Finite Group Databasebeta

← #36365530

Subgroups — #36365530

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#26012141C211yesmaximal, characteristic, complemented
2#26012132C211yesmaximal, characteristic, complemented
3#26011852C211yesmaximal, characteristic, complemented
4#26011819C211yesmaximal, characteristic, complemented
5#26011811C211yesmaximal, characteristic, complemented
6#26011780C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011635C211yesmaximal, characteristic, complemented
10#26011624C211yesmaximal, characteristic, complemented
11#26011600C211yesmaximal, characteristic, complemented
12#26011562C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825—21no—
18C22×D692C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583809C2211yescharacteristic, complemented
22#19149678C2211yescharacteristic, complemented
23#19149668C2211yescharacteristic
24#19149664—21no—
25#19149661—21no—
26#19149660—21no—
27#19149656—21no—
28#19149654—21no—
29#19149654C2211yescharacteristic
30#19149650C2211yescharacteristic, complemented
31#19149649—21no—
32D4×Dic173C2211yescharacteristic, complemented
33#19149641C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35C2×C4⋊Dic173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37C22⋊C4×D173—21no—
38#19149610C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—41no—
62#14583825—44no—
63C2×Dic346—21no—
64C2×D692—22no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—41no—
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173D411yescharacteristic
73C22×Dic173—21no—
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79#14583811—22no—
80#14583811—41no—
81#14583809—41no—
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
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#14583803D411yescharacteristic
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103SD16×C173—41no—
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173C2×D411yescharacteristic
118Dic346—42no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—42no—
123D692—42no—
124D692—81no—
125D692#4112yescomplemented
126C22⋊D173—22no—
127C22⋊D173—42no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—42no—
131C2×Dic173—42no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—42no—
138C4×D173—42no—
139C4×D173—81no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—42no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—22no—
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—84no—
175C4×C173—21no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C2×#4111yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—22no—
187C2×C173—81no—
188C2×C173#54611yescharacteristic
189C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C173#57711yescharacteristic
191D173—41no—
192D173—42no—
193D173—81no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547—1731no—
197#546—1731no—
198#489—1731no—
199#483—1731no—
200#481—1731no—
201#476—1731no—
202#469—1731no—
203#465D17311yescharacteristic, complemented
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×#41—1731no—
213C22×D4—1731no—
214C2×SD16C2×D17311yescharacteristic, complemented
215C2×D8—1731no—
216C2×D8—1731no—
217C2×M16—1731no—
218C22×C8C2×D17311yescharacteristic
219C4⋊Q8—1731no—
220#124—1731no—
221#123—3461no—
222#123—3461no—
223#123—3461no—
224C22⋊Q8—3461no—
225C22⋊Q8C2×D17311yescharacteristic
226#121—3461no—
227#121—1731no—
228#121—1731no—
229#121C2×D17311yescharacteristic
230#120—3461no—
231C4×D4—1731no—
232C4×D4—1731no—
233C4×D4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—3461no—
237#106—1731no—
238#106—1731no—
239#106C2×D17311yescharacteristic, complemented
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103C2×D17311yescharacteristic, complemented
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102C2×D17311yescharacteristic, complemented
251C22⋊C8—1731no—
252C22⋊C8—1731no—
253#97—1731no—
254C24—3461no—
255#41—3462no—
256#41—6921no—
257#41—6924no—
258C2×Q8—3461no—
259C2×Q8C22×D17311yescharacteristic
260C2×D4—21no—
261C2×D4—3462no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
270C2×D4C22×D17311yescharacteristic
271C22×C4—3461no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4—1731no—
277C22×C4C22×D17311yescharacteristic
278SD16—41no—
279D8—3462no—
280D8—6921no—
281M16—6921no—
282C2×C8—41no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
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—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—1731no—
305C22⋊C4—1731no—
306C42—1731no—
307C23—21no—
308C23—3461no—
309C23—3461no—
310C23—13841no—
311C23—1731no—
312C23—6922no—
313C23—6922no—
314C23D4×D17311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6922no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—3462no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—13841no—
326D4—1732no—
327D4—6922no—
328D4—6922no—
329D4—6922no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—13841no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—1731no—
343C2×C4—6922no—
344C2×C4—6922no—
345C2×C4—6922no—
346C2×C4—6922no—
347C2×C4D4×D17311yescharacteristic
348C2×C4C23×D17311yescharacteristic
349C8—21no—
350C8—21no—
351C8—3461no—
352C8—3461no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—3461no—
359C22—3461no—
360C22—3461no—
361C22—3462no—
362C22—13844no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22#1914970612yes—
369C22C2×D4×D17311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3461no—
375C4—3461no—
376C4—3461no—
377C4—6921no—
378C4—6921no—
379C4#41×D17311yescharacteristic
380C4#1914972411yescharacteristic
381C2—22no—
382C2—81no—
383C2—13841no—
384C2—6921no—
385C2—6922no—
386C2#2601213011yescharacteristic
387C2C2×#1914970611yescharacteristic
388C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012141C2semidirect
2. #26012132C2semidirect
3. #26011852C2semidirect
4. #26011819C2semidirect
5. #26011811C2semidirect
6. #26011780C2semidirect
7. #26011743C2semidirect
8. #26011708C2semidirect
9. #26011635C2semidirect
10. #26011624C2semidirect
11. #26011600C2semidirect
12. #26011562C2semidirect
13. #26011491C2semidirect
14. C2×#19149601C2semidirect
15. #465×C173C2semidirect
21. C2×#14583809C22semidirect
22. #19149678C22semidirect
30. #19149650C22semidirect
32. D4×Dic173C22semidirect
36. C4⋊C4×D173C22semidirect
38. #19149610C22semidirect
39. #19149610C22semidirect
40. #19149608C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
52. C2×SD16×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
125. D692#41semidirect
194. C173#3306semidirect
203. #465D173semidirect
214. C2×SD16C2×D173semidirect
239. #106C2×D173semidirect
243. #103C2×D173semidirect
250. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121412
#260121322
#260118522
#260118192
#260118112
#260117802
#260117432
#260117082
#260116352
#260116242
#260116002
#260115622
#260114912
C2×#191496012
#465×C1732
#3306173