Finite Group Databasebeta

← #36365070

Subgroups — #36365070

Order 22144.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C22×#14583815C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012132C211yesmaximal, characteristic
5#26011884C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011695C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011646C211yesmaximal, characteristic, complemented
11#26011641C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19C22×D692C2211yescharacteristic
20#19149711—24no—
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815—24no—
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic
26C2×#14583815C2211yescharacteristic
27C2×#14583815C2211yescharacteristic
28C2×#14583813C2211yescharacteristic
29C2×#14583813C2211yescharacteristic
30C2×#14583809C2211yescharacteristic, complemented
31#19149678—21no—
32#19149677C2211yescharacteristic, complemented
33#19149668C2211yescharacteristic
34#19149664—21no—
35#19149659C2211yescharacteristic, complemented
36#19149657—21no—
37#19149656—21no—
38#19149654—21no—
39#19149654C2211yescharacteristic
40#19149649—21no—
41#19149644C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic
43C4×C22⋊D173C2211yescharacteristic
44#19149639—21no—
45C22⋊C4×D173—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic
50#19149599C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic, complemented
53#19149589C2211yescharacteristic, complemented
54C2×Q16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#103×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64C2×Dic346—21no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70Q8×D173—22no—
71Q8×D173—41no—
72C2×D692—22no—
73C2×D692—21no—
74C2×D692—21no—
75C2×D692—21no—
76C2×D692—41no—
77C2×D692C2311yescharacteristic
78C2×D692C2311yescharacteristic
79C2×C22⋊D173D411yescharacteristic
80C22×Dic173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85#14583816—22no—
86#14583816—41no—
87#14583815—22no—
88#14583815—22no—
89#14583815—22no—
90#14583815D414yescomplemented
91#14583813—22no—
92#14583813—41no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C4⋊Dic173—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113Q16×C173—22no—
114C2×C8×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C4⋊C4×C173—21no—
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—42no—
133Dic346—42no—
134Dic346C2×D412yes—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—42no—
139D692—42no—
140D692—81no—
141D692C2×D412yes—
142C22⋊D173—22no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—42no—
146C2×Dic173C2×D411yescharacteristic
147C2×Dic173C2×D411yescharacteristic
148C2×Dic173C2×D411yescharacteristic
149C4×D173—21no—
150C4×D173—21no—
151C4×D173—42no—
152C4×D173—42no—
153C4×D173—81no—
154#13063436—21no—
155#13063436—21no—
156C23×C173C2×D411yescharacteristic
157Q8×C173—21no—
158Q8×C173—21no—
159Q8×C173—21no—
160Q8×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173C2×D412yes—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173#13712yescomplemented
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—22no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—84no—
185C4×C173—21no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173C22×D411yescharacteristic
191C4×C173H(5,𝔽2)11yescharacteristic
192Dic173—21no—
193Dic173—22no—
194Dic173—21no—
195Dic173—21no—
196Dic173—41no—
197C2×C173—22no—
198C2×C173D4211yescharacteristic
199C2×C173C2×#13711yescharacteristic
200C2×C173#57711yescharacteristic
201D173—41no—
202D173—42no—
203D173—81no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8—1731no—
209#546—1731no—
210#497—1731no—
211#468—1731no—
212#467D17311yescharacteristic, complemented
213#464—1731no—
214#461—1731no—
215#452—1731no—
216#451—1731no—
217#450—1731no—
218#448—1731no—
219#440—1731no—
220#435—1731no—
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8—1731no—
224C22×D4—1731no—
225#137—3464no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×Q16C2×D17311yescharacteristic, complemented
229C2×SD16—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×M16—1731no—
237C22×C8C2×D17311yescharacteristic
238C4⋊Q8—3461no—
239C4⋊Q8—1731no—
240#124—1731no—
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8C2×D17311yescharacteristic
246C22⋊Q8C2×D17311yescharacteristic
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4—1731no—
254C2×C22⋊C4—3461no—
255#107C2×D17311yescharacteristic, complemented
256C4⋊C8—1731no—
257#103—1731no—
258#103—1731no—
259#103C2×D17311yescharacteristic, complemented
260#103C2×D17311yescharacteristic, complemented
261#102—1731no—
262#102—1731no—
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265#97—1731no—
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3462no—
271C2×Q8—3461no—
272C2×Q8—3462no—
273C2×Q8—1731no—
274C2×Q8—1731no—
275C2×Q8—6921no—
276C2×Q8—6921no—
277C2×Q8C22×D17311yescharacteristic
278C2×Q8C22×D17311yescharacteristic
279C2×D4—3462no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4—1731no—
286C2×D4—6921no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292C22×C4C22×D17311yescharacteristic
293Q16—22no—
294Q16—3462no—
295Q16—6921no—
296SD16—3462no—
297SD16—3462no—
298SD16—3462no—
299SD16—3462no—
300SD16—1734no—
301SD16—6921no—
302M16—3462no—
303C2×C8—22no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—3461no—
311C4⋊C4—3461no—
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—1731no—
315C4⋊C4C22×D17311yescharacteristic
316C4⋊C4C22×D17311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—1731no—
325C22⋊C4—1731no—
326C42—3461no—
327C42—1731no—
328C23—3461no—
329C23—3461no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333C23—6922no—
334C23D4×D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—3462no—
346D4—3461no—
347D4—3461no—
348D4—3461no—
349D4—13841no—
350D4—1732no—
351D4—6922no—
352D4—6922no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—13841no—
364C2×C4—1731no—
365C2×C4—1731no—
366C2×C4—1731no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4—6922no—
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—3461no—
374C8—3461no—
375C8D4×D17312yes—
376C22—21no—
377C22—21no—
378C22—3461no—
379C22—3461no—
380C22—3461no—
381C22—3462no—
382C22—13844no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22#1914971112yescomplemented
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3461no—
396C4—3461no—
397C4—3461no—
398C4—3462no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972511yescharacteristic
402C2—22no—
403C2—13841no—
404C2—6921no—
405C2—6922no—
406C2#2601212511yescharacteristic
407C2C2×#1914971111yescharacteristic
408C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#26012132C2
hypercenter#467D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C22×#14583815C2semidirect
3. #26012154C2semidirect
5. #26011884C2semidirect
6. #26011738C2semidirect
7. #26011716C2semidirect
8. #26011695C2semidirect
9. #26011656C2semidirect
10. #26011646C2semidirect
11. #26011641C2semidirect
12. #26011631C2semidirect
13. #26011590C2semidirect
14. #26011562C2semidirect
15. #467×C173C2semidirect
18. C2×Q8×D173C22semidirect
21. C2×#14583816C22semidirect
22. C2×#14583816C22semidirect
30. C2×#14583809C22semidirect
32. #19149677C22semidirect
35. #19149659C22semidirect
41. #19149644C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
48. #19149602C22semidirect
52. #19149591C22semidirect
53. #19149589C22semidirect
54. C2×Q16×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
60. #103×C173C22semidirect
90. #14583815D4semidirect
173. C22×C173#137semidirect
204. C173#3266semidirect
212. #467D173semidirect
228. C2×Q16C2×D173semidirect
255. #107C2×D173semidirect
259. #103C2×D173semidirect
260. #103C2×D173semidirect
388. C22#19149711semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C22×#145838152
#260121542
#260121322
#260118842
#260117382
#260117162
#260116952
#260116562
#260116462
#260116412
#260116312
#260115902
#260115622
#467×C1732
#3266173