Finite Group Databasebeta

← #36365612

Subgroups — #36365612

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012176C211yesmaximal, characteristic
2#26012154C211yesmaximal, characteristic
3#26011858C211yesmaximal, characteristic, complemented
4#26011835C211yesmaximal, characteristic, complemented
5#26011801C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011649C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011597C211yesmaximal, characteristic, complemented
12#26011559C211yesmaximal, characteristic
13#26011498C211yesmaximal, characteristic, complemented
14C2×#19149602C211yesmaximal, characteristic, complemented
15#483×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149678—21no—
24#19149677C2211yescharacteristic
25#19149674—21no—
26#19149670C2211yescharacteristic
27#19149669—21no—
28#19149662—21no—
29#19149659C2211yescharacteristic
30#19149659C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149657—21no—
33#19149656—21no—
34#19149644C2211yescharacteristic
35#19149644C2211yescharacteristic
36C4×C22⋊D173C2211yescharacteristic
37#19149639—21no—
38#19149638—21no—
39#19149637C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—22no—
61C2×Dic346—21no—
62C2×Dic346—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×C22⋊D173D411yescharacteristic
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72Dic692—22no—
73Dic692—41no—
74#14583816—41no—
75#14583809—41no—
76C2×#13063436—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—42no—
118Dic346—42no—
119Dic346#4112yes—
120C22⋊D173—22no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—42no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132#13063436—21no—
133#13063436—21no—
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—42no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173#13712yes—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C4×C173—21no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173C2×#4111yescharacteristic
161C4×C173H(5,𝔽2)11yescharacteristic
162Dic173—21no—
163Dic173—22no—
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—22no—
170C2×C173#54611yescharacteristic
171C2×C173C2×#13711yescharacteristic
172C2×C173#57811yescharacteristic
173D173—41no—
174C173#331211yescharacteristic, complemented, Sylow
175#3312—1731nomaximal, Sylow
176#552—1731no—
177D4×Q8—1731no—
178#491—1731no—
179#485—1731no—
180#483D17311yescharacteristic, complemented
181#479—1731no—
182#470—1731no—
183#467—1731no—
184#462—1731no—
185#451—1731no—
186#451—1731no—
187#441—1731no—
188#435—1731no—
189#419—1731no—
190C2×#103—1731no—
191C22×Q8—1731no—
192C22×Q8—1731no—
193C2×Q16—1731no—
194C2×Q16—1731no—
195C2×Q16—1731no—
196C2×M16—1731no—
197C22×C8—1731no—
198C4⋊Q8—3461no—
199C4⋊Q8—1731no—
200#126—3461no—
201#125—3461no—
202#125—1731no—
203#123—3461no—
204C22⋊Q8—3461no—
205C22⋊Q8—3461no—
206C22⋊Q8—1731no—
207C22⋊Q8—1731no—
208C22⋊Q8—1731no—
209C22⋊Q8C2×D17311yescharacteristic
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—1731no—
214#117—3461no—
215#117—1731no—
216C2×C4⋊C4C2×D17311yescharacteristic
217#107—1731no—
218#106C2×D17311yescharacteristic
219#106C2×D17311yescharacteristic
220C4⋊C8—1731no—
221#103—1731no—
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#103C2×D17311yescharacteristic
231C22⋊C8—1731no—
232C22⋊C8C2×D17311yescharacteristic
233#97—1731no—
234C2×Q8—3461no—
235C2×Q8—3462no—
236C2×Q8—3461no—
237C2×Q8—3462no—
238C2×Q8—1731no—
239C2×Q8—1731no—
240C2×Q8—1731no—
241C2×Q8—6921no—
242C2×Q8—6921no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—1731no—
245C22×C4—21no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—1731no—
249C22×C4C22×D17311yescharacteristic
250Q16—3462no—
251Q16—6921no—
252Q16—6921no—
253M16—6921no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8—6921no—
257C2×C8C22×D17311yescharacteristic
258C2×C8C22×D17311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—1731no—
281C22⋊C4—1731no—
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42—1731no—
286C23—1731no—
287C23D4×D17311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3461no—
291Q8—3461no—
292Q8—3461no—
293Q8—3461no—
294Q8—3461no—
295Q8—1732no—
296Q8—6922no—
297Q8—6922no—
298D4—3462no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—6922no—
316C2×C4—6922no—
317C2×C4D4×D17311yescharacteristic
318C2×C4C23×D17311yescharacteristic
319C8—21no—
320C8—21no—
321C8—3461no—
322C8—3461no—
323C22—21no—
324C22—21no—
325C22—3461no—
326C22—3461no—
327C22—3461no—
328C22#1914970812yes—
329C22C2×D4×D17311yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3461no—
336C4—3461no—
337C4—3461no—
338C4—3462no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4#41×D17311yescharacteristic
343C4#1914972511yescharacteristic
344C2—22no—
345C2—6921no—
346C2#2601213311yescharacteristic
347C2C2×#1914970811yescharacteristic
348C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#483D173
hypocenterC173#3312
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #26011858C2semidirect
4. #26011835C2semidirect
5. #26011801C2semidirect
6. #26011750C2semidirect
7. #26011730C2semidirect
8. #26011700C2semidirect
9. #26011649C2semidirect
10. #26011648C2semidirect
11. #26011597C2semidirect
13. #26011498C2semidirect
14. C2×#19149602C2semidirect
15. #483×C173C2semidirect
174. C173#3312semidirect
180. #483D173semidirect

Maximal subgroups

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

subgroupindex
#260121762
#260121542
#260118582
#260118352
#260118012
#260117502
#260117302
#260117002
#260116492
#260116482
#260115972
#260115592
#260114982
C2×#191496022
#483×C1732
#3312173