Finite Group Databasebeta

← #36365419

Subgroups — #36365419

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012193C211yesmaximal, characteristic, complemented
2#26012132C211yesmaximal, characteristic, complemented
3#26011847C211yesmaximal, characteristic, complemented
4#26011824C211yesmaximal, characteristic, complemented
5#26011817C211yesmaximal, characteristic, complemented
6#26011790C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011635C211yesmaximal, characteristic, complemented
10#26011624C211yesmaximal, characteristic, complemented
11#26011571C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic, complemented
21#19149672—21no—
22#19149670C2211yescharacteristic, complemented
23#19149668C2211yescharacteristic
24#19149664—21no—
25#19149663—21no—
26#19149654—21no—
27#19149654C2211yescharacteristic
28#19149651—21no—
29#19149650C2211yescharacteristic, complemented
30#19149649—21no—
31D4×Dic173—21no—
32D4×Dic173C2211yescharacteristic, complemented
33#19149641C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149637C2211yescharacteristic, complemented
36C2×C4⋊Dic173C2211yescharacteristic
37C22⋊C4×D173—21no—
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, 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
51C8×Dic173C2211yescharacteristic, 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—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—41no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583811—22no—
77#14583811—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102SD16×C173—41no—
103C2×C8×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117D692—21no—
118D692—21no—
119D692—21no—
120D692—42no—
121D692—42no—
122D692—81no—
123D692#4112yes—
124C22⋊D173—22no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—81no—
129C2×Dic173—81no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136C4×D173—81no—
137#13063436—21no—
138#13063436—21no—
139C23×C173—21no—
140C23×C173C2×D411yescharacteristic
141Q8×C173—21no—
142Q8×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—42no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173#13512yes—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—22no—
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—84no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173C2×#4111yescharacteristic
177C4×C173H(5,𝔽2)11yescharacteristic
178Dic173—21no—
179Dic173—21no—
180Dic173—21no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—22no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173C2×#13511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—41no—
189D173—42no—
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
195#488—1731no—
196#484—1731no—
197#482—1731no—
198#478—1731no—
199#466—1731no—
200#465D17311yescharacteristic, complemented
201#463—1731no—
202#449—1731no—
203#447—1731no—
204C4×D8—1731no—
205C8×D4—1731no—
206#416—1731no—
207C2×#102—1731no—
208C2×#41—1731no—
209C22×D4—1731no—
210C2×SD16C2×D17311yescharacteristic, complemented
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8—1731no—
214C22×C8C2×D17311yescharacteristic
215#126—3461no—
216#125—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8C2×D17311yescharacteristic
221#121—3461no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121C2×D17311yescharacteristic
226#120—3461no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4—1731no—
231#117—1731no—
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
234#107—1731no—
235#107—1731no—
236#106C2×D17311yescharacteristic, complemented
237C4⋊C8—1731no—
238#103—1731no—
239#103—1731no—
240#103C2×D17311yescharacteristic, complemented
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102C2×D17311yescharacteristic, complemented
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250C4×C8—1731no—
251C24—3461no—
252#41—3462no—
253#41—6921no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—21no—
256C2×D4—3462no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—6921no—
265C2×D4C22×D17311yescharacteristic
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—1731no—
271C22×C4C22×D17311yescharacteristic
272SD16—41no—
273D8—3462no—
274D8—6921no—
275C2×C8—41no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8—6921no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—1731no—
300C22⋊C4—1731no—
301C42—3461no—
302C42—1731no—
303C23—21no—
304C23—3461no—
305C23—3461no—
306C23—13841no—
307C23—1731no—
308C23—6922no—
309C23—6922no—
310C23D4×D17311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—3462no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—13841no—
321D4—13841no—
322D4—1732no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—6922no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—3461no—
353C22—3461no—
354C22—3461no—
355C22—3462no—
356C22—13844no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22—6921no—
362C22#1914970012yes—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3461no—
369C4—3461no—
370C4—3461no—
371C4—6921no—
372C4—6921no—
373C4#41×D17311yescharacteristic
374C4#1914972411yescharacteristic
375C2—22no—
376C2—81no—
377C2—13841no—
378C2—6921no—
379C2—6922no—
380C2#2601213011yescharacteristic
381C2C2×#1914970011yescharacteristic
382C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3297
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. #26012193C2semidirect
2. #26012132C2semidirect
3. #26011847C2semidirect
4. #26011824C2semidirect
5. #26011817C2semidirect
6. #26011790C2semidirect
7. #26011726C2semidirect
8. #26011708C2semidirect
9. #26011635C2semidirect
10. #26011624C2semidirect
11. #26011571C2semidirect
12. C8×C22⋊D173C2semidirect
13. #26011486C2semidirect
14. C2×#19149601C2semidirect
15. #465×C173C2semidirect
20. C2×C8×D173C22semidirect
22. #19149670C22semidirect
29. #19149650C22semidirect
32. D4×Dic173C22semidirect
35. #19149637C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149608C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
52. C2×SD16×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
191. C173#3297semidirect
200. #465D173semidirect
210. C2×SD16C2×D173semidirect
236. #106C2×D173semidirect
240. #103C2×D173semidirect
247. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121932
#260121322
#260118472
#260118242
#260118172
#260117902
#260117262
#260117082
#260116352
#260116242
#260115712
C8×C22⋊D1732
#260114862
C2×#191496012
#465×C1732
#3297173