Finite Group Databasebeta

← #36366300

Subgroups — #36366300

Order 22144.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012221C211yesmaximal, characteristic, complemented
2#26012210C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic
4#26011840C211yesmaximal, characteristic, complemented
5#26011779C211yesmaximal, characteristic
6#26011774C211yesmaximal, characteristic, complemented
7#26011773C211yesmaximal, characteristic, complemented
8#26011712C211yesmaximal, characteristic
9#26011710C211yesmaximal, characteristic
10#26011700C211yesmaximal, characteristic, complemented
11#26011619C211yesmaximal, characteristic, complemented
12#26011599C211yesmaximal, characteristic, complemented
13#26011595C211yesmaximal, characteristic, complemented
14#26011582C211yesmaximal, characteristic
15#489×C173C211yesmaximal, characteristic
16C2×Dic692C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19#19149678—21no—
20#19149678C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677—21no—
23#19149677—21no—
24#19149677C2211yescharacteristic
25#19149672—21no—
26#19149672—21no—
27#19149666C2211yescharacteristic
28#19149656—21no—
29Q8×Dic173C2211yescharacteristic
30C4×Dic346C2211yescharacteristic
31C4×Dic346C2211yescharacteristic
32C4×Dic346C2211yescharacteristic
33#19149644—21no—
34#19149644C2211yescharacteristic
35#19149644C2211yescharacteristic
36D4×Dic173C2211yescharacteristic
37#19149635—21no—
38#19149611C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346D411yescharacteristic
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C22×Dic173—21no—
65Dic692—22no—
66#14583816—22no—
67#14583814—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
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#14583805—21no—
82#14583805D411yescharacteristic
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C4×Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173C2311yescharacteristic
93C4×Dic173C2311yescharacteristic
94C2×Q8×C173—21no—
95C2×Q8×C173C2311yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C42×C173C2311yescharacteristic
105Dic346—21no—
106Dic346—22no—
107Dic346—21no—
108Dic346—21no—
109Dic346—81no—
110Dic346—81no—
111Dic346#4112yes—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—42no—
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121#13063436—21no—
122#13063436—21no—
123C23×C173—21no—
124Q8×C173—21no—
125Q8×C173—21no—
126Q8×C173—81no—
127D4×C173—21no—
128D4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173C2×D411yescharacteristic
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2411yescharacteristic
135C8×C173—21no—
136C8×C173—21no—
137C22×C173—41no—
138C22×C173—41no—
139C22×C173—41no—
140C22×C173C22×D411yescharacteristic
141C4×C173—21no—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173—41no—
146C4×C173C2×#4111yescharacteristic
147C4×C1732−1+411yescharacteristic
148Dic173—22no—
149Dic173—21no—
150Dic173—21no—
151Dic173—21no—
152Dic173—41no—
153Dic173—41no—
154Dic173—41no—
155Dic173#13712yes—
156C2×C173—81no—
157C2×C173#54811yescharacteristic
158C2×C173C2×#13711yescharacteristic
159C2×C173#57711yescharacteristic
160C173#336911yescharacteristic, complemented, Sylow
161#3369—1731nomaximal, Sylow
162Q82—1731no—
163#556—1731no—
164#489—1731no—
165#489D17311yescharacteristic
166#487—1731no—
167#476—1731no—
168#475—1731no—
169#475—1731no—
170#464—1731no—
