Finite Group Databasebeta

← #36365596

Subgroups — #36365596

Order 22144.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012215C211yesmaximal, characteristic, complemented
2#26012136C211yesmaximal, characteristic
3#26011853C211yesmaximal, characteristic, complemented
4#26011835C211yesmaximal, characteristic, complemented
5#26011817C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011714C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011561C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#476×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic, complemented
23#19149674—21no—
24#19149674—21no—
25#19149668—21no—
26#19149667C2211yescharacteristic
27#19149663—21no—
28#19149659C2211yescharacteristic
29#19149659C2211yescharacteristic
30#19149659C2211yescharacteristic, complemented
31#19149654C2211yescharacteristic
32#19149653—21no—
33#19149649—21no—
34Q8×Dic173—21no—
35C4×Dic346C2211yescharacteristic, complemented
36C4×D692C2211yescharacteristic
37#19149637C2211yescharacteristic
38#19149637C2211yescharacteristic, complemented
39C2×#14583803—21no—
40#19149612C2211yescharacteristic, complemented
41#19149604C2211yescharacteristic, complemented
42#19149602C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149590C2211yescharacteristic, complemented
52C4⋊Q8×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60#14583825—22no—
61#14583825—41no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692—21no—
65C2×D692D411yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69D4×D173—22no—
70D4×D173—41no—
71C22×Dic173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74Dic692—41no—
75#14583815—41no—
76#14583814—22no—
77#14583809—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173D411yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173—21no—
100C2×Q8×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—42no—
113C22×D173—42no—
114C22×D173—81no—
115C22×D173C2×D411yescharacteristic
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—21no—
121D692—21no—
122D692—21no—
123D692—41no—
124D692—81no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—42no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137#13063436—21no—
138#13063436—21no—
139C23×C173—21no—
140Q8×C173—42no—
141D4×C173—21no—
142D4×C173#4112yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—81no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—22no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—84no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—42no—
179C2×C173#54611yescharacteristic
180C2×C173#57511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—41no—
183D173—41no—
184D173—81no—
185C173#331111yescharacteristic, complemented, Sylow
186#3311—1731nomaximal, Sylow
187#556—1731no—
188#546—1731no—
189#489—1731no—
190#485—1731no—
191#482—1731no—
192#476D17311yescharacteristic, complemented
193#470—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198#440—1731no—
199#435—1731no—
200#418—1731no—
201#416—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×M16—1731no—
208C22×C8—1731no—
209C4⋊Q8C2×D17311yescharacteristic, complemented
210#126—3461no—
211#126—3461no—
212#124—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8—1731no—
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8—3461no—
222C4×Q8—1731no—
223C4×D4—1731no—
224C4×D4C2×D17311yescharacteristic
225#117—1731no—
226#117—1731no—
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106C2×D17311yescharacteristic, complemented
230#106C2×D17311yescharacteristic, complemented
231C4⋊C8C2×D17311yescharacteristic, complemented
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244#97—1731no—
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—21no—
249C2×Q8—1731no—
250C2×Q8—1731no—
251C2×D4—3461no—
252C2×D4—3462no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4—6921no—
258C2×D4C22×D17311yescharacteristic
259C22×C4—21no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1731no—
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266M16—6921no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4C22×D17311yescharacteristic
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—3461no—
294C42—3461no—
295C42—3461no—
296C42C22×D17311yescharacteristic
297C23—21no—
298C23—3461no—
299C23—13841no—
300C23—1731no—
301C23—1731no—
302C23—6922no—
303C23—6922no—
304Q8—42no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309D4—21no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—6921no—
314D4—6922no—
315D4—6922no—
316D4#1458382712yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—3461no—
342C22—3461no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—3461no—
347C22—13844no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3461no—
358C4—3461no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×#1458382711yescharacteristic
363C4#1914972511yescharacteristic
364C2—42no—
365C2—13841no—
366C2—6921no—
367C2—6921no—
368C2#2601213211yescharacteristic
369C2#2601228711yescharacteristic
370C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#476×C173C2
socleC22×C173C22×D4
prime radical#26012136C2
hypercenter#476D173
hypocenterC173#3311
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #26012215C2semidirect
3. #26011853C2semidirect
4. #26011835C2semidirect
5. #26011817C2semidirect
6. #26011750C2semidirect
7. #26011721C2semidirect
8. #26011714C2semidirect
9. #26011654C2semidirect
10. #26011643C2semidirect
11. #26011591C2semidirect
12. #26011561C2semidirect
13. #26011494C2semidirect
14. #26011484C2semidirect
15. #476×C173C2semidirect
18. C2×Dic692C22semidirect
22. C2×C8×D173C22semidirect
30. #19149659C22semidirect
35. C4×Dic346C22semidirect
38. #19149637C22semidirect
40. #19149612C22semidirect
41. #19149604C22semidirect
42. #19149602C22semidirect
43. #19149602C22semidirect
44. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149590C22semidirect
52. C4⋊Q8×C173C22semidirect
54. #106×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
185. C173#3311semidirect
192. #476D173semidirect
209. C4⋊Q8C2×D173semidirect
229. #106C2×D173semidirect
230. #106C2×D173semidirect
231. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122152
#260121362
#260118532
#260118352
#260118172
#260117502
#260117212
#260117142
#260116542
#260116432
#260115912
#260115612
#260114942
#260114842
#476×C1732
#3311173