Finite Group Databasebeta

← #36364472

Subgroups — #36364472

Order 22144.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012070C211yesmaximal, characteristic, complemented
2#26012069C211yesmaximal, characteristic, complemented
3#26011822C211yesmaximal, characteristic, complemented
4#26011820C211yesmaximal, characteristic, complemented
5#26011808C211yesmaximal, characteristic, complemented
6#26011807C211yesmaximal, characteristic, complemented
7#26011748C211yesmaximal, characteristic, complemented
8#26011741C211yesmaximal, characteristic, complemented
9#26011719C211yesmaximal, characteristic, complemented
10#26011717C211yesmaximal, characteristic, complemented
11#26011653C211yesmaximal, characteristic, complemented
12#26011650C211yesmaximal, characteristic, complemented
13#26011633C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149659C2211yescharacteristic
23#19149658—21no—
24#19149658C2211yescharacteristic
25#19149656—21no—
26#19149655C2211yescharacteristic
27#19149655C2211yescharacteristic
28#19149653—21no—
29#19149652—21no—
30#19149651—21no—
31#19149651—21no—
32#19149651—21no—
33#19149650—21no—
34#19149650C2211yescharacteristic
35#19149650C2211yescharacteristic
36#19149650C2211yescharacteristic
37C2×C4⋊Dic173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346D411yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692D411yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C2×C22⋊D173D411yescharacteristic
72C22×Dic173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583816—41no—
78#14583814—41no—
79#14583809—41no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C8×D173—41no—
83C4⋊Dic173—21no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173C2×D411yescharacteristic
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—41no—
119D692—41no—
120C22⋊D173—41no—
121C22⋊D173—41no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—42no—
131C2×Dic173—81no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—81no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—42no—
151D4×C173—81no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—42no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173C22×D411yescharacteristic
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173H(5,𝔽2)11yescharacteristic
181C4×C173H(5,𝔽2)11yescharacteristic
182Dic173—21no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173—81no—
191C2×C173#53411yescharacteristic
192C2×C173#57711yescharacteristic
193C2×C173#57711yescharacteristic
194D173—41no—
195D173—41no—
196C173#320411yescharacteristic, complemented, Sylow
197#3204—1731nomaximal, Sylow
198#537—1731no—
199#537—1731no—
200#483—1731no—
201#483—1731no—
202#481—1731no—
203#481—1731no—
204#470—1731no—
205#469—1731no—
206#465—1731no—
207#465—1731no—
208#452—1731no—
209#452—1731no—
210#449—1731no—
211#449D17311yescharacteristic, complemented
212#408—1731no—
213C2×#41—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C2×Q16—1731no—
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×D8C2×D17311yescharacteristic
220C2×M16—1731no—
221C22×C8—1731no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8—1731no—
228#121—3461no—
229#121—3461no—
230#121—3461no—
231#121—3461no—
232#121—3461no—
233#121—3461no—
234#121—1731no—
235#121—1731no—
236#121—1731no—
237#121C2×D17311yescharacteristic
238C2×C4⋊C4—1731no—
239C2×C4⋊C4—1731no—
240#106—1731no—
241#106—1731no—
242#106—1731no—
243#106—1731no—
244#103—1731no—
245#103—1731no—
246#103—1731no—
247#103C2×D17311yescharacteristic
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8C2×D17311yescharacteristic
256#41—41no—
257#41—41no—
258#41—6921no—
259#41—6921no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8—1731no—
264C2×Q8C22×D17311yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4C22×D17311yescharacteristic
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4C22×D17311yescharacteristic
283Q16—6921no—
284SD16—41no—
285SD16—6921no—
286D8—41no—
287M16—6921no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—6921no—
291C2×C8C22×D17311yescharacteristic
292C2×C8C22×D17311yescharacteristic
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—1731no—
300C4⋊C4—1731no—
301C4⋊C4—1731no—
302C4⋊C4C22×D17311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C23—21no—
312C23—21no—
313C23—1731no—
314C23—1731no—
315C23D4×D17311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—3461no—
319Q8—3461no—
320Q8—6921no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—13841no—
328D4—13841no—
329D4—13841no—
330D4—6921no—
331D4—6921no—
332D4—6921no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—13841no—
348C2×C4—13841no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—6922no—
352C2×C4D4×D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3461no—
357C8—3461no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—3461no—
372C22—3461no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—3461no—
378C4—3461no—
379C4—6921no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—6921no—
389C2—6921no—
390C2#2601205111yescharacteristic
391C2#2601229811yescharacteristic
392C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3204
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012070C2semidirect
2. #26012069C2semidirect
3. #26011822C2semidirect
4. #26011820C2semidirect
5. #26011808C2semidirect
6. #26011807C2semidirect
7. #26011748C2semidirect
8. #26011741C2semidirect
9. #26011719C2semidirect
10. #26011717C2semidirect
11. #26011653C2semidirect
12. #26011650C2semidirect
13. #26011633C2semidirect
14. #26011456C2semidirect
15. #449×C173C2semidirect
196. C173#3204semidirect
211. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260120702
#260120692
#260118222
#260118202
#260118082
#260118072
#260117482
#260117412
#260117192
#260117172
#260116532
#260116502
#260116332
#260114562
#449×C1732
#3204173