Finite Group Databasebeta

← #36365640

Subgroups — #36365640

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012196C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic, complemented
4#26011826C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011732C211yesmaximal, characteristic, complemented
9#26011712C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011588C211yesmaximal, characteristic, complemented
13#26011562C211yesmaximal, characteristic, complemented
14#26011489C211yesmaximal, characteristic, complemented
15C2×#102×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic, complemented
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583809C2211yescharacteristic, complemented
21#19149678C2211yescharacteristic, complemented
22#19149669—21no—
23#19149666C2211yescharacteristic
24#19149660—21no—
25#19149659C2211yescharacteristic, complemented
26#19149658—21no—
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149651—21no—
31#19149650C2211yescharacteristic
32#19149648—21no—
33#19149644C2211yescharacteristic, complemented
34D4×Dic173C2211yescharacteristic
35C4×C22⋊D173—21no—
36C4×C22⋊D173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic, complemented
38C2×C22⋊Dic173—21no—
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149612C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C2×C4⋊C4×C173C2211yescharacteristic
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173C2311yescharacteristic
71Dic692—41no—
72#14583814—22no—
73#14583814—41no—
74#14583809—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173—21no—
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⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173D411yescharacteristic
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C24×C173—21no—
98C2×D4×C173—22no—
99C2×D4×C173—21no—
100C2×D4×C173—41no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173C2311yescharacteristic
105C2×C8×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—81no—
117C22⋊D173—22no—
118C22⋊D173—42no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—81no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173—21no—
136C23×C173—42no—
137C23×C173—42no—
138C23×C173—81no—
139C23×C173C2×D411yescharacteristic
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—42no—
144D4×C173#4112yes—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—22no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—84no—
163C22×C173#13712yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—21no—
175Dic173—21no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—42no—
182C2×C173—81no—
183C2×C173#54611yescharacteristic
184C2×C173C2×#13711yescharacteristic
185C2×C173#57611yescharacteristic
186D173—41no—
187C173#331411yescharacteristic, complemented, Sylow
188#3314—1731nomaximal, Sylow
189#554—1731no—
190#546—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#474—1731no—
195#468—1731no—
196#467—1731no—
197#464—1731no—
198#452—1731no—
199#450—1731no—
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#102D17311yescharacteristic, complemented
204C2×#41—1731no—
205C22×D4C2×D17311yescharacteristic
206C2×Q16—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×M16—1731no—
210C22×C8C2×D17311yescharacteristic
211C4⋊Q8—1731no—
212#125—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—1731no—
218C22⋊Q8—1731no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#120—3461no—
223C4×Q8—1731no—
224C4×D4—3461no—
225C4×D4—1731no—
226C4×D4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4C2×D17311yescharacteristic
229C2×C22⋊C4—3461no—
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241#102C2×D17311yescharacteristic
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246#97—1731no—
247C24—21no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8—1731no—
252C2×Q8—1731no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—1731no—
259C2×D4C22×D17311yescharacteristic
260C2×D4C22×D17311yescharacteristic
261C22×C4—21no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266C22×C4C22×D17311yescharacteristic
267Q16—6921no—
268SD16—3462no—
269SD16—6921no—
270M16—6921no—
271C2×C8—41no—
272C2×C8—1731no—
273C2×C8—1731no—
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4C22×D17311yescharacteristic
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—1731no—
296C22⋊C4—1731no—
297C42—3461no—
298C42—1731no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—1731no—
305C23D4×D17311yescharacteristic
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—3462no—
316D4—13841no—
317D4—6922no—
318D4#1458382612yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—13841no—
331C2×C4—13841no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—6922no—
336C2×C4D4×D17311yescharacteristic
337C2×C4C23×D17311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3461no—
341C8—3461no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—3461no—
352C22—3461no—
353C22—3461no—
354C22#1914970812yes—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3461no—
360C4—3461no—
361C4—3461no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4C2×#1458382611yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—6921no—
372C2#2601213111yescharacteristic
373C2C2×#1914970811yescharacteristic
374C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#102×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenterC2×#102D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #26012196C2semidirect
3. #26011849C2semidirect
4. #26011826C2semidirect
5. #26011803C2semidirect
6. #26011769C2semidirect
7. #26011736C2semidirect
8. #26011732C2semidirect
9. #26011712C2semidirect
10. #26011654C2semidirect
11. #26011642C2semidirect
12. #26011588C2semidirect
13. #26011562C2semidirect
14. #26011489C2semidirect
15. C2×#102×C173C2semidirect
17. C2×Dic692C22semidirect
20. C2×#14583809C22semidirect
21. #19149678C22semidirect
25. #19149659C22semidirect
33. #19149644C22semidirect
37. C4⋊C4×D173C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149612C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
187. C173#3314semidirect
203. C2×#102D173semidirect

Maximal subgroups

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

subgroupindex
#260121962
#260121352
#260118492
#260118262
#260118032
#260117692
#260117362
#260117322
#260117122
#260116542
#260116422
#260115882
#260115622
#260114892
C2×#102×C1732
#3314173