Finite Group Databasebeta

← #36356193

Subgroups — #36356193

Order 22144.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149639C211yesmaximal, characteristic
2#26011384C211yesmaximal, characteristic, complemented
3#26011375C211yesmaximal, characteristic, complemented
4#26011373C211yesmaximal, characteristic, complemented
5#26011367C211yesmaximal, characteristic
6#26011367C211yesmaximal, characteristic
7#26011365C211yesmaximal, characteristic, complemented
8#26011362C211yesmaximal, characteristic, complemented
9#26011358C211yesmaximal, characteristic, complemented
10#26011356C211yesmaximal, characteristic, complemented
11#26011352C211yesmaximal, characteristic, complemented
12#26011349C211yesmaximal, characteristic, complemented
13C4×#14583803C211yesmaximal, characteristic, complemented
14#95×D173C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21#19149639C418yescomplemented
22C2×C4⋊Dic173C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×C4×Dic173C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33#19149586C2211yescharacteristic
34#19149586C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C22⋊C4×C173C2211yescharacteristic
47C2×C22⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C42×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C23×D173C2311yescharacteristic
53C2×C22⋊D173—22no—
54C2×C22⋊D173—22no—
55C2×C22⋊D173C2×C414yes—
56C22×Dic173—41no—
57C22×Dic173—41no—
58C22×Dic173C2×C414yescomplemented
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C2×C4×D173—22no—
67C2×C4×D173—22no—
68C2×C4×D173—41no—
69C2×C4×D173—41no—
70C2×C4×D173—41no—
71C2×C4×D173—41no—
72C2×C4×D173—41no—
73C2×C4×D173—41no—
74C2×C4×D173C2×C414yes—
75C4⋊Dic173—41no—
76#14583805—41no—
77#14583805—41no—
78#14583805C2×C414yescomplemented
79C22⋊Dic173—41no—
80C22⋊Dic173—41no—
81#14583803—22no—
82#14583803—22no—
83#14583803C2×C414yes—
84C4×Dic173—41no—
85C4×Dic173C2×C414yescomplemented
86C24×C173C2311yescharacteristic
87C22×C4×C173C2311yescharacteristic
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—41no—
94C22⋊C4×C173—41no—
95C22⋊C4×C173—41no—
96C22⋊C4×C173C2×C414yescomplemented
97C42×C173—41no—
98C22×D173—21no—
99C22×D173—21no—
100C22×D173—21no—
101C22×D173—21no—
102C22×D173C22×C412yes—
103C22×D173#4112yes—
104C22×D173#4112yes—
105C22⋊D173—28no—
106C2×Dic173—21no—
107C2×Dic173—21no—
108C2×Dic173—21no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—44no—
125C2×Dic173C22×C412yes—
126C2×Dic173C22×C412yes—
127C2×Dic173C22×C412yes—
128C4×D173—44no—
129C4×D173—44no—
130C4×D173—44no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173—21no—
136C23×C173—21no—
137C23×C173C22×C412yes—
138C23×C173C2411yescharacteristic
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173C22×C412yes—
156C2×C4×C173C22×C412yes—
157C22×C173—22no—
158C22×C173—22no—
159C22×C173—22no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C23×C411yescharacteristic
165C22×C173C2×#4111yescharacteristic
166C22×C173C2×#4111yescharacteristic
167C22×C173H(5,𝔽2)11yescharacteristic
168C22×C173H(5,𝔽2)11yescharacteristic
169C22×C173H(5,𝔽2)11yescharacteristic
170C22×C1732−1+411yescharacteristic
171C2×D173—22no—
172C2×D173—22no—
173C2×D173—22no—
174C2×D173—22no—
175C2×D173—22no—
176C2×D173—22no—
177C2×D173#11714yes—
178C4×C173—22no—
179C4×C173—22no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173—41no—
184Dic173—22no—
185Dic173—22no—
186Dic173—22no—
187Dic173—41no—
188Dic173—41no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—42no—
192C2×C173C2×#11711yescharacteristic
193C2×C173#51811yescharacteristic
194C2×C173#52011yescharacteristic
195C2×C173#53811yescharacteristic
196C2×C173#53811yescharacteristic
197C2×C173#53911yescharacteristic
198C2×C173#53911yescharacteristic
199D173—24no—
200C173#231811yescharacteristic, complemented, Sylow
201#2318—1731nomaximal, Sylow
202C2×C4×D4—1731no—
203#388—1731no—
204#388D17311yescharacteristic, complemented
205#387—1731no—
206#387—1731no—
207#386—1731no—
