Finite Group Databasebeta

← #36363677

Subgroups — #36363677

Order 22144.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012026C211yesmaximal, characteristic, complemented
2C4×#14583825C211yesmaximal, characteristic, complemented
3#26011834C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011714C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011654C211yesmaximal, characteristic, complemented
9#26011637C211yesmaximal, characteristic
10#26011600C211yesmaximal, characteristic, complemented
11#26011597C211yesmaximal, characteristic, complemented
12#26011590C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14#26011547C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21#19149674—21no—
22#19149667C2211yescharacteristic
23#19149664—21no—
24#19149659C2211yescharacteristic
25#19149654C2211yescharacteristic
26C4×Dic346C2211yescharacteristic
27#19149644C2211yescharacteristic
28#19149644C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30#19149641—21no—
31#19149641C2211yescharacteristic
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149637—21no—
35#19149637C2211yescharacteristic
36#19149635—21no—
37C2×C4×Dic173—21no—
38C42×D173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173D411yescharacteristic
70C2×C4×D173D411yescharacteristic
71C2×C4×D173C2311yescharacteristic
72Dic692—22no—
73#14583815—22no—
74#14583814—22no—
75#14583811—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346#4112yes—
114D692—21no—
115D692—21no—
116D692—42no—
117C22⋊D173—24no—
118C2×Dic173—24no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—42no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—41no—
129C4×D173—41no—
130C4×D173—42no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134D4×C173—21no—
135D4×C173#4112yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—42no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173C22×D411yescharacteristic
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173C2×#4111yescharacteristic
161C4×C173C2×#4111yescharacteristic
162Dic173—21no—
163Dic173—22no—
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168C2×C173—42no—
169C2×C173#52511yescharacteristic
170C2×C173#57511yescharacteristic
171C2×C173#57511yescharacteristic
172D173—41no—
173D173—81no—
174C173#311111yescharacteristic, complemented, Sylow
175#3111—1731nomaximal, Sylow
176#529—1731no—
177C4×#41—1731no—
178#485—1731no—
179#482—1731no—
180#479—1731no—
181#478D17311yescharacteristic, complemented
182#464—1731no—
183#463—1731no—
184#452—1731no—
185#449—1731no—
186#442—1731no—
187#441—1731no—
188#440—1731no—
189#440—1731no—
190#432—1731no—
191C2×#41—1731no—
192C2×Q16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×D8—1731no—
196#126—3461no—
197#125C2×D17311yescharacteristic
198#124—1731no—
199#123—3461no—
200C22⋊Q8—1731no—
201#121—1731no—
202C4×Q8—1731no—
203C4×Q8—1731no—
204C4×Q8—1731no—
205C4×D4—3461no—
206C4×D4—3461no—
207C4×D4—3461no—
208C4×D4—1731no—
209C4×D4—1731no—
210C4×D4C2×D17311yescharacteristic
211#117—3461no—
212#117—3461no—
213#117—1731no—
214C2×C42—3461no—
215C2×C42—1731no—
216#107—1731no—
217#107C2×D17311yescharacteristic
218#106—1731no—
219#106C2×D17311yescharacteristic
220C4⋊C8—1731no—
221C4⋊C8C2×D17311yescharacteristic
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#102—1731no—
227#102—1731no—
228#102C2×D17311yescharacteristic
229#102C2×D17311yescharacteristic
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232#97—1731no—
233#97—1731no—
234#41—3462no—
235#41—3462no—
236C2×Q8—1731no—
237C2×Q8—1731no—
238C2×D4—3461no—
239C2×D4—3461no—
240C2×D4—1731no—
241C2×D4C22×D17311yescharacteristic
242C22×C4—21no—
243C22×C4—3461no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—1731no—
247C22×C4—1731no—
248C22×C4—1731no—
249Q16—3462no—
250SD16—3462no—
251SD16—3462no—
252D8—3462no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8C22×D17311yescharacteristic
256C2×C8C22×D17311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C4⋊C4C22×D17311yescharacteristic
266C4⋊C4C22×D17311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C42—3461no—
276C42—3461no—
277C42—3461no—
278C42—3461no—
279C42—1731no—
280C42—1731no—
281C42—1731no—
282C42C22×D17311yescharacteristic
283C23—21no—
284C23—3461no—
285C23—1731no—
286Q8—3461no—
287Q8—3461no—
288Q8—3461no—
289Q8—1732no—
290D4—21no—
291D4—3464no—
292D4—3461no—
293D4—3461no—
294D4—6922no—
295D4#1458382712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—3464no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—1731no—
307C2×C4—1731no—
308C2×C4—1731no—
309C2×C4—6921no—
310C2×C4—6921no—
311C2×C4—6922no—
312C2×C4—6922no—
313C2×C4D4×D17311yescharacteristic
314C2×C4D4×D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3461no—
319C8—3461no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—3461no—
324C22—3461no—
325C22—3461no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22C2×D4×D17311yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3461no—
336C4—3462no—
337C4—3461no—
338C4—3461no—
339C4—6921no—
340C4—6921no—
341C4#41×D17311yescharacteristic
342C4C2×#1458382711yescharacteristic
343C2—42no—
344C2—13841no—
345C2—6921no—
346C2#2601200411yescharacteristic
347C2#2601228711yescharacteristic
348C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radical#26011637C2
hypercenter#478D173
hypocenterC173#3111
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012026C2semidirect
2. C4×#14583825C2semidirect
3. #26011834C2semidirect
4. #26011817C2semidirect
5. #26011795C2semidirect
6. #26011714C2semidirect
7. #26011706C2semidirect
8. #26011654C2semidirect
10. #26011600C2semidirect
11. #26011597C2semidirect
12. #26011590C2semidirect
13. #26011588C2semidirect
14. #26011547C2semidirect
15. #478×C173C2semidirect
174. C173#3111semidirect
181. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260120262
C4×#145838252
#260118342
#260118172
#260117952
#260117142
#260117062
#260116542
#260116372
#260116002
#260115972
#260115902
#260115882
#260115472
#478×C1732
#3111173