Finite Group Databasebeta

← #36365648

Subgroups — #36365648

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#26012199C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011734C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011711C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12#26011559C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583816C2211yescharacteristic, complemented
18C2×#14583809C2211yescharacteristic, complemented
19C22×#13063436C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149669—21no—
22#19149666C2211yescharacteristic
23#19149660—21no—
24#19149659—21no—
25#19149658C2211yescharacteristic
26#19149656—21no—
27#19149655C2211yescharacteristic, complemented
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149648—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149639—21no—
35C2×C4⋊Dic173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37C2×C22⋊Dic173—21no—
38#19149613C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C22×Dic173—21no—
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173C2311yescharacteristic
71#14583816—41no—
72#14583809—41no—
73C2×#13063436—41no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
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⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C24×C173—21no—
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173—22no—
99C2×D4×C173—21no—
100C2×D4×C173—41no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104SD16×C173—22no—
105SD16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—81no—
115C22⋊D173—22no—
116C22⋊D173—42no—
117C22⋊D173—81no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—42no—
123C2×Dic173—81no—
124C2×Dic173—81no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—81no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173—42no—
136C23×C173—42no—
137C23×C173—81no—
138C23×C173C2×D411yescharacteristic
139Q8×C173—21no—
140Q8×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—42no—
145D4×C173#4112yes—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
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#450D17311yescharacteristic, complemented
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#102—1731no—
204C2×#41—1731no—
205C22×D4C2×D17311yescharacteristic
206C2×Q16—1731no—
207C2×SD16C2×D17311yescharacteristic
208C2×SD16C2×D17311yescharacteristic
209C2×M16—1731no—
210C22×C8—1731no—
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⋊Q8C2×D17311yescharacteristic
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⋊C4—1731no—
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#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8C2×D17311yescharacteristic
246#97—1731no—
247C24—21no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8—1731no—
252C2×Q8C22×D17311yescharacteristic
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—3461no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266C22×C4C22×D17311yescharacteristic
267Q16—6921no—
268SD16—22no—
269SD16—41no—
270M16—6921no—
271C2×C8—1731no—
272C2×C8—1731no—
273C2×C8—6921no—
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⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
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—21no—
307Q8—21no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—3462no—
316D4—13841no—
317D4—6922no—
318D4#1458382512yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—13841no—
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×#1458382511yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—6921no—
372C2#2601212811yescharacteristic
373C2C2×#1914970811yescharacteristic
374C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenter#450D173
hypocenterC173#3314
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. #26012199C2semidirect
3. #26011849C2semidirect
4. #26011827C2semidirect
5. #26011803C2semidirect
6. #26011769C2semidirect
7. #26011734C2semidirect
8. #26011730C2semidirect
9. #26011711C2semidirect
10. #26011650C2semidirect
11. #26011592C2semidirect
12. #26011559C2semidirect
13. #26011489C2semidirect
14. C2×#19149600C2semidirect
15. #450×C173C2semidirect
17. C2×#14583816C22semidirect
18. C2×#14583809C22semidirect
20. #19149678C22semidirect
27. #19149655C22semidirect
31. Q8×Dic173C22semidirect
36. C4⋊C4×D173C22semidirect
38. #19149613C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149603C22semidirect
45. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
187. C173#3314semidirect
199. #450D173semidirect

Maximal subgroups

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

subgroupindex
#260121992
#260121352
#260118492
#260118272
#260118032
#260117692
#260117342
#260117302
#260117112
#260116502
#260115922
#260115592
#260114892
C2×#191496002
#450×C1732
#3314173