Finite Group Databasebeta

← #36363349

Subgroups — #36363349

Order 22144.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012044C211yesmaximal, characteristic, complemented
2#26011959C211yesmaximal, characteristic
3#26011801C211yesmaximal, characteristic, complemented
4#26011797C211yesmaximal, characteristic, complemented
5#26011794C211yesmaximal, characteristic, complemented
6#26011790C211yesmaximal, characteristic, complemented
7#26011785C211yesmaximal, characteristic, complemented
8#26011780C211yesmaximal, characteristic, complemented
9#26011775C211yesmaximal, characteristic, complemented
10#26011770C211yesmaximal, characteristic, complemented
11#26011527C211yesmaximal, characteristic, complemented
12C2×#19149608C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic
14#26011485C211yesmaximal, characteristic
15#427×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×#13063436C2211yescharacteristic
18#19149678C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149670C2211yescharacteristic
21#19149670C2211yescharacteristic
22#19149656—21no—
23#19149656—21no—
24C4×Dic346—21no—
25C4×Dic346—21no—
26#19149644C2211yescharacteristic
27#19149644C2211yescharacteristic
28#19149641C2211yescharacteristic
29#19149641C2211yescharacteristic
30#19149639—21no—
31#19149639—21no—
32#19149638C2211yescharacteristic
33#19149636—21no—
34C2×C4×Dic173C2211yescharacteristic
35C42×D173—21no—
36#19149612C2211yescharacteristic
37#19149612C2211yescharacteristic
38#19149610C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52C22×C8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#106×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59#14583826—22no—
60#14583826Q814yescomplemented
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C22×Dic173D411yescharacteristic
67C22×Dic173D411yescharacteristic
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×#13063436—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173D411yescharacteristic
91C4×Dic173D411yescharacteristic
92C4×Dic173C2311yescharacteristic
93C4×Dic173C2311yescharacteristic
94C22×C4×C173C2311yescharacteristic
95C2×C8×C173—22no—
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C4⋊C4×C173C2311yescharacteristic
99C4⋊C4×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C42×C173—21no—
105C42×C173—21no—
106C22×D173—21no—
107Dic346—21no—
108Dic346—42no—
109Dic346—42no—
110Dic346C2×Q812yes—
111D692—21no—
112D692C2×Q812yes—
113C22⋊D173—24no—
114C2×Dic173—21no—
115C2×Dic173—22no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—41no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—24no—
124C4×D173—42no—
125C4×D173—42no—
126#13063436—21no—
127#13063436—21no—
128C23×C173C2×D411yescharacteristic
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2×Q812yes—
135C2×C4×C173#4112yes—
136C2×C4×C173C2411yescharacteristic
137C8×C173—21no—
138C8×C173—21no—
139C22×C173—21no—
140C22×C173—21no—
141C22×C173C22⋊Q812yes—
142C22×C173C22×D411yescharacteristic
143C2×D173—22no—
144C2×D173—41no—
145C2×D173—41no—
146C4×C173—41no—
147C4×C173—41no—
148C4×C173—41no—
149C4×C173—41no—
150C4×C173C22⋊Q812yes—
151C4×C173C22×Q811yescharacteristic
152C4×C173C2×#4111yescharacteristic
153Dic173—22no—
154Dic173—21no—
155Dic173—21no—
156Dic173—21no—
157Dic173—41no—
158Dic173—41no—
159Dic173#13512yes—
160C2×C173—22no—
161C2×C173C2×C22⋊Q811yescharacteristic
162C2×C173C2×#13511yescharacteristic
163C2×C173#57511yescharacteristic
164D173—42no—
165C173#307611yescharacteristic, complemented, Sylow
166#3076—1731nomaximal, Sylow
167#533—1731no—
168C4×#41—1731no—
169#479—1731no—
170#479—1731no—
171#478—1731no—
172#478—1731no—
173#477—1731no—
174#476—1731no—
175#475—1731no—
