Finite Group Databasebeta

← #36363760

Subgroups — #36363760

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012092C211yesmaximal, characteristic, complemented
2#26011973C211yesmaximal, characteristic, complemented
3#26011808C211yesmaximal, characteristic, complemented
4#26011806C211yesmaximal, characteristic, complemented
5#26011796C211yesmaximal, characteristic, complemented
6#26011794C211yesmaximal, characteristic, complemented
7#26011689C211yesmaximal, characteristic, complemented
8#26011685C211yesmaximal, characteristic, complemented
9#26011639C211yesmaximal, characteristic, complemented
10#26011635C211yesmaximal, characteristic, complemented
11#26011604C211yesmaximal, characteristic, complemented
12C4×#14583813C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011553C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×D1384C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149675C2211yescharacteristic
21#19149670C2211yescharacteristic
22#19149662—21no—
23#19149661—21no—
24#19149654C2211yescharacteristic
25#19149653—21no—
26#19149652—21no—
27#19149650C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×D692—21no—
30C4×D692C2211yescharacteristic
31C4×D692C2211yescharacteristic
32#19149641—21no—
33#19149641C2211yescharacteristic
34#19149636—21no—
35#19149636C2211yescharacteristic
36C4⋊C4×D173—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61C2×D692—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583813—22no—
75D1384—22no—
76#14583811—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111D692—24no—
112D692—21no—
113D692—21no—
114D692—21no—
115D692—81no—
116D692#4112yes—
117C22⋊D173—81no—
118C22⋊D173—81no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—24no—
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—81no—
129C4×D173—81no—
130C4×D173—81no—
131C4×D173—81no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135Q8×C173—21no—
136Q8×C173#4112yes—
137D4×C173—21no—
138D4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—81no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—22no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C4×C173—22no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173C2×#4111yescharacteristic
167C4×C173C2×#4111yescharacteristic
168Dic173—21no—
169Dic173—21no—
170Dic173—21no—
171Dic173—41no—
172Dic173—41no—
173C2×C173—81no—
174C2×C173#52511yescharacteristic
175C2×C173#57611yescharacteristic
176C2×C173#57711yescharacteristic
177D173—41no—
178D173—42no—
179D173—81no—
180C173#311911yescharacteristic, complemented, Sylow
181#3119—1731nomaximal, Sylow
182#540—1731no—
183#520—1731no—
184#481—1731no—
185#480—1731no—
186#479—1731no—
187#478—1731no—
188#460—1731no—
189#459—1731no—
190#449—1731no—
191#449—1731no—
192#442—1731no—
193#440D17311yescharacteristic, complemented
194C4×SD16—1731no—
195C4×D8—1731no—
196#433—1731no—
197C2×#41—1731no—
198C2×SD16—1731no—
199C2×SD16C2×D17311yescharacteristic
200C2×D8—1731no—
201C2×D8—1731no—
202#127—1731no—
203#125—1731no—
204#123—3461no—
205#123—3461no—
206#121—3461no—
207#121—3461no—
208#121—1731no—
209#121—1731no—
210C4×Q8—1731no—
211C4×Q8C2×D17311yescharacteristic
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4C2×D17311yescharacteristic
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—3461no—
221C2×C4⋊C4—1731no—
222#107—1731no—
223#107C2×D17311yescharacteristic
224#106—1731no—
225#106—1731no—
226C4⋊C8—1731no—
227C4⋊C8—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102C2×D17311yescharacteristic
236C22⋊C8—1731no—
237C22⋊C8—1731no—
238#97C2×D17311yescharacteristic
239C4×C8—1731no—
240#41—3462no—
241#41—3462no—
242C2×Q8C22×D17311yescharacteristic
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—1731no—
258SD16—22no—
259SD16—3462no—
260D8—3462no—
261D8—3462no—
262C2×C8—1731no—
263C2×C8—1731no—
264C2×C8C22×D17311yescharacteristic
265C2×C8C22×D17311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—21no—
286C42—3461no—
287C42—1731no—
288C42C22×D17311yescharacteristic
289C23—21no—
290C23—3461no—
291C23—3461no—
292C23—1731no—
293Q8—21no—
294Q8#1458382612yes—
295D4—21no—
296D4—21no—
297D4—3464no—
298D4—3461no—
299D4—3461no—
300D4—3461no—
301D4—13841no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305D4—1732no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—3464no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4D4×D17311yescharacteristic
323C2×C4D4×D17311yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—21no—
326C8—21no—
327C8—3461no—
328C8—3461no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—3461no—
333C22—3462no—
334C22—3461no—
335C22—3461no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22C2×D4×D17311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3461no—
348C4—3461no—
349C4—3461no—
350C4—6921no—
351C4—6921no—
352C4#41×D17311yescharacteristic
353C4C2×#1458382611yescharacteristic
354C2—81no—
355C2—13841no—
356C2—6921no—
357C2—6922no—
358C2#2601200011yescharacteristic
359C2#2601229511yescharacteristic
360C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
hypercenter#440D173
hypocenterC173#3119
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #26012092C2semidirect
2. #26011973C2semidirect
3. #26011808C2semidirect
4. #26011806C2semidirect
5. #26011796C2semidirect
6. #26011794C2semidirect
7. #26011689C2semidirect
8. #26011685C2semidirect
9. #26011639C2semidirect
10. #26011635C2semidirect
11. #26011604C2semidirect
12. C4×#14583813C2semidirect
13. C4×#14583811C2semidirect
14. #26011553C2semidirect
15. #440×C173C2semidirect
180. C173#3119semidirect
193. #440D173semidirect

Maximal subgroups

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

subgroupindex
#260120922
#260119732
#260118082
#260118062
#260117962
#260117942
#260116892
#260116852
#260116392
#260116352
#260116042
C4×#145838132
C4×#145838112
#260115532
#440×C1732
#3119173