Finite Group Databasebeta

← #36363861

Subgroups — #36363861

Order 22144.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012102C211yesmaximal, characteristic
2#26011967C211yesmaximal, characteristic
3#26011831C211yesmaximal, characteristic
4#26011830C211yesmaximal, characteristic
5#26011799C211yesmaximal, characteristic, complemented
6#26011798C211yesmaximal, characteristic, complemented
7#26011698C211yesmaximal, characteristic, complemented
8#26011645C211yesmaximal, characteristic
9#26011645C211yesmaximal, characteristic
10#26011598C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12Q16×Dic173C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic
15#462×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19#19149674—21no—
20#19149674—21no—
21#19149671C2211yescharacteristic
22#19149669—21no—
23#19149669—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27Q8×Dic173C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×Dic346—21no—
30C4×Dic346C2211yescharacteristic
31#19149644—21no—
32#19149644C2211yescharacteristic
33#19149637—21no—
34#19149637C2211yescharacteristic
35#19149636—21no—
36#19149636C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C4⋊Q8×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61Q8×D173—22no—
62Q8×D173—22no—
63C2×C4×D173—21no—
64C2×C4×D173C2311yescharacteristic
65#14583816—22no—
66#14583816—22no—
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
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#14583803D411yescharacteristic
87#14583803D411yescharacteristic
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98Q16×C173—22no—
99Q16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—24no—
110Dic346—21no—
111Dic346#4112yes—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—24no—
120C4×D173—21no—
121C4×D173—21no—
122#13063436—21no—
123#13063436—21no—
124Q8×C173—21no—
125Q8×C173—21no—
126Q8×C173—21no—
127Q8×C173—21no—
128Q8×C173—21no—
129Q8×C173#4112yes—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2×D411yescharacteristic
136C2×C4×C173C2411yescharacteristic
137C8×C173—21no—
138C8×C173—21no—
139C22×C173C22×D411yescharacteristic
140C2×D173—21no—
141C2×D173—21no—
142C2×D173—21no—
143C4×C173—22no—
144C4×C173—21no—
145C4×C173—21no—
146C4×C173—41no—
147C4×C173—41no—
148C4×C173—41no—
149C4×C173C2×#4111yescharacteristic
150C4×C173C2×#4111yescharacteristic
151Dic173—22no—
152Dic173—21no—
153Dic173—21no—
154Dic173—21no—
155Dic173—41no—
156Dic173—41no—
157Dic173—41no—
158C2×C173#52511yescharacteristic
159C2×C173#57811yescharacteristic
160C2×C173#57811yescharacteristic
161D173—41no—
162C173#312711yescharacteristic, complemented, Sylow
163#3127—1731nomaximal, Sylow
164#541—1731no—
165#519—1731no—
166#485—1731no—
167#485—1731no—
168#479—1731no—
169#479—1731no—
170#462—1731no—
171#462D17311yescharacteristic, complemented
172#451—1731no—
173#451—1731no—
174#441—1731no—
175#441—1731no—
176C4×Q16—1731no—
177C4×Q16—1731no—
178#433—1731no—
179C22×Q8—1731no—
180C2×Q16—1731no—
181C2×Q16—1731no—
182C2×Q16C2×D17311yescharacteristic
183C2×Q16C2×D17311yescharacteristic
184C4⋊Q8C2×D17311yescharacteristic
185#126—3461no—
186#126—3461no—
187#125—3461no—
188#125—3461no—
189#125—1731no—
190C22⋊Q8—1731no—
191C22⋊Q8—1731no—
192C4×Q8—3461no—
193C4×Q8—3461no—
194C4×Q8—1731no—
195C4×Q8—1731no—
196C4×Q8—1731no—
197C4×Q8—1731no—
198C4×Q8—1731no—
199C4×Q8C2×D17311yescharacteristic
200#117—3461no—
201#117—3461no—
202#117—1731no—
203#117—1731no—
204#107—1731no—
205#107—1731no—
206#106—1731no—
207#106—1731no—
208C4⋊C8—1731no—
209C4⋊C8C2×D17311yescharacteristic
210#103—1731no—
211#103—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103C2×D17311yescharacteristic
217#103C2×D17311yescharacteristic
218C22⋊C8—1731no—
219C22⋊C8—1731no—
220#97—1731no—
221C4×C8—1731no—
222C2×Q8—3461no—
223C2×Q8—3462no—
224C2×Q8—3462no—
225C2×Q8—1731no—
226C2×Q8C22×D17311yescharacteristic
227C2×Q8C22×D17311yescharacteristic
228C2×Q8C22×D17311yescharacteristic
229C22×C4—3461no—
230C22×C4—1731no—
231Q16—22no—
232Q16—22no—
233Q16—3462no—
234Q16—3462no—
235C2×C8—1731no—
236C2×C8—1731no—
237C2×C8C22×D17311yescharacteristic
238C2×C8C22×D17311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—1731no—
252C4⋊C4—1731no—
253C4⋊C4—1731no—
254C4⋊C4—1731no—
255C4⋊C4—1731no—
256C4⋊C4C22×D17311yescharacteristic
257C22⋊C4—3461no—
258C22⋊C4—3461no—
259C22⋊C4—3461no—
260C22⋊C4—1731no—
261C22⋊C4—1731no—
262C42—21no—
263C42—3461no—
264C42—3461no—
265C42—3461no—
266C42—3461no—
267C42—3461no—
268C42—3461no—
269C42—1731no—
270C42C22×D17311yescharacteristic
271C23—1731no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—3464no—
278Q8—3461no—
279Q8—1732no—
280Q8#1458382512yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3464no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—3461no—
291C2×C4—3461no—
292C2×C4—1731no—
293C2×C4—1731no—
294C2×C4—1731no—
295C2×C4D4×D17311yescharacteristic
296C2×C4D4×D17311yescharacteristic
297C2×C4C23×D17311yescharacteristic
298C8—21no—
299C8—21no—
300C8—3461no—
301C8—3461no—
302C22—3461no—
303C22—3461no—
304C22—3461no—
305C22C2×D4×D17311yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—3462no—
313C4—3461no—
314C4—3461no—
315C4—3461no—
316C4—6921no—
317C4—6921no—
318C4—6921no—
319C4#41×D17311yescharacteristic
320C4C2×#1458382511yescharacteristic
321C2—6921no—
322C2#2601200111yescharacteristic
323C2#2601230411yescharacteristic
324C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#462×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#462D173
hypocenterC173#3127
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
5. #26011799C2semidirect
6. #26011798C2semidirect
7. #26011698C2semidirect
10. #26011598C2semidirect
11. #26011595C2semidirect
12. Q16×Dic173C2semidirect
13. #26011583C2semidirect
15. #462×C173C2semidirect
162. C173#3127semidirect
171. #462D173semidirect

Maximal subgroups

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

subgroupindex
#260121022
#260119672
#260118312
#260118302
#260117992
#260117982
#260116982
#260116452
#260115982
#260115952
Q16×Dic1732
#260115832
#260115512
#462×C1732
#3127173