Finite Group Databasebeta

← #36361880

Subgroups — #36361880

Order 22144.

582 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×#19149674C211yesmaximal, characteristic, complemented
2C22×C4×Dic173C211yesmaximal, characteristic, complemented
3#26011383C211yesmaximal, characteristic, complemented
4#26011381C211yesmaximal, characteristic, complemented
5#26011380C211yesmaximal, characteristic, complemented
6#26011377C211yesmaximal, characteristic, complemented
7#26011374C211yesmaximal, characteristic, complemented
8#26011372C211yesmaximal, characteristic, complemented
9#26011370C211yesmaximal, characteristic
10#26011368C211yesmaximal, characteristic
11#26011368C211yesmaximal, characteristic
12#26011366C211yesmaximal, characteristic, complemented
13#26011363C211yesmaximal, characteristic, complemented
14#26011358C211yesmaximal, characteristic, complemented
15#385×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C23×Dic173C2211yescharacteristic
19#19149674—24no—
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×C22⋊Dic173C2211yescharacteristic
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×C4×Dic173—24no—
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35C2×C4×Dic173C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46C23×C4×C173C2211yescharacteristic
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53C23×D173C2311yescharacteristic
54C2×C22⋊D173—22no—
55C2×C22⋊D173—22no—
56C2×C22⋊D173—22no—
57C22×Dic173—22no—
58C22×Dic173—22no—
59C22×Dic173—22no—
60C22×Dic173—22no—
61C22×Dic173—22no—
62C22×Dic173—22no—
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C22×Dic173C2311yescharacteristic
70C4⋊Dic173—22no—
71C4⋊Dic173—22no—
72#14583805—22no—
73#14583805—22no—
74C22⋊Dic173—22no—
75#14583803—22no—
76#14583803—22no—
77#14583803—22no—
78#14583803—22no—
79#14583803—22no—
80#14583803—22no—
81C4×Dic173—22no—
82C4×Dic173—22no—
83C4×Dic173—22no—
84C4×Dic173D414yescomplemented
85C24×C173C2311yescharacteristic
86C22×C4×C173—22no—
87C22×C4×C173—22no—
88C22×C4×C173—22no—
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C22×C4×C173C2311yescharacteristic
95C4⋊C4×C173—22no—
96C4⋊C4×C173—22no—
97C22⋊C4×C173—22no—
98C22⋊C4×C173—22no—
99C22×D173—21no—
100C22×D173—21no—
101C22×D173—21no—
102C22×D173—21no—
103C22×D173—21no—
104C22×D173—21no—
105C22×D173—21no—
106C22⋊D173—44no—
107C2×Dic173—28no—
108C2×Dic173—28no—
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—21no—
125C2×Dic173—21no—
126C2×Dic173C2×D412yes—
127C2×Dic173C2×D412yes—
128C2×Dic173#4112yes—
129C2×Dic173#4112yes—
130C23×C173—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173#4112yes—
135C23×C173#4112yes—
136C23×C173#4112yes—
137C23×C173C2411yescharacteristic
138C2×C4×C173—28no—
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×C173C2×D412yes—
156C2×C4×C173#4112yes—
157C22×C173—22no—
158C22×C173—22no—
159C22×C173—22no—
160C22×C173—22no—
161C22×C173—22no—
162C22×C173—22no—
163C22×C173#12614yescomplemented
164C22×C173C22×D411yescharacteristic
165C22×C173C2×#4111yescharacteristic
166C22×C173C2×#4111yescharacteristic
167C22×C173C2×#4111yescharacteristic
168C22×C173C2×#4111yescharacteristic
169C22×C173C2×#4111yescharacteristic
170C22×C173C2×#4111yescharacteristic
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
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—22no—
188Dic173—41no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—24no—
192C2×C173#52511yescharacteristic
193C2×C173#52511yescharacteristic
194C2×C173#52511yescharacteristic
195C2×C173C2×#12611yescharacteristic
196C2×C173#52911yescharacteristic
197C2×C173#52911yescharacteristic
198C2×C173#52911yescharacteristic
199D173—81no—
200C173#285211yescharacteristic, complemented, Sylow
201#2852—1731nomaximal, Sylow
202C2×#126—1731no—
203C22×C42—1731no—
204#388—1731no—
