Finite Group Databasebeta

← #36363809

Subgroups — #36363809

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#26012101C211yesmaximal, characteristic, complemented
2#26011966C211yesmaximal, characteristic
3#26011833C211yesmaximal, characteristic, complemented
4#26011829C211yesmaximal, characteristic, complemented
5#26011793C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011694C211yesmaximal, characteristic, complemented
8#26011694C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic
10#26011592C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011582C211yesmaximal, characteristic, complemented
13SD16×Dic173C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic
16C2×#14583814C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18#19149677C2211yescharacteristic, complemented
19#19149672—21no—
20#19149672—21no—
21#19149670—21no—
22#19149670C2211yescharacteristic, complemented
23#19149669—21no—
24#19149658C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic, complemented
26#19149644C2211yescharacteristic, complemented
27D4×Dic173—21no—
28D4×Dic173—21no—
29D4×Dic173C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33#19149638C2211yescharacteristic
34#19149638C2211yescharacteristic
35C2×C22⋊Dic173—21no—
36C2×C22⋊Dic173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C22×Dic173—21no—
61C22×Dic173—21no—
62C22×Dic173—21no—
63C22×Dic173—21no—
64#14583814—22no—
65#14583814—22no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C4⋊Dic173—21no—
69C4⋊Dic173—21no—
70C4⋊Dic173D411yescharacteristic
71C4⋊Dic173D411yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173D411yescharacteristic
87C22⋊Dic173D411yescharacteristic
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C4×Dic173C2311yescharacteristic
93C24×C173—21no—
94C2×Q8×C173C2311yescharacteristic
95C2×D4×C173—22no—
96C2×D4×C173—22no—
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101SD16×C173—22no—
102SD16×C173—22no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108Dic346—21no—
109Dic346—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—81no—
115C2×Dic173—81no—
116C2×Dic173—81no—
117C2×Dic173—81no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122#13063436—21no—
123#13063436—21no—
124C23×C173—24no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173—81no—
128C23×C173—81no—
129C23×C173C2×D411yescharacteristic
130Q8×C173—21no—
131Q8×C173—21no—
132D4×C173—24no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173#4112yes—
136D4×C173#4112yes—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—21no—
146C22×C173—22no—
147C22×C173—22no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—84no—
155C22×C173C22×D411yescharacteristic
156C4×C173—21no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173C2×#4111yescharacteristic
160C4×C173C2×#4111yescharacteristic
161Dic173—21no—
162Dic173—21no—
163Dic173—21no—
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—41no—
170C2×C173—42no—
171C2×C173—42no—
172C2×C173#52511yescharacteristic
173C2×C173#57711yescharacteristic
174C2×C173#57711yescharacteristic
175C173#312311yescharacteristic, complemented, Sylow
176#3123—1731nomaximal, Sylow
177#541—1731no—
178#518—1731no—
179#485—1731no—
180#485—1731no—
181#478—1731no—
182#478—1731no—
183#461—1731no—
184#461—1731no—
185#450—1731no—
186#450D17311yescharacteristic
187#440—1731no—
188#440—1731no—
189C4×SD16—1731no—
190C4×SD16—1731no—
191#433—1731no—
192C22×D4C2×D17311yescharacteristic
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×SD16C2×D17311yescharacteristic
196C2×SD16C2×D17311yescharacteristic
197C4⋊Q8—1731no—
198#126—3461no—
199#126—3461no—
200#125—3461no—
201#125—3461no—
202#125—1731no—
203C22⋊Q8—1731no—
204C22⋊Q8C2×D17311yescharacteristic
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×D4—3461no—
208C4×D4—3461no—
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212C4×D4—1731no—
213#117—1731no—
214#117—1731no—
215C2×C22⋊C4—3461no—
216C2×C22⋊C4—3461no—
217#107—1731no—
218#107—1731no—
219#106—1731no—
220#106—1731no—
221C4⋊C8—1731no—
222C4⋊C8—1731no—
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⋊C8C2×D17311yescharacteristic
233#97—1731no—
234C4×C8—1731no—
235C24—21no—
236C2×Q8—1731no—
237C2×Q8C22×D17311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D17311yescharacteristic
242C2×D4C22×D17311yescharacteristic
243C22×C4—3461no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4C22×D17311yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—3462no—
251SD16—3462no—
252C2×C8—1731no—
253C2×C8—1731no—
254C2×C8C22×D17311yescharacteristic
255C2×C8C22×D17311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3461no—
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⋊C4—1731no—
268C4⋊C4—1731no—
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—
283C42—1731no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D17311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3461no—
293Q8—3461no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1458382512yes—
298D4#1458382512yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—13841no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—13841no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4D4×D17311yescharacteristic
316C2×C4C23×D17311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3461no—
320C8—3461no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D17311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—3461no—
336C4—3461no—
337C4—3461no—
338C4—3461no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4—6921no—
343C4C2×#1458382511yescharacteristic
344C4C2×#1458382511yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2601199711yescharacteristic
349C2#2601229711yescharacteristic
350C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC22×D4×C173C22
hypercenter#450D173
hypocenterC173#3123
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012101C2semidirect
3. #26011833C2semidirect
4. #26011829C2semidirect
5. #26011793C2semidirect
6. #26011789C2semidirect
7. #26011694C2semidirect
8. #26011694C2semidirect
10. #26011592C2semidirect
11. #26011588C2semidirect
12. #26011582C2semidirect
13. SD16×Dic173C2semidirect
14. #26011549C2semidirect
18. #19149677C22semidirect
22. #19149670C22semidirect
25. Q8×Dic173C22semidirect
26. #19149644C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149603C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
175. C173#3123semidirect

Maximal subgroups

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

subgroupindex
#260121012
#260119662
#260118332
#260118292
#260117932
#260117892
#260116942
#260116422
#260115922
#260115882
#260115822
SD16×Dic1732
#260115492
#450×C1732
#3123173