Finite Group Databasebeta

← #36366249

Subgroups — #36366249

Order 22144.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012217C211yesmaximal, characteristic, complemented
2#26012213C211yesmaximal, characteristic
3#26011847C211yesmaximal, characteristic
4#26011846C211yesmaximal, characteristic
5#26011840C211yesmaximal, characteristic, complemented
6#26011798C211yesmaximal, characteristic, complemented
7#26011797C211yesmaximal, characteristic, complemented
8#26011770C211yesmaximal, characteristic
9#26011715C211yesmaximal, characteristic
10#26011713C211yesmaximal, characteristic
11#26011699C211yesmaximal, characteristic, complemented
12C8×Dic346C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011576C211yesmaximal, characteristic
15C4×Q16×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18#19149678—21no—
19#19149678C2211yescharacteristic
20#19149677C2211yescharacteristic
21#19149674—21no—
22#19149673—21no—
23#19149671—21no—
24#19149671C2211yescharacteristic
25#19149670—21no—
26#19149670C2211yescharacteristic
27#19149668C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149657—21no—
30Q8×Dic173C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32C4×Dic346—21no—
33C4×Dic346C2211yescharacteristic
34#19149644C2211yescharacteristic
35#19149641C2211yescharacteristic
36#19149636—21no—
37#19149612C2211yescharacteristic
38#19149610C2211yescharacteristic
39C4⋊#13063436C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149599C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346D411yescharacteristic
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C4×D173—21no—
64#14583816—22no—
65#14583813—22no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C4⋊Dic173—21no—
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173D411yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82#14583805D411yescharacteristic
83#14583803—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C4×Dic173C2311yescharacteristic
93C2×Q8×C173C2311yescharacteristic
94C2×Q8×C173C2311yescharacteristic
95Q16×C173—22no—
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C4⋊C4×C173—21no—
99C4⋊C4×C173—21no—
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C42×C173—21no—
103C42×C173—21no—
104C42×C173C2311yescharacteristic
105C22×D173—21no—
106Dic346—22no—
107Dic346—21no—
108Dic346—21no—
109Dic346—42no—
110D692—21no—
111D692—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173C2×D411yescharacteristic
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—42no—
120#13063436—21no—
121#13063436—21no—
122Q8×C173—21no—
123Q8×C173—21no—
124Q8×C173—21no—
125Q8×C173#4112yes—
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173C2×D411yescharacteristic
131C2×C4×C173C2×D411yescharacteristic
132C2×C4×C173C2411yescharacteristic
133C8×C173—21no—
134C8×C173—21no—
135C22×C173C22×D411yescharacteristic
136C2×D173—41no—
137C2×D173—41no—
138C2×D173—41no—
139C4×C173—22no—
140C4×C173—21no—
141C4×C173—21no—
142C4×C173—41no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173C2×#4111yescharacteristic
146C4×C1732−1+411yescharacteristic
147Dic173—21no—
148Dic173—21no—
149Dic173—21no—
150Dic173—41no—
151Dic173—41no—
152Dic173—41no—
153Dic173#13512yes—
154C2×C173#54811yescharacteristic
155C2×C173C2×#13511yescharacteristic
156C2×C173#57811yescharacteristic
157D173—81no—
158C173#336411yescharacteristic, complemented, Sylow
159#3364—1731nomaximal, Sylow
160#557—1731no—
161#556—1731no—
162#488—1731no—
163#488—1731no—
164#487—1731no—
165#479—1731no—
166#479—1731no—