176#474—1731no—
177#427—1731no—
178#427D17311yescharacteristic, complemented
179C2×C4⋊C8—1731no—
180#416—1731no—
181#416—1731no—
182C2×#41—1731no—
183C22×C8—1731no—
184C22×C8C2×D17311yescharacteristic
185C4⋊Q8—1731no—
186C4⋊Q8—1731no—
187#125—1731no—
188#125—1731no—
189C22⋊Q8—3461no—
190C22⋊Q8—3461no—
191C4×Q8—3461no—
192C4×Q8—3461no—
193C4×Q8—1731no—
194C4×Q8—1731no—
195C4×D4—3461no—
196C4×D4—3461no—
197C4×D4—1731no—
198C4×D4—1731no—
199#117—3461no—
200#117—1731no—
201#117C2×D17311yescharacteristic
202#117C2×D17311yescharacteristic
203C2×C42—3461no—
204C2×C42—1731no—
205#107—1731no—
206#107—1731no—
207#107C2×D17311yescharacteristic
208#107C2×D17311yescharacteristic
209#106—1731no—
210#106—1731no—
211#106C2×D17311yescharacteristic
212#106C2×D17311yescharacteristic
213C4⋊C8—1731no—
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216C4⋊C8—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#102—1731no—
222#102—1731no—
223#102—1731no—
224#102—1731no—
225#41—3462no—
226#41—1734no—
227C2×Q8—3461no—
228C2×Q8—3461no—
229C2×Q8—1731no—
230C2×D4—3461no—
231C2×D4—1731no—
232C22×C4—3461no—
233C22×C4—3461no—
234C22×C4—3461no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4C22×D17311yescharacteristic
238C2×C8—22no—
239C2×C8—3462no—
240C2×C8—1731no—
241C2×C8—1731no—
242C2×C8C22×D17311yescharacteristic
243C2×C8C22×D17311yescharacteristic
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—1731no—
253C4⋊C4—1731no—
254C4⋊C4C22×D17311yescharacteristic
255C4⋊C4C22×D17311yescharacteristic
256C4⋊C4C22×D17311yescharacteristic
257C4⋊C4C22×D17311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—3461no—
261C22⋊C4—3461no—
262C22⋊C4—3461no—
263C42—21no—
264C42—21no—
265C42—3461no—
266C42—3461no—
267C42—3461no—
268C42—1731no—
269C42—1731no—
270C42—1731no—
271C42—1731no—
272C23—3461no—
273C23D4×D17311yescharacteristic
274Q8—3461no—
275Q8—1732no—
276Q8—6922no—
277Q8—6922no—
278D4—3464no—
279D4—3461no—
280D4—1732no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3461no—
286C2×C4—3462no—
287C2×C4—3464no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—1731no—
291C2×C4—1731no—
292C2×C4—1731no—
293C2×C4—1731no—
294C2×C4—6922no—
295C2×C4—6922no—
296C2×C4—6921no—
297C2×C4D4×D17311yescharacteristic
298C2×C4Q8×D17312yes—
299C2×C4#1458382612yes—
300C2×C4C23×D17311yescharacteristic
301C8—21no—
302C8—21no—
303C8—3461no—
304C8—3461no—
305C22—21no—
306C22—21no—
307C22—3462no—
308C22—6921no—
309C22—6921no—
310C22#1914965712yes—
311C22C2×D4×D17311yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—3462no—
317C4—3461no—
318C4—3461no—
319C4—3461no—
320C4—1732no—
321C4—6921no—
322C4—6921no—
323C4#1914965712yes—
324C4C2×Q8×D17311yescharacteristic
325C4C2×#1458382611yescharacteristic
326C2—22no—
327C2—6922no—
328C2C2×#1914965711yescharacteristic
329C2#135×D17311yescharacteristic
330C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#427×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#427D173
hypocenterC173#3076
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #26012044C2semidirect
3. #26011801C2semidirect
4. #26011797C2semidirect
5. #26011794C2semidirect
6. #26011790C2semidirect
7. #26011785C2semidirect
8. #26011780C2semidirect
9. #26011775C2semidirect
10. #26011770C2semidirect
11. #26011527C2semidirect
12. C2×#19149608C2semidirect
15. #427×C173C2semidirect
60. #14583826Q8semidirect
165. C173#3076semidirect
178. #427D173semidirect

Maximal subgroups

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

subgroupindex
#260120442
#260119592
#260118012
#260117972
#260117942
#260117902
#260117852
#260117802
#260117752
#260117702
#260115272
C2×#191496082
#260114882
#260114852
#427×C1732
#3076173