Finite Group Databasebeta

← #36365500

Subgroups — #36365500

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012209C211yesmaximal, characteristic, complemented
2#26012159C211yesmaximal, characteristic
3#26011836C211yesmaximal, characteristic, complemented
4#26011818C211yesmaximal, characteristic, complemented
5#26011775C211yesmaximal, characteristic, complemented
6#26011721C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011690C211yesmaximal, characteristic, complemented
9#26011655C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011585C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011485C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149675—21no—
25#19149675C2211yescharacteristic
26#19149673—21no—
27#19149672—21no—
28#19149665—21no—
29#19149659C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149654C2211yescharacteristic
32#19149652—21no—
33#19149651—21no—
34C4×Dic346—21no—
35#19149644C2211yescharacteristic
36#19149644C2211yescharacteristic
37C4×C22⋊D173C2211yescharacteristic
38#19149639—21no—
39#19149637C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#121×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173D411yescharacteristic
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76Dic692—22no—
77#14583815—41no—
78#14583814—41no—
79C2×#13063436—41no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C8×D173—41no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
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×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346#4112yes—
118D692—21no—
119D692—21no—
120D692—81no—
121C22⋊D173—22no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—42no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—81no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—22no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173#13512yes—
177C2×C173—41no—
178C2×C173—81no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—41no—
183D173—81no—
184C173#330411yescharacteristic, complemented, Sylow
185#3304—1731nomaximal, Sylow
186#556—1731no—
187#550—1731no—
188#486—1731no—
189#482—1731no—
190#482D17311yescharacteristic, complemented
191#475—1731no—
192#465—1731no—
193#465—1731no—
194#460—1731no—
195#452—1731no—
196#452—1731no—
197C4×Q16—1731no—
198C8×D4—1731no—
199#416—1731no—
200#416—1731no—
201C2×#41—1731no—
202C2×#41—1731no—
203C2×Q16—1731no—
204C2×SD16—1731no—
205C2×SD16—1731no—
206C22×C8—1731no—
207C22×C8—1731no—
208C4⋊Q8—1731no—
209#127—3461no—
210#127—1731no—
211#126—3461no—
212#126—3461no—
213#124—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8—3461no—
221C4×Q8—1731no—
222C4×Q8—1731no—
223C4×D4—3461no—
224C4×D4—1731no—
225#117—1731no—
226#117C2×D17311yescharacteristic
227#107C2×D17311yescharacteristic
228#106—1731no—
229#106C2×D17311yescharacteristic
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102C2×D17311yescharacteristic
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8C2×D17311yescharacteristic
243C4×C8—1731no—
244#41—3462no—
245#41—3462no—
246#41—6921no—
247#41—6921no—
248C2×Q8—3461no—
249C2×Q8—1731no—
250C2×Q8—1731no—
251C2×D4—21no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4C22×D17311yescharacteristic
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264Q16—3462no—
265SD16—6921no—
266SD16—6921no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C4⋊C4C22×D17311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—1731no—
290C22⋊C4—1731no—
291C42—21no—
292C42—3461no—
293C42—3461no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—3461no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—3461no—
301Q8—3461no—
302Q8—3461no—
303Q8—1732no—
304Q8—6922no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—3462no—
309D4—3461no—
310D4—3461no—
311D4—13841no—
312D4—13841no—
313D4—6922no—
314D4—6922no—
315D4—6922no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—3462no—
357C4—1732no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972511yescharacteristic
362C2—41no—
363C2—81no—
364C2—13841no—
365C2—6921no—
366C2#2601213311yescharacteristic
367C2#135×D17311yescharacteristic
368C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
prime radical#26012159C2
hypercenter#482D173
hypocenterC173#3304
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. #26012209C2semidirect
3. #26011836C2semidirect
4. #26011818C2semidirect
5. #26011775C2semidirect
6. #26011721C2semidirect
7. #26011718C2semidirect
8. #26011690C2semidirect
9. #26011655C2semidirect
10. #26011654C2semidirect
11. #26011585C2semidirect
12. #26011554C2semidirect
13. #26011485C2semidirect
14. #26011484C2semidirect
15. #482×C173C2semidirect
184. C173#3304semidirect
190. #482D173semidirect

Maximal subgroups

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

subgroupindex
#260122092
#260121592
#260118362
#260118182
#260117752
#260117212
#260117182
#260116902
#260116552
#260116542
#260115852
#260115542
#260114852
#260114842
#482×C1732
#3304173