Finite Group Databasebeta

← #36365532

Subgroups — #36365532

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012193C211yesmaximal, characteristic, complemented
2#26012140C211yesmaximal, characteristic, complemented
3#26011858C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011822C211yesmaximal, characteristic, complemented
6#26011793C211yesmaximal, characteristic, complemented
7#26011735C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011603C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149677C2211yescharacteristic
21#19149672—21no—
22#19149670C2211yescharacteristic
23#19149663—21no—
24#19149662—21no—
25#19149660—21no—
26#19149658C2211yescharacteristic
27#19149655C2211yescharacteristic
28#19149651—21no—
29#19149651—21no—
30#19149650C2211yescharacteristic
31D4×Dic173—21no—
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149637C2211yescharacteristic
36C2×C4⋊Dic173C2211yescharacteristic
37C2×#14583805—21no—
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173D411yescharacteristic
65C22×Dic173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583814—41no—
71#14583810—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C8×D173—41no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96#41×C173—22no—
97#41×C173—41no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173C2311yescharacteristic
105SD16×C173—41no—
106D8×C173—22no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114C22⋊D173—41no—
115C22⋊D173—42no—
116C22⋊D173—81no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C2×Dic173—81no—
124C2×Dic173—81no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130#13063436—21no—
131#13063436—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—42no—
141D4×C173—42no—
142D4×C173#4112yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—42no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—22no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173H(5,𝔽2)11yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—42no—
179C2×C173—81no—
180C2×C173#54611yescharacteristic
181C2×C173#57511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—41no—
184C173#330711yescharacteristic, complemented, Sylow
185#3307—1731nomaximal, Sylow
186#553—1731no—
187#547—1731no—
188#491—1731no—
189#485—1731no—
190#483—1731no—
191#478—1731no—
192#468—1731no—
193#465—1731no—
194#461—1731no—
195#449—1731no—
196#449D17311yescharacteristic, complemented
197#442—1731no—
198#435—1731no—
199#417—1731no—
200#416—1731no—
201C2×#41—1731no—
202C2×#41C2×D17311yescharacteristic
203C2×SD16—1731no—
204C2×SD16C2×D17311yescharacteristic
205C2×D8C2×D17311yescharacteristic
206C2×M16—1731no—
207C22×C8—1731no—
208C4⋊Q8—1731no—
209#126—3461no—
210#125—1731no—
211#123—3461no—
212#123—3461no—
213#123—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×D4—3461no—
221C4×D4—1731no—
222C4×D4—1731no—
223C4×D4—1731no—
224#117—1731no—
225C2×C4⋊C4—3461no—
226C2×C4⋊C4—1731no—
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8C2×D17311yescharacteristic
243#97—1731no—
244#41—22no—
245#41—41no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4C22×D17311yescharacteristic
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—41no—
265SD16—6921no—
266D8—22no—
267M16—6921no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4—1731no—
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—21no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3461no—
303Q8—3461no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—42no—
309D4—13841no—
310D4—6921no—
311D4—6922no—
312D4#1458382512yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—13841no—
325C2×C4—13841no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3461no—
347C22—3461no—
348C22—3461no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4C2×#1458382511yescharacteristic
361C4#1914972411yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—6921no—
366C2#2601212811yescharacteristic
367C2#2601228711yescharacteristic
368C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3307
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012193C2semidirect
2. #26012140C2semidirect
3. #26011858C2semidirect
4. #26011830C2semidirect
5. #26011822C2semidirect
6. #26011793C2semidirect
7. #26011735C2semidirect
8. #26011717C2semidirect
9. #26011694C2semidirect
10. #26011633C2semidirect
11. #26011603C2semidirect
12. #26011565C2semidirect
13. #26011492C2semidirect
14. #26011484C2semidirect
15. #449×C173C2semidirect
184. C173#3307semidirect
196. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260121932
#260121402
#260118582
#260118302
#260118222
#260117932
#260117352
#260117172
#260116942
#260116332
#260116032
#260115652
#260114922
#260114842
#449×C1732
#3307173