Finite Group Databasebeta

← #36365576

Subgroups — #36365576

Order 22144.

404 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#26012191C211yesmaximal, characteristic, complemented
2#26012161C211yesmaximal, characteristic, complemented
3#26011855C211yesmaximal, characteristic, complemented
4#26011814C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011799C211yesmaximal, characteristic, complemented
7#26011741C211yesmaximal, characteristic, complemented
8#26011724C211yesmaximal, characteristic, complemented
9#26011633C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011561C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149675—21no—
22#19149672—21no—
23#19149671C2211yescharacteristic
24#19149662—21no—
25#19149652—21no—
26#19149651—21no—
27#19149651—21no—
28#19149650C2211yescharacteristic
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic
31#19149650C2211yescharacteristic
32C4×Dic346C2211yescharacteristic
33C4×D692C2211yescharacteristic
34D4×Dic173—21no—
35C4×C22⋊D173—21no—
36#19149637C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54#127×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346C2311yescharacteristic
64C2×D692D411yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75C2×C4×D173C2311yescharacteristic
76#14583814—22no—
77#14583809—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173—21no—
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104D8×C173—41no—
105D8×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173C2×D411yescharacteristic
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346#4112yes—
115D692—41no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C22⋊D173—42no—
119C22⋊D173—81no—
120C22⋊D173—81no—
121C22⋊D173—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—42no—
127C2×Dic173—42no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—81no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173—21no—
139C23×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—42no—
147D4×C173—81no—
148C2×C4×C173—21no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173C2×#4111yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—22no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—81no—
183C2×C173—81no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173#57511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—41no—
189D173—41no—
190C173#330911yescharacteristic, complemented, Sylow
191#3309—1731nomaximal, Sylow
192#553—1731no—
193#550—1731no—
194#490—1731no—
195#482—1731no—
196#480—1731no—
197#479—1731no—
198#469—1731no—
199#466—1731no—
200#459D17311yescharacteristic, complemented
201#449—1731no—
202#449—1731no—
203#440—1731no—
204#435—1731no—
205#417—1731no—
206#416—1731no—
207C2×#41—1731no—
208C2×#41—1731no—
209C2×SD16—1731no—
210C2×D8C2×D17311yescharacteristic
211C2×D8C2×D17311yescharacteristic
212C2×M16—1731no—
213C22×C8—1731no—
214#127—3461no—
215#127C2×D17311yescharacteristic
216#126—3461no—
217#125—1731no—
218#123—3461no—
219#121—3461no—
220#121—3461no—
221#121—3461no—
222#121—1731no—
223#121—1731no—
224#121—1731no—
225#121—1731no—
226C4×Q8—1731no—
227C4×D4—3461no—
228C4×D4—3461no—
229C4×D4—1731no—
230C4×D4C2×D17311yescharacteristic
231#117—1731no—
232C2×C4⋊C4—1731no—
233#107—1731no—
234#107—1731no—
235#106—1731no—
236C4⋊C8C2×D17311yescharacteristic
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102C2×D17311yescharacteristic
246#102C2×D17311yescharacteristic
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249#97—1731no—
250#41—3462no—
251#41—3462no—
252#41—6921no—
253#41—6921no—
254C2×Q8—1731no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—1731no—
264C2×D4C22×D17311yescharacteristic
265C2×D4C22×D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4—1731no—
273SD16—3462no—
274D8—41no—
275D8—41no—
276M16—6921no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8—6921no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4—1731no—
289C4⋊C4—1731no—
290C4⋊C4C22×D17311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—1731no—
298C22⋊C4—1731no—
299C42—3461no—
300C42—3461no—
301C42C22×D17311yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1731no—
306C23—1731no—
307Q8—3461no—
308Q8—1732no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—81no—
317D4—13841no—
318D4—13841no—
319D4—13841no—
320D4—6921no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4D4×D17311yescharacteristic
340C2×C4D4×D17311yescharacteristic
341C2×C4C23×D17311yescharacteristic
342C8—21no—
343C8—21no—
344C8—3461no—
345C8—3461no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—3461no—
359C22—3461no—
360C22—3461no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—3462no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4#41×D17311yescharacteristic
372C4#1914972411yescharacteristic
373C2—81no—
374C2—81no—
375C2—81no—
376C2—6921no—
377C2—6921no—
378C2#2601213811yescharacteristic
379C2#2601228711yescharacteristic
380C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3309
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012191C2semidirect
2. #26012161C2semidirect
3. #26011855C2semidirect
4. #26011814C2semidirect
5. #26011803C2semidirect
6. #26011799C2semidirect
7. #26011741C2semidirect
8. #26011724C2semidirect
9. #26011633C2semidirect
10. #26011633C2semidirect
11. #26011591C2semidirect
12. #26011561C2semidirect
13. #26011489C2semidirect
14. #26011484C2semidirect
15. #459×C173C2semidirect
190. C173#3309semidirect
200. #459D173semidirect

Maximal subgroups

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

subgroupindex
#260121912
#260121612
#260118552
#260118142
#260118032
#260117992
#260117412
#260117242
#260116332
#260115912
#260115612
#260114892
#260114842
#459×C1732
#3309173