Finite Group Databasebeta

← #36362821

Subgroups — #36362821

Order 22144.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#14583826C211yesmaximal, characteristic
2C2×#19149710C212yesmaximal, complemented
3#26011651C214yesmaximal, complemented
4#26011646C212yesmaximal, complemented
5#26011641C212yesmaximal, complemented
6#26011490C212yesmaximal, complemented
7#26011455C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C173C211yesmaximal, characteristic, complemented
9C2×#14583826—21no—
10C2×#14583826—21no—
11C2×#14583826—28no—
12C2×#14583826C2212yes—
13C2×#14583826C2212yes—
14C22×Dic346C2211yescharacteristic
15C22×D692C2211yescharacteristic
16C22×C22⋊D173—21no—
17C22×C4×D173—21no—
18#19149710—28no—
19C2×#14583816C2214yescomplemented
20C2×#14583813C2214yescomplemented
21C2×#14583810C2212yescomplemented
22#19149602C2214yescomplemented
23#19149599C2214yescomplemented
24C22⋊#13063436C2212yes—
25C22⋊#13063436C2212yescomplemented
26C22×Q8×C173C2211yescharacteristic, complemented
27C23×C4×C173C2211yescharacteristic
28C22⋊Q8×C173—21no—
29C22⋊Q8×C173—21no—
30C22⋊Q8×C173C2212yes—
31C22⋊Q8×C173C2212yescomplemented
32C2×C4⋊C4×C173—21no—
33C2×C4⋊C4×C173C2211yescharacteristic, complemented
34C2×C22⋊C4×C173—21no—
35C23×D173—21no—
36#14583826—28no—
37#14583826—28no—
38#14583826—24no—
39#14583826—24no—
40#14583826—24no—
41#14583826D418yes—
42C2×Dic346—24no—
43C2×Dic346—21no—
44C2×Dic346C2312yes—
45C2×D692—24no—
46C2×D692—21no—
47C2×D692C2312yes—
48C2×C22⋊D173—28no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—22no—
51C22×Dic173—21no—
52C2×C4×D173—28no—
53C2×C4×D173—22no—
54C2×C4×D173—22no—
55#14583816—28no—
56#14583813—28no—
57#14583810—24no—
58C2×#13063436C2314yes—
59C24×C173D411yescharacteristic
60C2×Q8×C173—21no—
61C2×Q8×C173—24no—
62C2×Q8×C173C2312yes—
63C22×C4×C173—24no—
64C22×C4×C173—21no—
65C22×C4×C173—21no—
66C22×C4×C173D411yescharacteristic
67C22×C4×C173D411yescharacteristic
68C22×C4×C173D411yescharacteristic
69C22×C4×C173C2311yescharacteristic
70C22×C4×C173C2311yescharacteristic
71C22×C4×C173C2311yescharacteristic
72C4⋊C4×C173—22no—
73C4⋊C4×C173—22no—
74C4⋊C4×C173—21no—
75C4⋊C4×C173C2312yes—
76C22⋊C4×C173—22no—
77C22⋊C4×C173—22no—
78C22×D173—24no—
79C22×D173—22no—
80C22×D173—42no—
81C22×D173—42no—
82Dic346—24no—
83Dic346—22no—
84Dic346C2×D414yes—
85D692—24no—
86D692—22no—
87D692C2×D414yes—
88C22⋊D173—28no—
89C22⋊D173—28no—
90C2×Dic173—24no—
91C2×Dic173—22no—
92C4×D173—28no—
93C4×D173—28no—
94#13063436—24no—
95C23×C173—22no—
96C23×C173—22no—
97C23×C173—41no—
98C23×C173C2×D411yescharacteristic
99C23×C173C2×D411yescharacteristic
100C23×C173C2×D411yescharacteristic
101Q8×C173—24no—
102Q8×C173—42no—
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—22no—
110C2×C4×C173—42no—
111C2×C4×C173—42no—
112C2×C4×C173C2×D414yes—
113C2×C4×C173C2×D411yescharacteristic
114C2×C4×C173C2×D411yescharacteristic
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2411yescharacteristic
117C22×C173—21no—
118C22×C173—21no—
119C22×C173—21no—
120C22×C173—21no—
121C22×C173—21no—
122C22×C173—21no—
123C22×C173—21no—
124C22×C173—44no—
125C22×C173#12012yes—
126C22×C173#13712yescomplemented
127C22×C173C22×D411yescharacteristic
128C2×D173—24no—
129C2×D173—48no—
130C2×D173—42no—
131C2×D173—42no—
132C4×C173—21no—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173—42no—
136C4×C173#12012yes—
137C4×C173C22×D412yes—
138Dic173—24no—
139C2×C173—22no—
140C2×C173—22no—
141C2×C173—41no—
142C2×C173C2×#12011yescharacteristic
143C2×C173C2×#13711yescharacteristic
144C2×C173#57511yescharacteristic
145D173—44no—
146C173#300711yescharacteristic, complemented, Sylow
147#3007—1731nomaximal, Sylow
