Finite Group Databasebeta

← #36364411

Subgroups — #36364411

Order 22144.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012070C211yesmaximal, characteristic, complemented
2#26012058C211yesmaximal, characteristic, complemented
3#26011833C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011808C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011741C211yesmaximal, characteristic, complemented
8#26011736C211yesmaximal, characteristic, complemented
9#26011724C211yesmaximal, characteristic, complemented
10#26011719C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011633C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic, complemented
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583809C2211yescharacteristic, complemented
21C2×C8×D173C2211yescharacteristic, complemented
22#19149675—21no—
23#19149666—21no—
24#19149659C2211yescharacteristic, complemented
25#19149658C2211yescharacteristic
26#19149656—21no—
27#19149652—21no—
28#19149651—21no—
29#19149650—21no—
30#19149650C2211yescharacteristic
31#19149650C2211yescharacteristic
32#19149650C2211yescharacteristic
33#19149650C2211yescharacteristic
34#19149650C2211yescharacteristic, complemented
35#19149648—21no—
36#19149648—21no—
37#19149638C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346D411yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692D411yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173D411yescharacteristic
73C2×C22⋊D173D411yescharacteristic
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77C2×C4×D173C2311yescharacteristic
78#14583814—41no—
79#14583814—41no—
80#14583809—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97#14583803—21no—
98C4×Dic173—21no—
99C24×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108D8×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173C2×D411yescharacteristic
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—41no—
119D692—41no—
120C22⋊D173—41no—
121C22⋊D173—41no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—42no—
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—21no—
139C4×D173—81no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173—21no—
144C23×C173—21no—
145C23×C173—42no—
146C23×C173—81no—
147C23×C173—81no—
148C23×C173C2×D411yescharacteristic
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—42no—
156D4×C173—81no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—84no—
177C22×C173C22×D411yescharacteristic
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—21no—
184C4×C173—21no—
185C4×C173—41no—
186C4×C173H(5,𝔽2)11yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—21no—
190Dic173—41no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173—81no—
197C2×C173—81no—
198C2×C173#53411yescharacteristic
199C2×C173#57611yescharacteristic
200C2×C173#57711yescharacteristic
201D173—41no—
202D173—41no—
203C173#319811yescharacteristic, complemented, Sylow
204#3198—1731nomaximal, Sylow
205#537—1731no—
206#535—1731no—
207#485—1731no—
208#482—1731no—
209#481—1731no—
210#480—1731no—
211#469—1731no—
212#468—1731no—
213#466—1731no—
214#465—1731no—
215#450—1731no—
216#449—1731no—
217#449—1731no—
218#447D17311yescharacteristic, complemented
219#408—1731no—
220C2×#41—1731no—
221C2×#41—1731no—
222C22×D4C2×D17311yescharacteristic
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×D8C2×D17311yescharacteristic
226C2×D8C2×D17311yescharacteristic
227C2×M16—1731no—
228C22×C8—1731no—
229#127—3461no—
230#124—3461no—
231C22⋊Q8—3461no—
232C22⋊Q8—1731no—
233C22⋊Q8—1731no—
234#121—3461no—
235#121—3461no—
236#121—3461no—
237#121—1731no—
238#121—1731no—
239#121—1731no—
240#121—1731no—
241#121—1731no—
242#121C2×D17311yescharacteristic
243#120—3461no—
244#120—3461no—
245#117—1731no—
246C2×C4⋊C4—1731no—
247#107—1731no—
248#107—1731no—
249#106—1731no—
250#106—1731no—
251#103—1731no—
252#103—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102C2×D17311yescharacteristic
258#102C2×D17311yescharacteristic
259C22⋊C8—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262C22⋊C8C2×D17311yescharacteristic
263C24—21no—
264#41—6921no—
265#41—6921no—
266#41—6921no—
267#41—6921no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—1731no—
283C2×D4C22×D17311yescharacteristic
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4—1731no—
290C22×C4C22×D17311yescharacteristic
291SD16—6921no—
292SD16—6921no—
293D8—41no—
294D8—41no—
295M16—6921no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—6921no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—1731no—
305C4⋊C4—1731no—
306C4⋊C4—1731no—
307C4⋊C4C22×D17311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C42—3461no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—1731no—
325C23—1731no—
326C23D4×D17311yescharacteristic
327Q8—3461no—
328Q8—3461no—
329Q8—6921no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—81no—
338D4—13841no—
339D4—13841no—
340D4—13841no—
341D4—13841no—
342D4—13841no—
343D4—6921no—
344D4—6921no—
345D4—6921no—
346D4—6922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—13841no—
359C2×C4—1731no—
360C2×C4—1731no—
361C2×C4—6922no—
362C2×C4D4×D17311yescharacteristic
363C2×C4C23×D17311yescharacteristic
364C8—21no—
365C8—21no—
366C8—3461no—
367C8—3461no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—3461no—
382C22—3461no—
383C22—3461no—
384C22—3461no—
385C22—3461no—
386C22—3461no—
387C22C2×D4×D17311yescharacteristic
388C4—21no—
389C4—41no—
390C4—3461no—
391C4—3461no—
392C4—6921no—
393C4—6921no—
394C4—6921no—
395C4—6921no—
396C4#1914972411yescharacteristic
397C4#1914972411yescharacteristic
398C2—41no—
399C2—81no—
400C2—81no—
401C2—81no—
402C2—6921no—
403C2—6921no—
404C2#2601205111yescharacteristic
405C2#2601229211yescharacteristic
406C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3198
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012070C2semidirect
2. #26012058C2semidirect
3. #26011833C2semidirect
4. #26011812C2semidirect
5. #26011808C2semidirect
6. #26011803C2semidirect
7. #26011741C2semidirect
8. #26011736C2semidirect
9. #26011724C2semidirect
10. #26011719C2semidirect
11. #26011642C2semidirect
12. #26011633C2semidirect
13. #26011633C2semidirect
14. #26011456C2semidirect
15. #447×C173C2semidirect
17. C2×#14583825C22semidirect
20. C2×#14583809C22semidirect
21. C2×C8×D173C22semidirect
24. #19149659C22semidirect
34. #19149650C22semidirect
38. C4⋊C4×D173C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
203. C173#3198semidirect
218. #447D173semidirect

Maximal subgroups

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

subgroupindex
#260120702
#260120582
#260118332
#260118122
#260118082
#260118032
#260117412
#260117362
#260117242
#260117192
#260116422
#260116332
#260114562
#447×C1732
#3198173