Finite Group Databasebeta

← #36364529

Subgroups — #36364529

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012026C211yesmaximal, characteristic, complemented
2#124×D173C211yesmaximal, characteristic, complemented
3#26011875C211yesmaximal, characteristic, complemented
4#26011873C211yesmaximal, characteristic, complemented
5#26011848C211yesmaximal, characteristic, complemented
6#26011716C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011709C211yesmaximal, characteristic, complemented
9#26011705C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011643C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011626C211yesmaximal, characteristic
14#26011542C211yesmaximal, characteristic, complemented
15#488×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583815C2211yescharacteristic
21C2×#14583815C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic
23C2×#14583813C2211yescharacteristic, complemented
24C2×D1384C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26#19149674—21no—
27#19149668C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149667C2211yescharacteristic, complemented
30#19149666C2211yescharacteristic
31#19149665—21no—
32#19149665—21no—
33#19149664—21no—
34#19149659C2211yescharacteristic
35#19149654C2211yescharacteristic
36#19149644C2211yescharacteristic, complemented
37#19149641—21no—
38#19149641C2211yescharacteristic, complemented
39#19149637—21no—
40C22⋊C4×D173—21no—
41C22⋊C4×D173—21no—
42C42×D173C2211yescharacteristic
43#19149608C2211yescharacteristic, complemented
44#19149608C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic, complemented
52#125×C173C2211yescharacteristic, complemented
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic, complemented
55#103×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63Q8×D173—41no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69D4×D173—41no—
70D4×D173—41no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173D411yescharacteristic
74C2×C4×D173D411yescharacteristic
75C2×C4×D173C2311yescharacteristic
76Dic692—41no—
77#14583816—41no—
78#14583815—41no—
79#14583815—41no—
80#14583814—41no—
81#14583813—41no—
82D1384—41no—
83#14583811—41no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C4⋊Dic173—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99C4×Dic173—21no—
100C4×Dic173D411yescharacteristic
101C4×Dic173D411yescharacteristic
102C4×Dic173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173—42no—
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—42no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130C22⋊D173—42no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—22no—
137C4×D173—21no—
138C4×D173—21no—
139C4×D173—41no—
140C4×D173—42no—
141C4×D173—42no—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—84no—
170C2×D173#13512yes—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173H(5,𝔽2)11yescharacteristic
177C4×C173H(5,𝔽2)11yescharacteristic
178Dic173—21no—
179Dic173—21no—
180Dic173—21no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—81no—
184C2×C173#53511yescharacteristic
185C2×C173C2×#13511yescharacteristic
186C2×C173#57511yescharacteristic
187D173—22no—
188D173—81no—
189D173—81no—
190C173#321111yescharacteristic, complemented, Sylow
191#3211—1731nomaximal, Sylow
192#529—1731no—
193C2×#124—1731no—
194#495—1731no—
195#495—1731no—
196#488—1731no—
197#488D17311yescharacteristic, complemented
198#464—1731no—
199#464—1731no—
200#463—1731no—
201#463—1731no—
202#451—1731no—
203#450—1731no—
204#448—1731no—
205#447—1731no—
206#431—1731no—
207C22×Q8—1731no—
208C22×D4—1731no—
209C2×Q16—1731no—
210C2×Q16—1731no—
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×SD16—1731no—
215C2×D8—1731no—
216C2×D8—1731no—
217#126—3461no—
218#125C2×D17311yescharacteristic, complemented
