Finite Group Databasebeta

← #36365538

Subgroups — #36365538

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012194C211yesmaximal, characteristic, complemented
2#26012144C211yesmaximal, characteristic, complemented
3#26011830C211yesmaximal, characteristic, complemented
4#26011820C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011740C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011693C211yesmaximal, characteristic, complemented
9#26011639C211yesmaximal, characteristic, complemented
10#26011639C211yesmaximal, characteristic, complemented
11#26011604C211yesmaximal, characteristic, complemented
12#26011564C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#26011486C211yesmaximal, characteristic, complemented
15#491×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149674—21no—
24#19149664—21no—
25#19149664—21no—
26#19149664—21no—
27#19149655C2211yescharacteristic
28#19149655C2211yescharacteristic
29#19149654—21no—
30#19149654—21no—
31#19149654C2211yescharacteristic
32#19149654C2211yescharacteristic
33C4×D692C2211yescharacteristic
34C4×D692C2211yescharacteristic
35C4×D692C2211yescharacteristic
36#19149641—21no—
37#19149637C2211yescharacteristic
38C4⋊C4×D173—21no—
39C4⋊C4×D173C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#125×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61#14583827—41no—
62#14583827—41no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583813—41no—
78#14583813—41no—
79D1384—22no—
80#14583809—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173D411yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98#14583803D411yescharacteristic
99C4×Dic173—21no—
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113C22×D173C2×D411yescharacteristic
114C22×D173C2×D411yescharacteristic
115D692—21no—
116D692—21no—
117D692—21no—
118D692—41no—
119D692—42no—
120D692—42no—
121D692—42no—
122D692—81no—
123D692#4112yes—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—42no—
133C4×D173—42no—
134C4×D173—42no—
135C4×D173—81no—
136C4×D173—81no—
137#13063436—21no—
138#13063436—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—22no—
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177C2×C173#54611yescharacteristic
178C2×C173#57511yescharacteristic
179C2×C173#57711yescharacteristic
180D173—41no—
181D173—41no—
182D173—42no—
183D173—81no—
184C173#330711yescharacteristic, complemented, Sylow
185#3307—1731nomaximal, Sylow
186#553—1731no—
187#547—1731no—
188#491D17311yescharacteristic, complemented
189#485—1731no—
190#483—1731no—
191#478—1731no—
192#468—1731no—
193#465—1731no—
194#461—1731no—
195#449—1731no—
196#449—1731no—
197#442—1731no—
198#435—1731no—
199#417—1731no—
200#416—1731no—
201C2×#41—1731no—
202C2×#41—1731no—
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C2×M16—1731no—
207C22×C8—1731no—
208C4⋊Q8C2×D17311yescharacteristic
209#126—3461no—
210#125C2×D17311yescharacteristic
211#123—3461no—
212#123—3461no—
213#123—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121—1731no—
220C4×D4—3461no—
221C4×D4—1731no—
222C4×D4—1731no—
223C4×D4—1731no—
224#117—1731no—
225C2×C4⋊C4—3461no—
226C2×C4⋊C4—1731no—
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103C2×D17311yescharacteristic
234#103C2×D17311yescharacteristic
235#103C2×D17311yescharacteristic
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243#97C2×D17311yescharacteristic
244#41—3462no—
245#41—3462no—
246#41—6921no—
247#41—6921no—
248C2×Q8C22×D17311yescharacteristic
249C2×Q8C22×D17311yescharacteristic
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—1731no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4—1731no—
264SD16—6921no—
265SD16—6921no—
266D8—3462no—
267M16—6921no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4C22×D17311yescharacteristic
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42C22×D17311yescharacteristic
296C23—3461no—
297C23—3461no—
298C23—1731no—
299C23—1731no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—3461no—
305D4—3461no—
306D4—3461no—
307D4—13841no—
308D4—1732no—
309D4—6921no—
310D4—6922no—
311D4—6922no—
312D4—6922no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—13841no—
324C2×C4—13841no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4—6922no—
328C2×C4—6922no—
329C2×C4—6922no—
330C2×C4D4×D17311yescharacteristic
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—3461no—
338C22—3461no—
339C22—3461no—
340C22—3461no—
341C22—3461no—
342C22—3461no—
343C22—3462no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3461no—
357C4—3461no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972511yescharacteristic
362C2—13841no—
363C2—6921no—
364C2—6921no—
365C2—6922no—
366C2#2601213411yescharacteristic
367C2#2601229011yescharacteristic
368C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#491×C173C2
socleC22×C173C22×D4
hypercenter#491D173
hypocenterC173#3307
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26012194C2semidirect
2. #26012144C2semidirect
3. #26011830C2semidirect
4. #26011820C2semidirect
5. #26011788C2semidirect
6. #26011740C2semidirect
7. #26011723C2semidirect
8. #26011693C2semidirect
9. #26011639C2semidirect
10. #26011639C2semidirect
11. #26011604C2semidirect
12. #26011564C2semidirect
13. #26011491C2semidirect
14. #26011486C2semidirect
15. #491×C173C2semidirect
184. C173#3307semidirect
188. #491D173semidirect

Maximal subgroups

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

subgroupindex
#260121942
#260121442
#260118302
#260118202
#260117882
#260117402
#260117232
#260116932
#260116392
#260116042
#260115642
#260114912
#260114862
#491×C1732
#3307173