205#388—1731no—
206#388—1731no—
207#387—1731no—
208#387—1731no—
209#387—1731no—
210#386—1731no—
211#386—1731no—
212#386—1731no—
213#385—1731no—
214#385—1731no—
215#385D17311yescharacteristic, complemented
216#383—1731no—
217C22×D4—1731no—
218C23×C4—1731no—
219C23×C4—1731no—
220C23×C4C2×D17311yescharacteristic
221#126—3464no—
222C2×C4⋊C4—1731no—
223C2×C4⋊C4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4—1731no—
226C2×C4⋊C4C2×D17311yescharacteristic
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×C42—3464no—
238C2×C42—1731no—
239C2×C42—1731no—
240C2×C42—1731no—
241C2×C42—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#95—1731no—
252#95C2×D17311yescharacteristic
253#95C2×D17311yescharacteristic
254C24—1731no—
255C24C22×D17311yescharacteristic
256C2×D4—3462no—
257C2×D4—3462no—
258C2×D4—3462no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—3462no—
263C22×C4—3462no—
264C22×C4—3462no—
265C22×C4—3462no—
266C22×C4—3462no—
267C22×C4—3462no—
268C22×C4—1731no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—1731no—
275C22×C4C22×D17311yescharacteristic
276C22×C4C22×D17311yescharacteristic
277C22×C4C22×D17311yescharacteristic
278C22×C4C22×D17311yescharacteristic
279C22×C4C22×D17311yescharacteristic
280C22×C4C22×D17311yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—3462no—
284C4⋊C4—3462no—
285C4⋊C4—3462no—
286C4⋊C4—3462no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—3462no—
290C22⋊C4—3462no—
291C22⋊C4—3462no—
292C22⋊C4—3462no—
293C22⋊C4—3462no—
294C22⋊C4—3462no—
295C22⋊C4—3462no—
296C42—3462no—
297C42—3462no—
298C42—3462no—
299C42—1734no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—3461no—
305C23—3461no—
306C23—3461no—
307C23—3461no—
308C23—3461no—
309C23—3461no—
310C23—3461no—
311C23#1458382512yes—
312C23#1458382612yes—
313C23#1458382712yes—
314C23C23×D17311yescharacteristic
315D4—6924no—
316C2×C4—28no—
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—21no—
332C2×C4—21no—
333C2×C4—3468no—
334C2×C4—3468no—
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—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—1732no—
353C2×C4—1732no—
354C2×C4—1732no—
355C2×C4—1732no—
356C2×C4D4×D17312yes—
357C2×C4#1458382512yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22#1914967414yescomplemented
372C22C2×D4×D17311yescharacteristic
373C22#41×D17311yescharacteristic
374C22#41×D17311yescharacteristic
375C22C2×#1458382511yescharacteristic
376C22C2×#1458382511yescharacteristic
377C22C2×#1458382611yescharacteristic
378C22C2×#1458382711yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3462no—
386C4—3462no—
387C4—3462no—
388C4—3462no—
389C4—6921no—
390C4—6921no—
391C4—6921no—
392C2—24no—
393C2—13841no—
394C2#2601199711yescharacteristic
395C2#2601200011yescharacteristic
396C2#2601200411yescharacteristic
397C2C2×#1914967411yescharacteristic
398C2#2601201811yescharacteristic
399C2#2601202111yescharacteristic
400C2#2601202711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#385×C173C2
prime radicalC22×C22⋊D173C22
hypercenter#385D173
hypocenterC173#2852
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#19149674C2semidirect
2. C22×C4×Dic173C2semidirect
3. #26011383C2semidirect
4. #26011381C2semidirect
5. #26011380C2semidirect
6. #26011377C2semidirect
7. #26011374C2semidirect
8. #26011372C2semidirect
12. #26011366C2semidirect
13. #26011363C2semidirect
14. #26011358C2semidirect
15. #385×C173C2semidirect
84. C4×Dic173D4semidirect
163. C22×C173#126semidirect
200. C173#2852semidirect
215. #385D173semidirect
371. C22#19149674semidirect

Maximal subgroups

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

subgroupindex
C2×#191496742
C22×C4×Dic1732
#260113832
#260113812
#260113802
#260113772
#260113742
#260113722
#260113702
#260113682
#260113662
#260113632
#260113582
#385×C1732
#2852173