Finite Group Databasebeta

← #36362818

Subgroups — #36362818

Order 22144.

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

normal subgroups only

This group has 754 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
2C2×#19149708C212yesmaximal, complemented
3C22⋊Q8×D173C211yesmaximal, characteristic, complemented
4#26011654C212yesmaximal, complemented
5#26011650C212yesmaximal, complemented
6#26011649C211yesmaximal, characteristic, complemented
7#26011646C211yesmaximal, characteristic, complemented
8#26011642C211yesmaximal, characteristic, complemented
9#26011489C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C2×#14583825—28no—
13C2×#14583825—21no—
14C2×#14583825—21no—
15C2×#14583825C2212yes—
16C2×#14583825C2212yes—
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19C22×C22⋊D173—21no—
20C23×Dic173—21no—
21C22×C4×D173C2211yescharacteristic
22#19149708—28no—
23C2×Dic692C2212yescomplemented
24C2×#14583816C2212yescomplemented
25C2×#14583814C2212yescomplemented
26C2×#14583809C2212yescomplemented
27#19149659C2211yescharacteristic, complemented
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149656—21no—
31#19149655C2211yescharacteristic, complemented
32C4⋊C4×D173—21no—
33C4⋊C4×D173C2211yescharacteristic, complemented
34C22⋊C4×D173—21no—
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#14583825—28no—
47#14583825—28no—
48#14583825—24no—
49#14583825—24no—
50#14583825—24no—
51#14583825D418yes—
52C2×Dic346—24no—
53C2×Dic346—21no—
54C2×Dic346—21no—
55C2×Dic346C2311yescharacteristic
56C2×Dic346C2312yes—
57Q8×D173—24no—
58C2×C22⋊D173—28no—
59C2×C22⋊D173—22no—
60C2×C22⋊D173—22no—
61C22×Dic173—28no—
62C22×Dic173—22no—
63C22×Dic173—22no—
64C22×Dic173—21no—
65C22×Dic173D411yescharacteristic
66C2×C4×D173—24no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173D411yescharacteristic
70C2×C4×D173D411yescharacteristic
71C2×C4×D173C2311yescharacteristic
72C2×C4×D173C2311yescharacteristic
73Dic692—24no—
74#14583816—24no—
75#14583814—24no—
76#14583809—24no—
77C2×#13063436C2312yes—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—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—41no—
100C22×D173C2×D411yescharacteristic
101C22×D173C2×D411yescharacteristic
102Dic346—24no—
103Dic346—22no—
104Dic346—22no—
105Dic346—42no—
106Dic346C2×D414yes—
107C22⋊D173—28no—
108C22⋊D173—28no—
109C2×Dic173—24no—
110C2×Dic173—28no—
111C2×Dic173—22no—
112C2×Dic173—22no—
113C2×Dic173—28no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C4×D173—22no—
119C4×D173—22no—
120C4×D173—22no—
121C4×D173—42no—
122C4×D173—42no—
123C4×D173C2×D414yes—
124#13063436—22no—
125C23×C173—24no—
126C23×C173—22no—
127C23×C173—42no—
128C23×C173—42no—
129C23×C173C2×D411yescharacteristic
130Q8×C173—22no—
131D4×C173—24no—
132D4×C173—22no—
133D4×C173C2×D414yes—
134C2×C4×C173—22no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2411yescharacteristic
139C8×C173—22no—
140C22×C173—21no—
141C22×C173—21no—
142C22×C173—24no—
143C22×C173—48no—
144C22×C173—42no—
145C22×C173—42no—
146C22×C173#13712yescomplemented
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—44no—
154C2×D173#12012yes—
155C4×C173—21no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173C22×D412yes—
159Dic173—21no—
160Dic173—24no—
161Dic173—41no—
162Dic173—41no—
163Dic173#12012yes—
164C2×C173—22no—
165C2×C173—44no—
166C2×C173C2×#12011yescharacteristic
167C2×C173C2×#13711yescharacteristic
168C2×C173#57511yescharacteristic
169D173—22no—
170D173—41no—
171C173#300711yescharacteristic, complemented, Sylow
172#3007—1731nomaximal, Sylow
173C22×#41—1731no—
174C2×#137—1732no—
175C2×C22⋊Q8—1731no—
176#452—1732no—
177#452—1732no—
178#451—1731no—
179#451—1731no—
180#450—1731no—
181#450D17311yescharacteristic, complemented
182#417—1732no—
183#407—1731no—
184C2×#41—3468no—
