Finite Group Databasebeta

← #36364437

Subgroups — #36364437

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012067C211yesmaximal, characteristic, complemented
2#26012057C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011815C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011721C211yesmaximal, characteristic, complemented
9#26011719C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011643C211yesmaximal, characteristic, complemented
12#26011632C211yesmaximal, characteristic, complemented
13#26011629C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic, complemented
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583809C2211yescharacteristic, complemented
24C2×C8×D173C2211yescharacteristic, complemented
25#19149679—21no—
26#19149675—21no—
27#19149667—21no—
28#19149665—21no—
29#19149663—21no—
30#19149663—21no—
31#19149659C2211yescharacteristic
32#19149659C2211yescharacteristic
33#19149659C2211yescharacteristic, complemented
34#19149659C2211yescharacteristic, complemented
35#19149654C2211yescharacteristic
36#19149653C2211yescharacteristic
37#19149650C2211yescharacteristic
38#19149649—21no—
39#19149647—21no—
40#19149637C2211yescharacteristic
41#19149612C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
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—41no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346—41no—
65C2×Dic346D411yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692—21no—
69C2×D692D411yescharacteristic
70C2×D692C2311yescharacteristic
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×D173C2311yescharacteristic
80C2×C4×D173C2311yescharacteristic
81#14583815—41no—
82#14583815—41no—
83#14583814—41no—
84#14583814—41no—
85#14583809—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—41no—
89C4⋊Dic173—21no—
90C4⋊Dic173—21no—
91C4⋊Dic173—21no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94C22⋊Dic173—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102C4×Dic173—21no—
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×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—21no—
123Dic346—21no—
124Dic346—41no—
125Dic346—42no—
126D692—21no—
127D692—21no—
128D692—41no—
129D692—81no—
130C22⋊D173—41no—
131C22⋊D173—41no—
132C22⋊D173—42no—
133C22⋊D173—81no—
134C22⋊D173—81no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—42no—
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—21no—
144C4×D173—21no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173C2×D411yescharacteristic
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—81no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—84no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173H(5,𝔽2)11yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—41no—
190C2×C173—81no—
191C2×C173#53411yescharacteristic
192C2×C173#57711yescharacteristic
193C2×C173#57711yescharacteristic
194D173—41no—
195D173—41no—
196D173—81no—
197C173#320011yescharacteristic, complemented, Sylow
198#3200—1731nomaximal, Sylow
199#536—1731no—
200#535—1731no—
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482D17311yescharacteristic, complemented
205#468—1731no—
206#468—1731no—
207#465—1731no—
208#465—1731no—
209#450—1731no—
210#450—1731no—
211#448—1731no—
212#448—1731no—
213#408—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×D4—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×SD16—1731no—
221C2×M16—1731no—
222C22×C8—1731no—
223C4⋊Q8—3461no—
224#127—3461no—
225#124—3461no—
226#124—3461no—
227#123—3461no—
228#123—3461no—
229C22⋊Q8—1731no—
230C22⋊Q8—1731no—
231C22⋊Q8—1731no—
232C22⋊Q8—1731no—
233#121—1731no—
234#121—1731no—
235#121—1731no—
236#121C2×D17311yescharacteristic
237#120—3461no—
238#120—3461no—
239#117—1731no—
240#117C2×D17311yescharacteristic, complemented
241#107—1731no—
242#107C2×D17311yescharacteristic, complemented
243#106—1731no—
244#106C2×D17311yescharacteristic, complemented
245#103—1731no—
246#103—1731no—
247#103—1731no—
248#103—1731no—
249#102—1731no—
250#102—1731no—
251#102C2×D17311yescharacteristic
252#102C2×D17311yescharacteristic
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8C2×D17311yescharacteristic, complemented
257C24—3461no—
258#41—6921no—
259#41—6921no—
260C2×Q8—3461no—
261C2×Q8—1731no—
262C2×Q8—1731no—
263C2×Q8—1731no—
264C2×Q8—6921no—
265C2×Q8—6921no—
266C2×D4—21no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4—6921no—
276C2×D4—6921no—
277C2×D4C22×D17311yescharacteristic
278C22×C4—3461no—
279C22×C4—1731no—
280C22×C4—1731no—
281C22×C4C22×D17311yescharacteristic
282SD16—6921no—
283SD16—6921no—
284SD16—6921no—
285SD16—6921no—
286M16—6921no—
287C2×C8—1731no—
288C2×C8—1731no—
289C2×C8—6921no—
290C2×C8C22×D17311yescharacteristic
291C2×C8C22×D17311yescharacteristic
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4C22×D17311yescharacteristic
298C4⋊C4C22×D17311yescharacteristic
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—21no—
311C42—3461no—
312C23—21no—
313C23—3461no—
314C23—13841no—
315C23—13841no—
316C23—1731no—
317C23—1731no—
318C23—6922no—
319C23D4×D17311yescharacteristic
320Q8—3461no—
321Q8—3461no—
322Q8—3461no—
323Q8—3461no—
324Q8—6921no—
325Q8—6922no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—3461no—
330D4—3461no—
331D4—13841no—
332D4—13841no—
333D4—13841no—
334D4—6921no—
335D4—6921no—
336D4—6921no—
337D4—6922no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—1731no—
351C2×C4—1731no—
352C2×C4—6922no—
353C2×C4D4×D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—21no—
356C8—21no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—13844no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4#1914972511yescharacteristic
386C4#1914972511yescharacteristic
387C2—41no—
388C2—81no—
389C2—13841no—
390C2—6921no—
391C2—6921no—
392C2#2601205211yescharacteristic
393C2#2601229811yescharacteristic
394C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
prime radical#26012057C2
hypercenter#482D173
hypocenterC173#3200
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. #26012067C2semidirect
3. #26011835C2semidirect
4. #26011834C2semidirect
5. #26011815C2semidirect
6. #26011738C2semidirect
7. #26011736C2semidirect
8. #26011721C2semidirect
9. #26011719C2semidirect
10. #26011643C2semidirect
11. #26011643C2semidirect
12. #26011632C2semidirect
13. #26011629C2semidirect
14. #26011456C2semidirect
15. #482×C173C2semidirect
17. C22×Dic346C22semidirect
23. C2×#14583809C22semidirect
24. C2×C8×D173C22semidirect
33. #19149659C22semidirect
34. #19149659C22semidirect
41. #19149612C22semidirect
42. #19149610C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
49. C22⋊#13063436C22semidirect
53. #117×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
58. C22⋊C8×C173C22semidirect
197. C173#3200semidirect
204. #482D173semidirect
240. #117C2×D173semidirect
242. #107C2×D173semidirect
244. #106C2×D173semidirect
256. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120672
#260120572
#260118352
#260118342
#260118152
#260117382
#260117362
#260117212
#260117192
#260116432
#260116322
#260116292
#260114562
#482×C1732
#3200173