Finite Group Databasebeta

← #36365593

Subgroups — #36365593

Order 22144.

402 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#26011853C211yesmaximal, characteristic, complemented
3#26011832C211yesmaximal, characteristic, complemented
4#26011815C211yesmaximal, characteristic, complemented
5#26011778C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011495C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#556×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
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
47#19149590C2211yescharacteristic, complemented
48C4⋊Q8×C173C2211yescharacteristic, complemented
49#126×C173—21no—
50#126×C173—21no—
51#124×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53C22⋊Q8×C173C2211yescharacteristic, complemented
54C4×Q8×C173—21no—
55C4×Q8×C173C2211yescharacteristic, complemented
56C4×D4×C173C2211yescharacteristic
57#117×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic, complemented
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—
75#14583810—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×C173—21no—
89C2×Q8×C173C2311yescharacteristic
90C2×D4×C173D411yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173—21no—
97C4⋊C4×C173D411yescharacteristic
98C4⋊C4×C173C2311yescharacteristic
99C4⋊C4×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173D411yescharacteristic
106C22⋊C4×C173D411yescharacteristic
107C42×C173—21no—
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—
137Q8×C173—42no—
138D4×C173—41no—
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—21no—
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—21no—
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—41no—
179C2×C173—41no—
180C2×C173#54611yescharacteristic
181C2×C173#57511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—42no—
184D173—81no—
185C173#331111yescharacteristic, complemented, Sylow
186#3311—1731nomaximal, Sylow
187#556D17311yescharacteristic, complemented
188#546—1731no—
189#489—1731no—
190#485—1731no—
191#482—1731no—
192#476—1731no—
193#470—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198#440—1731no—
199#435—1731no—
200#418—1731no—
201#416—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×M16—1731no—
208C22×C8—1731no—
209C4⋊Q8C2×D17311yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8C2×D17311yescharacteristic
217C22⋊Q8C2×D17311yescharacteristic, complemented
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8—21no—
222C4×Q8C2×D17311yescharacteristic, complemented
223C4×D4—1731no—
224C4×D4C2×D17311yescharacteristic
225#117C2×D17311yescharacteristic
226#117C2×D17311yescharacteristic, complemented
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—
244#97—1731no—
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—21no—
249C2×Q8—1731no—
250C2×Q8C22×D17311yescharacteristic
251C2×D4—3462no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4—6921no—
258C2×D4C22⋊D17311yescharacteristic
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266M16—6921no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8—6921no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3461no—
277C4⋊C4—1731no—
278C4⋊C4C22⋊D17311yescharacteristic
279C4⋊C4C22×D17311yescharacteristic
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4C22⋊D17311yescharacteristic
292C22⋊C4C22⋊D17311yescharacteristic
293C42—21no—
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—42no—
307Q8—3461no—
308Q8—3461no—
309D4—41no—
310D4—3461no—
311D4—3461no—
312D4—3461no—
313D4—13841no—
314D4—1732no—
315D4—6922no—
316D4—6922no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
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—21no—
344C22—3462no—
345C22—13844no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
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—41no—
365C2—41no—
366C2—13841no—
367C2—6922no—
368C2#2601212911yescharacteristic
369C2#2601228911yescharacteristic
370C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#556×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenter#556D173
hypocenterC173#3311
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #26011853C2semidirect
3. #26011832C2semidirect
4. #26011815C2semidirect
5. #26011778C2semidirect
6. #26011747C2semidirect
7. #26011722C2semidirect
8. #26011713C2semidirect
9. #26011651C2semidirect
10. #26011641C2semidirect
11. #26011589C2semidirect
12. #26011563C2semidirect
13. #26011495C2semidirect
14. #26011485C2semidirect
15. #556×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. #19149590C22semidirect
48. C4⋊Q8×C173C22semidirect
53. C22⋊Q8×C173C22semidirect
55. C4×Q8×C173C22semidirect
58. #117×C173C22semidirect
185. C173#3311semidirect
187. #556D173semidirect
209. C4⋊Q8C2×D173semidirect
217. C22⋊Q8C2×D173semidirect
222. C4×Q8C2×D173semidirect
226. #117C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121372
#260118532
#260118322
#260118152
#260117782
#260117472
#260117222
#260117132
#260116512
#260116412
#260115892
#260115632
#260114952
#260114852
#556×C1732
#3311173