Finite Group Databasebeta

← #36356539

Subgroups — #36356539

Order 22144.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149649C211yesmaximal, characteristic
2C2×#19149641C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D173C211yesmaximal, characteristic, complemented
4C22×#14583803C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D173C211yesmaximal, characteristic, complemented
6#26011392C211yesmaximal, characteristic, complemented
7#26011381C211yesmaximal, characteristic, complemented
8#26011380C211yesmaximal, characteristic, complemented
9#26011369C211yesmaximal, characteristic, complemented
10#26011369C211yesmaximal, characteristic, complemented
11#26011368C211yesmaximal, characteristic, complemented
12#26011362C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic, complemented
15#385×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C22×D692C2211yescharacteristic
19C22×C22⋊D173C2211yescharacteristic
20C23×Dic173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22C22×C4×D173C2211yescharacteristic
23C22×C4×D173C2211yescharacteristic, complemented
24#19149649C418yescomplemented
25#19149641—24no—
26C4×C22⋊D173—24no—
27C2×#14583805C2211yescharacteristic, complemented
28C2×C22⋊Dic173C2211yescharacteristic
29C2×#14583803—24no—
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C2×#14583803C2211yescharacteristic, complemented
39C2×#14583803C2211yescharacteristic, complemented
40C2×#14583803C2211yescharacteristic, complemented
41C22⋊C4×D173—24no—
42C2×C4×Dic173C2211yescharacteristic, complemented
43C2×C4×Dic173C2211yescharacteristic, complemented
44C2×C4×Dic173C2211yescharacteristic, complemented
45#19149586C2211yescharacteristic, complemented
46#19149586C2211yescharacteristic, complemented
47#19149586C2211yescharacteristic, complemented
48#19149586C2211yescharacteristic, complemented
49C23×C4×C173C2211yescharacteristic
50C2×C4⋊C4×C173C2211yescharacteristic, complemented
51C2×C4⋊C4×C173C2211yescharacteristic, complemented
52C2×C22⋊C4×C173C2211yescharacteristic
53C2×C22⋊C4×C173C2211yescharacteristic
54#95×C173C2211yescharacteristic, complemented
55#95×C173C2211yescharacteristic, complemented
56C23×D173—22no—
57C23×D173—22no—
58C23×D173—22no—
59C23×D173—22no—
60C23×D173—41no—
61C23×D173—41no—
62C23×D173C2×C414yescomplemented
63C23×D173C2311yescharacteristic
64C23×D173C2311yescharacteristic
65C23×D173C2311yescharacteristic
66C2×D692—22no—
67C2×D692—22no—
68C2×D692—41no—
69C2×D692—41no—
70C2×D692C2×C414yescomplemented
71C2×D692C2×C414yescomplemented
72C2×C22⋊D173—22no—
73C2×C22⋊D173—22no—
74C2×C22⋊D173C2×C414yescomplemented
75C22×Dic173—22no—
76C22×Dic173—22no—
77C22×Dic173—41no—
78C22×Dic173C2311yescharacteristic
79C22×Dic173C2311yescharacteristic
80C22×Dic173C2311yescharacteristic
81C22×Dic173C2311yescharacteristic
82C22×Dic173C2311yescharacteristic
83C2×C4×D173—22no—
84C2×C4×D173—22no—
85C2×C4×D173—22no—
86C2×C4×D173—22no—
87C2×C4×D173—22no—
88C2×C4×D173—22no—
89C2×C4×D173—41no—
90C2×C4×D173—41no—
91C2×C4×D173—41no—
92#14583805—22no—
93C22⋊Dic173D414yescomplemented
94#14583803—22no—
95#14583803—22no—
96#14583803—22no—
97#14583803—22no—
98#14583803—22no—
99#14583803—22no—
100#14583803—22no—
101#14583803—41no—
102#14583803C2×C414yescomplemented
103#14583803C2×C414yescomplemented
