Finite Group Databasebeta

← #36363925

Subgroups — #36363925

Order 22144.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012118C211yesmaximal, characteristic, complemented
2#26012061C211yesmaximal, characteristic
3#26011860C211yesmaximal, characteristic, complemented
4#26011858C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011845C211yesmaximal, characteristic, complemented
7#26011844C211yesmaximal, characteristic, complemented
8#26011843C211yesmaximal, characteristic, complemented
9#26011840C211yesmaximal, characteristic, complemented
10#26011498C211yesmaximal, characteristic, complemented
11#26011489C211yesmaximal, characteristic
12#26011484C211yesmaximal, characteristic
13#103×D173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#489×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149679C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677—21no—
23#19149677C2211yescharacteristic
24#19149671C2211yescharacteristic
25#19149670C2211yescharacteristic
26#19149669—21no—
27#19149666C2211yescharacteristic
28#19149661—21no—
29#19149661—21no—
30#19149657—21no—
31#19149656—21no—
32#19149656—21no—
33#19149655—21no—
34#19149637C2211yescharacteristic
35#19149636C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149604C2211yescharacteristic
38#19149604C2211yescharacteristic
39#19149604C2211yescharacteristic
40#19149604C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149589C2211yescharacteristic
50C4×#13063436C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65Q8×D173—22no—
66Q8×D173—22no—
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583809—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—22no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—42no—
114Dic346—81no—
115C22⋊D173—42no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—42no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—42no—
125C4×D173—81no—
126C4×D173#4112yes—
127C4×D173#4112yes—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—81no—
134D4×C173—21no—
135D4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173#12412yes—
151C4×C173—21no—
152C4×C173—21no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173C2×#4111yescharacteristic
157C4×C173C2×#4111yescharacteristic
158Dic173—21no—
159Dic173—21no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163Dic173—41no—
164Dic173#12412yes—
165C2×C173—81no—
166C2×C173C2×#12411yescharacteristic
167C2×C173#57711yescharacteristic
168C2×C173#57811yescharacteristic
169D173—22no—
170C173#313711yescharacteristic, complemented, Sylow
171#3137—1731nomaximal, Sylow
172#544—1731no—
173#536—1731no—
174#491—1731no—
175#491—1731no—
176#489—1731no—
177#489D17311yescharacteristic, complemented
178#488—1731no—
179#488—1731no—
180#487—1731no—
181#487—1731no—
182#419—1731no—
183#417—1731no—
184#416—1731no—
185C2×#103—1731no—
186#405—1731no—
187C2×#41—1731no—
188C22×Q8—1731no—
189C2×M16—1731no—
190C22×C8—1731no—
191C4⋊Q8—3461no—
192C4⋊Q8—1731no—
193C4⋊Q8—1731no—
194C4⋊Q8—1731no—
195C4⋊Q8C2×D17311yescharacteristic
196#125—3461no—
197#125—1731no—
198#125—1731no—
199#124—1731no—
200#124C2×D17311yescharacteristic
201#123—3461no—
202#123—3461no—
203C22⋊Q8—3461no—
204C22⋊Q8—3461no—
205C22⋊Q8—3461no—
206C22⋊Q8—3461no—
207#117—1731no—
208#117—1731no—
209C2×C4⋊C4—1731no—
210#103—1731no—
211#103—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#103C2×D17311yescharacteristic
222#102—1731no—
223#102—1731no—
224#102C2×D17311yescharacteristic
225#102C2×D17311yescharacteristic
226#97—1731no—
227#97C2×D17311yescharacteristic
228C4×C8—1731no—
229C4×C8—1731no—
230#41—3462no—
231#41—3462no—
232C2×Q8—21no—
233C2×Q8—3461no—
234C2×Q8—3461no—
235C2×Q8—3462no—
236C2×Q8—3462no—
237C2×Q8—1731no—
238C2×Q8—1731no—
239C2×Q8C22×D17311yescharacteristic
240C2×D4—3461no—
241C2×D4C22×D17311yescharacteristic
242C22×C4—3461no—
243C22×C4—3461no—
244C22×C4—3461no—
245C22×C4—1731no—
246M16—3462no—
247C2×C8—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
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⋊C4—1731no—
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—1731no—
275C22⋊C4—1731no—
276C42—3461no—
277C42—1731no—
278C42C22×D17311yescharacteristic
279C23—21no—
280C23—1731no—
281Q8—21no—
282Q8—21no—
283Q8—81no—
284Q8—3461no—
285Q8—3461no—
286Q8—3461no—
287Q8—3461no—
288Q8—13841no—
289Q8—6922no—
290D4—21no—
291D4—21no—
292D4—6922no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—13841no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—1732no—
305C2×C4—1732no—
306C2×C4—6922no—
307C2×C4—6922no—
308C2×C4D4×D17311yescharacteristic
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3461no—
314C8—3461no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—3461no—
319C22—3461no—
320C22—1732no—
321C22C2×D4×D17311yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3461no—
328C4—3461no—
329C4—1732no—
330C4—6921no—
331C4—6921no—
332C4—6921no—
333C4—6921no—
334C4#41×D17311yescharacteristic
335C4#41×D17311yescharacteristic
336C2—81no—
337C2—3462no—
338C2#124×D17311yescharacteristic
339C2#2601230111yescharacteristic
340C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#489×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583825C22
hypercenter#489D173
hypocenterC173#3137
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #26012118C2semidirect
3. #26011860C2semidirect
4. #26011858C2semidirect
5. #26011849C2semidirect
6. #26011845C2semidirect
7. #26011844C2semidirect
8. #26011843C2semidirect
9. #26011840C2semidirect
10. #26011498C2semidirect
13. #103×D173C2semidirect
14. #26011448C2semidirect
15. #489×C173C2semidirect
170. C173#3137semidirect
177. #489D173semidirect

Maximal subgroups

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

subgroupindex
#260121182
#260120612
#260118602
#260118582
#260118492
#260118452
#260118442
#260118432
#260118402
#260114982
#260114892
#260114842
#103×D1732
#260114482
#489×C1732
#3137173