Finite Group Databasebeta

← #36365563

Subgroups — #36365563

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#26012162C211yesmaximal, characteristic, complemented
2#26011857C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic, complemented
8#26011686C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011634C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#553×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149676C2211yescharacteristic
24#19149675—21no—
25#19149653C2211yescharacteristic
26#19149653C2211yescharacteristic
27#19149652—21no—
28#19149652—21no—
29C4×Dic346C2211yescharacteristic
30#19149641—21no—
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149602C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149600C2211yescharacteristic
39#19149599C2211yescharacteristic
40#19149599C2211yescharacteristic
41#19149599C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48#126×C173—21no—
49#125×C173C2211yescharacteristic
50#123×C173—21no—
51#121×C173—21no—
52#121×C173C2211yescharacteristic
53#121×C173C2211yescharacteristic
54C4×D4×C173—21no—
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57#117×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61#14583826—41no—
62#14583826—41no—
63C2×Dic346C2311yescharacteristic
64C2×D692—21no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583814—22no—
75#14583811—41no—
76#14583811—41no—
77#14583810—41no—
78C2×#13063436—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C2×D4×C173—21no—
89C2×D4×C173—21no—
90C2×D4×C173D411yescharacteristic
91C2×D4×C173C2311yescharacteristic
92C22×C4×C173—21no—
93C22×C4×C173—21no—
94C22×C4×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96C4⋊C4×C173—21no—
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173D411yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113Dic346—21no—
114Dic346#4112yes—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—42no—
120D692—81no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C4×D173—42no—
128C4×D173—42no—
129#13063436—21no—
130#13063436—21no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134C23×C173C2×D411yescharacteristic
135C23×C173C2×D411yescharacteristic
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—41no—
139D4×C173—42no—
140D4×C173—81no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—81no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—21no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173C2×#4111yescharacteristic
179C4×C173H(5,𝔽2)11yescharacteristic
180Dic173—22no—
181Dic173—41no—
182C2×C173—41no—
183C2×C173—41no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173#57511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—81no—
189D173—81no—
190C173#330911yescharacteristic, complemented, Sylow
191#3309—1731nomaximal, Sylow
192#553D17311yescharacteristic, complemented
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#449—1731no—
203#440—1731no—
204#435—1731no—
205#417—1731no—
206#416—1731no—
207C2×#41—1731no—
208C2×#41—1731no—
209C2×SD16—1731no—
210C2×D8—1731no—
211C2×D8—1731no—
212C2×M16—1731no—
213C22×C8—1731no—
214#127—3461no—
215#127—1731no—
216#126—21no—
217#125C2×D17311yescharacteristic
218#123—21no—
219#121—21no—
220#121—3461no—
221#121—3461no—
222#121—1731no—
223#121—1731no—
224#121C2×D17311yescharacteristic
225#121C2×D17311yescharacteristic
226C4×Q8—1731no—
227C4×D4—21no—
228C4×D4—3461no—
229C4×D4C2×D17311yescharacteristic
230C4×D4C2×D17311yescharacteristic
231#117C2×D17311yescharacteristic
232C2×C4⋊C4C2×D17311yescharacteristic
233#107—1731no—
234#107—1731no—
235#106—1731no—
236C4⋊C8—1731no—
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#102—1731no—
246#102—1731no—
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×D4—1731no—
265C2×D4C22⋊D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273SD16—3462no—
274D8—6921no—
275D8—6921no—
276M16—6921no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8—1731no—
281C2×C8—6921no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—3461no—
286C4⋊C4—1731no—
287C4⋊C4C22⋊D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C4⋊C4C22×D17311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4C22⋊D17311yescharacteristic
298C22⋊C4C22⋊D17311yescharacteristic
299C42—21no—
300C42—3461no—
301C42C22×D17311yescharacteristic
302C23—21no—
303C23—3461no—
304C23—3461no—
305C23C2×C22⋊D17311yescharacteristic
306C23C2×C22⋊D17311yescharacteristic
307Q8—3461no—
308Q8—1732no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—3461no—
315D4—3461no—
316D4—3461no—
317D4—3461no—
318D4—13841no—
319D4—13841no—
320D4—13841no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—6922no—
336C2×C4—6922no—
337C2×C4C2×C22⋊D17311yescharacteristic
338C2×C4C2×C22⋊D17311yescharacteristic
339C2×C4C2×C22⋊D17311yescharacteristic
340C2×C4C2×C22⋊D17311yescharacteristic
341C2×C4C23×D17311yescharacteristic
342C8—3461no—
343C8—3461no—
344C8—3461no—
345C8—3461no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C22×C22⋊D17311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3462no—
370C4—6921no—
371C4#41×D17311yescharacteristic
372C4#1914972511yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—13841no—
377C2—13841no—
378C2#2601212911yescharacteristic
379C2#2601228911yescharacteristic
380C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#553×C173C2
socleC22×C173C22×D4
hypercenter#553D173
hypocenterC173#3309
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26012162C2semidirect
2. #26011857C2semidirect
3. #26011815C2semidirect
4. #26011804C2semidirect
5. #26011795C2semidirect
6. #26011742C2semidirect
7. #26011725C2semidirect
8. #26011686C2semidirect
9. #26011634C2semidirect
10. #26011634C2semidirect
11. #26011591C2semidirect
12. #26011563C2semidirect
13. #26011490C2semidirect
14. #26011485C2semidirect
15. #553×C173C2semidirect
190. C173#3309semidirect
192. #553D173semidirect

Maximal subgroups

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

subgroupindex
#260121622
#260118572
#260118152
#260118042
#260117952
#260117422
#260117252
#260116862
#260116342
#260115912
#260115632
#260114902
#260114852
#553×C1732
#3309173