Finite Group Databasebeta

← #36364127

Subgroups — #36364127

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012025C211yesmaximal, characteristic, complemented
2#26011890C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011848C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic
8#26011722C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic, complemented
10C4×#14583816C211yesmaximal, characteristic, complemented
11C4×#14583813C211yesmaximal, characteristic, complemented
12C4×#14583814C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14C2×C4×#13063436C211yesmaximal, characteristic, complemented
15#529×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22#19149673—21no—
23#19149671C2211yescharacteristic
24#19149667C2211yescharacteristic
25#19149662—21no—
26#19149658C2211yescharacteristic
27#19149653C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29C4×D692C2211yescharacteristic
30C4×C22⋊D173—21no—
31#19149636—21no—
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34#19149610C2211yescharacteristic
35#19149610C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C4×#13063436C2211yescharacteristic
45C4×#13063436C2211yescharacteristic
46C4×#13063436C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48#126×C173—21no—
49#125×C173C2211yescharacteristic
50#124×C173C2211yescharacteristic
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55C4×D4×C173—21no—
56C4×D4×C173C2211yescharacteristic
57#117×C173—21no—
58C2×C42×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C4×D173—21no—
63#14583816—22no—
64#14583814—22no—
65#14583813—22no—
66#14583811—22no—
67C2×#13063436—22no—
68C2×#13063436—22no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80#14583803—21no—
81#14583803—21no—
82#14583803—21no—
83#14583803—21no—
84C4×Dic173—21no—
85C2×Q8×C173C2311yescharacteristic
86C2×D4×C173—21no—
87C2×D4×C173C2311yescharacteristic
88C22×C4×C173—21no—
89C22×C4×C173D411yescharacteristic
90C22×C4×C173D411yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C4⋊C4×C173—21no—
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173C2311yescharacteristic
97C4⋊C4×C173C2311yescharacteristic
98C22⋊C4×C173—21no—
99C22⋊C4×C173—21no—
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C42×C173—21no—
104C42×C173D411yescharacteristic
105C42×C173D411yescharacteristic
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346—21no—
111D692—21no—
112D692—21no—
113C22⋊D173—42no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C4×D173—42no—
118#13063436—21no—
119#13063436—21no—
120#13063436#4112yes—
121#13063436#4112yes—
122C23×C173—21no—
123C23×C173C2×D411yescharacteristic
124Q8×C173—21no—
125Q8×C173—21no—
126D4×C173—21no—
127D4×C173—21no—
128D4×C173—42no—
129C2×C4×C173—22no—
130C2×C4×C173—22no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—42no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C22×C173—21no—
144C22×C173—21no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173#13512yes—
149C22×C173C22×D411yescharacteristic
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173#13512yes—
161C4×C173C2×#4111yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173—22no—
167C2×C173—81no—
168C2×C173#53211yescharacteristic
169C2×C173C2×#13511yescharacteristic
170C2×C173C2×#13511yescharacteristic
171D173—81no—
172C173#316811yescharacteristic, complemented, Sylow
173#3168—1731nomaximal, Sylow
174#529—1731no—
175#529D17311yescharacteristic, complemented
176#499—1731no—
177#495—1731no—
178#488—1731no—
179#488—1731no—
180#467—1731no—
181#466—1731no—