167#474—1731no—
168#464—1731no—
169#464—1731no—
170#462—1731no—
171C8×Q8—1731no—
172C4×Q16—1731no—
173C4×Q16D17311yescharacteristic, complemented
174C4×SD16—1731no—
175C2×Q16—1731no—
176C2×Q16C2×D17311yescharacteristic
177C2×SD16—1731no—
178C4⋊Q8—3461no—
179C4⋊Q8—1731no—
180C4⋊Q8—1731no—
181#126—3461no—
182#126—3461no—
183#125—3461no—
184#125—3461no—
185#125—1731no—
186#125—1731no—
187#124—1731no—
188#124—1731no—
189C22⋊Q8—3461no—
190C4×Q8—3461no—
191C4×Q8—1731no—
192C4×Q8—1731no—
193C4×Q8—1731no—
194C4×Q8—1731no—
195C4×Q8C2×D17311yescharacteristic
196C4×Q8C2×D17311yescharacteristic
197C4×D4—1731no—
198#117—3461no—
199#107—1731no—
200#107C2×D17311yescharacteristic
201#106—1731no—
202C4⋊C8—1731no—
203C4⋊C8—1731no—
204C4⋊C8—1731no—
205#103—1731no—
206#103—1731no—
207#103—1731no—
208#103—1731no—
209#103—1731no—
210#103C2×D17311yescharacteristic
211#103C2×D17311yescharacteristic
212#102—1731no—
213#102—1731no—
214#102—1731no—
215C4×C8—1731no—
216C4×C8—1731no—
217C4×C8C2×D17311yescharacteristic
218C2×Q8—3461no—
219C2×Q8—1731no—
220C2×Q8—1731no—
221C2×Q8C22×D17311yescharacteristic
222C2×Q8C22×D17311yescharacteristic
223C2×D4—1731no—
224C22×C4—3461no—
225Q16—22no—
226Q16—3462no—
227SD16—3462no—
228C2×C8—1731no—
229C2×C8—1731no—
230C2×C8C22×D17311yescharacteristic
231C2×C8C22×D17311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—3461no—
235C4⋊C4—3461no—
236C4⋊C4—3461no—
237C4⋊C4—3461no—
238C4⋊C4—3461no—
239C4⋊C4—3461no—
240C4⋊C4—3461no—
241C4⋊C4—3461no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—1731no—
245C4⋊C4—1731no—
246C4⋊C4—1731no—
247C4⋊C4—1731no—
248C4⋊C4—1731no—
249C4⋊C4C22×D17311yescharacteristic
250C4⋊C4C22×D17311yescharacteristic
251C22⋊C4—3461no—
252C22⋊C4—3461no—
253C22⋊C4—3461no—
254C22⋊C4—3461no—
255C22⋊C4—3461no—
256C42—21no—
257C42—21no—
258C42—3461no—
259C42—3461no—
260C42—3461no—
261C42—1731no—
262C42—1731no—
263C42C22×D17311yescharacteristic
264C23—3461no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—3462no—
269Q8—3461no—
270Q8—3461no—
271Q8—6922no—
272Q8#1458382612yes—
273D4—3461no—
274D4—3461no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—3461no—
280C2×C4—3461no—
281C2×C4—3461no—
282C2×C4—1731no—
283C2×C4—1731no—
284C2×C4—1731no—
285C2×C4—1731no—
286C2×C4—6922no—
287C2×C4D4×D17311yescharacteristic
288C2×C4D4×D17311yescharacteristic
289C2×C4C23×D17311yescharacteristic
290C8—21no—
291C8—21no—
292C8—3461no—
293C8—3461no—
294C22—6921no—
295C22—6921no—
296C22—6921no—
297C22C2×D4×D17311yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—3461no—
305C4—3461no—
306C4—3461no—
307C4—1732no—
308C4—6921no—
309C4—6921no—
310C4—6921no—
311C4C2×#1458382611yescharacteristic
312C4#1914972611yescharacteristic
313C2—13841no—
314C2#2601215011yescharacteristic
315C2#135×D17311yescharacteristic
316C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×Q16×C173C2
socleC22×C173C22×D4
prime radicalC2×D692C23
hypercenterC4×Q16D173
hypocenterC173#3364
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #26012217C2semidirect
5. #26011840C2semidirect
6. #26011798C2semidirect
7. #26011797C2semidirect
11. #26011699C2semidirect
12. C8×Dic346C2semidirect
13. #26011583C2semidirect
15. C4×Q16×C173C2semidirect
158. C173#3364semidirect
173. C4×Q16D173semidirect

Maximal subgroups

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

subgroupindex
#260122172
#260122132
#260118472
#260118462
#260118402
#260117982
#260117972
#260117702
#260117152
#260117132
#260116992
C8×Dic3462
#260115832
#260115762
C4×Q16×C1732
#3364173