104#14583803D414yescomplemented
105C4×Dic173—22no—
106C4×Dic173—22no—
107C4×Dic173—41no—
108C24×C173C2311yescharacteristic
109C22×C4×C173—22no—
110C22×C4×C173—22no—
111C22×C4×C173—41no—
112C22×C4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114C22×C4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116C22×C4×C173C2311yescharacteristic
117C22×C4×C173C2311yescharacteristic
118C4⋊C4×C173—22no—
119C4⋊C4×C173—41no—
120C22⋊C4×C173—22no—
121C22⋊C4×C173C2×C414yescomplemented
122C22×D173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—28no—
127C22×D173—21no—
128C22×D173—21no—
129C22×D173—21no—
130C22×D173—21no—
131C22×D173—21no—
132C22×D173—21no—
133C22×D173—21no—
134C22×D173—21no—
135C22×D173—21no—
136C22×D173—21no—
137C22×D173—21no—
138C22×D173—21no—
139C22×D173—28no—
140C22×D173—44no—
141C22×D173—44no—
142C22×D173—44no—
143C22×D173—44no—
144C22×D173—44no—
145C22×D173C22×C412yes—
146C22×D173C22×C412yes—
147C22×D173C22×C412yes—
148C22×D173C2×D412yes—
149C22×D173#4112yes—
150D692—28no—
151D692—44no—
152C22⋊D173—28no—
153C2×Dic173—21no—
154C2×Dic173—21no—
155C2×Dic173—21no—
156C2×Dic173—21no—
157C2×Dic173—21no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173—21no—
161C2×Dic173—21no—
162C2×Dic173—21no—
163C2×Dic173—21no—
164C2×Dic173—44no—
165C2×Dic173C22×C412yes—
166C2×Dic173C2×D412yes—
167C2×Dic173C2×D412yes—
168C2×Dic173C2×D412yes—
169C4×D173—44no—
170C4×D173—44no—
171C4×D173—44no—
172C23×C173—21no—
173C23×C173—21no—
174C23×C173—21no—
175C23×C173—21no—
176C23×C173C22×C412yes—
177C23×C173C2×D412yes—
178C23×C173#4112yes—
179C23×C173C2411yescharacteristic
180C2×C4×C173—21no—
181C2×C4×C173—21no—
182C2×C4×C173—21no—
183C2×C4×C173—21no—
184C2×C4×C173—21no—
185C2×C4×C173—21no—
186C2×C4×C173—21no—
187C2×C4×C173—21no—
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173—21no—
194C2×C4×C173—21no—
195C2×C4×C173—44no—
196C2×C4×C173C22×C412yes—
197C2×C4×C173C22×C412yes—
198C2×C4×C173C2×D412yes—
199C22×C173—22no—
200C22×C173—22no—
201C22×C173—22no—
202C22×C173—22no—
203C22×C173—22no—
204C22×C173—22no—
205C22×C173C4×D414yescomplemented
206C22×C173C23×C411yescharacteristic
207C22×C173C22×D411yescharacteristic
208C22×C173C22×D411yescharacteristic
209C22×C173C2×#4111yescharacteristic
210C22×C173C2×#4111yescharacteristic
211C22×C173H(5,𝔽2)11yescharacteristic
212C22×C173H(5,𝔽2)11yescharacteristic
213C2×D173—22no—
214C2×D173—22no—
215C2×D173—22no—
216C2×D173—22no—
217C2×D173—22no—
218C2×D173—22no—
219C2×D173—22no—
220C2×D173—22no—
221C2×D173—22no—
222C2×D173—22no—
223C2×D173—22no—
224C2×D173—22no—
225C2×D173—416no—
226C2×D173—41no—
227C2×D173—41no—
228C2×D173—41no—
229C2×D173—41no—
230C2×D173—41no—
231C2×D173—41no—
232C2×D173—41no—
233C2×D173—41no—
234C2×D173C4×D414yescomplemented
235C4×C173—22no—
236C4×C173—22no—
237C4×C173—22no—
238C4×C173—41no—
239C4×C173—41no—
240C4×C173—41no—
241Dic173—22no—
242Dic173—22no—
243Dic173—22no—
244Dic173—22no—
245Dic173—41no—
246C2×C173—24no—
247C2×C173C2×C4×D411yescharacteristic
248C2×C173C2×C4×D411yescharacteristic
249C2×C173#51811yescharacteristic
250C2×C173D4211yescharacteristic
251C2×C173#54611yescharacteristic
252C2×C173#54611yescharacteristic
253C2×C173#55111yescharacteristic
254D173—24no—
255D173—42no—
256D173—42no—
257C173#235811yescharacteristic, complemented, Sylow
