Finite Group Databasebeta

← #36365569

Subgroups — #36365569

Order 22144.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012187C211yesmaximal, characteristic, complemented
2#26011855C211yesmaximal, characteristic, complemented
3#26011812C211yesmaximal, characteristic, complemented
4#26011802C211yesmaximal, characteristic, complemented
5#26011798C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic, complemented
8#26011683C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14#26011487C211yesmaximal, characteristic, complemented
15#550×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×#14583811C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583810C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21#19149674—21no—
22#19149671C2211yescharacteristic
23#19149661—21no—
24#19149653C2211yescharacteristic
25#19149653C2211yescharacteristic
26#19149651—21no—
27C4×D692C2211yescharacteristic
28#19149639—21no—
29#19149638C2211yescharacteristic
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C2×#41×C173C2211yescharacteristic
50#127×C173—21no—
51#127×C173C2211yescharacteristic
52#121×C173—21no—
53#121×C173—21no—
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56C4×Q8×C173C2211yescharacteristic
57C4×D4×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C22×Dic173—21no—
63C2×C4×D173—21no—
64#14583813—22no—
65#14583811—41no—
66#14583811—41no—
67#14583810—41no—
68C2×#13063436—41no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80#14583803—21no—
81#14583803—21no—
82#14583803—21no—
83C4×Dic173—21no—
84#41×C173—22no—
85#41×C173—22no—
86#41×C173—41no—
87#41×C173—41no—
88C2×Q8×C173C2311yescharacteristic
89C2×D4×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173—21no—
95C2×D4×C173D411yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173—21no—
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173D411yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112D692—21no—
113D692—21no—
114C22⋊D173—42no—
115C22⋊D173—81no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—81no—
120C4×D173—81no—
121#13063436—21no—
122#13063436—21no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173C2×D411yescharacteristic
128C23×C173C2×D411yescharacteristic
129Q8×C173—21no—
130Q8×C173#4112yes—
131D4×C173—21no—
132D4×C173—21no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—41no—
136D4×C173—42no—
137D4×C173—42no—
138D4×C173—42no—
139D4×C173—81no—
140D4×C173—81no—
141D4×C173—81no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—42no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—22no—
174C4×C173—21no—
175C4×C173—21no—
176C4×C173—41no—
177C4×C173C2×#4111yescharacteristic
178C4×C173H(5,𝔽2)11yescharacteristic
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—41no—
183C2×C173—41no—
184C2×C173—81no—
185C2×C173—81no—
186C2×C173#54611yescharacteristic
187C2×C173#57511yescharacteristic
188C2×C173#57611yescharacteristic
189D173—81no—
190C173#330911yescharacteristic, complemented, Sylow
191#3309—1731nomaximal, Sylow
192#553—1731no—
193#550D17311yescharacteristic, complemented
194#490—1731no—
195#482—1731no—
196#480—1731no—
197#479—1731no—
198#469—1731no—
199#466—1731no—
200#459—1731no—
201#449—1731no—
202#449—1731no—
203#440—1731no—
204#435—1731no—
205#417—1731no—
206#416—1731no—
207C2×#41C2×D17311yescharacteristic
208C2×#41C2×D17311yescharacteristic
209C2×SD16—1731no—
210C2×D8—1731no—
211C2×D8—1731no—
212C2×M16—1731no—
213C22×C8—1731no—
214#127—21no—
215#127C2×D17311yescharacteristic
216#126—3461no—
217#125—1731no—
218#123—3461no—
219#121—21no—
220#121—21no—
221#121—3461no—
222#121—1731no—
223#121—1731no—
224#121C2×D17311yescharacteristic
225#121C2×D17311yescharacteristic
226C4×Q8C2×D17311yescharacteristic
227C4×D4—21no—
228C4×D4—3461no—
229C4×D4—1731no—
230C4×D4C2×D17311yescharacteristic
231#117—1731no—
232C2×C4⋊C4—1731no—
233#107—1731no—
234#107—1731no—
235#106—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249#97—1731no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—1731no—
264C2×D4C22⋊D17311yescharacteristic
265C2×D4C22×D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273SD16—3462no—
274D8—6921no—
275D8—6921no—
276M16—6921no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8—1731no—
281C2×C8—6921no—
282C4⋊C4—21no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4—1731no—
289C4⋊C4C22⋊D17311yescharacteristic
290C4⋊C4C22×D17311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4C22⋊D17311yescharacteristic
298C22⋊C4C22⋊D17311yescharacteristic
299C42—21no—
300C42—3461no—
301C42C22×D17311yescharacteristic
302C23—21no—
303C23—21no—
304C23—3461no—
305C23C2×C22⋊D17311yescharacteristic
306C23C2×C22⋊D17311yescharacteristic
307Q8—21no—
308Q8#1458382512yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—3461no—
321D4—3461no—
322D4—13841no—
323D4—6922no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4C2×C22⋊D17311yescharacteristic
338C2×C4C2×C22⋊D17311yescharacteristic
339C2×C4C2×C22⋊D17311yescharacteristic
340C2×C4C2×C22⋊D17311yescharacteristic
341C2×C4C23×D17311yescharacteristic
342C8—3461no—
343C8—3461no—
344C8—3461no—
345C8—3461no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C22×C22⋊D17311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4C2×#1458382511yescharacteristic
372C4#1914972411yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—13841no—
378C2#2601213511yescharacteristic
379C2#2601228911yescharacteristic
380C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#550×C173C2
socleC22×C173C22×D4
hypercenter#550D173
hypocenterC173#3309
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26011855C2semidirect
3. #26011812C2semidirect
4. #26011802C2semidirect
5. #26011798C2semidirect
6. #26011742C2semidirect
7. #26011725C2semidirect
8. #26011683C2semidirect
9. #26011636C2semidirect
10. #26011636C2semidirect
11. #26011589C2semidirect
12. #26011563C2semidirect
13. #26011492C2semidirect
14. #26011487C2semidirect
15. #550×C173C2semidirect
190. C173#3309semidirect
193. #550D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260118552
#260118122
#260118022
#260117982
#260117422
#260117252
#260116832
#260116362
#260115892
#260115632
#260114922
#260114872
#550×C1732
#3309173