148C22×#41—1731no—
149C2×#137—1732no—
150C2×C22⋊Q8D17311yescharacteristic, complemented
151#452—1734no—
152#451—1732no—
153#450—1732no—
154#417—1732no—
155#407—1731no—
156C2×#41—3468no—
157C2×#41—3461no—
158C2×#41—3461no—
159C2×#41—1732no—
160C2×#41—1732no—
161C22×Q8—1731no—
162C22×Q8C2×D17311yescharacteristic, complemented
163C22×D4—3461no—
164C22×D4—1731no—
165C23×C4—3461no—
166C23×C4C2×D17311yescharacteristic
167#137—3468no—
168C2×Q16—1734no—
169C2×SD16—1734no—
170C2×M16—1732no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D17312yes—
174C22⋊Q8C2×D17312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D17311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1734no—
179#102—1734no—
180C22⋊C8—1732no—
181C22⋊C8—1732no—
182C24—3461no—
183C24C22⋊D17311yescharacteristic
184#41—3468no—
185#41—3468no—
186#41—3464no—
187#41—3464no—
188#41—3464no—
189#41—1738no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—3464no—
193C2×Q8—3461no—
194C2×Q8—1732no—
195C2×Q8C22×D17312yes—
196C2×D4—3464no—
197C2×D4—3468no—
198C2×D4—3462no—
199C2×D4—3462no—
200C2×D4—3461no—
201C2×D4—1732no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—3468no—
206C22×C4—3461no—
207C22×C4—3462no—
208C22×C4—3462no—
209C22×C4C22⋊D17311yescharacteristic
210C22×C4C22⋊D17311yescharacteristic
211C22×C4C22⋊D17311yescharacteristic
212C22×C4C22×D17311yescharacteristic
213C22×C4C22×D17311yescharacteristic
214C22×C4C22×D17311yescharacteristic
215Q16—3468no—
216SD16—3468no—
217M16—3464no—
218C2×C8—1734no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D17312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3464no—
229C23—3462no—
230C23—6922no—
231C23—6922no—
232C23C2×C22⋊D17311yescharacteristic
233C23C2×C22⋊D17311yescharacteristic
234C23C2×C22⋊D17311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—3464no—
238Q8—3462no—
239Q8—1734no—
240D4—3464no—
241D4—3468no—
242D4—3468no—
243D4—3462no—
244D4—1734no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—3468no—
255C2×C4—3464no—
256C2×C4—3462no—
257C2×C4—3468no—
258C2×C4D4×D17314yes—
259C2×C4C2×C22⋊D17311yescharacteristic
260C2×C4C2×C22⋊D17311yescharacteristic
261C2×C4C2×C22⋊D17311yescharacteristic
262C2×C4C23×D17311yescharacteristic
263C8—3464no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—3464no—
273C22—6928no—
274C22—6922no—
275C22—6922no—
276C22#1914964712yes—
277C22#1914971012yescomplemented
278C22C22×C22⋊D17311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—3464no—
284C4#1914964712yes—
285C4C2×D4×D17312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—6924no—
290C2C2×#1914964711yescharacteristic
291C2C2×#1914971011yescharacteristic
292C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×C22⋊Q8×C173C2
socleC22×C173C22×D4
prime radicalC22×#14583826C2
hypercenterC2×C22⋊Q8D173
hypocenterC173#3007
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19149710C2semidirect
3. #26011651C2semidirect
4. #26011646C2semidirect
5. #26011641C2semidirect
6. #26011490C2semidirect
7. #26011455C2semidirect
8. C2×C22⋊Q8×C173C2semidirect
19. C2×#14583816C22semidirect
20. C2×#14583813C22semidirect
21. C2×#14583810C22semidirect
22. #19149602C22semidirect
23. #19149599C22semidirect
25. C22⋊#13063436C22semidirect
26. C22×Q8×C173C22semidirect
31. C22⋊Q8×C173C22semidirect
33. C2×C4⋊C4×C173C22semidirect
126. C22×C173#137semidirect
146. C173#3007semidirect
150. C2×C22⋊Q8D173semidirect
162. C22×Q8C2×D173semidirect
174. C22⋊Q8C2×D173semidirect
176. C2×C4⋊C4C2×D173semidirect
277. C22#19149710semidirect

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191497102
#260116512
#260116462
#260116412
#260114902
#260114552
C2×C22⋊Q8×C1732
#3007173