258#2358—1731nomaximal, Sylow
259C2×#120—1731no—
260C2×C4×D4—1731no—
261C2×C4×D4—1731no—
262C22×C22⋊C4—1731no—
263C22×C22⋊C4—1731no—
264#390—1731no—
265#388—1731no—
266#388—1731no—
267#386—1731no—
268#386—1731no—
269#386—1731no—
270#385—1731no—
271#385D17311yescharacteristic, complemented
272C4×C22⋊C4—1731no—
273C4×C22⋊C4—1731no—
274C25—1731no—
275C22×D4—1731no—
276C22×D4—1731no—
277C22×D4—1731no—
278C23×C4—1731no—
279C23×C4—1731no—
280C23×C4—1731no—
281C23×C4—1731no—
282C23×C4C2×D17311yescharacteristic
283#120—1738no—
284C4×D4—3464no—
285C4×D4—3464no—
286C2×C4⋊C4—1731no—
287C2×C4⋊C4C2×D17311yescharacteristic, complemented
288C2×C4⋊C4C2×D17311yescharacteristic, complemented
289C2×C22⋊C4—3464no—
290C2×C22⋊C4—3464no—
291C2×C22⋊C4—1731no—
292C2×C22⋊C4—1731no—
293C2×C22⋊C4—1731no—
294C2×C22⋊C4—1731no—
295C2×C22⋊C4—1731no—
296C2×C22⋊C4—1731no—
297C2×C22⋊C4—1731no—
298C2×C22⋊C4—1731no—
299C2×C22⋊C4—1731no—
300C2×C22⋊C4—1731no—
301C2×C22⋊C4—1731no—
302C2×C22⋊C4—1731no—
303C2×C22⋊C4C2×D17311yescharacteristic
304C2×C22⋊C4C2×D17311yescharacteristic
305C2×C42—1731no—
306C2×C42—1731no—
307C2×C42—1731no—
308#95—1731no—
309#95—1731no—
310#95—1731no—
311#95—1731no—
312#95C2×D17311yescharacteristic, complemented
313#95C2×D17311yescharacteristic, complemented
314C24—3462no—
315C24—3462no—
316C24—3462no—
317C24—3462no—
318C24—1731no—
319C24—1731no—
320C24—1731no—
321C24—1734no—
322C24—6921no—
323C24—6921no—
324C24C22×D17311yescharacteristic
325C2×D4—3462no—
326C2×D4—3462no—
327C2×D4—3462no—
328C2×D4—3462no—
329C2×D4—1734no—
330C2×D4—1734no—
331C2×D4—1734no—
332C2×D4—6921no—
333C2×D4—6921no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—3462no—
338C22×C4—3462no—
339C22×C4—3462no—
340C22×C4—3462no—
341C22×C4—3462no—
342C22×C4—3462no—
343C22×C4—3462no—
344C22×C4—3462no—
345C22×C4—1731no—
346C22×C4—1731no—
347C22×C4—1731no—
348C22×C4—1731no—
349C22×C4—1731no—
350C22×C4—6921no—
351C22×C4—6921no—
352C22×C4—6921no—
353C22×C4—6921no—
354C22×C4C22×D17311yescharacteristic
355C22×C4C22×D17311yescharacteristic
356C22×C4C22×D17311yescharacteristic
357C22×C4C22×D17311yescharacteristic
358C22×C4C22×D17311yescharacteristic
359C22×C4C22×D17311yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—3462no—
363C22⋊C4—22no—
364C22⋊C4—3462no—
365C22⋊C4—3462no—
366C22⋊C4—3462no—
367C22⋊C4—3462no—
368C22⋊C4—3462no—
369C22⋊C4—3462no—
370C22⋊C4—3462no—
371C22⋊C4—1734no—
372C22⋊C4—1734no—
373C22⋊C4—1734no—
374C22⋊C4—1734no—
375C22⋊C4—6921no—
376C22⋊C4C4×D17314yescomplemented
377C42—3462no—
378C42—3462no—
379C42—6921no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—3468no—
385C23—3461no—
386C23—3461no—
387C23—3461no—
388C23—3461no—
389C23—3468no—
390C23—3461no—
391C23—3461no—
392C23—3461no—
393C23—3461no—
394C23—3461no—
395C23—3461no—
396C23—3461no—
397C23—3461no—
398C23—3461no—
399C23—3461no—
400C23—3461no—
401C23—3461no—
402C23—1732no—
403C23—1732no—
404C23—1732no—
405C23—1732no—
406C23—1732no—
407C23—6924no—
408C23—6924no—
409C23—6924no—
410C23—6924no—
411C23—6924no—
412C23C2×C4×D17312yes—
413C23D4×D17312yes—
414C23#1458382712yes—
415C23C23×D17311yescharacteristic
416D4—3468no—
417D4—3468no—
418D4—6924no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—3461no—
436C2×C4—3461no—
437C2×C4—3461no—
