Finite Group Databasebeta

← #36356295

Subgroups — #36356295

Order 22144.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149655C211yesmaximal, characteristic
2#26011398C211yesmaximal, characteristic, complemented
3#26011388C211yesmaximal, characteristic, complemented
4#26011386C211yesmaximal, characteristic, complemented
5#26011376C211yesmaximal, characteristic, complemented
6#26011375C211yesmaximal, characteristic, complemented
7#26011374C211yesmaximal, characteristic, complemented
8#26011373C211yesmaximal, characteristic, complemented
9#26011365C211yesmaximal, characteristic
10#26011362C211yesmaximal, characteristic
11#26011350C211yesmaximal, characteristic
12C4×#14583803C211yesmaximal, characteristic
13#26011338C211yesmaximal, characteristic
14#95×D173C211yesmaximal, characteristic
15#391×C173C211yesmaximal, characteristic, complemented
16C22×C4×D173C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149655C418yescomplemented
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C22×Q8×C173C2211yescharacteristic
46C2×C4⋊C4×C173C2211yescharacteristic
47C2×C42×C173C2211yescharacteristic
48#95×C173C2211yescharacteristic
49#95×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C23×D173C2311yescharacteristic
53C22×Dic173C2311yescharacteristic
54C22×Dic173C2311yescharacteristic
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C2×C4×D173—22no—
61C2×C4×D173—41no—
62C2×C4×D173—41no—
63C2×C4×D173—41no—
64C2×C4×D173—41no—
65C2×C4×D173—41no—
66C2×C4×D173—41no—
67C2×C4×D173C2×C414yes—
68C2×C4×D173Q814yescomplemented
69C4⋊Dic173—41no—
70C4⋊Dic173C2×C414yescomplemented
71#14583805—41no—
72#14583805—41no—
73#14583805—41no—
74#14583805—41no—
75#14583805C2×C414yescomplemented
76#14583805C2×C414yescomplemented
77#14583803—22no—
78#14583803—22no—
79#14583803C2×C414yes—
80#14583803C2×C414yes—
81C4×Dic173—41no—
82C4×Dic173—41no—
83C4×Dic173—41no—
84C2×Q8×C173—41no—
85C2×Q8×C173—41no—
86C2×Q8×C173C2×C414yescomplemented
87C22×C4×C173C2311yescharacteristic
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C4⋊C4×C173—41no—
95C42×C173—41no—
96C22×D173—21no—
97C22×D173—21no—
98C22×D173—21no—
99C22×D173—21no—
100C22×D173C22×C412yes—
101C22×D173C2×Q812yes—
102C22×D173#4112yes—
103C2×Dic173—21no—
104C2×Dic173—21no—
105C2×Dic173—21no—
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×Dic173C22×C412yes—
121C2×Dic173C22×C412yes—
122C2×Dic173C22×C412yes—
123C2×Dic173C2×Q812yes—
124C4×D173—44no—
125C4×D173—44no—
126C4×D173—44no—
127C23×C173C2411yescharacteristic
128Q8×C173—44no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×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×C173—21no—
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×C173C22×C412yes—
147C2×C4×C173C22×C412yes—
148C2×C4×C173C22×C412yes—
149C2×C4×C173C2×Q812yes—
150C22×C173C23×C411yescharacteristic
151C22×C173C22×Q811yescharacteristic
152C22×C173C2×#4111yescharacteristic
153C22×C173H(5,𝔽2)11yescharacteristic
154C22×C173H(5,𝔽2)11yescharacteristic
155C22×C173H(5,𝔽2)11yescharacteristic
156C22×C1732−1+411yescharacteristic
157C2×D173—22no—
158C2×D173—22no—
159C2×D173—22no—
160C2×D173—22no—
161C2×D173—22no—
162C2×D173—22no—
163C2×D173C4×Q814yes—
164C4×C173—22no—
165C4×C173—22no—
166C4×C173—22no—
167C4×C173—22no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171Dic173—22no—
172Dic173—22no—
173Dic173—22no—
174Dic173—22no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173C2×C4×Q811yescharacteristic
179C2×C173#51811yescharacteristic
180C2×C173#52011yescharacteristic
181C2×C173#53811yescharacteristic
182C2×C173#53911yescharacteristic
183C2×C173#54311yescharacteristic
184C2×C173#54411yescharacteristic
185D173—24no—
186C173#232911yescharacteristic, complemented, Sylow
187#2329—1731nomaximal, Sylow
188C2×C22⋊Q8—1731no—
189#391—1731no—
190#391D17311yescharacteristic, complemented
191#389—1731no—
