Finite Group Databasebeta

← #36361852

Subgroups — #36361852

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×#19149660C211yesmaximal, characteristic, complemented
3C2×#19149647C211yesmaximal, characteristic
4#26011367C214yesmaximal, complemented
5#26011362C214yesmaximal, complemented
6#26011354C211yesmaximal, characteristic, complemented
7#26011352C211yesmaximal, characteristic, complemented
8#26011346C211yesmaximal, characteristic, complemented
9C2×#123×C173C211yesmaximal, characteristic, complemented
10C24×D173C2211yescharacteristic
11C22×C22⋊D173C2212yes—
12C23×Dic173C2211yescharacteristic
13C22×C4×D173—24no—
14C22×C4×D173—24no—
15C22×C4×D173—24no—
16C22×C4×D173C2214yes—
17#19149660—24no—
18#19149647—24no—
19C2×#14583805C2212yescomplemented
20C2×C22⋊Dic173C2211yescharacteristic
21C2×C22⋊Dic173C2211yescharacteristic
22C2×C22⋊Dic173C2212yescomplemented
23C2×#14583803C2212yes—
24C2×#14583803C2214yes—
25#19149586C2214yescomplemented
26#19149586C2214yescomplemented
27C22×D4×C173C2211yescharacteristic
28C23×C4×C173C2211yescharacteristic
29#123×C173—24no—
30C2×C4⋊C4×C173C2212yescomplemented
31C2×C22⋊C4×C173C2211yescharacteristic
32C2×C22⋊C4×C173C2212yescomplemented
33C23×D173—28no—
34C23×D173—22no—
35C23×D173—22no—
36C23×D173D414yes—
37C23×D173C2312yes—
38C2×C22⋊D173—24no—
39C2×C22⋊D173—28no—
40C22×Dic173—22no—
41C22×Dic173—22no—
42C22×Dic173D414yescomplemented
43C22×Dic173C2312yes—
44C22×Dic173C2312yes—
45C22×Dic173C2312yes—
46C2×C4×D173—232no—
47C2×C4×D173—28no—
48C2×C4×D173—28no—
49C2×C4×D173—28no—
50#14583805—24no—
51C22⋊Dic173—24no—
52C22⋊Dic173—22no—
53C22⋊Dic173—22no—
54#14583803—24no—
55#14583803—28no—
56C24×C173C2311yescharacteristic
57C24×C173C2311yescharacteristic
58C2×D4×C173—22no—
59C2×D4×C173—22no—
60C2×D4×C173—22no—
61C22×C4×C173—22no—
62C22×C4×C173—22no—
63C22×C4×C173D414yescomplemented
64C22×C4×C173C2311yescharacteristic
65C22×C4×C173C2312yes—
66C22×C4×C173C2312yes—
67C4⋊C4×C173—24no—
68C22⋊C4×C173—22no—
69C22⋊C4×C173—24no—
70C22×D173—216no—
71C22×D173—216no—
72C22×D173—28no—
73C22×D173—22no—
74C22×D173—22no—
75C22×D173—24no—
76C22×D173—216no—
77C22×D173C2×D414yes—
78C22×D173#4118yes—
79C22⋊D173—48no—
80C2×Dic173—28no—
81C2×Dic173—22no—
82C2×Dic173—22no—
83C2×Dic173—22no—
84C2×Dic173—22no—
85C2×Dic173—24no—
86C2×Dic173—24no—
87C2×Dic173C2×D414yes—
88C4×D173—232no—
89C23×C173—24no—
90C23×C173—21no—
91C23×C173—21no—
92C23×C173—21no—
93C23×C173—24no—
94C23×C173C2×D412yes—
95C23×C173C2×D412yes—
96C23×C173C2×D412yes—
97C23×C173C2411yescharacteristic
98D4×C173—44no—
99C2×C4×C173—28no—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—21no—
104C2×C4×C173—21no—
105C2×C4×C173—21no—
106C2×C4×C173—24no—
107C2×C4×C173C2×D414yes—
108C22×C173—24no—
109C22×C173—24no—
110C22×C173—24no—
111C22×C173—41no—
112C22×C173—42no—
113C22×C173—42no—
114C22×C173—42no—
115C22×C173#12014yes—
116C22×C173C22×D411yescharacteristic
117C22×C173C22×D411yescharacteristic
118C22×C173C22×D411yescharacteristic
119C22×C173C2×#4114yes—
120C2×D173—232no—
121C2×D173—24no—
122C2×D173—24no—
123C2×D173—28no—
124C2×D173—28no—
125C2×D173#12318yescomplemented
126C4×C173—24no—
127C4×C173—41no—
128C4×C173—42no—
129Dic173—24no—
130Dic173—42no—
131Dic173—42no—
132C2×C173—24no—
133C2×C173—81no—
134C2×C173C2×#12011yescharacteristic
135C2×C173C2×#12312yes—
136C2×C173#52512yes—
137C2×C173#52512yes—
138D173—28no—
