Finite Group Databasebeta

← #36364410

Subgroups — #36364410

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#26012057C211yesmaximal, characteristic, complemented
3#26011835C211yesmaximal, characteristic, complemented
4#26011813C211yesmaximal, characteristic, complemented
5#26011811C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011725C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011637C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011459C211yesmaximal, characteristic, complemented
15#469×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic, complemented
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C22×#13063436C2211yescharacteristic, complemented
23#19149675—21no—
24#19149665—21no—
25#19149659C2211yescharacteristic
26#19149658C2211yescharacteristic, complemented
27#19149656—21no—
28#19149654C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149652—21no—
31#19149651—21no—
32#19149650—21no—
33#19149650C2211yescharacteristic
34#19149650C2211yescharacteristic
35#19149649—21no—
36#19149647—21no—
37#19149637C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic, complemented
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic, complemented
56#106×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62#14583825—41no—
63#14583825—41no—
64C2×Dic346D411yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C2×C22⋊D173D411yescharacteristic
76D4×D173—41no—
77D4×D173—41no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82#14583815—41no—
83#14583814—41no—
84#14583811—41no—
85C2×#13063436—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173—21no—
89C4⋊Dic173C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100C4×Dic173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106D8×C173—41no—
107M16×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122Dic346—41no—
123D692—21no—
124D692—21no—
125D692—41no—
126D692—81no—
127C22⋊D173—41no—
128C22⋊D173—41no—
129C22⋊D173—42no—
130C22⋊D173—42no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C22⋊D173—81no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—42no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—21no—
144C4×D173—81no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173C2×D411yescharacteristic
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—81no—
155D4×C173—81no—
156C2×C4×C173—21no—
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×C173C22×D411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C4×C173—21no—
185C4×C173—41no—
186C4×C173—41no—
187C4×C173H(5,𝔽2)11yescharacteristic
188C4×C173H(5,𝔽2)11yescharacteristic
189Dic173—21no—
190Dic173—21no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173—81no—
197C2×C173#53411yescharacteristic
198C2×C173#57611yescharacteristic
199C2×C173#57711yescharacteristic
200D173—41no—
201D173—41no—
202D173—81no—
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#469D17311yescharacteristic, complemented
212#468—1731no—
213#466—1731no—
214#465—1731no—
215#450—1731no—
216#449—1731no—
217#449—1731no—
218#447—1731no—
219#408—1731no—
220C2×#41—1731no—
221C2×#41—1731no—
222C22×D4—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×D8—1731no—
226C2×D8C2×D17311yescharacteristic
227C2×M16C2×D17311yescharacteristic, complemented
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#121C2×D17311yescharacteristic
242#121C2×D17311yescharacteristic, complemented
243#120—3461no—
244#120—3461no—
245#117—1731no—
246C2×C4⋊C4—1731no—
247#107—1731no—
248#107—1731no—
249#106—1731no—
250#106C2×D17311yescharacteristic, complemented
251#103—1731no—
252#103—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102C2×D17311yescharacteristic
258#102C2×D17311yescharacteristic, complemented
259C22⋊C8—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262C22⋊C8—1731no—
263C24—3461no—
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—3461no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—1731no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—6921no—
283C2×D4—6921no—
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—6921no—
295M16—41no—
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⋊C4C22×D17311yescharacteristic
307C4⋊C4C22×D17311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
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—3461no—
321C23—13841no—
322C23—13841no—
323C23—1731no—
324C23—1731no—
325C23—6922no—
326C23D4×D17311yescharacteristic
327Q8—3461no—
328Q8—3461no—
329Q8—6921no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—81no—
335D4—81no—
336D4—3461no—
337D4—3461no—
338D4—13841no—
339D4—13841no—
340D4—13841no—
341D4—13841no—
342D4—6921no—
343D4—6921no—
344D4—6921no—
345D4—6922no—
346D4—6922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
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—3461no—
378C22—3461no—
379C22—3461no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22—13844no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22C2×D4×D17311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3461no—
392C4—3461no—
393C4—6921no—
394C4—6921no—
395C4—6921no—
396C4#1914972411yescharacteristic
397C4#1914972511yescharacteristic
398C2—41no—
399C2—81no—
400C2—81no—
401C2—13841no—
402C2—6921no—
403C2—6921no—
404C2#2601204911yescharacteristic
405C2#2601229511yescharacteristic
406C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3198
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #26012070C2semidirect
2. #26012057C2semidirect
3. #26011835C2semidirect
4. #26011813C2semidirect
5. #26011811C2semidirect
6. #26011803C2semidirect
7. #26011737C2semidirect
8. #26011725C2semidirect
9. #26011718C2semidirect
10. #26011643C2semidirect
11. #26011637C2semidirect
12. #26011633C2semidirect
13. #26011623C2semidirect
14. #26011459C2semidirect
15. #469×C173C2semidirect
17. C2×#14583825C22semidirect
22. C22×#13063436C22semidirect
26. #19149658C22semidirect
38. C4⋊C4×D173C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149610C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
46. #19149600C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
53. C2×M16×C173C22semidirect
55. #121×C173C22semidirect
56. #106×C173C22semidirect
58. #102×C173C22semidirect
203. C173#3198semidirect
211. #469D173semidirect
227. C2×M16C2×D173semidirect
242. #121C2×D173semidirect
250. #106C2×D173semidirect
258. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120702
#260120572
#260118352
#260118132
#260118112
#260118032
#260117372
#260117252
#260117182
#260116432
#260116372
#260116332
#260116232
#260114592
#469×C1732
#3198173