192#389—1731no—
193#387—1731no—
194#387—1731no—
195#387—1731no—
196#387—1731no—
197#385—1731no—
198#385—1731no—
199#380—1731no—
200C4×C22⋊C4—1731no—
201C4×C22⋊C4—1731no—
202C2×#95—1731no—
203C22×Q8C2×D17311yescharacteristic
204C23×C4—1731no—
205C23×C4—1731no—
206C23×C4—1731no—
207C22⋊Q8—1738no—
208C2×C4⋊C4—1731no—
209C2×C4⋊C4—1731no—
210C2×C4⋊C4—1731no—
211C2×C4⋊C4—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4—1731no—
215C2×C4⋊C4—1731no—
216C2×C4⋊C4C2×D17311yescharacteristic
217C2×C22⋊C4—1731no—
218C2×C22⋊C4—1731no—
219C2×C22⋊C4—1731no—
220C2×C22⋊C4—1731no—
221C2×C42—1731no—
222C2×C42—1731no—
223C2×C42—1731no—
224C2×C42C2×D17311yescharacteristic
225#95—1731no—
226#95—1731no—
227#95—1731no—
228#95—1731no—
229#95—1731no—
230#95—1731no—
231#95—1731no—
232#95—1731no—
233#95—1731no—
234#95—1731no—
235#95C2×D17311yescharacteristic
236#95C2×D17311yescharacteristic
237#95C2×D17311yescharacteristic
238#95C2×D17311yescharacteristic
239C24—1731no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8C4×D17314yescomplemented
243C22×C4—3462no—
244C22×C4—1731no—
245C22×C4—1731no—
246C22×C4—1731no—
247C22×C4—1731no—
248C22×C4—1731no—
249C22×C4—1731no—
250C22×C4—1731no—
251C22×C4—1734no—
252C22×C4—1734no—
253C22×C4—6921no—
254C22×C4—6921no—
255C22×C4—6921no—
256C22×C4—6921no—
257C22×C4—6921no—
258C22×C4—6921no—
259C22×C4C22×D17311yescharacteristic
260C22×C4C22×D17311yescharacteristic
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263C22×C4C22×D17311yescharacteristic
264C22×C4C22×D17311yescharacteristic
265C22×C4C22×D17311yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—1734no—
268C4⋊C4—1734no—
269C4⋊C4—1734no—
270C4⋊C4—6921no—
271C4⋊C4—6921no—
272C4⋊C4—6921no—
273C4⋊C4—6921no—
274C4⋊C4—6921no—
275C22⋊C4—3462no—
276C22⋊C4—3462no—
277C22⋊C4—1734no—
278C22⋊C4—1734no—
279C42—41no—
280C42—6921no—
281C42—6921no—
282C42—6921no—
283C23—3461no—
284C23—3461no—
285C23—3461no—
286C23—3461no—
287C23—1732no—
288C23—1732no—
289C23—1732no—
290C23C23×D17311yescharacteristic
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—1732no—
327C2×C4—1732no—
328C2×C4—1732no—
329C2×C4—1732no—
330C2×C4—6924no—
331C2×C4—6924no—
332C2×C4—6924no—
333C2×C4C2×C4×D17312yes—
334C2×C4C2×C4×D17312yes—
335C2×C4C2×C4×D17312yes—
336C2×C4Q8×D17312yes—
337C22—3462no—
338C22—3462no—
339C22—3462no—
340C22—3462no—
341C22—3462no—
342C22—3462no—
343C22—1734no—
344C22C22×C4×D17311yescharacteristic
345C22C2×Q8×D17311yescharacteristic
346C22#41×D17311yescharacteristic
347C22#1914972411yescharacteristic
348C22#1914972411yescharacteristic
349C22#1914972511yescharacteristic
350C22#1914972711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3462no—
359C4—3462no—
360C4—3462no—
361C4—3462no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C2—3464no—
366C2C4×Q8×D17311yescharacteristic
367C2#2601196411yescharacteristic
368C2#2601197311yescharacteristic
369C2#2601207511yescharacteristic
370C2#2601208711yescharacteristic
371C2#2601211611yescharacteristic
372C2#2601211811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#391×C173C2
socle, cosocleC23×C173C24
prime radicalC23×D173C23
hypercenter#391D173
hypocenterC173#2329
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #26011398C2semidirect
3. #26011388C2semidirect
4. #26011386C2semidirect
5. #26011376C2semidirect
6. #26011375C2semidirect
7. #26011374C2semidirect
8. #26011373C2semidirect
15. #391×C173C2semidirect
19. #19149655C4semidirect
68. C2×C4×D173Q8semidirect
70. C4⋊Dic173C2×C4semidirect
75. #14583805C2×C4semidirect
76. #14583805C2×C4semidirect
86. C2×Q8×C173C2×C4semidirect
186. C173#2329semidirect
190. #391D173semidirect
242. C2×Q8C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496552
#260113982
#260113882
#260113862
#260113762
#260113752
#260113742
#260113732
#260113652
#260113622
#260113502
C4×#145838032
#260113382
#95×D1732
#391×C1732
#2329173