Finite Group Databasebeta

← #36365463

Subgroups — #36365463

Order 22144.

398 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#26012189C211yesmaximal, characteristic, complemented
2#26012147C211yesmaximal, characteristic, complemented
3#26011844C211yesmaximal, characteristic, complemented
4#26011835C211yesmaximal, characteristic, complemented
5#26011801C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011637C211yesmaximal, characteristic, complemented
10#26011632C211yesmaximal, characteristic, complemented
11#26011578C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14C2×#19149602C211yesmaximal, characteristic, complemented
15#481×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149674—21no—
24#19149670C2211yescharacteristic
25#19149666C2211yescharacteristic
26#19149665—21no—
27#19149664—21no—
28#19149659C2211yescharacteristic
29#19149656—21no—
30#19149654C2211yescharacteristic
31#19149653—21no—
32#19149653C2211yescharacteristic
33#19149651—21no—
34#19149644C2211yescharacteristic
35D4×Dic173—21no—
36#19149641C2211yescharacteristic
37C4×C22⋊D173—21no—
38C4×C22⋊D173C2211yescharacteristic
39#19149637C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75#14583815—22no—
76#14583815—41no—
77#14583811—41no—
78C2×#13063436—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C8×D173—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346#4112yes—
118D692—21no—
119D692—21no—
120D692—81no—
121C22⋊D173—22no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—42no—
128C2×Dic173—42no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—81no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173C2×D411yescharacteristic
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—81no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—22no—
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—22no—
178C2×C173—81no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—41no—
183D173—81no—
184C173#330111yescharacteristic, complemented, Sylow
185#3301—1731nomaximal, Sylow
186#553—1731no—
187#548—1731no—
188#488—1731no—
189#485—1731no—
190#481D17311yescharacteristic, complemented
191#479—1731no—
192#466—1731no—
193#465—1731no—
194#463—1731no—
195#449—1731no—
196#448—1731no—
197C4×SD16—1731no—
198C8×D4—1731no—
199#416—1731no—
200C2×#103—1731no—
201C2×#41—1731no—
202C22×Q8—1731no—
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C22×C8—1731no—
207C22×C8—1731no—
208#126—3461no—
209#125—1731no—
210#124—3461no—
211#124—1731no—
212#123—3461no—
213C22⋊Q8—3461no—
214C22⋊Q8—1731no—
215#121—3461no—
216#121—3461no—
217#121—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—3461no—
223C4×D4—1731no—
224C4×D4—1731no—
225#117—1731no—
226C2×C4⋊C4C2×D17311yescharacteristic
227#107—1731no—
228#106C2×D17311yescharacteristic
229#106C2×D17311yescharacteristic
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#102C2×D17311yescharacteristic
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8C2×D17311yescharacteristic
243C4×C8—1731no—
244#41—3462no—
245#41—6921no—
246C2×Q8—3462no—
247C2×Q8—3461no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×Q8—6921no—
251C2×D4—21no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—3462no—
265SD16—6921no—
266D8—6921no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C4⋊C4C22×D17311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—3461no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—3461no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—3461no—
301Q8—3461no—
302Q8—3461no—
303Q8—1732no—
304Q8—6922no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—3462no—
309D4—3461no—
310D4—3461no—
311D4—13841no—
312D4—6922no—
313D4—6922no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—13841no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3461no—
343C22—3461no—
344C22—3461no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22#1914970012yes—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—3461no—
357C4—3462no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972511yescharacteristic
362C2—22no—
363C2—81no—
364C2—13841no—
365C2—6921no—
366C2#2601213311yescharacteristic
367C2C2×#1914970011yescharacteristic
368C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#481×C173C2
socleC22×C173C22×D4
hypercenter#481D173
hypocenterC173#3301
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #26012189C2semidirect
2. #26012147C2semidirect
3. #26011844C2semidirect
4. #26011835C2semidirect
5. #26011801C2semidirect
6. #26011725C2semidirect
7. #26011721C2semidirect
8. #26011705C2semidirect
9. #26011637C2semidirect
10. #26011632C2semidirect
11. #26011578C2semidirect
12. #26011554C2semidirect
13. #26011484C2semidirect
14. C2×#19149602C2semidirect
15. #481×C173C2semidirect
184. C173#3301semidirect
190. #481D173semidirect

Maximal subgroups

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

subgroupindex
#260121892
#260121472
#260118442
#260118352
#260118012
#260117252
#260117212
#260117052
#260116372
#260116322
#260115782
#260115542
#260114842
C2×#191496022
#481×C1732
#3301173