438C2×C4—3461no—
439C2×C4—3461no—
440C2×C4—3461no—
441C2×C4—3461no—
442C2×C4—3461no—
443C2×C4—3461no—
444C2×C4—3461no—
445C2×C4—3461no—
446C2×C4—1732no—
447C2×C4—1732no—
448C2×C4—1732no—
449C2×C4—1732no—
450C2×C4—6924no—
451C2×C4—6924no—
452C2×C4—6924no—
453C2×C4—6924no—
454C2×C4C2×C4×D17312yes—
455C2×C4C2×C4×D17312yes—
456C2×C4D4×D17312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—3462no—
464C22—3462no—
465C22—3462no—
466C22—3462no—
467C22—3462no—
468C22—3462no—
469C22—3462no—
470C22—3462no—
471C22—3462no—
472C22—3462no—
473C22—3462no—
474C22—3462no—
475C22—1734no—
476C22—69216no—
477C22—6921no—
478C22—6921no—
479C22—6921no—
480C22—6921no—
481C22—6921no—
482C22—6921no—
483C22—6921no—
484C22—6921no—
485C22#1914964114yescomplemented
486C22C22×C4×D17311yescharacteristic
487C22C2×D4×D17311yescharacteristic
488C22C2×D4×D17311yescharacteristic
489C22#41×D17311yescharacteristic
490C22C2×#1458382711yescharacteristic
491C22#1914972411yescharacteristic
492C22#1914972511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—3462no—
500C4—3462no—
501C4—3462no—
502C4—3462no—
503C4—6921no—
504C2—24no—
505C2—3464no—
506C2—6922no—
507C2—6922no—
508C2C4×D4×D17311yescharacteristic
509C2C2×#1914964111yescharacteristic
510C2#2601196311yescharacteristic
511C2#2601212311yescharacteristic
512C2#2601213211yescharacteristic
513C2#2601213311yescharacteristic
514C2#2601216811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#385×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149649C2
hypercenter#385D173
hypocenterC173#2358
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#19149641C2semidirect
3. C2×C4×C22⋊D173C2semidirect
4. C22×#14583803C2semidirect
5. C2×C22⋊C4×D173C2semidirect
6. #26011392C2semidirect
7. #26011381C2semidirect
8. #26011380C2semidirect
9. #26011369C2semidirect
10. #26011369C2semidirect
11. #26011368C2semidirect
12. #26011362C2semidirect
13. C22⋊C4×Dic173C2semidirect
14. #26011338C2semidirect
15. #385×C173C2semidirect
23. C22×C4×D173C22semidirect
24. #19149649C4semidirect
27. C2×#14583805C22semidirect
38. C2×#14583803C22semidirect
39. C2×#14583803C22semidirect
40. C2×#14583803C22semidirect
42. C2×C4×Dic173C22semidirect
43. C2×C4×Dic173C22semidirect
44. C2×C4×Dic173C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
47. #19149586C22semidirect
48. #19149586C22semidirect
50. C2×C4⋊C4×C173C22semidirect
51. C2×C4⋊C4×C173C22semidirect
54. #95×C173C22semidirect
55. #95×C173C22semidirect
62. C23×D173C2×C4semidirect
70. C2×D692C2×C4semidirect
71. C2×D692C2×C4semidirect
74. C2×C22⋊D173C2×C4semidirect
93. C22⋊Dic173D4semidirect
102. #14583803C2×C4semidirect
103. #14583803C2×C4semidirect
104. #14583803D4semidirect
121. C22⋊C4×C173C2×C4semidirect
205. C22×C173C4×D4semidirect
234. C2×D173C4×D4semidirect
257. C173#2358semidirect
271. #385D173semidirect
287. C2×C4⋊C4C2×D173semidirect
288. C2×C4⋊C4C2×D173semidirect
312. #95C2×D173semidirect
313. #95C2×D173semidirect
376. C22⋊C4C4×D173semidirect
485. C22#19149641semidirect

Maximal subgroups

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

subgroupindex
C2×#191496492
C2×#191496412
C2×C4×C22⋊D1732
C22×#145838032
C2×C22⋊C4×D1732
#260113922
#260113812
#260113802
#260113692
#260113682
#260113622
C22⋊C4×Dic1732
#260113382
#385×C1732
#2358173