Finite Group Databasebeta

← #36365524

Subgroups — #36365524

Order 22144.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012145C211yesmaximal, characteristic, complemented
2#26011850C211yesmaximal, characteristic, complemented
3#26011819C211yesmaximal, characteristic, complemented
4#26011807C211yesmaximal, characteristic, complemented
5#26011781C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011625C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583813C2211yescharacteristic, complemented
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583810C2211yescharacteristic, complemented
21C22×#13063436C2211yescharacteristic
22#19149679C2211yescharacteristic, complemented
23#19149663—21no—
24#19149663—21no—
25#19149659—21no—
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic, complemented
28C4×D692C2211yescharacteristic, complemented
29C2×C4⋊Dic173C2211yescharacteristic
30C2×C4⋊Dic173C2211yescharacteristic, complemented
31#19149610C2211yescharacteristic, complemented
32#19149610C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149603C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149599C2211yescharacteristic, complemented
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47#19149590C2211yescharacteristic, complemented
48C2×#41×C173C2211yescharacteristic
49C22×D4×C173C2211yescharacteristic
50#124×C173C2211yescharacteristic
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173—21no—
54#121×C173C2211yescharacteristic
55#120×C173—21no—
56C4×D4×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58C2×C22⋊C4×C173—21no—
59#14583825—44no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173—21no—
67#14583813—41no—
68#14583811—22no—
69#14583811—41no—
70#14583810—41no—
71C2×#13063436—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85C24×C173—21no—
86#41×C173—22no—
87#41×C173—41no—
88C2×Q8×C173C2311yescharacteristic
89C2×D4×C173—22no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173—41no—
94C2×D4×C173D411yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C22×C4×C173—21no—
98C22×C4×C173—21no—
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113Dic346—42no—
114D692—21no—
115D692—21no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C4×D173—42no—
124#13063436—21no—
125#13063436—21no—
126#13063436—21no—
127#13063436—21no—
128C23×C173—21no—
129C23×C173—21no—
130C23×C173—42no—
131C23×C173—42no—
132C23×C173—81no—
133C23×C173C2×D411yescharacteristic
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—22no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142D4×C173—42no—
143D4×C173—81no—
144D4×C173#4112yescomplemented
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—42no—
152C2×C4×C173—81no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—22no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—84no—
170C22×C173AGL(1,ℤ/8)12yes—
171C22×C173C22×D411yescharacteristic
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C4×C173—21no—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173C2×#4111yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—22no—
187C2×C173—41no—
188C2×C173—42no—
189C2×C173—81no—
190C2×C173#54611yescharacteristic
191C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C173#57711yescharacteristic
193D173—81no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547—1731no—
197#546D17311yescharacteristic, complemented
198#489—1731no—
199#483—1731no—
200#481—1731no—
201#476—1731no—
202#469—1731no—
203#465—1731no—
204#463—1731no—
205#449—1731no—
206#447—1731no—
207#442—1731no—
208#435—1731no—
209#417—1731no—
210C2×#102—1731no—
211C2×#41—3461no—
212C2×#41C2×D17311yescharacteristic
213C22×D4C2×D17311yescharacteristic
214C2×SD16—1731no—
215C2×D8—1731no—
216C2×D8—1731no—
217C2×M16—1731no—
218C22×C8—1731no—
219C4⋊Q8—1731no—
220#124C2×D17311yescharacteristic
221#123—21no—
222#123—3461no—
223#123—3461no—
224C22⋊Q8—3461no—
225C22⋊Q8C2×D17311yescharacteristic
226#121—21no—
227#121—1731no—
228#121—1731no—
229#121C2×D17311yescharacteristic
230#120—21no—
231C4×D4—1731no—
232C4×D4C2×D17311yescharacteristic
233C4×D4C2×D17311yescharacteristic
234C2×C4⋊C4—1731no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—21no—
237#106—1731no—
238#106—1731no—
239#106—1731no—
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251C22⋊C8—1731no—
252C22⋊C8—1731no—
253#97—1731no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—6924no—
258C2×Q8—3461no—
259C2×Q8C22×D17311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—1731no—
268C2×D4C22⋊D17311yescharacteristic
269C2×D4C22×D17311yescharacteristic
270C2×D4C22×D17311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4C22×D17311yescharacteristic
277C22×C4C22×D17311yescharacteristic
278SD16—6921no—
279D8—3462no—
280D8—6921no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—1731no—
286C2×C8—6921no—
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4C22⋊D17311yescharacteristic
295C4⋊C4C22×D17311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4C22⋊D17311yescharacteristic
305C22⋊C4C22⋊D17311yescharacteristic
306C42C22×D17311yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—3461no—
313C23C2×C22⋊D17311yescharacteristic
314C23C2×C22⋊D17311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6922no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3461no—
326D4—3461no—
327D4—6922no—
328D4—6922no—
329D4#1458382512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—6922no—
342C2×C4—6922no—
343C2×C4—6922no—
344C2×C4C2×C22⋊D17311yescharacteristic
345C2×C4C2×C22⋊D17311yescharacteristic
346C2×C4C2×C22⋊D17311yescharacteristic
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C23×D17311yescharacteristic
349C8—3461no—
350C8—3461no—
351C8—3461no—
352C8—3461no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22#1914970412yes—
369C22C22×C22⋊D17311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—6921no—
377C4—6921no—
378C4—6921no—
379C4C2×#1458382511yescharacteristic
380C4#1914972411yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—13841no—
386C2#2601213511yescharacteristic
387C2C2×#1914970411yescharacteristic
388C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #26012145C2semidirect
2. #26011850C2semidirect
3. #26011819C2semidirect
4. #26011807C2semidirect
5. #26011781C2semidirect
6. #26011742C2semidirect
7. #26011722C2semidirect
8. #26011703C2semidirect
9. #26011636C2semidirect
10. #26011625C2semidirect
11. #26011601C2semidirect
12. #26011563C2semidirect
13. #26011492C2semidirect
14. C2×#19149600C2semidirect
15. #546×C173C2semidirect
17. C2×#14583813C22semidirect
20. C2×#14583810C22semidirect
22. #19149679C22semidirect
27. #19149653C22semidirect
28. C4×D692C22semidirect
30. C2×C4⋊Dic173C22semidirect
31. #19149610C22semidirect
32. #19149610C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149603C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
44. #19149599C22semidirect
46. C22⋊#13063436C22semidirect
47. #19149590C22semidirect
144. D4×C173#41semidirect
194. C173#3306semidirect
197. #546D173semidirect
329. D4#14583825semidirect

Maximal subgroups

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

subgroupindex
#260121452
#260118502
#260118192
#260118072
#260117812
#260117422
#260117222
#260117032
#260116362
#260116252
#260116012
#260115632
#260114922
C2×#191496002
#546×C1732
#3306173