Finite Group Databasebeta

← #36363721

Subgroups — #36363721

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic, complemented
2C4×#14583825C211yesmaximal, characteristic
3#26011834C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011785C211yesmaximal, characteristic, complemented
6#26011779C211yesmaximal, characteristic, complemented
7#26011700C211yesmaximal, characteristic, complemented
8#26011694C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic
10#26011650C211yesmaximal, characteristic
11#26011599C211yesmaximal, characteristic, complemented
12#26011596C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011546C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677C2211yescharacteristic
22#19149669—21no—
23#19149659C2211yescharacteristic
24#19149657—21no—
25#19149655C2211yescharacteristic
26C4×Dic346—21no—
27C4×Dic346C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29#19149644C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31C4×C22⋊D173—21no—
32#19149639—21no—
33#19149637C2211yescharacteristic
34#19149636—21no—
35#19149635—21no—
36C2×C4×Dic173—21no—
37C42×D173C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C22×Dic173—21no—
66C22×Dic173—21no—
67C2×C4×D173D411yescharacteristic
68C2×C4×D173D411yescharacteristic
69C2×C4×D173C2311yescharacteristic
70Dic692—22no—
71#14583816—22no—
72#14583814—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173D411yescharacteristic
94C4×Dic173D411yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—42no—
113Dic346#4112yes—
114C22⋊D173—24no—
115C2×Dic173—24no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—42no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—21no—
125C4×D173—21no—
126C4×D173—22no—
127C4×D173—41no—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133D4×C173—21no—
134D4×C173#4112yes—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—42no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C22×C173—22no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C4×C173—21no—
152C4×C173—21no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173C2×#4111yescharacteristic
157C4×C173C2×#4111yescharacteristic
158Dic173—21no—
159Dic173—22no—
160Dic173—21no—
161Dic173—41no—
162Dic173—41no—
163Dic173—41no—
164Dic173#13712yes—
165C2×C173—42no—
166C2×C173#52511yescharacteristic
167C2×C173C2×#13711yescharacteristic
168C2×C173#57511yescharacteristic
169D173—41no—
170C173#311611yescharacteristic, complemented, Sylow
171#3116—1731nomaximal, Sylow
172#533—1731no—
173C4×#41—1731no—
174#485—1731no—
175#485—1731no—
176#477—1731no—
177#476—1731no—
178#462—1731no—
179#461—1731no—
180#452—1731no—
181#452—1731no—
182#441—1731no—
183#441—1731no—
184#440—1731no—
185#440D17311yescharacteristic, complemented
186#432—1731no—
187C2×#41—1731no—
188C2×Q16—1731no—
189C2×Q16—1731no—
190C2×SD16—1731no—
191C2×SD16C2×D17311yescharacteristic
192C4⋊Q8—1731no—
193C4⋊Q8—1731no—
194#125—3461no—
195C22⋊Q8—3461no—
196C22⋊Q8—1731no—
197C22⋊Q8—1731no—
198C4×Q8—3461no—
199C4×Q8—1731no—
200C4×Q8—1731no—
201C4×Q8—1731no—
202C4×Q8C2×D17311yescharacteristic
203C4×D4—3461no—
204C4×D4—3461no—
205C4×D4—1731no—
206C4×D4C2×D17311yescharacteristic
207#117—3461no—
208#117—3461no—
209#117—1731no—
210C2×C42—3461no—
211C2×C42—1731no—
212#107—1731no—
213#107C2×D17311yescharacteristic
214#106—1731no—
215#106—1731no—
216C4⋊C8—1731no—
217C4⋊C8—1731no—
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#103—1731no—
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102C2×D17311yescharacteristic
226C22⋊C8—1731no—
227C22⋊C8—1731no—
228#97—1731no—
229#97C2×D17311yescharacteristic
230#41—3462no—
231#41—3462no—
232C2×Q8—3461no—
233C2×Q8—1731no—
234C2×Q8—1731no—
235C2×Q8C22×D17311yescharacteristic
236C2×D4—3461no—
237C2×D4C22×D17311yescharacteristic
238C22×C4—21no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1731no—
244Q16—3462no—
245Q16—3462no—
246SD16—22no—
247SD16—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C42—21no—
271C42—3461no—
272C42—3461no—
273C42—3461no—
274C42—3461no—
275C42—1731no—
276C42—1731no—
277C42—1731no—
278C42C22×D17311yescharacteristic
279C23—21no—
280C23—1731no—
281Q8—21no—
282Q8—21no—
283Q8—3461no—
284Q8—3461no—
285Q8—3461no—
286Q8—1732no—
287Q8—6922no—
288D4—21no—
289D4—3464no—
290D4#1458382612yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—3462no—
296C2×C4—3464no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—1731no—
306C2×C4—6921no—
307C2×C4—6922no—
308C2×C4D4×D17311yescharacteristic
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3461no—
314C8—3461no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—3461no—
319C22—3461no—
320C22—3461no—
321C22C2×D4×D17311yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3461no—
328C4—3462no—
329C4—3461no—
330C4—1732no—
331C4—6921no—
332C4—6921no—
333C4—6921no—
334C4#41×D17311yescharacteristic
335C4C2×#1458382611yescharacteristic
336C2—42no—
337C2—6921no—
338C2#2601200011yescharacteristic
339C2#137×D17311yescharacteristic
340C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583825C22
hypercenter#440D173
hypocenterC173#3116
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #26012045C2semidirect
3. #26011834C2semidirect
4. #26011830C2semidirect
5. #26011785C2semidirect
6. #26011779C2semidirect
7. #26011700C2semidirect
8. #26011694C2semidirect
11. #26011599C2semidirect
12. #26011596C2semidirect
13. #26011591C2semidirect
14. #26011546C2semidirect
15. #440×C173C2semidirect
170. C173#3116semidirect
185. #440D173semidirect

Maximal subgroups

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

subgroupindex
#260120452
C4×#145838252
#260118342
#260118302
#260117852
#260117792
#260117002
#260116942
#260116542
#260116502
#260115992
#260115962
#260115912
#260115462
#440×C1732
#3116173