Finite Group Databasebeta

← #36362782

Subgroups — #36362782

Order 22144.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14583825C211yesmaximal, characteristic, complemented
2C2×#19149700C212yesmaximal, complemented
3#26012004C211yesmaximal, characteristic, complemented
4#26011650C212yesmaximal, complemented
5#26011646C211yesmaximal, characteristic, complemented
6#26011637C212yesmaximal, complemented
7#26011632C211yesmaximal, characteristic, complemented
8#26011625C211yesmaximal, characteristic, complemented
9#26011484C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic, complemented
13C2×#14583825—28no—
14C2×#14583825—21no—
15C2×#14583825—21no—
16C2×#14583825C2212yes—
17C2×#14583825C2212yes—
18C22×Dic346C2211yescharacteristic
19C22×C22⋊D173—21no—
20C23×Dic173—21no—
21C22×C4×D173C2211yescharacteristic
22#19149700—28no—
23C2×#14583816C2212yescomplemented
24C2×#14583815C2212yescomplemented
25C2×#14583811C2212yescomplemented
26C2×C8×D173C2212yescomplemented
27#19149664—21no—
28#19149655C2211yescharacteristic, complemented
29#19149654C2211yescharacteristic, complemented
30#19149653C2211yescharacteristic
31#19149649—21no—
32#19149641—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic, complemented
35#19149604C2212yescomplemented
36#19149602C2212yescomplemented
37#19149600C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic
39#19149591C2212yescomplemented
40C22×D4×C173C2211yescharacteristic
41C2×SD16×C173C2212yescomplemented
42C22⋊Q8×C173C2211yescharacteristic
43#102×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic
45C23×D173D411yescharacteristic
46#14583827—24no—
47#14583825—28no—
48#14583825—28no—
49#14583825—24no—
50#14583825—24no—
51#14583825—24no—
52#14583825D418yescomplemented
53C2×Dic346—24no—
54C2×Dic346—21no—
55C2×Dic346C2312yes—
56C2×D692—21no—
57C2×D692—21no—
58C2×D692C2311yescharacteristic
59C2×C22⋊D173—28no—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—22no—
62C2×C22⋊D173—21no—
63C22×Dic173—28no—
64C22×Dic173—22no—
65C22×Dic173—22no—
66C22×Dic173—21no—
67C22×Dic173D411yescharacteristic
68C2×C4×D173—24no—
69C2×C4×D173—21no—
70C2×C4×D173D411yescharacteristic
71C2×C4×D173D411yescharacteristic
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583816—24no—
75#14583815—24no—
76#14583811—24no—
77C2×#13063436C2312yes—
78C8×D173—24no—
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583803—21no—
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85C4×Dic173—21no—
86C24×C173—21no—
87C2×Q8×C173C2311yescharacteristic
88C2×D4×C173—24no—
89C2×D4×C173—21no—
90C2×D4×C173C2312yes—
91C22×C4×C173C2311yescharacteristic
92SD16×C173—24no—
93C2×C8×C173C2312yes—
94C4⋊C4×C173—21no—
95C4⋊C4×C173C2311yescharacteristic
96C22⋊C4×C173—21no—
97C22×D173—22no—
98C22×D173—22no—
99C22×D173—21no—
100C22×D173—41no—
101C22×D173C2×D411yescharacteristic
102C22×D173C2×D411yescharacteristic
103Dic346—24no—
104Dic346—22no—
105Dic346C2×D414yes—
106D692—22no—
107D692—42no—
108D692—42no—
109C22⋊D173—28no—
110C22⋊D173—28no—
111C22⋊D173—42no—
112C2×Dic173—24no—
113C2×Dic173—28no—
114C2×Dic173—22no—
115C2×Dic173—22no—
116C2×Dic173—28no—
117C2×Dic173—21no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C4×D173—22no—
121C4×D173—22no—
122C4×D173—22no—
123C4×D173—42no—
124C4×D173C2×D414yes—
125#13063436—22no—
126C23×C173—24no—
127C23×C173—22no—
128C23×C173—42no—
129C23×C173—42no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—22no—
132D4×C173—24no—
133D4×C173—22no—
134D4×C173C2×D414yes—
135C2×C4×C173—22no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—22no—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—24no—
144C22×C173—48no—
145C22×C173—42no—
146C22×C173—42no—
147C22×C173#13512yescomplemented
148C22×C173C22×D411yescharacteristic
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—44no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173#12012yes—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C22×D412yes—
163Dic173—21no—
164Dic173—24no—
165Dic173—41no—
166Dic173#12012yes—
167C2×C173—22no—
168C2×C173—44no—
169C2×C173C2×#12011yescharacteristic
170C2×C173C2×#13511yescharacteristic
171C2×C173#57511yescharacteristic
172D173—22no—
173D173—41no—
174D173—81no—
175C173#300211yescharacteristic, complemented, Sylow
