Finite Group Databasebeta

← #36356705

Subgroups — #36356705

Order 22144.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149661C211yesmaximal, characteristic
2#26011391C211yesmaximal, characteristic
3#26011389C211yesmaximal, characteristic, complemented
4#26011388C211yesmaximal, characteristic, complemented
5#26011385C211yesmaximal, characteristic, complemented
6#26011383C211yesmaximal, characteristic, complemented
7#26011382C211yesmaximal, characteristic, complemented
8#26011374C211yesmaximal, characteristic, complemented
9#26011372C211yesmaximal, characteristic, complemented
10#26011368C211yesmaximal, characteristic
11#26011366C211yesmaximal, characteristic, complemented
12#26011363C211yesmaximal, characteristic, complemented
13#26011350C211yesmaximal, characteristic, complemented
14#26011348C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149661C418yescomplemented
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×C4⋊Dic173C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×C4×Dic173C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
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—41no—
54C2×C22⋊D173—41no—
55C2×C22⋊D173C2×C414yes—
56C22×Dic173—41no—
57C22×Dic173—41no—
58C22×Dic173—41no—
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C2×C4×D173—41no—
67C2×C4×D173—41no—
68C2×C4×D173C2×C414yescomplemented
69C4⋊Dic173—41no—
70C4⋊Dic173—41no—
71C4⋊Dic173C2×C414yescomplemented
72#14583805—41no—
73#14583805—41no—
74#14583805—41no—
75#14583805—41no—
76#14583805C2×C414yescomplemented
77C22⋊Dic173—41no—
78C22⋊Dic173C2×C414yes—
79#14583803—41no—
80#14583803—41no—
81#14583803—41no—
82#14583803—41no—
83#14583803C2×C414yescomplemented
84C4×Dic173—41no—
85C4×Dic173—41no—
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×C414yes—
97C42×C173—41no—
98C22×D173—21no—
99C22×D173—21no—
100C22×D173—21no—
101C22×D173—21no—
102C22×D173—21no—
103C22×D173—21no—
104C22×D173C22×C412yes—
105C22⋊D173—44no—
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—
129C23×C173—21no—
130C23×C173—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173C22×C412yes—
136C23×C173C2411yescharacteristic
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×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×C173C22×C412yes—
154C2×C4×C173C22×C412yes—
155C22×C173—22no—
156C22×C173—22no—
157C22×C173—22no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173C23×C411yescharacteristic
163C22×C173H(5,𝔽2)11yescharacteristic
164C22×C173H(5,𝔽2)11yescharacteristic
165C22×C173H(5,𝔽2)11yescharacteristic
166C22×C173H(5,𝔽2)11yescharacteristic
167C22×C1732−1+411yescharacteristic
168C22×C1732−1+411yescharacteristic
169C2×D173—22no—
170C2×D173—22no—
171C2×D173—22no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C4×C173—22no—
177C4×C173—22no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182Dic173—22no—
183Dic173—22no—
184Dic173—22no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—42no—
190C2×C173#51811yescharacteristic
191C2×C173#52011yescharacteristic
192C2×C173#52011yescharacteristic
193C2×C173#56011yescharacteristic
194C2×C173#56211yescharacteristic
195C2×C173#56211yescharacteristic
196C2×C173#56311yescharacteristic
197D173—42no—
198C173#237611yescharacteristic, complemented, Sylow
199#2376—1731nomaximal, Sylow
200C2×#123—1731no—
201#390—1731no—
202#389—1731no—
203#389—1731no—
204#388—1731no—
205#388—1731no—
206#388—1731no—
207#388D17311yescharacteristic, complemented
208#387—1731no—
209#387—1731no—
210#386—1731no—
211#385—1731no—
212#385—1731no—
213#380—1731no—
214#380—1731no—
215C22×D4—1731no—