139C173#284911yescharacteristic, complemented, Sylow
140#2849—1731nomaximal, Sylow
141C24×C4—1731no—
142C2×#123—1731no—
143C2×#123D17311yescharacteristic, complemented
144C2×#120—1731no—
145#386—1734no—
146#385—1734no—
147#381—1731no—
148#381—1731no—
149#379—1731no—
150C25—1731no—
151C22×D4—1732no—
152C22×D4C2×D17311yescharacteristic
153C23×C4—3464no—
154C23×C4—3464no—
155C23×C4—3464no—
156C23×C4—1734no—
157C23×C4—1731no—
158C23×C4C2×D17311yescharacteristic
159#123—24no—
160#123—3464no—
161#120—3464no—
162C2×C4⋊C4—1732no—
163C2×C4⋊C4C2×D17312yescomplemented
164C2×C22⋊C4—1731no—
165C2×C22⋊C4—1731no—
166C2×C22⋊C4—1732no—
167C2×C22⋊C4—1732no—
168C2×C22⋊C4—1734no—
169C2×C22⋊C4C2×D17311yescharacteristic
170C2×C22⋊C4C2×D17312yescomplemented
171#95—1734no—
172#95—1734no—
173C24—3468no—
174C24—3462no—
175C24—3462no—
176C24—1734no—
177C24—1732no—
178C24C22×D17311yescharacteristic
179C24C22×D17311yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—3464no—
184C2×D4—3468no—
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—1732no—
195C22×C4—1734no—
196C22×C4—1732no—
197C22×C4C22⋊D17314yescomplemented
198C22×C4C22×D17312yes—
199C22×C4C22×D17311yescharacteristic
200C22×C4C22×D17312yes—
201C4⋊C4—24no—
202C4⋊C4—3464no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—3464no—
206C22⋊C4—3468no—
207C22⋊C4—3462no—
208C22⋊C4—3462no—
209C22⋊C4—3464no—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—34616no—
216C23—34616no—
217C23—34616no—
218C23—3468no—
219C23—3462no—
220C23—3462no—
221C23—3464no—
222C23—1734no—
223C23—1738no—
224C23D4×D17312yes—
225C23D4×D17312yes—
226C23C2×C22⋊D17312yes—
227C23C23×D17311yescharacteristic
228D4—44no—
229D4—6928no—
230C2×C4—28no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—34632no—
239C2×C4—3468no—
240C2×C4—3462no—
241C2×C4—3462no—
242C2×C4—3462no—
243C2×C4—3462no—
244C2×C4—3464no—
245C2×C4—3464no—
246C2×C4—1734no—
247C2×C4C2×C22⋊D17314yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—34632no—
256C22—3464no—
257C22—3464no—
258C22—3468no—
259C22—3468no—
260C22—1738no—
261C22#1914964714yes—
262C22C2×D4×D17311yescharacteristic
263C22C2×D4×D17311yescharacteristic
264C22C22×C22⋊D17311yescharacteristic
265C22#41×D17314yes—
266C4—24no—
267C4—41no—
268C4—42no—
269C4—3464no—
270C4—6922no—
271C4—6922no—
272C2—24no—
273C2—81no—
274C2—3468no—
275C2C2×#1914964711yescharacteristic
276C2#123×D17312yes—
277C2#2601199812yes—
278C2#2601199912yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
FittingC2×#123×C173C2
prime radicalC2×#19149647C2
hypercenterC2×#123D173
hypocenterC173#2849
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×D173C2semidirect
2. C2×#19149660C2semidirect
4. #26011367C2semidirect
5. #26011362C2semidirect
6. #26011354C2semidirect
7. #26011352C2semidirect
8. #26011346C2semidirect
9. C2×#123×C173C2semidirect
19. C2×#14583805C22semidirect
22. C2×C22⋊Dic173C22semidirect
25. #19149586C22semidirect
26. #19149586C22semidirect
30. C2×C4⋊C4×C173C22semidirect
32. C2×C22⋊C4×C173C22semidirect
42. C22×Dic173D4semidirect
63. C22×C4×C173D4semidirect
125. C2×D173#123semidirect
139. C173#2849semidirect
143. C2×#123D173semidirect
163. C2×C4⋊C4C2×D173semidirect
170. C2×C22⋊C4C2×D173semidirect
197. C22×C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1732
C2×#191496602
C2×#191496472
#260113672
#260113622
#260113542
#260113522
#260113462
C2×#123×C1732
#2849173