208#386—1731no—
209#385—1731no—
210#385—1731no—
211#383—1731no—
212#382—1731no—
213#381—1731no—
214#380—1731no—
215C4×C22⋊C4—1731no—
216C2×#95—1731no—
217C22×D4—1731no—
218C23×C4—1731no—
219C23×C4—1731no—
220C23×C4—1731no—
221C23×C4—1731no—
222C4×D4—1738no—
223C2×C4⋊C4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4—1731no—
226C2×C4⋊C4—1731no—
227C2×C4⋊C4C2×D17311yescharacteristic
228C2×C22⋊C4—1731no—
229C2×C22⋊C4—1731no—
230C2×C22⋊C4—1731no—
231C2×C22⋊C4—1731no—
232C2×C22⋊C4—1731no—
233C2×C22⋊C4C2×D17311yescharacteristic
234C2×C22⋊C4C2×D17311yescharacteristic
235C2×C22⋊C4C2×D17311yescharacteristic
236C2×C42—1731no—
237C2×C42—1731no—
238C2×C42C2×D17311yescharacteristic
239#95—1731no—
240#95—1731no—
241#95—1731no—
242#95—1731no—
243#95—1731no—
244#95—1731no—
245#95—1731no—
246#95—1731no—
247#95—1731no—
248#95—1731no—
249#95—1731no—
250#95—1731no—
251#95C2×D17311yescharacteristic
252#95C2×D17311yescharacteristic
253C24—1731no—
254C24C22×D17311yescharacteristic
255C2×D4—3462no—
256C2×D4—3462no—
257C2×D4—1734no—
258C22×C4—3462no—
259C22×C4—3462no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4—1731no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4—1734no—
268C22×C4—1734no—
269C22×C4—6921no—
270C22×C4—6921no—
271C22×C4—6921no—
272C22×C4—6921no—
273C22×C4—6921no—
274C22×C4—6921no—
275C22×C4—6921no—
276C22×C4—6921no—
277C22×C4C22×D17311yescharacteristic
278C22×C4C22×D17311yescharacteristic
279C22×C4C22×D17311yescharacteristic
280C22×C4C22×D17311yescharacteristic
281C22×C4C22×D17311yescharacteristic
282C22×C4C22×D17311yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—1734no—
285C4⋊C4—6921no—
286C4⋊C4—6921no—
287C4⋊C4—6921no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—3462no—
291C22⋊C4—3462no—
292C22⋊C4—1734no—
293C22⋊C4—6921no—
294C22⋊C4—6921no—
295C22⋊C4C4×D17314yescomplemented
296C42—41no—
297C42—1734no—
298C42—6921no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—3461no—
306C23—3461no—
307C23—3461no—
308C23—3461no—
309C23—1732no—
310C23—1732no—
311C23—1732no—
312C23C2×C4×D17312yes—
313C23C23×D17311yescharacteristic
314D4—3468no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—1732no—
350C2×C4—1732no—
351C2×C4—1732no—
352C2×C4—6924no—
353C2×C4—6924no—
354C2×C4—6924no—
355C2×C4—6924no—
356C2×C4C2×C4×D17312yes—
357C2×C4C2×C4×D17312yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3462no—
366C22—3462no—
367C22—3462no—
368C22—3462no—
369C22—3462no—
370C22—3462no—
371C22—1734no—
372C22C22×C4×D17311yescharacteristic
373C22#41×D17311yescharacteristic
374C22#41×D17311yescharacteristic
375C22#1914972411yescharacteristic
376C22#1914972411yescharacteristic
377C22#1914972511yescharacteristic
378C22#1914972611yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3462no—
386C4—3462no—
387C4—3462no—
388C4—6921no—
389C4—6921no—
390C4—6921no—
391C4—6921no—
392C2—42no—
393C2—3464no—
394C2#117×D17311yescharacteristic
395C2#2601196311yescharacteristic
396C2#2601197011yescharacteristic
397C2#2601207311yescharacteristic
398C2#2601207511yescharacteristic
399C2#2601208211yescharacteristic
400C2#2601208411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC22×C22⋊D173C22
hypercenter#388D173
hypocenterC173#2318
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #26011384C2semidirect
3. #26011375C2semidirect
4. #26011373C2semidirect
7. #26011365C2semidirect
8. #26011362C2semidirect
9. #26011358C2semidirect
10. #26011356C2semidirect
11. #26011352C2semidirect
12. #26011349C2semidirect
13. C4×#14583803C2semidirect
14. #95×D173C2semidirect
15. #388×C173C2semidirect
21. #19149639C4semidirect
58. C22×Dic173C2×C4semidirect
78. #14583805C2×C4semidirect
85. C4×Dic173C2×C4semidirect
96. C22⋊C4×C173C2×C4semidirect
200. C173#2318semidirect
204. #388D173semidirect
295. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496392
#260113842
#260113752
#260113732
#260113672
#260113652
#260113622
#260113582
#260113562
#260113522
#260113492
C4×#145838032
#95×D1732
#388×C1732
#2318173