Finite Group Databasebeta

← #36364319

Subgroups — #36364319

Order 22144.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊Q8×D173C211yesmaximal, characteristic, complemented
2C2×#19149654C211yesmaximal, characteristic
3#26011821C211yesmaximal, characteristic, complemented
4#26011811C212yesmaximal, complemented
5#26011723C212yesmaximal, complemented
6#26011721C212yesmaximal, complemented
7#26011644C211yesmaximal, characteristic, complemented
8#26011641C211yesmaximal, characteristic, complemented
9#26011631C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#483×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic, complemented
13C22×D692—21no—
14C22×D692C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16C2×#14583815C2212yescomplemented
17C2×#14583813C2212yescomplemented
18C2×C8×D173C2212yescomplemented
19#19149659C2211yescharacteristic, complemented
20#19149658C2211yescharacteristic
21#19149657—21no—
22#19149656—21no—
23#19149655C2211yescharacteristic, complemented
24#19149654—21no—
25#19149654—21no—
26#19149654C2212yes—
27#19149654C2212yes—
28C4⋊C4×D173—21no—
29C4⋊C4×D173C2211yescharacteristic, complemented
30C2×#14583803—21no—
31C22⋊C4×D173—21no—
32#19149610C2212yescomplemented
33#19149602C2212yescomplemented
34#19149601C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yescomplemented
38C22⋊Q8×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#106×C173C2212yescomplemented
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173—21no—
44C23×D173D411yescharacteristic
45C2×Dic346—21no—
46C2×Dic346C2311yescharacteristic
47Q8×D173—42no—
48C2×D692—22no—
49C2×D692—22no—
50C2×D692—21no—
51C2×D692—42no—
52C2×D692—44no—
53C2×D692C2312yes—
54C22×Dic173D411yescharacteristic
55C2×C4×D173—21no—
56C2×C4×D173—21no—
57C2×C4×D173—42no—
58C2×C4×D173D411yescharacteristic
59C2×C4×D173D411yescharacteristic
60C2×C4×D173C2311yescharacteristic
61C2×C4×D173C2311yescharacteristic
62#14583815—42no—
63#14583813—42no—
64C2×#13063436C2312yes—
65C8×D173—42no—
66C4⋊Dic173—21no—
67C4⋊Dic173C2311yescharacteristic
68#14583805—21no—
69#14583805—21no—
70#14583805—21no—
71C22⋊Dic173—21no—
72#14583803—21no—
73#14583803—21no—
74#14583803—22no—
75#14583803—22no—
76C2×Q8×C173C2311yescharacteristic
77C22×C4×C173—21no—
78C22×C4×C173C2311yescharacteristic
79C2×C8×C173C2312yes—
80C4⋊C4×C173—21no—
81C4⋊C4×C173—21no—
82C4⋊C4×C173C2311yescharacteristic
83C4⋊C4×C173C2312yes—
84C22⋊C4×C173—21no—
85C22×D173—22no—
86C22×D173—41no—
87C22×D173—41no—
88C22×D173—42no—
89C22×D173—42no—
90C22×D173—42no—
91C22×D173SD1614yescomplemented
92C22×D173C2×D411yescharacteristic
93C22×D173C2×D411yescharacteristic
94Dic346—22no—
95Dic346—42no—
96D692—22no—
97D692—22no—
98D692—42no—
99D692—44no—
100D692—44no—
101C2×Dic173—21no—
102C2×Dic173—21no—
103C2×Dic173—41no—
104C2×Dic173C2×D411yescharacteristic
105C2×Dic173C2×D411yescharacteristic
106C4×D173—22no—
107C4×D173—22no—
108C4×D173—41no—
109C4×D173—41no—
110C4×D173—42no—
111C4×D173—42no—
112#13063436—22no—
113C23×C173C2×D411yescharacteristic
114Q8×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—21no—
117C2×C4×C173—21no—
118C2×C4×C173—22no—
119C2×C4×C173—42no—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2411yescharacteristic
122C8×C173—22no—
123C22×C173—21no—
124C22×C173—21no—
125C22×C173C2×SD1612yes—
126C22×C173C22×D411yescharacteristic
127C2×D173—21no—
128C2×D173—21no—
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—44no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—42no—
135C2×D173—84no—
136C2×D173C2×SD1612yes—
137C2×D173C2×SD1612yes—
138C4×C173—21no—
139C4×C173—41no—
140C4×C173—42no—
141C4×C173—41no—
142C4×C173H(5,𝔽2)12yes—
143Dic173—21no—
144Dic173—21no—
145Dic173—41no—
146Dic173—41no—
147C2×C173—22no—
148C2×C173#53411yescharacteristic
149C2×C173C22×SD1611yescharacteristic
150C2×C173#57511yescharacteristic
151D173—22no—
152D173—22no—
153D173—82no—
