×D173—24no—
16C22×C4×D173—24no—
17C22×C4×D173C2214yes—
18#19149663—24no—
19#19149662—24no—
20#19149649—24no—
21C2×C4⋊Dic173C2212yescomplemented
22C2×#14583805C2212yescomplemented
23C2×C22⋊Dic173C2211yescharacteristic
24C2×#14583803C2212yes—
25C2×#14583803C2214yes—
26C2×#14583803C2212yescomplemented
27C2×#14583803C2212yescomplemented
28#19149586C2214yescomplemented
29C23×C4×C173C2211yescharacteristic
30C2×C22⋊C4×C173C2211yescharacteristic
31C2×C22⋊C4×C173C2211yescharacteristic
32#95×C173C2214yescomplemented
33C23×D173—28no—
34C23×D173—22no—
35C23×D173—22no—
36C23×D173D414yes—
37C23×D173C2311yescharacteristic
38C23×D173C2312yes—
39C2×D692—24no—
40C2×D692—28no—
41C2×C22⋊D173—22no—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44C22×Dic173—22no—
45C22×Dic173—22no—
46C22×Dic173D414yescomplemented
47C22×Dic173C2312yes—
48C22×Dic173C2311yescharacteristic
49C22×Dic173C2312yes—
50C2×C4×D173—232no—
51C2×C4×D173—28no—
52C2×C4×D173—28no—
53C2×C4×D173—28no—
54C4⋊Dic173—24no—
55#14583805—24no—
56C22⋊Dic173—22no—
57#14583803—24no—
58#14583803—28no—
59#14583803—24no—
60#14583803—24no—
61C24×C173C2311yescharacteristic
62C22×C4×C173—22no—
63C22×C4×C173—22no—
64C22×C4×C173D414yescomplemented
65C22×C4×C173C2312yes—
66C22×C4×C173C2312yes—
67C22×C4×C173C2312yes—
68C22⋊C4×C173—22no—
69C22⋊C4×C173—22no—
70C22×D173—216no—
71C22×D173—216no—
72C22×D173—28no—
73C22×D173—22no—
74C22×D173—22no—
75C22×D173—24no—
76C22×D173—216no—
77C22×D173—21no—
78C22×D173—21no—
79C22×D173—21no—
80C22×D173—24no—
81C22×D173C2×D414yes—
82C22×D173#4118yes—
83D692—48no—
84C22⋊D173—44no—
85C2×Dic173—28no—
86C2×Dic173—22no—
87C2×Dic173—22no—
88C2×Dic173—21no—
89C2×Dic173—21no—
90C2×Dic173—21no—
91C2×Dic173—22no—
92C2×Dic173—24no—
93C2×Dic173C2×D414yes—
94C4×D173—232no—
95C23×C173—24no—
96C23×C173C2×D412yes—
97C23×C173C2×D412yes—
98C23×C173C2×D412yes—
99C23×C173C2411yescharacteristic
100C2×C4×C173—28no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—24no—
106C2×C4×C173—24no—
107C2×C4×C173C2×D414yes—
108C22×C173—24no—
109C22×C173—24no—
110C22×C173—24no—
111C22×C173#12014yes—
112C22×C173C22×D411yescharacteristic
113C22×C173C22×D411yescharacteristic
114C22×C173C22×D411yescharacteristic
115C22×C173C2×#4114yes—
116C2×D173—232no—
117C2×D173—24no—
118C2×D173—24no—
119C2×D173—28no—
120C2×D173—28no—
121C2×D173—41no—
122C2×D173—42no—
123C2×D173—42no—
124C2×D173—42no—
125C2×D173#12318yescomplemented
126C4×C173—24no—
127C4×C173—42no—
128C4×C173—42no—
129Dic173—24no—
130Dic173—41no—
131Dic173—42no—
132C2×C173—24no—
133C2×C173C2×#12011yescharacteristic
134C2×C173C2×#12312yes—
135C2×C173#52512yes—
136C2×C173#52512yes—
137D173—28no—
138D173—81no—
139C173#284911yescharacteristic, complemented, Sylow
140#2849—1731nomaximal, Sylow
141C24×C4—1731no—
142C2×#123—1731no—
143C2×#123—1731no—
144C2×#120—1731no—
145#386—1734no—
146#385—1734no—
147#381—1731no—
148#381D17311yescharacteristic, complemented
149#379—1731no—
150C25—1731no—
151C22×D4—1731no—
152C22×D4—1732no—
153C23×C4—3464no—
154C23×C4—3464no—
155C23×C4—3464no—
156C23×C4—1734no—
157C23×C4—1731no—
158C23×C4C2×D17311yescharacteristic
