Finite Group Databasebeta

← #36356264

Subgroups — #36356264

Order 22144.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149651C211yesmaximal, characteristic
2C2×D4×Dic173C211yesmaximal, characteristic, complemented
3C2×#19149639C211yesmaximal, characteristic, complemented
4#26011385C211yesmaximal, characteristic, complemented
5#26011380C211yesmaximal, characteristic, complemented
6#26011378C211yesmaximal, characteristic, complemented
7#26011373C211yesmaximal, characteristic, complemented
8#26011370C211yesmaximal, characteristic, complemented
9#26011368C211yesmaximal, characteristic, complemented
10#26011367C211yesmaximal, characteristic, complemented
11#26011365C211yesmaximal, characteristic, complemented
12#26011363C211yesmaximal, characteristic, complemented
13#26011349C211yesmaximal, characteristic, complemented
14C2×#19149586C211yesmaximal, characteristic, complemented
15#386×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C23×Dic173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic
20C23×Dic173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22#19149651C418yescomplemented
23D4×Dic173—24no—
24#19149639—24no—
25C2×C4⋊Dic173C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C2×#14583805C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×C22⋊Dic173C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47C22×D4×C173C2211yescharacteristic
48C23×C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54C23×D173C2311yescharacteristic
55C2×C22⋊D173—22no—
56C2×C22⋊D173—22no—
57C2×C22⋊D173—41no—
58C2×C22⋊D173—41no—
59C2×C22⋊D173C2×C414yescomplemented
60C2×C22⋊D173C2×C414yescomplemented
61C22×Dic173—22no—
62C22×Dic173—41no—
63C22×Dic173—41no—
64C22×Dic173—41no—
65C22×Dic173—41no—
66C22×Dic173—41no—
67C22×Dic173—41no—
68C22×Dic173C2×C414yescomplemented
69C22×Dic173D414yescomplemented
70C22×Dic173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C2×C4×D173—22no—
78C2×C4×D173—41no—
79C2×C4×D173—41no—
80C4⋊Dic173—22no—
81#14583805—22no—
82#14583805—41no—
83#14583805—41no—
84#14583805C2×C414yescomplemented
85C22⋊Dic173—22no—
86C22⋊Dic173—22no—
87C22⋊Dic173C2×C414yescomplemented
88#14583803—22no—
89#14583803—41no—
90#14583803—41no—
91#14583803C2×C414yescomplemented
92C4×Dic173—22no—
93C4×Dic173—22no—
94C24×C173C2311yescharacteristic
95C24×C173C2311yescharacteristic
96C2×D4×C173—22no—
97C2×D4×C173—22no—
98C2×D4×C173C2×C414yescomplemented
99C22×C4×C173—22no—
100C22×C4×C173—41no—
101C22×C4×C173—41no—
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22⋊C4×C173—22no—
108C22⋊C4×C173—41no—
109C22⋊C4×C173—41no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173C22×C412yes—
117C22⋊D173—28no—
118C22⋊D173—44no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—44no—
136C2×Dic173—44no—
137C2×Dic173—44no—
138C2×Dic173C22×C412yes—
139C2×Dic173C22×C412yes—
140C2×Dic173C22×C412yes—
141C2×Dic173C2×D412yes—
142C2×Dic173C2×D412yes—
143C4×D173—44no—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173—21no—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—21no—
154C23×C173C22×C412yes—
155C23×C173C22×C412yes—
156C23×C173C2×D412yes—
157C23×C173#4112yes—
158C23×C173C2411yescharacteristic
159D4×C173—28no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173—44no—
175C2×C4×C173C22×C412yes—
176C22×C173—22no—
177C22×C173—22no—
178C22×C173—22no—
179C22×C173—22no—
180C22×C173—22no—
181C22×C173—22no—
182C22×C173—22no—
183C22×C173—22no—
184C22×C173—22no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173C4×D414yes—
190C22×C173C23×C411yescharacteristic
191C22×C173C22×D411yescharacteristic
192C22×C173C2×#4111yescharacteristic
193C22×C173H(5,𝔽2)11yescharacteristic
194C22×C173H(5,𝔽2)11yescharacteristic
195C22×C173H(5,𝔽2)11yescharacteristic
196C22×C1732−1+411yescharacteristic
197C2×D173—22no—
198C2×D173—22no—
199C2×D173—22no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C4×C173—22no—
205C4×C173—41no—
206C4×C173—41no—
207C4×C173—41no—
208C4×C173—41no—
209Dic173—22no—
210Dic173—22no—
211Dic173—22no—
212Dic173—22no—
213Dic173—22no—
214Dic173—41no—
215Dic173—41no—
216C2×C173—24no—
217C2×C173—42no—
218C2×C173C2×C4×D411yescharacteristic
219C2×C173#51811yescharacteristic
220C2×C173#52011yescharacteristic
221C2×C173#53411yescharacteristic
222C2×C173#53711yescharacteristic
223C2×C173#53811yescharacteristic
224C2×C173#53911yescharacteristic
225D173—42no—
226C173#232511yescharacteristic, complemented, Sylow
227#2325—1731nomaximal, Sylow
228C2×#121—1731no—
229C2×C4×D4—1731no—
230C2×C4×D4—1731no—
231#388—1731no—
232#388—1731no—
233#387—1731no—
234#387—1731no—
235#386—1731no—
236#386—1731no—
237#386—1731no—
238#386D17311yescharacteristic, complemented
