Finite Group Databasebeta

← #36364267

Subgroups — #36364267

Order 22144.

366 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012043C211yesmaximal, characteristic, complemented
2#26012027C211yesmaximal, characteristic
3#26011899C211yesmaximal, characteristic, complemented
4#26011884C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011852C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011722C211yesmaximal, characteristic
9#26011720C211yesmaximal, characteristic
10#26011598C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14C2×#19149589C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677C2211yescharacteristic
22#19149674—21no—
23#19149670C2211yescharacteristic
24#19149669—21no—
25#19149668C2211yescharacteristic
26#19149663—21no—
27#19149658C2211yescharacteristic
28#19149656—21no—
29#19149653C2211yescharacteristic
30Q8×Dic173C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32#19149644—21no—
33#19149644C2211yescharacteristic
34#19149641C2211yescharacteristic
35#19149639—21no—
36#19149637—21no—
37#19149635—21no—
38C2×C4×Dic173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149589C2211yescharacteristic
49#19149589C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C22×Dic173D411yescharacteristic
64C22×Dic173D411yescharacteristic
65C2×C4×D173—21no—
66#14583816—22no—
67#14583815—22no—
68#14583813—22no—
69C2×#13063436—22no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173D411yescharacteristic
92C4×Dic173D411yescharacteristic
93C4×Dic173C2311yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C22×C4×C173C2311yescharacteristic
98Q16×C173—22no—
99C2×C8×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22×D173—21no—
109Dic346—21no—
110Dic346—21no—
111Dic346—42no—
112D692—21no—
113D692—21no—
114C22⋊D173—42no—
115C2×Dic173—22no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—41no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—42no—
125#13063436—21no—
126#13063436#4112yes—
127C23×C173C2×D411yescharacteristic
128Q8×C173—21no—
129Q8×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—21no—
141C8×C173#4112yes—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173#13712yes—
145C22×C173C22×D411yescharacteristic
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—21no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173C2×#4111yescharacteristic
155C4×C173C2×#4111yescharacteristic
156Dic173—21no—
157Dic173—21no—
158Dic173—21no—
159Dic173—21no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163C2×C173—22no—
164C2×C173#53211yescharacteristic
165C2×C173C2×#13711yescharacteristic
166C2×C173#57511yescharacteristic
167D173—81no—
168C173#318111yescharacteristic, complemented, Sylow
169#3181—1731nomaximal, Sylow
170#533—1731no—
171#529—1731no—
172#501—1731no—
173#497—1731no—
174#491—1731no—
175#489—1731no—
176#467—1731no—
177#467D17311yescharacteristic, complemented
178#465—1731no—
179#465—1731no—
180#441—1731no—
181#441—1731no—
182#440—1731no—
183#440—1731no—
184C2×#97—1731no—
185C2×Q16—1731no—
186C2×Q16C2×D17311yescharacteristic
187C2×SD16—1731no—
188C2×SD16—1731no—
189C22×C8—1731no—
190C22×C8C2×D17311yescharacteristic
191C4⋊Q8—1731no—
192C4⋊Q8—1731no—
193#126—3461no—
194#125—3461no—
195#125—1731no—
196#124—1731no—
197#123—3461no—
198C22⋊Q8—3461no—
199C22⋊Q8—1731no—
200C22⋊Q8C2×D17311yescharacteristic
201C22⋊Q8C2×D17311yescharacteristic
202#121—1731no—
203C4×Q8—3461no—
204C4×Q8—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×D4—3461no—
208C4×D4—1731no—
209#117—3461no—
210#117—3461no—
211C2×C42—1731no—
212#107—1731no—
213#107C2×D17311yescharacteristic
214#106—1731no—
215#106—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#97—1731no—
225#97—1731no—
226#97—1731no—
227#97—1731no—
228C2×Q8—3461no—
229C2×Q8—1731no—
230C2×Q8C22×D17311yescharacteristic
231C2×Q8C22×D17311yescharacteristic
232C2×D4—3461no—
233C2×D4—1731no—
234C22×C4—3461no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4C22×D17311yescharacteristic
238Q16—22no—
239Q16—3462no—
240SD16—3462no—
241SD16—3462no—
242C2×C8—22no—
243C2×C8—3462no—
244C2×C8—1731no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C2×C8C22×D17311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4C22×D17311yescharacteristic
262C4⋊C4C22×D17311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C42—3461no—
271C42—3461no—
272C42—3461no—
273C42—1731no—
274C42—1731no—
275C42—1731no—
276C42—1731no—
277C23—3461no—
278C23D4×D17311yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—3461no—
284Q8—3461no—
285Q8—6922no—
286D4—3461no—
287D4—3461no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3462no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—1731no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—6922no—
304C2×C4—6921no—
305C2×C4D4×D17311yescharacteristic
306C2×C4C23×D17311yescharacteristic
307C8—21no—
308C8—3461no—
309C8—1732no—
310C8#1458382512yes—
311C22—21no—
312C22—21no—
313C22—6921no—
314C22—6921no—
315C22—6921no—
316C22#1914971112yes—
317C22C2×D4×D17311yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3461no—
324C4—3461no—
325C4—3461no—
326C4—3461no—
327C4—6921no—
328C4—6921no—
329C4—6921no—
330C4#41×D17311yescharacteristic
331C4C2×#1458382511yescharacteristic
332C2—22no—
333C2—13841no—
334C2#2601203611yescharacteristic
335C2C2×#1914971111yescharacteristic
336C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#19149653C22
hypercenter#467D173
hypocenterC173#3181
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26012043C2semidirect
3. #26011899C2semidirect
4. #26011884C2semidirect
5. #26011858C2semidirect
6. #26011852C2semidirect
7. #26011730C2semidirect
10. #26011598C2semidirect
11. #26011595C2semidirect
12. #26011594C2semidirect
13. #26011590C2semidirect
14. C2×#19149589C2semidirect
15. #467×C173C2semidirect
168. C173#3181semidirect
177. #467D173semidirect

Maximal subgroups

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

subgroupindex
#260120432
#260120272
#260118992
#260118842
#260118582
#260118522
#260117302
#260117222
#260117202
#260115982
#260115952
#260115942
#260115902
C2×#191495892
#467×C1732
#3181173