154C173#318711yescharacteristic, complemented, Sylow
155#3187—1731nomaximal, Sylow
156C2×C22⋊Q8—1731no—
157C2×#121—1731no—
158#483—1731no—
159#483D17311yescharacteristic, complemented
160#481—1732no—
161#465—1732no—
162#465—1732no—
163#450—1731no—
164#450—1731no—
165#448—1732no—
166C2×C22⋊C8—1731no—
167C22×Q8—1731no—
168C22×D4—3461no—
169C22×D4—1731no—
170C23×C4—1731no—
171C2×SD16—1732no—
172C2×SD16—1732no—
173C22×C8—1732no—
174C22⋊Q8—3461no—
175C22⋊Q8—3461no—
176C22⋊Q8—1731no—
177C22⋊Q8—1731no—
178C22⋊Q8—1731no—
179C22⋊Q8C2×D17311yescharacteristic
180#121—3461no—
181#121—3461no—
182#121—1732no—
183#121—1732no—
184C2×C4⋊C4—3461no—
185C2×C4⋊C4—1731no—
186C2×C4⋊C4C2×D17311yescharacteristic
187C2×C22⋊C4—3461no—
188C2×C22⋊C4—3461no—
189#106—1732no—
190#106C2×D17312yescomplemented
191#103—1732no—
192#103C2×D17312yescomplemented
193#102—1732no—
194#102—1732no—
195C22⋊C8—1731no—
196C22⋊C8—1732no—
197C22⋊C8C2×D17311yescharacteristic
198C24—3461no—
199C24—1731no—
200C2×Q8—3461no—
201C2×Q8—1731no—
202C2×Q8—6922no—
203C2×Q8C22×D17311yescharacteristic
204C2×D4—3462no—
205C2×D4—3462no—
206C2×D4—3461no—
207C2×D4—1732no—
208C2×D4—6922no—
209C2×D4—6924no—
210C22×C4—21no—
211C22×C4—3461no—
212C22×C4—3461no—
213C22×C4—1731no—
214C22×C4—1731no—
215C22×C4—1731no—
216C22×C4—1731no—
217C22×C4—1731no—
218C22×C4—6922no—
219C22×C4C22×D17311yescharacteristic
220SD16—6922no—
221SD16—6922no—
222C2×C8—1732no—
223C2×C8—6922no—
224C2×C8C22×D17312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—3461no—
228C4⋊C4—3461no—
229C4⋊C4—3461no—
230C4⋊C4—3461no—
231C4⋊C4—1731no—
232C4⋊C4C22×D17311yescharacteristic
233C4⋊C4C22×D17312yes—
234C22⋊C4—21no—
235C22⋊C4—3461no—
236C22⋊C4—3461no—
237C22⋊C4—3462no—
238C22⋊C4—3462no—
239C22⋊C4—3461no—
240C23—3462no—
241C23—1731no—
242C23—1731no—
243C23—1734no—
244C23—6921no—
245C23—6921no—
246C23—6922no—
247C23—6922no—
248C23—6922no—
249C23D4×D17311yescharacteristic
250Q8—22no—
251Q8—3462no—
252Q8—6922no—
253D4—3462no—
254D4—3462no—
255D4—6922no—
256D4—6924no—
257D4—6924no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—3462no—
264C2×C4—3462no—
265C2×C4—3461no—
266C2×C4—3461no—
267C2×C4—1731no—
268C2×C4—1731no—
269C2×C4—6921no—
270C2×C4—6921no—
271C2×C4—6922no—
272C2×C4—6922no—
273C2×C4—6921no—
274C2×C4D4×D17311yescharacteristic
275C2×C4C23×D17311yescharacteristic
276C8—22no—
277C8—3462no—
278C22—21no—
279C22—21no—
280C22—3461no—
281C22—3461no—
282C22—3461no—
283C22—3461no—
284C22—13844no—
285C22—1732no—
286C22—1732no—
287C22—6924no—
288C22—6922no—
289C22—6922no—
290C22—6922no—
291C22SD16×D17312yes—
292C22C2×D4×D17311yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3461no—
298C4—3461no—
299C4—6921no—
300C4—6921no—
301C4#1914972512yes—
302C2—22no—
303C2—3462no—
304C2—3462no—
305C2—13842no—
306C2#2601205211yescharacteristic
307C2C2×SD16×D17311yescharacteristic
308C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149654C2
hypercenter#483D173
hypocenterC173#3187
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D173C2semidirect
3. #26011821C2semidirect
4. #26011811C2semidirect
5. #26011723C2semidirect
6. #26011721C2semidirect
7. #26011644C2semidirect
8. #26011641C2semidirect
9. #26011631C2semidirect
10. C22⋊C8×D173C2semidirect
11. #483×C173C2semidirect
12. C2×Q8×D173C22semidirect
16. C2×#14583815C22semidirect
17. C2×#14583813C22semidirect
18. C2×C8×D173C22semidirect
19. #19149659C22semidirect
23. #19149655C22semidirect
29. C4⋊C4×D173C22semidirect
32. #19149610C22semidirect
33. #19149602C22semidirect
34. #19149601C22semidirect
35. #19149599C22semidirect
37. #19149591C22semidirect
40. #106×C173C22semidirect
41. #103×C173C22semidirect
91. C22×D173SD16semidirect
154. C173#3187semidirect
159. #483D173semidirect
190. #106C2×D173semidirect
192. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
C2×#191496542
#260118212
#260118112
#260117232
#260117212
#260116442
#260116412
#260116312
C22⋊C8×D1732
#483×C1732
#3187173