239#385—1731no—
240#385—1731no—
241#380—1731no—
242C2×#95—1731no—
243C22×D4—1731no—
244C22×D4—1731no—
245C22×D4C2×D17311yescharacteristic
246C23×C4—1731no—
247C23×C4—1731no—
248C23×C4—1731no—
249C23×C4—1731no—
250C23×C4C2×D17311yescharacteristic
251#121—1738no—
252C4×D4—3464no—
253C4×D4—3464no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
257C2×C4⋊C4—1731no—
258C2×C4⋊C4—1731no—
259C2×C22⋊C4—1731no—
260C2×C22⋊C4—1731no—
261C2×C22⋊C4—1731no—
262C2×C22⋊C4—1731no—
263C2×C22⋊C4—1731no—
264C2×C22⋊C4—1731no—
265C2×C22⋊C4—1731no—
266C2×C22⋊C4C2×D17311yescharacteristic
267C2×C22⋊C4C2×D17311yescharacteristic
268C2×C22⋊C4C2×D17311yescharacteristic
269C2×C42—1731no—
270C2×C42—1731no—
271#95—1731no—
272#95—1731no—
273#95—1731no—
274#95—1731no—
275#95—1731no—
276#95—1731no—
277#95—1731no—
278#95—1731no—
279#95C2×D17311yescharacteristic
280#95C2×D17311yescharacteristic
281C24—1731no—
282C24C22×D17311yescharacteristic
283C24C22×D17311yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—3462no—
287C2×D4—3462no—
288C2×D4—1734no—
289C2×D4—1734no—
290C2×D4—6921no—
291C2×D4—6921no—
292C2×D4C4×D17314yescomplemented
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—3462no—
297C22×C4—3462no—
298C22×C4—1731no—
299C22×C4—1731no—
300C22×C4—1731no—
301C22×C4—1731no—
302C22×C4—1731no—
303C22×C4—1731no—
304C22×C4—1731no—
305C22×C4—1734no—
306C22×C4—1734no—
307C22×C4—6921no—
308C22×C4—6921no—
309C22×C4—6921no—
310C22×C4—6921no—
311C22×C4—6921no—
312C22×C4—6921no—
313C22×C4—6921no—
314C22×C4—6921no—
315C22×C4C22×D17311yescharacteristic
316C22×C4C22×D17311yescharacteristic
317C22×C4C22×D17311yescharacteristic
318C22×C4C22×D17311yescharacteristic
319C22×C4C22×D17311yescharacteristic
320C4⋊C4—3462no—
321C4⋊C4—3462no—
322C4⋊C4—1734no—
323C4⋊C4—6921no—
324C4⋊C4—6921no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—3462no—
329C22⋊C4—3462no—
330C22⋊C4—3462no—
331C22⋊C4—1734no—
332C22⋊C4—1734no—
333C22⋊C4—6921no—
334C22⋊C4—6921no—
335C42—3462no—
336C42—3462no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—3461no—
348C23—3461no—
349C23—3461no—
350C23—3461no—
351C23—3461no—
352C23—3461no—
353C23—1732no—
354C23C2×C4×D17312yes—
355C23C2×C4×D17312yes—
356C23D4×D17312yes—
357C23#1458382512yes—
358C23C23×D17311yescharacteristic
359D4—28no—
360D4—3468no—
361D4—6924no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—3461no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3461no—
386C2×C4—3461no—
387C2×C4—3461no—
388C2×C4—3461no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—1732no—
394C2×C4—1732no—
395C2×C4—1732no—
396C2×C4—1732no—
397C2×C4—1732no—
398C2×C4—6924no—
399C2×C4—6924no—
400C2×C4—6924no—
401C2×C4—6924no—
402C2×C4C2×C4×D17312yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3462no—
417C22—3462no—
418C22—3462no—
419C22—6921no—
420C22—6921no—
421C22—6921no—
422C22—6921no—
423C22#1914963914yes—
424C22C22×C4×D17311yescharacteristic
425C22C2×D4×D17311yescharacteristic
426C22C2×#1458382511yescharacteristic
427C22#1914972411yescharacteristic
428C22#1914972411yescharacteristic
429C22#1914972511yescharacteristic
430C22#1914972611yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—3462no—
437C4—3462no—
438C4—3462no—
439C4—3462no—
440C4—3462no—
441C4—6921no—
442C4—6921no—
443C2—24no—
444C2—42no—
445C2—6922no—
446C2C2×#1914963911yescharacteristic
447C2#2601196311yescharacteristic
448C2#2601197011yescharacteristic
449C2#2601204911yescharacteristic
450C2#2601206811yescharacteristic
451C2#2601207611yescharacteristic
452C2#2601208511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#386×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149651C2
hypercenter#386D173
hypocenterC173#2325
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic173C2semidirect
3. C2×#19149639C2semidirect
4. #26011385C2semidirect
5. #26011380C2semidirect
6. #26011378C2semidirect
7. #26011373C2semidirect
8. #26011370C2semidirect
9. #26011368C2semidirect
10. #26011367C2semidirect
11. #26011365C2semidirect
12. #26011363C2semidirect
13. #26011349C2semidirect
14. C2×#19149586C2semidirect
15. #386×C173C2semidirect
22. #19149651C4semidirect
59. C2×C22⋊D173C2×C4semidirect
60. C2×C22⋊D173C2×C4semidirect
68. C22×Dic173C2×C4semidirect
69. C22×Dic173D4semidirect
84. #14583805C2×C4semidirect
87. C22⋊Dic173C2×C4semidirect
91. #14583803C2×C4semidirect
98. C2×D4×C173C2×C4semidirect
226. C173#2325semidirect
238. #386D173semidirect
292. C2×D4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496512
C2×D4×Dic1732
C2×#191496392
#260113852
#260113802
#260113782
#260113732
#260113702
#260113682
#260113672
#260113652
#260113632
#260113492
C2×#191495862
#386×C1732
#2325173