Finite Group Databasebeta

← #36363856

Subgroups — #36363856

Order 22144.

382 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#26012097C211yesmaximal, characteristic
2#26011967C211yesmaximal, characteristic, complemented
3#26011818C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011692C211yesmaximal, characteristic, complemented
7#26011691C211yesmaximal, characteristic, complemented
8#26011631C211yesmaximal, characteristic, complemented
9#26011631C211yesmaximal, characteristic, complemented
10#26011594C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011578C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#479×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21#19149676C2211yescharacteristic
22#19149664—21no—
23#19149664—21no—
24#19149654—21no—
25#19149654—21no—
26#19149654C2211yescharacteristic
27#19149654C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×Dic346—21no—
30C4×Dic346C2211yescharacteristic
31#19149644—21no—
32#19149644C2211yescharacteristic
33#19149641C2211yescharacteristic
34#19149641C2211yescharacteristic
35#19149637—21no—
36#19149637C2211yescharacteristic
37#19149636—21no—
38#19149636C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149602C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61Q8×D173—22no—
62Q8×D173—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583815—22no—
74#14583815—22no—
75#14583813—22no—
76#14583813—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803D411yescharacteristic
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111Dic346—24no—
112Dic346—21no—
113Dic346#4112yes—
114D692—21no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—81no—
119D692—81no—
120D692—81no—
121D692—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—24no—
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—81no—
131C4×D173—81no—
132#13063436—21no—
133#13063436—21no—
134Q8×C173—21no—
135Q8×C173#4112yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173C22×D411yescharacteristic
146C2×D173—21no—
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—41no—
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—22no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—22no—
164Dic173—21no—
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168C2×C173#52511yescharacteristic
169C2×C173#57711yescharacteristic
170C2×C173#57711yescharacteristic
171D173—41no—
172D173—81no—
173D173—81no—
174C173#312611yescharacteristic, complemented, Sylow
175#3126—1731nomaximal, Sylow
176#540—1731no—
177#519—1731no—
178#482—1731no—
179#482—1731no—
180#479—1731no—
181#479D17311yescharacteristic, complemented
182#460—1731no—
183#460—1731no—
184#448—1731no—
185#448—1731no—
186#440—1731no—
187#440—1731no—
188C4×SD16—1731no—
189C4×SD16—1731no—
190#433—1731no—
191C22×Q8—1731no—
192C2×SD16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×SD16—1731no—
196#127—1731no—
197#125C2×D17311yescharacteristic
198#123—3461no—
199#123—3461no—
200#121—3461no—
201#121—3461no—
202#121—1731no—
203#121—1731no—
204C4×Q8—3461no—
205C4×Q8—3461no—
206C4×Q8—1731no—
207C4×Q8—1731no—
208C4×Q8—1731no—
209C4×Q8C2×D17311yescharacteristic
210C4×D4—1731no—
211C4×D4—1731no—
212#117—3461no—
213#117—3461no—
214#117—1731no—
215#117—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#103C2×D17311yescharacteristic
225#103C2×D17311yescharacteristic
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232#97—1731no—
233C4×C8—1731no—
234C2×Q8—3461no—
235C2×Q8—3462no—
236C2×Q8—3462no—
237C2×Q8—1731no—
238C2×Q8C22×D17311yescharacteristic
239C2×D4—3461no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—1731no—
244C2×D4—1731no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—1731no—
249SD16—3462no—
250SD16—3462no—
251SD16—3462no—
252SD16—3462no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8C22×D17311yescharacteristic
256C2×C8C22×D17311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4C22×D17311yescharacteristic
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—1731no—
275C22⋊C4—1731no—
276C42—21no—
277C42—3461no—
278C42—3461no—
279C42—3461no—
280C42—3461no—
281C42—1731no—
282C42C22×D17311yescharacteristic
283C23—3461no—
284C23—3461no—
285C23—1731no—
286Q8—21no—
287Q8—3464no—
288Q8—3461no—
289Q8—1732no—
290Q8#1458382712yes—
291D4—3461no—
292D4—3461no—
293D4—3461no—
294D4—3461no—
295D4—13841no—
296D4—13841no—
297D4—13841no—
298D4—13841no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—3464no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—1731no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4D4×D17311yescharacteristic
314C2×C4D4×D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3461no—
319C8—3461no—
320C22—3461no—
321C22—3461no—
322C22—3461no—
323C22—6921no—
324C22—6921no—
325C22—6921no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22C2×D4×D17311yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—3462no—
337C4—3461no—
338C4—3461no—
339C4—3461no—
340C4—6921no—
341C4#41×D17311yescharacteristic
342C4C2×#1458382711yescharacteristic
343C2—13841no—
344C2—13841no—
345C2—6921no—
346C2#2601200411yescharacteristic
347C2#2601229811yescharacteristic
348C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#479×C173C2
socleC22×C173C22×D4
prime radical#26012097C2
hypercenter#479D173
hypocenterC173#3126
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. #26011967C2semidirect
3. #26011818C2semidirect
4. #26011817C2semidirect
5. #26011795C2semidirect
6. #26011692C2semidirect
7. #26011691C2semidirect
8. #26011631C2semidirect
9. #26011631C2semidirect
10. #26011594C2semidirect
11. #26011590C2semidirect
12. #26011578C2semidirect
13. #26011576C2semidirect
14. #26011551C2semidirect
15. #479×C173C2semidirect
174. C173#3126semidirect
181. #479D173semidirect

Maximal subgroups

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

subgroupindex
#260120972
#260119672
#260118182
#260118172
#260117952
#260116922
#260116912
#260116312
#260115942
#260115902
#260115782
#260115762
#260115512
#479×C1732
#3126173