219#124—3461no—
220#124—3461no—
221#124—1731no—
222#124—1731no—
223#124—1731no—
224#124—1731no—
225#124C2×D17311yescharacteristic
226#123—3461no—
227C22⋊Q8—1731no—
228#121—1731no—
229C4×Q8—1731no—
230C4×D4—3461no—
231C4×D4—1731no—
232#117—3461no—
233C2×C22⋊C4—3461no—
234C2×C22⋊C4—3461no—
235C2×C42—1731no—
236C4⋊C8—1731no—
237C4⋊C8—1731no—
238#103—1731no—
239#103—1731no—
240#103C2×D17311yescharacteristic, complemented
241#103C2×D17311yescharacteristic, complemented
242#102—1731no—
243#102—1731no—
244#102C2×D17311yescharacteristic
245#102C2×D17311yescharacteristic
246C22⋊C8—1731no—
247C22⋊C8—1731no—
248C4×C8—1731no—
249C4×C8C2×D17311yescharacteristic, complemented
250C24—3461no—
251C2×Q8—3461no—
252C2×Q8—1731no—
253C2×Q8—1731no—
254C2×Q8—6921no—
255C2×Q8—6921no—
256C2×Q8C22×D17311yescharacteristic
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—1731no—
260C2×D4—1731no—
261C2×D4—6921no—
262C2×D4—6921no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—1731no—
267C22×C4—1731no—
268C22×C4—1731no—
269Q16—6921no—
270Q16—6921no—
271SD16—6921no—
272SD16—6921no—
273SD16—6921no—
274SD16—6921no—
275D8—6921no—
276D8—6921no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4C22×D17311yescharacteristic
286C4⋊C4C22×D17311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C42—3461no—
301C42—1731no—
302C42—1731no—
303C42—1731no—
304C42C22×D17311yescharacteristic
305C23—21no—
306C23—3461no—
307C23—3461no—
308C23—1731no—
309C23—6922no—
310C23—6922no—
311C23—6922no—
312Q8—21no—
313Q8—21no—
314Q8—3461no—
315Q8—3461no—
316Q8—3461no—
317Q8—3461no—
318Q8—6922no—
319D4—21no—
320D4—21no—
321D4—3461no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—6922no—
326D4—6922no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3462no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—1731no—
338C2×C4—6921no—
339C2×C4—6922no—
340C2×C4—6922no—
341C2×C4D4×D17311yescharacteristic
342C2×C4D4×D17311yescharacteristic
343C2×C4C23×D17311yescharacteristic
344C8—21no—
345C8—21no—
346C8—3461no—
347C8—3461no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3461no—
352C22—3461no—
353C22—13844no—
354C22—1732no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3461no—
368C4—3461no—
369C4—3461no—
370C4—6921no—
371C4—6921no—
372C4#1914972511yescharacteristic
373C4#1914972511yescharacteristic
374C2—81no—
375C2—3462no—
376C2—13841no—
377C2—13841no—
378C2#2601205911yescharacteristic
379C2#135×D17311yescharacteristic
380C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
prime radical#26011626C2
hypercenter#488D173
hypocenterC173#3211
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26012026C2semidirect
2. #124×D173C2semidirect
3. #26011875C2semidirect
4. #26011873C2semidirect
5. #26011848C2semidirect
6. #26011716C2semidirect
7. #26011712C2semidirect
8. #26011709C2semidirect
9. #26011705C2semidirect
10. #26011648C2semidirect
11. #26011643C2semidirect
12. #26011631C2semidirect
14. #26011542C2semidirect
15. #488×C173C2semidirect
18. C2×Dic692C22semidirect
19. C2×#14583816C22semidirect
21. C2×#14583815C22semidirect
23. C2×#14583813C22semidirect
29. #19149667C22semidirect
36. #19149644C22semidirect
38. #19149641C22semidirect
43. #19149608C22semidirect
44. #19149608C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
51. C4×#13063436C22semidirect
52. #125×C173C22semidirect
54. #103×C173C22semidirect
55. #103×C173C22semidirect
58. C4×C8×C173C22semidirect
190. C173#3211semidirect
197. #488D173semidirect
218. #125C2×D173semidirect
240. #103C2×D173semidirect
241. #103C2×D173semidirect
249. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120262
#124×D1732
#260118752
#260118732
#260118482
#260117162
#260117122
#260117092
#260117052
#260116482
#260116432
#260116312
#260116262
#260115422
#488×C1732
#3211173