185C2×#41—3461no—
186C2×#41—3461no—
187C2×#41—1732no—
188C2×#41—1732no—
189C22×Q8—1731no—
190C22×Q8—1731no—
191C22×D4—3461no—
192C22×D4C2×D17311yescharacteristic
193C23×C4—3461no—
194C23×C4—1731no—
195#137—3468no—
196C2×Q16—1732no—
197C2×Q16—1732no—
198C2×SD16—1732no—
199C2×SD16C2×D17312yescomplemented
200C2×M16—1732no—
201C22⋊Q8—3461no—
202C22⋊Q8—3461no—
203C22⋊Q8—1731no—
204C22⋊Q8—1731no—
205C22⋊Q8—1731no—
206C22⋊Q8C2×D17311yescharacteristic
207C2×C4⋊C4—3461no—
208C2×C4⋊C4—1731no—
209C2×C22⋊C4—3461no—
210#103—1732no—
211#103—1732no—
212#102—1732no—
213#102C2×D17312yescomplemented
214C22⋊C8—1731no—
215C22⋊C8—1732no—
216C22⋊C8C2×D17311yescharacteristic
217C24—21no—
218C24—1731no—
219#41—3468no—
220#41—3468no—
221#41—3464no—
222#41—3464no—
223#41—3464no—
224#41—1738no—
225C2×Q8—3464no—
226C2×Q8—3461no—
227C2×Q8—3461no—
228C2×Q8—3464no—
229C2×Q8—1731no—
230C2×Q8—1732no—
231C2×Q8C22×D17311yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—3468no—
235C2×D4—3462no—
236C2×D4—3462no—
237C2×D4C22×D17312yes—
238C22×C4—3464no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—3468no—
242C22×C4—3461no—
243C22×C4—3462no—
244C22×C4—3462no—
245C22×C4—1731no—
246C22×C4—1731no—
247C22×C4—1731no—
248C22×C4—1731no—
249C22×C4—1731no—
250C22×C4C22×D17311yescharacteristic
251Q16—3464no—
252Q16—3464no—
253SD16—24no—
254SD16—3464no—
255M16—3464no—
256C2×C8—1732no—
257C2×C8C22×D17312yes—
258C4⋊C4—21no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—3462no—
274C23—3462no—
275C23—1731no—
276C23—1731no—
277C23—6921no—
278C23D4×D17311yescharacteristic
279Q8—22no—
280Q8—3464no—
281Q8—3462no—
282Q8—3462no—
283Q8—1734no—
284Q8—6922no—
285D4—24no—
286D4—22no—
287D4—3468no—
288D4—3468no—
289D4D4×D17314yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—3462no—
294C2×C4—3462no—
295C2×C4—3464no—
296C2×C4—3468no—
297C2×C4—3462no—
298C2×C4—3462no—
299C2×C4—3468no—
300C2×C4—3462no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—1734no—
306C2×C4—6922no—
307C2×C4—6922no—
308C2×C4D4×D17311yescharacteristic
309C2×C4C23×D17311yescharacteristic
310C8—22no—
311C8—3462no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—3461no—
319C22—3461no—
320C22—3461no—
321C22—3461no—
322C22—3461no—
323C22—1732no—
324C22—6924no—
325C22#1914970812yescomplemented
326C22C2×D4×D17311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—3461no—
331C4—3464no—
332C4—1732no—
333C4—6921no—
334C4—6921no—
335C4C2×D4×D17312yes—
336C2—22no—
337C2—44no—
338C2—3462no—
339C2—6921no—
340C2#120×D17311yescharacteristic
341C2C2×#1914970811yescharacteristic
342C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC22×#14583825C2
hypercenter#450D173
hypocenterC173#3007
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
2. C2×#19149708C2semidirect
3. C22⋊Q8×D173C2semidirect
4. #26011654C2semidirect
5. #26011650C2semidirect
6. #26011649C2semidirect
7. #26011646C2semidirect
8. #26011642C2semidirect
9. #26011489C2semidirect
10. #26011453C2semidirect
11. #450×C173C2semidirect
18. C2×Q8×D173C22semidirect
23. C2×Dic692C22semidirect
24. C2×#14583816C22semidirect
25. C2×#14583814C22semidirect
26. C2×#14583809C22semidirect
27. #19149659C22semidirect
31. #19149655C22semidirect
33. C4⋊C4×D173C22semidirect
35. #19149604C22semidirect
36. #19149602C22semidirect
37. #19149600C22semidirect
39. #19149591C22semidirect
41. C2×SD16×C173C22semidirect
43. #102×C173C22semidirect
146. C22×C173#137semidirect
171. C173#3007semidirect
181. #450D173semidirect
199. C2×SD16C2×D173semidirect
213. #102C2×D173semidirect
325. C22#19149708semidirect

Maximal subgroups

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

subgroupindex
C22×#145838252
C2×#191497082
C22⋊Q8×D1732
#260116542
#260116502
#260116492
#260116462
#260116422
#260114892
#260114532
#450×C1732
#3007173