Finite Group Databasebeta

← #36365416

Subgroups — #36365416

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#26012138C211yesmaximal, characteristic, complemented
3#26011844C211yesmaximal, characteristic, complemented
4#26011828C211yesmaximal, characteristic, complemented
5#26011790C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011638C211yesmaximal, characteristic, complemented
10#26011626C211yesmaximal, characteristic, complemented
11#26011573C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011485C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic, complemented
19C2×D1384C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C22×#13063436C2211yescharacteristic, complemented
22C2×C8×D173C2211yescharacteristic
23#19149673—21no—
24#19149670C2211yescharacteristic, complemented
25#19149666C2211yescharacteristic
26#19149661—21no—
27#19149661—21no—
28#19149659C2211yescharacteristic
29#19149654C2211yescharacteristic
30#19149653C2211yescharacteristic, complemented
31#19149652—21no—
32#19149650—21no—
33#19149647—21no—
34C4×D692—21no—
35#19149641C2211yescharacteristic
36#19149641C2211yescharacteristic, complemented
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39C22⋊C4×D173—21no—
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic, complemented
52#121×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic, complemented
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70D4×D173—22no—
71D4×D173—41no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583815—41no—
78D1384—22no—
79#14583811—41no—
80C2×#13063436—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173—21no—
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173D411yescharacteristic
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803D411yescharacteristic
99C4×Dic173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
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
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—42no—
123D692#4112yes—
124C22⋊D173—22no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C22⋊D173—81no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136C4×D173—81no—
137C4×D173—81no—
138C4×D173—81no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142C23×C173C2×D411yescharacteristic
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—81no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—22no—
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—41no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173C2×#13511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—22no—
189D173—42no—
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
195#488—1731no—
196#484—1731no—
197#482D17311yescharacteristic, complemented
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#416—1731no—
207C2×#102—1731no—
208C2×#41—1731no—
209C22×D4—1731no—
210C2×SD16—1731no—
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8—1731no—
214C22×C8—1731no—
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#121C2×D17311yescharacteristic
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#107C2×D17311yescharacteristic, complemented
236#106C2×D17311yescharacteristic, complemented
237C4⋊C8—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
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⋊C8C2×D17311yescharacteristic, complemented
250C4×C8—1731no—
251C24—3461no—
252#41—3462no—
253#41—6921no—
254C2×Q8—1731no—
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—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—1731no—
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—3462no—
274D8—6921no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—6921no—
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⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
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—3461no—
312Q8—3461no—
313D4—21no—
314D4—21no—
315D4—81no—
316D4—3462no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—13841no—
321D4—1732no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—6922no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—3461no—
354C22—3461no—
355C22—3462no—
356C22—13844no—
357C22—1732no—
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—
373C4#41×D17311yescharacteristic
374C4#1914972511yescharacteristic
375C2—41no—
376C2—81no—
377C2—3462no—
378C2—13841no—
379C2—6922no—
380C2#2601213311yescharacteristic
381C2#135×D17311yescharacteristic
382C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
hypercenter#482D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #26012185C2semidirect
2. #26012138C2semidirect
3. #26011844C2semidirect
4. #26011828C2semidirect
5. #26011790C2semidirect
6. #26011725C2semidirect
7. #26011721C2semidirect
8. #26011705C2semidirect
9. #26011638C2semidirect
10. #26011626C2semidirect
11. #26011573C2semidirect
12. #26011554C2semidirect
13. #26011485C2semidirect
14. #102×D173C2semidirect
15. #482×C173C2semidirect
18. C2×#14583815C22semidirect
21. C22×#13063436C22semidirect
24. #19149670C22semidirect
30. #19149653C22semidirect
36. #19149641C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
44. #19149602C22semidirect
49. #19149599C22semidirect
51. C8×Dic173C22semidirect
53. #117×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
58. C22⋊C8×C173C22semidirect
191. C173#3297semidirect
197. #482D173semidirect
231. #117C2×D173semidirect
235. #107C2×D173semidirect
236. #106C2×D173semidirect
249. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121852
#260121382
#260118442
#260118282
#260117902
#260117252
#260117212
#260117052
#260116382
#260116262
#260115732
#260115542
#260114852
#102×D1732
#482×C1732
#3297173