Finite Group Databasebeta

← #36362780

Subgroups — #36362780

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×#14583827C211yesmaximal, characteristic, complemented
2C2×#19149700C212yesmaximal, complemented
3#26012001C211yesmaximal, characteristic, complemented
4#26011656C212yesmaximal, complemented
5#26011647C211yesmaximal, characteristic, complemented
6#26011644C211yesmaximal, characteristic, complemented
7#26011635C212yesmaximal, complemented
8#26011624C211yesmaximal, characteristic, complemented
9#26011486C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#448×C173C211yesmaximal, characteristic, complemented
12C2×#14583827—28no—
13C2×#14583827—21no—
14C2×#14583827—21no—
15C2×#14583827C2212yes—
16C2×#14583827C2212yes—
17C2×#14583825C2211yescharacteristic, complemented
18C22×D692—21no—
19C22×D692C2211yescharacteristic
20C22×C4×D173—21no—
21C22×C4×D173C2211yescharacteristic
22#19149700—28no—
23C2×#14583816C2212yescomplemented
24C2×#14583815C2212yescomplemented
25C2×#14583811C2212yescomplemented
26C2×C8×D173C2212yescomplemented
27#19149661—21no—
28#19149659C2211yescharacteristic, complemented
29#19149658C2211yescharacteristic
30#19149650C2211yescharacteristic, complemented
31#19149647—21no—
32D4×Dic173—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic, complemented
35#19149603C2212yescomplemented
36#19149601C2212yescomplemented
37#19149599C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic
39#19149591C2212yescomplemented
40C22×Q8×C173C2211yescharacteristic
41C2×SD16×C173C2212yescomplemented
42#121×C173C2211yescharacteristic
43#103×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic
45C23×D173—21no—
46C23×D173D411yescharacteristic
47#14583827—28no—
48#14583827—28no—
49#14583827—24no—
50#14583827—24no—
51#14583827—24no—
52#14583827D418yescomplemented
53#14583825—24no—
54C2×Dic346C2311yescharacteristic
55C2×D692—28no—
56C2×D692—22no—
57C2×D692—22no—
58C2×D692—24no—
59C2×D692—21no—
60C2×D692C2312yes—
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C22×Dic173D411yescharacteristic
65C2×C4×D173—28no—
66C2×C4×D173—22no—
67C2×C4×D173—22no—
68C2×C4×D173—24no—
69C2×C4×D173D411yescharacteristic
70C2×C4×D173D411yescharacteristic
71C2×C4×D173C2311yescharacteristic
72C2×C4×D173C2311yescharacteristic
73#14583816—24no—
74#14583815—24no—
75#14583811—24no—
76C2×#13063436C2312yes—
77C8×D173—24no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83#14583803—21no—
84#14583803—21no—
85C4×Dic173—21no—
86C2×Q8×C173—24no—
87C2×Q8×C173—21no—
88C2×Q8×C173C2312yes—
89C2×D4×C173—21no—
90C2×D4×C173C2311yescharacteristic
91C22×C4×C173—21no—
92C22×C4×C173C2311yescharacteristic
93SD16×C173—24no—
94C2×C8×C173C2312yes—
95C4⋊C4×C173C2311yescharacteristic
96C22⋊C4×C173—21no—
97C22×D173—24no—
98C22×D173—22no—
99C22×D173—22no—
100C22×D173—22no—
101C22×D173—41no—
102C22×D173—42no—
103C22×D173—42no—
104C22×D173C2×D411yescharacteristic
105C22×D173C2×D411yescharacteristic
106Dic346—22no—
107D692—28no—
108D692—24no—
109D692—28no—
110D692—22no—
111D692C2×D414yes—
112C22⋊D173—42no—
113C22⋊D173—42no—
114C2×Dic173—22no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—42no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C4×D173—28no—
121C4×D173—22no—
122C4×D173—22no—
123C4×D173—28no—
124C4×D173—22no—
125C4×D173C2×D414yes—
126#13063436—22no—
127C23×C173—21no—
128C23×C173C2×D411yescharacteristic
129Q8×C173—24no—
130Q8×C173—22no—
131Q8×C173C2×D414yes—
132D4×C173—22no—
133D4×C173—42no—
134C2×C4×C173—24no—
135C2×C4×C173—22no—
136C2×C4×C173—22no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—22no—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173#13512yescomplemented
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—24no—
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—44no—
155C2×D173—48no—
156C2×D173—42no—
157C2×D173—42no—
158C2×D173#12012yes—
159C4×C173—24no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173C22×D412yes—
