Finite Group Databasebeta

← #36365392

Subgroups — #36365392

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012137C211yesmaximal, characteristic
2#26011848C211yesmaximal, characteristic, complemented
3#26011821C211yesmaximal, characteristic, complemented
4#26011815C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11C4×#14583813C211yesmaximal, characteristic, complemented
12D4×#13063436C211yesmaximal, characteristic, complemented
13#26011485C211yesmaximal, characteristic, complemented
14C2×#19149599C211yesmaximal, characteristic, complemented
15#552×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C22×#13063436C2211yescharacteristic, complemented
23#19149667C2211yescharacteristic
24#19149664—21no—
25#19149658C2211yescharacteristic
26#19149653C2211yescharacteristic
27#19149652—21no—
28#19149647—21no—
29C4×D692C2211yescharacteristic
30C22⋊C4×D173—21no—
31#19149612C2211yescharacteristic, complemented
32#19149610C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149603C2211yescharacteristic, complemented
36#19149602C2211yescharacteristic, complemented
37#19149602C2211yescharacteristic, complemented
38#19149602C2211yescharacteristic, complemented
39#19149599C2211yescharacteristic
40#19149599C2211yescharacteristic
41#19149599C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48#126×C173—21no—
49#125×C173—21no—
50#125×C173C2211yescharacteristic, complemented
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53C22⋊Q8×C173C2211yescharacteristic, complemented
54C4×Q8×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56#117×C173—21no—
57#117×C173C2211yescharacteristic, complemented
58C2×C4⋊C4×C173C2211yescharacteristic
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—41no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72#14583816—41no—
73#14583813—22no—
74#14583813—41no—
75C2×#13063436—41no—
76C2×#13063436—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88C2×Q8×C173C2311yescharacteristic
89C2×D4×C173D411yescharacteristic
90C22×C4×C173—21no—
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173—21no—
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173D411yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173D411yescharacteristic
107C22⋊C4×C173D411yescharacteristic
108C42×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—42no—
122D692#4112yes—
123C22⋊D173—42no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C4×D173—42no—
128C4×D173—81no—
129#13063436—21no—
130#13063436—21no—
131#13063436—21no—
132#13063436—21no—
133C23×C173C2×D411yescharacteristic
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—22no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—42no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C2×D173—22no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—84no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173C2×#4111yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—41no—
177Dic173—41no—
178C2×C173—22no—
179C2×C173—41no—
180C2×C173#54611yescharacteristic
181C2×C173C2×#13511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—42no—
184D173—81no—
185C173#329611yescharacteristic, complemented, Sylow
186#3296—1731nomaximal, Sylow
187#552D17311yescharacteristic, complemented
188#546—1731no—
189#488—1731no—
190#483—1731no—
191#482—1731no—
192#478—1731no—
193#467—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198C4×SD16—1731no—
199C8×D4—1731no—
200#416—1731no—
201C2×#102—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C22×C8—1731no—
208C22×C8—1731no—
209#126—21no—
210#125—21no—
211#125C2×D17311yescharacteristic, complemented
212#124—1731no—
213#123—21no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8C2×D17311yescharacteristic
217C22⋊Q8C2×D17311yescharacteristic, complemented
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8C2×D17311yescharacteristic, complemented
222C4×D4—1731no—
223C4×D4C2×D17311yescharacteristic
224#117—21no—
225#117C2×D17311yescharacteristic, complemented
226C2×C4⋊C4C2×D17311yescharacteristic
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106—1731no—
230#106—1731no—
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C4×C8—1731no—
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—3462no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—6921no—
257C2×D4C22⋊D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8—6921no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3461no—
279C4⋊C4—1731no—
280C4⋊C4C22⋊D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C4⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4C22⋊D17311yescharacteristic
293C22⋊C4C22⋊D17311yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D17311yescharacteristic
297C23—3461no—
298C23—3461no—
299C23—13841no—
300C23—6922no—
301C23—6922no—
302C23C2×C22⋊D17311yescharacteristic
303C23C2×C22⋊D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3461no—
307Q8—3461no—
308D4—22no—
309D4—3461no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—1732no—
314D4—6922no—
315D4—6922no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—13841no—
328C2×C4—6922no—
329C2×C4C2×C22⋊D17311yescharacteristic
330C2×C4C2×C22⋊D17311yescharacteristic
331C2×C4C2×C22⋊D17311yescharacteristic
332C2×C4C2×C22⋊D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—3461no—
335C8—3461no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3462no—
344C22—13844no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22—6921no—
350C22#1914969912yes—
351C22C22×C22⋊D17311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6921no—
361C4—6921no—
362C4#41×D17311yescharacteristic
363C4#1914972511yescharacteristic
364C2—22no—
365C2—41no—
366C2—13841no—
367C2—6922no—
368C2#2601212911yescharacteristic
369C2C2×#1914969911yescharacteristic
370C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#552×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenter#552D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26011848C2semidirect
3. #26011821C2semidirect
4. #26011815C2semidirect
5. #26011788C2semidirect
6. #26011730C2semidirect
7. #26011722C2semidirect
8. #26011713C2semidirect
9. #26011651C2semidirect
10. #26011641C2semidirect
11. C4×#14583813C2semidirect
12. D4×#13063436C2semidirect
13. #26011485C2semidirect
14. C2×#19149599C2semidirect
15. #552×C173C2semidirect
18. C2×#14583816C22semidirect
22. C22×#13063436C22semidirect
31. #19149612C22semidirect
32. #19149610C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149603C22semidirect
36. #19149602C22semidirect
37. #19149602C22semidirect
38. #19149602C22semidirect
46. C22⋊#13063436C22semidirect
47. C4×#13063436C22semidirect
50. #125×C173C22semidirect
53. C22⋊Q8×C173C22semidirect
54. C4×Q8×C173C22semidirect
57. #117×C173C22semidirect
185. C173#3296semidirect
187. #552D173semidirect
211. #125C2×D173semidirect
217. C22⋊Q8C2×D173semidirect
221. C4×Q8C2×D173semidirect
225. #117C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121372
#260118482
#260118212
#260118152
#260117882
#260117302
#260117222
#260117132
#260116512
#260116412
C4×#145838132
D4×#130634362
#260114852
C2×#191495992
#552×C1732
#3296173