Finite Group Databasebeta

← #36365533

Subgroups — #36365533

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#26012146C211yesmaximal, characteristic, complemented
2#26011861C211yesmaximal, characteristic, complemented
3#26011832C211yesmaximal, characteristic, complemented
4#26011821C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011735C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011696C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011634C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#553×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149679C2211yescharacteristic
24#19149661—21no—
25#19149661—21no—
26#19149658C2211yescharacteristic
27#19149658C2211yescharacteristic
28#19149650—21no—
29C4×D692C2211yescharacteristic
30C4⋊C4×D173—21no—
31#19149612C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149602C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48#126×C173—21no—
49#125×C173C2211yescharacteristic
50#123×C173—21no—
51#121×C173—21no—
52#121×C173C2211yescharacteristic
53#121×C173C2211yescharacteristic
54C4×D4×C173—21no—
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57#117×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61#14583826—41no—
62#14583826—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71#14583814—41no—
72#14583814—41no—
73#14583811—22no—
74#14583810—41no—
75C2×#13063436—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87#14583803—21no—
88C2×D4×C173—21no—
89C2×D4×C173—21no—
90C2×D4×C173D411yescharacteristic
91C2×D4×C173C2311yescharacteristic
92C22×C4×C173—21no—
93C22×C4×C173—21no—
94C22×C4×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96C4⋊C4×C173—21no—
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173D411yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116D692—21no—
117D692#4112yes—
118C22⋊D173—42no—
119C22⋊D173—42no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C4×D173—42no—
124C4×D173—42no—
125C4×D173—42no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131C23×C173C2×D411yescharacteristic
132C23×C173C2×D411yescharacteristic
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—41no—
136D4×C173—42no—
137D4×C173—81no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—81no—
146C2×C4×C173—81no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—22no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—41no—
179C2×C173—81no—
180C2×C173#54611yescharacteristic
181C2×C173#57511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—42no—
184C173#330711yescharacteristic, complemented, Sylow
185#3307—1731nomaximal, Sylow
186#553D17311yescharacteristic, complemented
187#547—1731no—
188#491—1731no—
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⋊Q8—1731no—
209#126—21no—
210#125C2×D17311yescharacteristic
211#123—21no—
212#123—3461no—
213#123—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—21no—
217#121—3461no—
218#121C2×D17311yescharacteristic
219#121C2×D17311yescharacteristic
220C4×D4—21no—
221C4×D4—1731no—
222C4×D4C2×D17311yescharacteristic
223C4×D4C2×D17311yescharacteristic
224#117C2×D17311yescharacteristic
225C2×C4⋊C4—3461no—
226C2×C4⋊C4C2×D17311yescharacteristic
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243#97—1731no—
244#41—3462no—
245#41—3462no—
246#41—6921no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4C22⋊D17311yescharacteristic
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4C22×D17311yescharacteristic
263C22×C4C22×D17311yescharacteristic
264SD16—6921no—
265SD16—6921no—
266D8—3462no—
267M16—6921no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8—1731no—
272C2×C8—6921no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4C22⋊D17311yescharacteristic
282C4⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4C22⋊D17311yescharacteristic
293C22⋊C4C22⋊D17311yescharacteristic
294C42—21no—
295C42C22×D17311yescharacteristic
296C23—21no—
297C23—3461no—
298C23C2×C22⋊D17311yescharacteristic
299C23C2×C22⋊D17311yescharacteristic
300Q8—3461no—
301Q8—3461no—
302Q8—3461no—
303Q8—3461no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—3461no—
310D4—1732no—
311D4—6922no—
312D4—6922no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—6922no—
326C2×C4—6922no—
327C2×C4—6922no—
328C2×C4C2×C22⋊D17311yescharacteristic
329C2×C4C2×C22⋊D17311yescharacteristic
330C2×C4C2×C22⋊D17311yescharacteristic
331C2×C4C2×C22⋊D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—3461no—
334C8—3461no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3462no—
347C22—6921no—
348C22—6921no—
349C22C22×C22⋊D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972511yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—6922no—
366C2#2601212911yescharacteristic
367C2#2601228911yescharacteristic
368C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#553×C173C2
socleC22×C173C22×D4
hypercenter#553D173
hypocenterC173#3307
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26012146C2semidirect
2. #26011861C2semidirect
3. #26011832C2semidirect
4. #26011821C2semidirect
5. #26011788C2semidirect
6. #26011735C2semidirect
7. #26011718C2semidirect
8. #26011696C2semidirect
9. #26011634C2semidirect
10. #26011634C2semidirect
11. #26011601C2semidirect
12. #26011563C2semidirect
13. #26011490C2semidirect
14. #26011485C2semidirect
15. #553×C173C2semidirect
184. C173#3307semidirect
186. #553D173semidirect

Maximal subgroups

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

subgroupindex
#260121462
#260118612
#260118322
#260118212
#260117882
#260117352
#260117182
#260116962
#260116342
#260116012
#260115632
#260114902
#260114852
#553×C1732
#3307173