171#464—1731no—
172#462—1731no—
173#446—1731no—
174#441—1731no—
175#441—1731no—
176C4×SD16—1731no—
177C2×Q16—1731no—
178C2×Q16—1731no—
179C2×SD16—1731no—
180C4⋊Q8—3461no—
181C4⋊Q8—3461no—
182C4⋊Q8—3461no—
183C4⋊Q8—1731no—
184C4⋊Q8—1731no—
185C4⋊Q8—1731no—
186C4⋊Q8C2×D17311yescharacteristic
187#126—3461no—
188#126—3461no—
189#124—1731no—
190#124C2×D17311yescharacteristic
191C22⋊Q8—3461no—
192C4×Q8—3461no—
193C4×Q8—1731no—
194C4×Q8—1731no—
195C4×Q8—1731no—
196C4×Q8—1731no—
197C4×Q8—1731no—
198C4×Q8—1731no—
199C4×D4—1731no—
200#117—3461no—
201#106—1731no—
202#106—1731no—
203#106—1731no—
204C4⋊C8—1731no—
205C4⋊C8—1731no—
206C4⋊C8—1731no—
207#103—1731no—
208#103—1731no—
209#103—1731no—
210#103—1731no—
211#103—1731no—
212#103C2×D17311yescharacteristic
213#103C2×D17311yescharacteristic
214#102—1731no—
215#102C2×D17311yescharacteristic
216#102C2×D17311yescharacteristic
217#97—1731no—
218#97C2×D17311yescharacteristic
219C4×C8—1731no—
220C2×Q8—21no—
221C2×Q8—3461no—
222C2×Q8—3461no—
223C2×Q8—1731no—
224C2×Q8—1731no—
225C2×Q8—1731no—
226C2×Q8C22×D17311yescharacteristic
227C2×D4C22×D17311yescharacteristic
228C22×C4—3461no—
229Q16—3462no—
230Q16—3462no—
231SD16—3462no—
232C2×C8—1731no—
233C2×C8—1731no—
234C2×C8C22×D17311yescharacteristic
235C2×C8C22×D17311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—3461no—
238C4⋊C4—3461no—
239C4⋊C4—3461no—
240C4⋊C4—3461no—
241C4⋊C4—3461no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—1731no—
247C4⋊C4—1731no—
248C4⋊C4—1731no—
249C4⋊C4—1731no—
250C4⋊C4—1731no—
251C4⋊C4C22×D17311yescharacteristic
252C4⋊C4C22×D17311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—3461no—
256C22⋊C4—3461no—
257C22⋊C4—3461no—
258C42—3461no—
259C42—3461no—
260C42—3461no—
261C42—3461no—
262C42—3461no—
263C42—1731no—
264C42—1731no—
265C42C22×D17311yescharacteristic
266C23—21no—
267Q8—21no—
268Q8—21no—
269Q8—81no—
270Q8—3462no—
271Q8—3461no—
272Q8—3461no—
273Q8—3461no—
274Q8—13841no—
275Q8—13841no—
276Q8—1732no—
277D4—21no—
278D4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3461no—
286C2×C4—1731no—
287C2×C4—1731no—
288C2×C4—1731no—
289C2×C4—1731no—
290C2×C4—6922no—
291C2×C4D4×D17311yescharacteristic
292C2×C4D4×D17311yescharacteristic
293C2×C4C23×D17311yescharacteristic
294C8—21no—
295C8—21no—
296C8—3461no—
297C8—3461no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D17311yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3461no—
308C4—3461no—
309C4—3461no—
310C4—3462no—
311C4—1732no—
312C4—6921no—
313C4—6921no—
314C4—6921no—
315C4#41×D17311yescharacteristic
316C4#1914972611yescharacteristic
317C2—81no—
318C2#2601214911yescharacteristic
319C2#137×D17311yescharacteristic
320C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#489×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×C173C23
hypercenter#489D173
hypocenterC173#3369
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #26012221C2semidirect
4. #26011840C2semidirect
6. #26011774C2semidirect
7. #26011773C2semidirect
10. #26011700C2semidirect
11. #26011619C2semidirect
12. #26011599C2semidirect
13. #26011595C2semidirect
160. C173#3369semidirect

Maximal subgroups

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

subgroupindex
#260122212
#260122102
#260118492
#260118402
#260117792
#260117742
#260117732
#260117122
#260117102
#260117002
#260116192
#260115992
#260115952
#260115822
#489×C1732
#3369173