159#123—3464no—
160#123—3464no—
161#120—3464no—
162C2×C4⋊C4—1732no—
163C2×C4⋊C4—1732no—
164C2×C22⋊C4—1731no—
165C2×C22⋊C4—1732no—
166C2×C22⋊C4—1732no—
167C2×C22⋊C4—1732no—
168C2×C22⋊C4—1734no—
169C2×C22⋊C4C2×D17311yescharacteristic
170C2×C22⋊C4C2×D17311yescharacteristic
171#95—1734no—
172#95C2×D17314yescomplemented
173C24—3468no—
174C24—3462no—
175C24—3462no—
176C24—1731no—
177C24—1734no—
178C24—1732no—
179C24C22×D17311yescharacteristic
180C2×D4—3464no—
181C2×D4—3468no—
182C2×D4—3462no—
183C2×D4—3462no—
184C2×D4—3462no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—34632no—
188C22×C4—3468no—
189C22×C4—3468no—
190C22×C4—3462no—
191C22×C4—3462no—
192C22×C4—3468no—
193C22×C4—1732no—
194C22×C4—1731no—
195C22×C4—1734no—
196C22×C4—1732no—
197C22×C4D69214yescomplemented
198C22×C4C22×D17312yes—
199C22×C4C22×D17312yes—
200C22×C4C22×D17312yes—
201C4⋊C4—3464no—
202C4⋊C4—3464no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—3464no—
206C22⋊C4—3468no—
207C22⋊C4—3464no—
208C22⋊C4—3462no—
209C22⋊C4—3464no—
210C23—24no—
211C23—34616no—
212C23—34616no—
213C23—34616no—
214C23—3468no—
215C23—3462no—
216C23—3462no—
217C23—3464no—
218C23—3461no—
219C23—3461no—
220C23—3461no—
221C23—3464no—
222C23—1734no—
223C23—1738no—
224C23D4×D17312yes—
225C23D4×D17312yes—
226C23C2×D69212yes—
227C23C23×D17311yescharacteristic
228D4—6928no—
229D4—6924no—
230C2×C4—28no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—34632no—
238C2×C4—3468no—
239C2×C4—3462no—
240C2×C4—3462no—
241C2×C4—3461no—
242C2×C4—3461no—
243C2×C4—3462no—
244C2×C4—3461no—
245C2×C4—3464no—
246C2×C4—1734no—
247C2×C4C2×D69214yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—34632no—
252C22—3464no—
253C22—3464no—
254C22—3468no—
255C22—3468no—
256C22—1738no—
257C22—6921no—
258C22—6922no—
259C22—6922no—
260C22—6922no—
261C22#1914964914yes—
262C22C2×D4×D17311yescharacteristic
263C22C2×D4×D17311yescharacteristic
264C22C22×D69211yescharacteristic
265C22#41×D17314yes—
266C4—24no—
267C4—42no—
268C4—42no—
269C4—3464no—
270C4—6921no—
271C4—6922no—
272C2—24no—
273C2—3468no—
274C2—13841no—
275C2C2×#1914964911yescharacteristic
276C2#123×D17312yes—
277C2#2601199912yes—
278C2#2601200312yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#381×C173C2
prime radicalC2×#19149649C2
hypercenter#381D173
hypocenterC173#2849
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×C4×D173C2semidirect
2. C2×#19149663C2semidirect
3. C2×#19149662C2semidirect
5. #26011369C2semidirect
6. #26011365C2semidirect
7. #26011352C2semidirect
8. #26011346C2semidirect
9. #381×C173C2semidirect
21. C2×C4⋊Dic173C22semidirect
22. C2×#14583805C22semidirect
26. C2×#14583803C22semidirect
27. C2×#14583803C22semidirect
28. #19149586C22semidirect
32. #95×C173C22semidirect
46. C22×Dic173D4semidirect
64. C22×C4×C173D4semidirect
125. C2×D173#123semidirect
139. C173#2849semidirect
148. #381D173semidirect
172. #95C2×D173semidirect
197. C22×C4D692semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1732
C2×#191496632
C2×#191496622
C2×#191496492
#260113692
#260113652
#260113522
#260113462
#381×C1732
#2849173