Finite Group Databasebeta

← #36356703

Subgroups — #36356703

Order 22144.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149649C211yesmaximal, characteristic
2#26011393C211yesmaximal, characteristic, complemented
3#26011392C211yesmaximal, characteristic, complemented
4#26011391C211yesmaximal, characteristic, complemented
5#26011384C211yesmaximal, characteristic, complemented
6#26011383C211yesmaximal, characteristic, complemented
7#26011382C211yesmaximal, characteristic, complemented
8#26011381C211yesmaximal, characteristic, complemented
9#26011380C211yesmaximal, characteristic, complemented
10#26011369C211yesmaximal, characteristic, complemented
11#26011369C211yesmaximal, characteristic, complemented
12#26011367C211yesmaximal, characteristic, complemented
13#26011346C211yesmaximal, characteristic, complemented
14#26011346C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C22×D692C2211yescharacteristic
19C22×C22⋊D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic, complemented
21#19149649C418yescomplemented
22C2×C4⋊Dic173C2211yescharacteristic, complemented
23C2×#14583805C2211yescharacteristic, complemented
24C2×C22⋊Dic173C2211yescharacteristic
25C2×C22⋊Dic173C2211yescharacteristic
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic, complemented
37C2×#14583803C2211yescharacteristic, complemented
38C2×#14583803C2211yescharacteristic, complemented
39C2×C4×Dic173C2211yescharacteristic, complemented
40C2×C4×Dic173C2211yescharacteristic, complemented
41#19149586C2211yescharacteristic, complemented
42#19149586C2211yescharacteristic, complemented
43#19149586C2211yescharacteristic, complemented
44#19149586C2211yescharacteristic, complemented
45C2×C4⋊C4×C173C2211yescharacteristic, complemented
46C2×C22⋊C4×C173C2211yescharacteristic
47C2×C22⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C42×C173C2211yescharacteristic, complemented
50#95×C173C2211yescharacteristic, complemented
51#95×C173C2211yescharacteristic, complemented
52C23×D173—41no—
53C23×D173—41no—
54C23×D173—41no—
55C23×D173—41no—
56C23×D173—41no—
57C23×D173—41no—
58C23×D173C2×C414yescomplemented
59C23×D173C2311yescharacteristic
60C23×D173C2311yescharacteristic
61C23×D173C2311yescharacteristic
62C2×D692—41no—
63C2×D692—41no—
64C2×D692—41no—
65C2×D692—41no—
66C2×D692C2×C414yes—
67C2×D692C2×C414yes—
68C2×C22⋊D173—41no—
69C2×C22⋊D173—41no—
70C2×C22⋊D173C2×C414yes—
71C22×Dic173C2311yescharacteristic
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C2×C4×D173—41no—
77C2×C4×D173—41no—
78C2×C4×D173—41no—
79C4⋊Dic173—41no—
80#14583805—41no—
81C22⋊Dic173—41no—
82C22⋊Dic173—41no—
83#14583803—41no—
84#14583803—41no—
85#14583803—41no—
86#14583803—41no—
87#14583803—41no—
88#14583803—41no—
89#14583803—41no—
90#14583803—41no—
91#14583803—41no—
92#14583803—41no—
93#14583803—41no—
94#14583803C2×C414yescomplemented
95#14583803C2×C414yescomplemented
96C4×Dic173—41no—
97C4×Dic173—41no—
98C24×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C4⋊C4×C173—41no—
106C22⋊C4×C173—41no—
107C22⋊C4×C173—41no—
108C22⋊C4×C173C2×C414yescomplemented
109C42×C173—41no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—21no—
128C22×D173—44no—
129C22×D173—44no—
130C22×D173—44no—
131C22×D173—44no—
132C22×D173—44no—
133C22×D173—44no—
134C22×D173—44no—
135C22×D173C22×C412yes—