176#3002—1731nomaximal, Sylow
177C22×#41—1731no—
178C2×#135—1732no—
179#525—1731no—
180#452—1732no—
181#451—1731no—
182#450D17311yescharacteristic, complemented
183#449—1732no—
184#448—1731no—
185#447—1731no—
186#416—1732no—
187C2×C22⋊C8—1731no—
188C2×#41—3468no—
189C2×#41—3461no—
190C2×#41—3461no—
191C2×#41—1731no—
192C2×#41—1732no—
193C2×#41—1732no—
194C22×Q8—1731no—
195C22×D4—3461no—
196C22×D4C2×D17311yescharacteristic
197C23×C4—3461no—
198C23×C4—1731no—
199#135—3468no—
200C2×Q16—1732no—
201C2×SD16—1732no—
202C2×SD16C2×D17312yescomplemented
203C2×D8—1732no—
204C22×C8—1732no—
205#123—3461no—
206C22⋊Q8—1731no—
207C22⋊Q8C2×D17311yescharacteristic
208#121—1731no—
209#121—1731no—
210#120—3461no—
211C4×D4—3461no—
212C4×D4—3461no—
213#117—1731no—
214#103—1732no—
215#103—1732no—
216#102—1732no—
217#102C2×D17312yescomplemented
218C22⋊C8—1731no—
219C22⋊C8—1732no—
220C22⋊C8C2×D17311yescharacteristic
221C24—21no—
222C24—1731no—
223#41—3468no—
224#41—3468no—
225#41—3464no—
226#41—3464no—
227#41—3464no—
228#41—3464no—
229#41—1738no—
230C2×Q8—3464no—
231C2×Q8—3461no—
232C2×Q8—1732no—
233C2×Q8C22×D17311yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—3461no—
237C2×D4—3461no—
238C2×D4—3468no—
239C2×D4—3462no—
240C2×D4—3462no—
241C2×D4—3461no—
242C2×D4—1731no—
243C2×D4C22×D17312yes—
244C22×C4—3464no—
245C22×C4—3461no—
246C22×C4—3468no—
247C22×C4—3461no—
248C22×C4—3462no—
249C22×C4—3462no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253C22×C4—1731no—
254C22×C4—1731no—
255C22×C4C22×D17311yescharacteristic
256Q16—3464no—
257SD16—24no—
258SD16—3464no—
259D8—3464no—
260C2×C8—3464no—
261C2×C8—1732no—
262C2×C8C22×D17312yes—
263C4⋊C4—21no—
264C4⋊C4—3461no—
265C4⋊C4—1731no—
266C4⋊C4C22×D17311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C42—3461no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—3462no—
278C23—3462no—
279C23—3461no—
280C23—1731no—
281C23—1731no—
282C23—6921no—
283C23D4×D17311yescharacteristic
284Q8—22no—
285Q8—3464no—
286Q8—3462no—
287Q8—1734no—
288D4—24no—
289D4—22no—
290D4—3468no—
291D4—3468no—
292D4—3462no—
293D4—6922no—
294D4—6922no—
295D4—6922no—
296D4D4×D17314yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3462no—
301C2×C4—3462no—
302C2×C4—3464no—
303C2×C4—3468no—
304C2×C4—3462no—
305C2×C4—3462no—
306C2×C4—3468no—
307C2×C4—3462no—
308C2×C4—3461no—
309C2×C4—1731no—
310C2×C4—1731no—
311C2×C4—1734no—
312C2×C4—6922no—
313C2×C4D4×D17311yescharacteristic
314C2×C4C23×D17311yescharacteristic
315C8—22no—
316C8—3462no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—3461no—
324C22—3461no—
325C22—3461no—
326C22—3461no—
327C22—3461no—
328C22—1732no—
329C22—6924no—
330C22—6921no—
331C22—6921no—
332C22—6921no—
333C22#1914970012yescomplemented
334C22C2×D4×D17311yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—3461no—
339C4—3464no—
340C4—1732no—
341C4—6921no—
342C4C2×D4×D17312yes—
343C2—22no—
344C2—44no—
345C2—3462no—
346C2—13841no—
347C2—6921no—
348C2#120×D17311yescharacteristic
349C2C2×#1914970011yescharacteristic
350C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3002
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14583825C2semidirect
2. C2×#19149700C2semidirect
3. #26012004C2semidirect
4. #26011650C2semidirect
5. #26011646C2semidirect
6. #26011637C2semidirect
7. #26011632C2semidirect
8. #26011625C2semidirect
9. #26011484C2semidirect
10. C22⋊C8×D173C2semidirect
11. #450×C173C2semidirect
12. C2×#14583827C22semidirect
23. C2×#14583816C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583811C22semidirect
26. C2×C8×D173C22semidirect
28. #19149655C22semidirect
29. #19149654C22semidirect
34. #19149637C22semidirect
35. #19149604C22semidirect
36. #19149602C22semidirect
37. #19149600C22semidirect
39. #19149591C22semidirect
41. C2×SD16×C173C22semidirect
43. #102×C173C22semidirect
52. #14583825D4semidirect
147. C22×C173#135semidirect
175. C173#3002semidirect
182. #450D173semidirect
202. C2×SD16C2×D173semidirect
217. #102C2×D173semidirect
333. C22#19149700semidirect

Maximal subgroups

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

subgroupindex
C22×#145838252
C2×#191497002
#260120042
#260116502
#260116462
#260116372
#260116322
#260116252
#260114842
C22⋊C8×D1732
#450×C1732
#3002173