216C23×C4—1731no—
217C23×C4—1731no—
218#123—1738no—
219C2×C4⋊C4—1731no—
220C2×C4⋊C4—1731no—
221C2×C4⋊C4—1731no—
222C2×C4⋊C4—1731no—
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⋊C4—1731no—
234C2×C22⋊C4—1731no—
235C2×C22⋊C4C2×D17311yescharacteristic
236C2×C22⋊C4C2×D17311yescharacteristic
237C2×C22⋊C4C2×D17311yescharacteristic
238C2×C42—1731no—
239C2×C42—1731no—
240C2×C42C2×D17311yescharacteristic
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#95C2×D17311yescharacteristic
250#95C2×D17311yescharacteristic
251C24—1731no—
252C24C22×D17311yescharacteristic
253C2×D4—1734no—
254C2×D4—6921no—
255C2×D4—6921no—
256C22×C4—1731no—
257C22×C4—1731no—
258C22×C4—1731no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4—1734no—
264C22×C4—6921no—
265C22×C4—6921no—
266C22×C4—6921no—
267C22×C4—6921no—
268C22×C4—6921no—
269C22×C4C22×D17311yescharacteristic
270C22×C4C22×D17311yescharacteristic
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273C22×C4C22×D17311yescharacteristic
274C22×C4C22×D17311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—1734no—
277C4⋊C4—1734no—
278C4⋊C4—6921no—
279C4⋊C4—6921no—
280C4⋊C4—6921no—
281C4⋊C4—6921no—
282C4⋊C4—6921no—
283C4⋊C4—6921no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—1734no—
287C22⋊C4—1734no—
288C22⋊C4—6921no—
289C22⋊C4—6921no—
290C22⋊C4—6921no—
291C22⋊C4—6921no—
292C22⋊C4—6921no—
293C22⋊C4C4×D17314yes—
294C42—41no—
295C42—6921no—
296C42—6921no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3461no—
304C23—3461no—
305C23—3461no—
306C23—3461no—
307C23—3461no—
308C23—3461no—
309C23—1732no—
310C23C2×C4×D17312yes—
311C23C23×D17311yescharacteristic
312D4—6924no—
313C2×C4—21no—
314C2×C4—21no—
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—3461no—
330C2×C4—3461no—
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—1732no—
348C2×C4—1732no—
349C2×C4—1732no—
350C2×C4—6924no—
351C2×C4—6924no—
352C2×C4C2×C4×D17312yes—
353C2×C4C2×C4×D17312yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3462no—
362C22—3462no—
363C22—3462no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22C22×C4×D17311yescharacteristic
369C22#1914972411yescharacteristic
370C22#1914972411yescharacteristic
371C22#1914972511yescharacteristic
372C22#1914972511yescharacteristic
373C22#1914972611yescharacteristic
374C22#1914972611yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3462no—
382C4—3462no—
383C4—3462no—
384C4—6921no—
385C4—6921no—
386C4—6921no—
387C4—6921no—
388C2—42no—
389C2—6922no—
390C2#2601196311yescharacteristic
391C2#2601197011yescharacteristic
392C2#2601197111yescharacteristic
393C2#2601223111yescharacteristic
394C2#2601223511yescharacteristic
395C2#2601223611yescharacteristic
396C2#2601224311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC22×C22⋊D173C22
hypercenter#388D173
hypocenterC173#2376
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #26011389C2semidirect
4. #26011388C2semidirect
5. #26011385C2semidirect
6. #26011383C2semidirect
7. #26011382C2semidirect
8. #26011374C2semidirect
9. #26011372C2semidirect
11. #26011366C2semidirect
12. #26011363C2semidirect
13. #26011350C2semidirect
14. #26011348C2semidirect
15. #388×C173C2semidirect
19. #19149661C4semidirect
68. C2×C4×D173C2×C4semidirect
71. C4⋊Dic173C2×C4semidirect
76. #14583805C2×C4semidirect
83. #14583803C2×C4semidirect
198. C173#2376semidirect
207. #388D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496612
#260113912
#260113892
#260113882
#260113852
#260113832
#260113822
#260113742
#260113722
#260113682
#260113662
#260113632
#260113502
#260113482
#388×C1732
#2376173