Finite Group Databasebeta

← #36365011

Subgroups — #36365011

Order 22144.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012104C211yesmaximal, characteristic, complemented
2#26012092C211yesmaximal, characteristic
3#26011801C211yesmaximal, characteristic, complemented
4#26011797C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011721C211yesmaximal, characteristic, complemented
9#26011719C211yesmaximal, characteristic, complemented
10#26011697C211yesmaximal, characteristic, complemented
11#26011696C211yesmaximal, characteristic, complemented
12#26011690C211yesmaximal, characteristic, complemented
13#26011690C211yesmaximal, characteristic, complemented
14#26011516C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×#14583815C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149675C2211yescharacteristic
24#19149674—21no—
25#19149674—21no—
26#19149670—21no—
27#19149670—21no—
28#19149670C2211yescharacteristic
29#19149662—21no—
30#19149661—21no—
31#19149659C2211yescharacteristic
32#19149659C2211yescharacteristic
33#19149654C2211yescharacteristic
34#19149653—21no—
35#19149652—21no—
36#19149650C2211yescharacteristic
37#19149644C2211yescharacteristic
38#19149644C2211yescharacteristic
39C4×D692C2211yescharacteristic
40#19149636C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×Dic346C2311yescharacteristic
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583815—41no—
69#14583815—41no—
70#14583814—41no—
71#14583814—41no—
72#14583809—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114D692—21no—
115D692—21no—
116D692—81no—
117C22⋊D173—81no—
118C22⋊D173—81no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—81no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132D4×C173—21no—
133D4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—81no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2411yescharacteristic
141C8×C173—21no—
142C8×C173—21no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173C22×D411yescharacteristic
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173H(5,𝔽2)11yescharacteristic
159C4×C1732−1+411yescharacteristic
160Dic173—21no—
161Dic173—21no—
162Dic173—21no—
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173—81no—
167C2×C173#53711yescharacteristic
168C2×C173#57711yescharacteristic
169C2×C173#57711yescharacteristic
170D173—41no—
171D173—81no—
172C173#326011yescharacteristic, complemented, Sylow
173#3260—1731nomaximal, Sylow
174#541—1731no—
175#540—1731no—
176#479—1731no—
177#479—1731no—
178#478—1731no—
179#478D17311yescharacteristic, complemented
180#468—1731no—
181#468—1731no—
182#465—1731no—
183#465—1731no—
184#461—1731no—
185#461—1731no—
186#460—1731no—
187#460—1731no—
188#424—1731no—
189C2×SD16—1731no—
190C2×SD16—1731no—
191C2×SD16—1731no—
192C2×SD16—1731no—
193C2×M16—1731no—
194C22×C8—1731no—
195C4⋊Q8—1731no—
196#127—1731no—
197#126—3461no—
198#126—3461no—
199#125—3461no—
200#125—3461no—
201#125—1731no—
202#125C2×D17311yescharacteristic
203#123—3461no—
204#123—3461no—
205C22⋊Q8—1731no—
206C22⋊Q8—1731no—
207#121—3461no—
208#121—3461no—
209#121—1731no—
210#121—1731no—
211C4×Q8—1731no—
212C4×Q8—1731no—
213C4×D4—1731no—
214C4×D4C2×D17311yescharacteristic
215#117—1731no—
216#107—1731no—
217#107C2×D17311yescharacteristic
218#106—1731no—
219#106C2×D17311yescharacteristic
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222C4⋊C8—1731no—
223C4⋊C8C2×D17311yescharacteristic
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#102—1731no—
229#102—1731no—
230#102C2×D17311yescharacteristic
231#102C2×D17311yescharacteristic
232C2×Q8—1731no—
233C2×Q8—1731no—
234C2×D4—3461no—
235C2×D4—3461no—
236C2×D4—3461no—
237C2×D4—3461no—
238C2×D4—1731no—
239C2×D4C22×D17311yescharacteristic
240C22×C4—21no—
241C22×C4—3461no—
242C22×C4—1731no—
243SD16—6921no—
244SD16—6921no—
245SD16—6921no—
246SD16—6921no—
247M16—6921no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8—6921no—
251C2×C8C22×D17311yescharacteristic
252C2×C8C22×D17311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
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—21no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—1731no—
276C22⋊C4—1731no—
277C42—3461no—
278C42—3461no—
279C42—1731no—
280C42C22×D17311yescharacteristic
281C23—21no—
282C23—3461no—
283C23—1731no—
284Q8—3461no—
285Q8—3461no—
286Q8—3461no—
287Q8—3461no—
288D4—21no—
289D4—21no—
290D4—3461no—
291D4—3461no—
292D4—13841no—
293D4—13841no—
294D4—13841no—
295D4—13841no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—13841no—
306C2×C4—1731no—
307C2×C4—1731no—
308C2×C4—1731no—
309C2×C4D4×D17311yescharacteristic
310C2×C4D4×D17311yescharacteristic
311C2×C4C23×D17311yescharacteristic
312C8—21no—
313C8—21no—
314C8—3461no—
315C8—3461no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—3461no—
320C22—3461no—
321C22—3461no—
322C22—6921no—
323C22—6921no—
324C22—6921no—
325C22C2×D4×D17311yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3461no—
332C4—3461no—
333C4—3461no—
334C4—6921no—
335C4—6921no—
336C4—6921no—
337C4#1914972511yescharacteristic
338C4#1914972711yescharacteristic
339C2—81no—
340C2—13841no—
341C2—6921no—
342C2#2601207111yescharacteristic
343C2#2601229811yescharacteristic
344C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radical#26012092C2
hypercenter#478D173
hypocenterC173#3260
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012104C2semidirect
3. #26011801C2semidirect
4. #26011797C2semidirect
5. #26011788C2semidirect
6. #26011738C2semidirect
7. #26011736C2semidirect
8. #26011721C2semidirect
9. #26011719C2semidirect
10. #26011697C2semidirect
11. #26011696C2semidirect
12. #26011690C2semidirect
13. #26011690C2semidirect
14. #26011516C2semidirect
15. #478×C173C2semidirect
172. C173#3260semidirect
179. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260120922
#260118012
#260117972
#260117882
#260117382
#260117362
#260117212
#260117192
#260116972
#260116962
#260116902
#260115162
#478×C1732
#3260173