182#465—1731no—
183#465—1731no—
184C4×Q16—1731no—
185C4×SD16—1731no—
186C4×SD16—1731no—
187C4×D8—1731no—
188C2×C4×C8—1731no—
189C2×Q16—1731no—
190C2×SD16—1731no—
191C2×SD16—1731no—
192C2×D8—1731no—
193C22×C8—1731no—
194C22×C8—1731no—
195#126—21no—
196#126—3461no—
197#125—1731no—
198#125C2×D17311yescharacteristic
199#124—1731no—
200#124C2×D17311yescharacteristic
201#123—21no—
202#123—3461no—
203C22⋊Q8—1731no—
204C22⋊Q8C2×D17311yescharacteristic
205#121—1731no—
206#121C2×D17311yescharacteristic
207C4×Q8—1731no—
208C4×Q8C2×D17311yescharacteristic
209C4×D4—21no—
210C4×D4—3461no—
211C4×D4—1731no—
212C4×D4C2×D17311yescharacteristic
213#117—21no—
214#117—3461no—
215C2×C42C2×D17311yescharacteristic
216#107—1731no—
217#107—1731no—
218#106—1731no—
219#106—1731no—
220#103—1731no—
221#103—1731no—
222#103—1731no—
223#103—1731no—
224#102—1731no—
225#102—1731no—
226#102—1731no—
227#102—1731no—
228C4×C8—1731no—
229C4×C8—1731no—
230C4×C8—1731no—
231C4×C8—1731no—
232C2×Q8—1731no—
233C2×Q8C22×D17311yescharacteristic
234C2×D4—21no—
235C2×D4—3461no—
236C2×D4—1731no—
237C2×D4C22×D17311yescharacteristic
238C22×C4—21no—
239C22×C4—3461no—
240C22×C4C22⋊D17311yescharacteristic
241C22×C4C22⋊D17311yescharacteristic
242C22×C4C22×D17311yescharacteristic
243Q16—3462no—
244SD16—3462no—
245SD16—3462no—
246D8—3462no—
247C2×C8—3462no—
248C2×C8—3462no—
249C2×C8—1731no—
250C2×C8—1731no—
251C2×C8—1731no—
252C2×C8—1731no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C42—21no—
276C42—3461no—
277C42C22⋊D17311yescharacteristic
278C42C22⋊D17311yescharacteristic
279C42C22×D17311yescharacteristic
280C42C22×D17311yescharacteristic
281C23—21no—
282C23—3461no—
283C23C2×C22⋊D17311yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—3461no—
287Q8—3461no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—3461no—
292D4—3461no—
293D4—6922no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—6922no—
306C2×C4C2×C22⋊D17311yescharacteristic
307C2×C4C2×C22⋊D17311yescharacteristic
308C2×C4C2×C22⋊D17311yescharacteristic
309C2×C4C2×C22⋊D17311yescharacteristic
310C2×C4C2×C22⋊D17311yescharacteristic
311C2×C4C23×D17311yescharacteristic
312C8—3461no—
313C8—3461no—
314C8—1732no—
315C8—1732no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—6921no—
322C22—6921no—
323C22—6921no—
324C22#1914969912yes—
325C22C22×C22⋊D17311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6921no—
334C4—6921no—
335C4—6921no—
336C4#1914969912yes—
337C4#41×D17311yescharacteristic
338C4#41×D17311yescharacteristic
339C2—22no—
340C2—81no—
341C2—13841no—
342C2#2601203711yescharacteristic
343C2C2×#1914969911yescharacteristic
344C2C2×#1914969911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#529×C173C2
socleC22×C173C22×D4
prime radical#26011725C2
hypercenter#529D173
hypocenterC173#3168
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #26012025C2semidirect
2. #26011890C2semidirect
3. #26011873C2semidirect
4. #26011848C2semidirect
5. #26011845C2semidirect
6. #26011730C2semidirect
8. #26011722C2semidirect
9. #26011718C2semidirect
10. C4×#14583816C2semidirect
11. C4×#14583813C2semidirect
12. C4×#14583814C2semidirect
13. C4×#14583811C2semidirect
14. C2×C4×#13063436C2semidirect
15. #529×C173C2semidirect
172. C173#3168semidirect
175. #529D173semidirect

Maximal subgroups

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

subgroupindex
#260120252
#260118902
#260118732
#260118482
#260118452
#260117302
#260117252
#260117222
#260117182
C4×#145838162
C4×#145838132
C4×#145838142
C4×#145838112
C2×C4×#130634362
#529×C1732
#3168173