Finite Group Databasebeta

← #36365573

Subgroups — #36365573

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#26012188C211yesmaximal, characteristic, complemented
2#26012160C211yesmaximal, characteristic, complemented
3#26011854C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011796C211yesmaximal, characteristic, complemented
7#26011741C211yesmaximal, characteristic, complemented
8#26011725C211yesmaximal, characteristic, complemented
9#26011684C211yesmaximal, characteristic, complemented
10#26011635C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12#26011559C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#26011487C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583811C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20#19149675—21no—
21#19149675C2211yescharacteristic
22#19149672—21no—
23#19149670C2211yescharacteristic
24#19149660—21no—
25#19149653—21no—
26#19149653C2211yescharacteristic
27#19149650—21no—
28#19149650—21no—
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149639—21no—
36#19149638C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—41no—
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583811—41no—
74#14583809—41no—
75C2×#13063436—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96#41×C173—22no—
97#41×C173—41no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173C2311yescharacteristic
105SD16×C173—22no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113D692—21no—
114D692—21no—
115D692—42no—
116C22⋊D173—41no—
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—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136C23×C173—21no—
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173#4112yes—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—42no—
145D4×C173—81no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—42no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
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—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—22no—
173C4×C173—41no—
174C4×C173C2×#4111yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—41no—
183C2×C173—81no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173#57511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—41no—
189D173—81no—
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#459—1731no—
201#449—1731no—
202#449D17311yescharacteristic, complemented
203#440—1731no—
204#435—1731no—
205#417—1731no—
206#416—1731no—
207C2×#41—1731no—
208C2×#41C2×D17311yescharacteristic
209C2×SD16C2×D17311yescharacteristic
210C2×D8—1731no—
211C2×D8C2×D17311yescharacteristic
212C2×M16—1731no—
213C22×C8—1731no—
214#127—3461no—
215#127—1731no—
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#121C2×D17311yescharacteristic
226C4×Q8—1731no—
227C4×D4—3461no—
228C4×D4—3461no—
229C4×D4—1731no—
230C4×D4—1731no—
231#117—1731no—
232C2×C4⋊C4—1731no—
233#107—1731no—
234#107—1731no—
235#106—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103C2×D17311yescharacteristic
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102C2×D17311yescharacteristic
247C22⋊C8—1731no—
248C22⋊C8C2×D17311yescharacteristic
249#97—1731no—
250#41—22no—
251#41—41no—
252#41—3462no—
253#41—6921no—
254C2×Q8C22×D17311yescharacteristic
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×D4—1731no—
265C2×D4C22×D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4C22×D17311yescharacteristic
273SD16—22no—
274D8—41no—
275D8—6921no—
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—
301C42—1731no—
302C23—21no—
303C23—21no—
304C23—3461no—
305C23—1731no—
306C23D4×D17311yescharacteristic
307Q8—21no—
308Q8#1458382512yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—81no—
315D4—3461no—
316D4—3461no—
317D4—13841no—
318D4—13841no—
319D4—13841no—
320D4—6922no—
321D4—6921no—
322D4—6922no—
323D4—6922no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—13841no—
336C2×C4—1731no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—6922no—
340C2×C4D4×D17311yescharacteristic
341C2×C4C23×D17311yescharacteristic
342C8—21no—
343C8—21no—
344C8—3461no—
345C8—3461no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—3461no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4C2×#1458382511yescharacteristic
372C4#1914972411yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—13841no—
377C2—6921no—
378C2#2601212811yescharacteristic
379C2#2601228811yescharacteristic
380C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3309
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012188C2semidirect
2. #26012160C2semidirect
3. #26011854C2semidirect
4. #26011816C2semidirect
5. #26011803C2semidirect
6. #26011796C2semidirect
7. #26011741C2semidirect
8. #26011725C2semidirect
9. #26011684C2semidirect
10. #26011635C2semidirect
11. #26011592C2semidirect
12. #26011559C2semidirect
13. #26011491C2semidirect
14. #26011487C2semidirect
15. #449×C173C2semidirect
190. C173#3309semidirect
202. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260121882
#260121602
#260118542
#260118162
#260118032
#260117962
#260117412
#260117252
#260116842
#260116352
#260115922
#260115592
#260114912
#260114872
#449×C1732
#3309173