Finite Group Databasebeta

← #36365414

Subgroups — #36365414

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012185C211yesmaximal, characteristic, complemented
2#26012129C211yesmaximal, characteristic, complemented
3#26011847C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011815C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011727C211yesmaximal, characteristic, complemented
8#26011723C211yesmaximal, characteristic, complemented
9#26011709C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011571C211yesmaximal, characteristic, complemented
13C8×C22⋊D173C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583813C2211yescharacteristic, complemented
18C2×D1384C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149673—21no—
22#19149670C2211yescharacteristic, complemented
23#19149668C2211yescharacteristic
24#19149662—21no—
25#19149661—21no—
26#19149655C2211yescharacteristic
27#19149654C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic, complemented
30#19149652—21no—
31#19149651—21no—
32#19149647—21no—
33C4×D692—21no—
34D4×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic, complemented
36C4×C22⋊D173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38C2×#14583803—21no—
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54#117×C173C2211yescharacteristic, complemented
55#103×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×D692—21no—
61C2×D692C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67D4×D173—22no—
68D4×D173—41no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583813—41no—
74D1384—41no—
75#14583811—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173D411yescharacteristic
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117D692—21no—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—42no—
122C22⋊D173—22no—
123C22⋊D173—42no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—81no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—81no—
135C4×D173—81no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×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—21no—
150C2×C4×C173—42no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—22no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—84no—
171C2×D173#13512yes—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173C2×#4111yescharacteristic
177C4×C173H(5,𝔽2)11yescharacteristic
178Dic173—21no—
179Dic173—21no—
180Dic173—21no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—42no—
184C2×C173—41no—
185C2×C173#54611yescharacteristic
186C2×C173C2×#13511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—22no—
189D173—81no—
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
195#488—1731no—
196#484—1731no—
197#482—1731no—
198#478—1731no—
199#466—1731no—
200#465—1731no—
201#463—1731no—
202#449—1731no—
203#447—1731no—
204C4×D8—1731no—
205C8×D4—1731no—
206#416D17311yescharacteristic, complemented
207C2×#102—1731no—
208C2×#41C2×D17311yescharacteristic
209C22×D4—1731no—
210C2×SD16—1731no—
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8—1731no—
214C22×C8C2×D17311yescharacteristic, complemented
215#126—3461no—
216#125—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8—1731no—
221#121—3461no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121—1731no—
226#120—3461no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4—1731no—
231#117C2×D17311yescharacteristic, complemented
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237C4⋊C8—1731no—
238#103—1731no—
239#103C2×D17311yescharacteristic, complemented
240#103C2×D17311yescharacteristic, complemented
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102C2×D17311yescharacteristic
247#102C2×D17311yescharacteristic
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250C4×C8—1731no—
251C24—3461no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—21no—
256C2×D4—3461no—
257C2×D4—3462no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—6921no—
265C2×D4C22×D17311yescharacteristic
266C22×C4—21no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—1731no—
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—3462no—
274D8—6921no—
275C2×C8—41no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8—6921no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—1731no—
300C22⋊C4—1731no—
301C42—21no—
302C42—1731no—
303C23—21no—
304C23—3461no—
305C23—3461no—
306C23—13841no—
307C23—1731no—
308C23—6922no—
309C23—6922no—
310C23D4×D17311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—3462no—
316D4—3461no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—13841no—
321D4—13841no—
322D4—6922no—
323D4—6922no—
324D4#1458382612yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4—13841no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—1731no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—3461no—
354C22—3461no—
355C22—13844no—
356C22—1732no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22—6921no—
362C22—6921no—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3461no—
369C4—3461no—
370C4—3461no—
371C4—6921no—
372C4—6921no—
373C4C2×#1458382611yescharacteristic
374C4#1914972411yescharacteristic
375C2—41no—
376C2—42no—
377C2—3462no—
378C2—13841no—
379C2—13841no—
380C2#2601213111yescharacteristic
381C2#135×D17311yescharacteristic
382C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
hypercenter#416D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #26012185C2semidirect
2. #26012129C2semidirect
3. #26011847C2semidirect
4. #26011825C2semidirect
5. #26011815C2semidirect
6. #26011789C2semidirect
7. #26011727C2semidirect
8. #26011723C2semidirect
9. #26011709C2semidirect
10. #26011636C2semidirect
11. #26011626C2semidirect
12. #26011571C2semidirect
13. C8×C22⋊D173C2semidirect
14. #102×D173C2semidirect
15. #416×C173C2semidirect
17. C2×#14583813C22semidirect
22. #19149670C22semidirect
29. #19149653C22semidirect
35. #19149641C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149610C22semidirect
42. #19149608C22semidirect
43. #19149603C22semidirect
45. #19149601C22semidirect
49. C22⋊#13063436C22semidirect
51. C8×Dic173C22semidirect
53. C22×C8×C173C22semidirect
54. #117×C173C22semidirect
55. #103×C173C22semidirect
56. #103×C173C22semidirect
191. C173#3297semidirect
206. #416D173semidirect
214. C22×C8C2×D173semidirect
231. #117C2×D173semidirect
239. #103C2×D173semidirect
240. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121852
#260121292
#260118472
#260118252
#260118152
#260117892
#260117272
#260117232
#260117092
#260116362
#260116262
#260115712
C8×C22⋊D1732
#102×D1732
#416×C1732
#3297173