Finite Group Databasebeta

← #36363817

Subgroups — #36363817

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012104C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011697C211yesmaximal, characteristic, complemented
7#26011696C211yesmaximal, characteristic, complemented
8#26011643C211yesmaximal, characteristic
9#26011643C211yesmaximal, characteristic
10#26011590C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011582C211yesmaximal, characteristic, complemented
13#26011578C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21#19149679C2211yescharacteristic, complemented
22#19149674—21no—
23#19149674—21no—
24#19149670—21no—
25#19149670—21no—
26#19149659C2211yescharacteristic
27#19149659C2211yescharacteristic
28#19149644C2211yescharacteristic, complemented
29#19149644C2211yescharacteristic, complemented
30C4×D692C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32#19149641C2211yescharacteristic
33C4×C22⋊D173—21no—
34#19149639—21no—
35#19149637C2211yescharacteristic
36#19149636C2211yescharacteristic
37C2×#14583803—21no—
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52#125×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64D4×D173—22no—
65D4×D173—22no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583815—22no—
71#14583815—22no—
72#14583814—22no—
73#14583814—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—24no—
110C22×D173—81no—
111C22×D173—81no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117D692—21no—
118D692#4112yes—
119C22⋊D173—24no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—81no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—81no—
130C4×D173—81no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134D4×C173—21no—
135D4×C173#4112yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—81no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173C22×D411yescharacteristic
149C2×D173—21no—
150C2×D173—22no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—84no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—21no—
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—42no—
170C2×C173#52511yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57711yescharacteristic
173D173—41no—
174D173—42no—
175C173#312311yescharacteristic, complemented, Sylow
176#3123—1731nomaximal, Sylow
177#541—1731no—
178#518—1731no—
179#485—1731no—
180#485—1731no—
181#478—1731no—
182#478D17311yescharacteristic, complemented
183#461—1731no—
184#461—1731no—
185#450—1731no—
186#450—1731no—
187#440—1731no—
188#440—1731no—
189C4×SD16—1731no—
190C4×SD16—1731no—
191#433—1731no—
192C22×D4—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C4⋊Q8—1731no—
198#126—3461no—
199#126—3461no—
200#125—3461no—
201#125—3461no—
202#125C2×D17311yescharacteristic, complemented
203C22⋊Q8—1731no—
204C22⋊Q8—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×D4—3461no—
208C4×D4—3461no—
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212C4×D4C2×D17311yescharacteristic
213#117—1731no—
214#117—1731no—
215C2×C22⋊C4—3461no—
216C2×C22⋊C4—3461no—
217#107—1731no—
218#107C2×D17311yescharacteristic, complemented
219#106—1731no—
220#106C2×D17311yescharacteristic, complemented
221C4⋊C8—1731no—
222C4⋊C8C2×D17311yescharacteristic, complemented
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#102—1731no—
228#102—1731no—
229#102C2×D17311yescharacteristic
230#102C2×D17311yescharacteristic
231C22⋊C8—1731no—
232C22⋊C8—1731no—
233#97—1731no—
234C4×C8—1731no—
235C24—3461no—
236C2×Q8—1731no—
237C2×Q8—1731no—
238C2×D4—3462no—
239C2×D4—3462no—
240C2×D4—3461no—
241C2×D4—1731no—
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—1731no—
248SD16—3462no—
249SD16—3462no—
250SD16—3462no—
251SD16—3462no—
252C2×C8—1731no—
253C2×C8—1731no—
254C2×C8C22×D17311yescharacteristic
255C2×C8C22×D17311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4—1731no—
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C4⋊C4C22×D17311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—1731no—
278C22⋊C4—1731no—
279C42—3461no—
280C42—3461no—
281C42—3461no—
282C42—1731no—
283C42C22×D17311yescharacteristic
284C23—21no—
285C23—3464no—
286C23—3461no—
287C23—13841no—
288C23—13841no—
289C23—1731no—
290Q8—3461no—
291Q8—3461no—
292Q8—3461no—
293Q8—3461no—
294D4—21no—
295D4—3464no—
296D4—3461no—
297D4—1732no—
298D4#1458382712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—13841no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4D4×D17311yescharacteristic
315C2×C4D4×D17311yescharacteristic
316C2×C4C23×D17311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3461no—
320C8—3461no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3461no—
325C22—3462no—
326C22—3461no—
327C22—3461no—
328C22—13844no—
329C22—6921no—
330C22—6921no—
331C22C2×D4×D17311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3461no—
338C4—3461no—
339C4—3461no—
340C4—6921no—
341C4—6921no—
342C4—6921no—
343C4#41×D17311yescharacteristic
344C4C2×#1458382711yescharacteristic
345C2—42no—
346C2—6921no—
347C2—6922no—
348C2#2601200411yescharacteristic
349C2#2601229811yescharacteristic
350C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#478D173
hypocenterC173#3123
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012104C2semidirect
3. #26011835C2semidirect
4. #26011834C2semidirect
5. #26011788C2semidirect
6. #26011697C2semidirect
7. #26011696C2semidirect
10. #26011590C2semidirect
11. #26011588C2semidirect
12. #26011582C2semidirect
13. #26011578C2semidirect
14. #26011551C2semidirect
15. #478×C173C2semidirect
21. #19149679C22semidirect
28. #19149644C22semidirect
29. #19149644C22semidirect
39. #19149612C22semidirect
40. #19149610C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149602C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
52. #125×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
175. C173#3123semidirect
182. #478D173semidirect
202. #125C2×D173semidirect
218. #107C2×D173semidirect
220. #106C2×D173semidirect
222. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260119632
#260118352
#260118342
#260117882
#260116972
#260116962
#260116432
#260115902
#260115882
#260115822
#260115782
#260115512
#478×C1732
#3123173