163Dic173—21no—
164Dic173—41no—
165Dic173—41no—
166Dic173#12012yes—
167C2×C173—22no—
168C2×C173—81no—
169C2×C173C2×#12011yescharacteristic
170C2×C173C2×#13511yescharacteristic
171C2×C173#57511yescharacteristic
172D173—22no—
173D173—41no—
174D173—44no—
175C173#300211yescharacteristic, complemented, Sylow
176#3002—1731nomaximal, Sylow
177C22×#41—1731no—
178C2×#135—1732no—
179#525—1731no—
180#452—1732no—
181#451—1731no—
182#450—1731no—
183#449—1732no—
184#448D17311yescharacteristic, complemented
185#447—1731no—
186#416—1732no—
187C2×C22⋊C8—1731no—
188C2×#41—3468no—
189C2×#41—3461no—
190C2×#41—3461no—
191C2×#41—1732no—
192C2×#41—1732no—
193C2×#41—1731no—
194C22×Q8C2×D17311yescharacteristic
195C22×D4—3461no—
196C22×D4—1731no—
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⋊Q8—1731no—
208#121—1731no—
209#121C2×D17311yescharacteristic
210#120—3461no—
211C4×D4—3461no—
212C4×D4—3461no—
213#117—1731no—
214#103—1732no—
215#103C2×D17312yescomplemented
216#102—1732no—
217#102—1732no—
218C22⋊C8—1731no—
219C22⋊C8—1732no—
220C22⋊C8C2×D17311yescharacteristic
221C24—3461no—
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—24no—
231C2×Q8—21no—
232C2×Q8—1731no—
233C2×Q8C22×D17312yes—
234C2×D4—21no—
235C2×D4—3468no—
236C2×D4—3462no—
237C2×D4—3462no—
238C2×D4—3464no—
239C2×D4—3461no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—1732no—
243C2×D4C22×D17311yescharacteristic
244C22×C4—21no—
245C22×C4—3468no—
246C22×C4—3462no—
247C22×C4—3462no—
248C22×C4—3464no—
249C22×C4—3461no—
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—3461no—
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—21no—
274C23—3464no—
275C23—3462no—
276C23—3462no—
277C23—3462no—
278C23—1731no—
279C23—1731no—
280C23—6921no—
281C23—6922no—
282C23—6922no—
283C23D4×D17311yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—3462no—
287Q8D4×D17314yes—
288D4—22no—
289D4—42no—
290D4—3468no—
291D4—3464no—
292D4—3468no—
293D4—3462no—
294D4—1734no—
295D4—6922no—
296D4—6922no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—3468no—
302C2×C4—3462no—
303C2×C4—3468no—
304C2×C4—3462no—
305C2×C4—3462no—
306C2×C4—3462no—
307C2×C4—3461no—
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—41no—
320C22—41no—
321C22—41no—
322C22—3461no—
323C22—3461no—
324C22—3461no—
325C22—3464no—
326C22—3461no—
327C22—3461no—
328C22—1732no—
329C22—6924no—
330C22—6928no—
331C22—6922no—
332C22—6922no—
333C22#1914970012yescomplemented
334C22C2×D4×D17311yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—3461no—
339C4—1732no—
340C4—6921no—
341C4—6921no—
342C4C2×D4×D17312yes—
343C2—22no—
344C2—81no—
345C2—3462no—
346C2—6921no—
347C2—6924no—
348C2#120×D17311yescharacteristic
349C2C2×#1914970011yescharacteristic
350C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
hypercenter#448D173
hypocenterC173#3002
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C2×#19149700C2semidirect
3. #26012001C2semidirect
4. #26011656C2semidirect
5. #26011647C2semidirect
6. #26011644C2semidirect
7. #26011635C2semidirect
8. #26011624C2semidirect
9. #26011486C2semidirect
10. C22⋊C8×D173C2semidirect
11. #448×C173C2semidirect
17. C2×#14583825C22semidirect
23. C2×#14583816C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583811C22semidirect
26. C2×C8×D173C22semidirect
28. #19149659C22semidirect
30. #19149650C22semidirect
34. #19149637C22semidirect
35. #19149603C22semidirect
36. #19149601C22semidirect
37. #19149599C22semidirect
39. #19149591C22semidirect
41. C2×SD16×C173C22semidirect
43. #103×C173C22semidirect
52. #14583827D4semidirect
146. C22×C173#135semidirect
175. C173#3002semidirect
184. #448D173semidirect
202. C2×SD16C2×D173semidirect
215. #103C2×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×#145838272
C2×#191497002
#260120012
#260116562
#260116472
#260116442
#260116352
#260116242
#260114862
C22⋊C8×D1732
#448×C1732
#3002173