136C22×D173C22×C412yes—
137C22×D173C22×C412yes—
138D692—44no—
139D692—44no—
140C22⋊D173—44no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—21no—
150C2×Dic173—21no—
151C2×Dic173—21no—
152C2×Dic173—21no—
153C2×Dic173—21no—
154C2×Dic173—21no—
155C2×Dic173C22×C412yes—
156C4×D173—44no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—21no—
160C23×C173—21no—
161C23×C173—21no—
162C23×C173—21no—
163C23×C173C22×C412yes—
164C23×C173C2411yescharacteristic
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—21no—
175C2×C4×C173—21no—
176C2×C4×C173—21no—
177C2×C4×C173—21no—
178C2×C4×C173—21no—
179C2×C4×C173—21no—
180C2×C4×C173—21no—
181C2×C4×C173C22×C412yes—
182C2×C4×C173C22×C412yes—
183C22×C173—22no—
184C22×C173—22no—
185C22×C173—22no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173C23×C411yescharacteristic
191C22×C173H(5,𝔽2)11yescharacteristic
192C22×C173H(5,𝔽2)11yescharacteristic
193C22×C173H(5,𝔽2)11yescharacteristic
194C22×C173H(5,𝔽2)11yescharacteristic
195C22×C173H(5,𝔽2)11yescharacteristic
196C22×C173H(5,𝔽2)11yescharacteristic
197C2×D173—22no—
198C2×D173—22no—
199C2×D173—22no—
200C2×D173—22no—
201C2×D173—22no—
202C2×D173—22no—
203C2×D173—22no—
204C2×D173—22no—
205C2×D173—22no—
206C2×D173—416no—
207C2×D173—41no—
208C2×D173—41no—
209C2×D173—41no—
210C2×D173—41no—
211C2×D173—41no—
212C2×D173—41no—
213C2×D173—41no—
214C2×D173—41no—
215C2×D173—41no—
216C2×D173—41no—
217C2×D173—41no—
218C2×D173—41no—
219C4×C173—22no—
220C4×C173—22no—
221C4×C173—41no—
222C4×C173—41no—
223C4×C173—41no—
224C4×C173—41no—
225Dic173—22no—
226Dic173—41no—
227Dic173—41no—
228Dic173—41no—
229Dic173—41no—
230C2×C173—42no—
231C2×C173#51811yescharacteristic
232C2×C173#51811yescharacteristic
233C2×C173#51811yescharacteristic
234C2×C173#56011yescharacteristic
235C2×C173#56011yescharacteristic
236C2×C173#56011yescharacteristic
237C2×C173H(3,𝔽4)11yescharacteristic
238D173—42no—
239D173—42no—
240D173—42no—
241C173#237511yescharacteristic, complemented, Sylow
242#2375—1731nomaximal, Sylow
243C2×#120—1731no—
244#390—1731no—
245#390—1731no—
246#390—1731no—
247#388—1731no—
248#388—1731no—
249#388—1731no—
250#388—1731no—
251#388—1731no—
252#388D17311yescharacteristic, complemented
253#386—1731no—
254#386—1731no—
255#386—1731no—
256#379—1731no—
257#379—1731no—
258C25—1731no—
259C22×D4—1731no—
260C22×D4—1731no—
261C22×D4—1731no—
262C23×C4—1731no—
263#120—1738no—
264C2×C4⋊C4—1731no—
265C2×C4⋊C4—1731no—
266C2×C4⋊C4C2×D17311yescharacteristic, complemented
267C2×C22⋊C4—1731no—
268C2×C22⋊C4—1731no—
269C2×C22⋊C4—1731no—
270C2×C22⋊C4—1731no—
271C2×C22⋊C4—1731no—
272C2×C22⋊C4—1731no—
273C2×C22⋊C4—1731no—
274C2×C22⋊C4—1731no—
275C2×C22⋊C4—1731no—
276C2×C22⋊C4—1731no—
277C2×C22⋊C4—1731no—
278C2×C22⋊C4—1731no—
279C2×C22⋊C4—1731no—
280C2×C22⋊C4—1731no—
281C2×C22⋊C4—1731no—
282C2×C22⋊C4C2×D17311yescharacteristic
283C2×C22⋊C4C2×D17311yescharacteristic
284C2×C22⋊C4C2×D17311yescharacteristic
285C2×C42—1731no—
286C2×C42—1731no—
287C2×C42C2×D17311yescharacteristic, complemented
288#95—1731no—
289#95—1731no—
290#95—1731no—
291#95—1731no—
292#95C2×D17311yescharacteristic, complemented
293#95C2×D17311yescharacteristic, complemented
294C24—1731no—
295C24—1731no—
296C24—1731no—
297C24—1734no—
298C24—6921no—
299C24—6921no—
300C24—6921no—
301C24—6921no—
302C24—6921no—
303C24—6921no—
304C24C22×D17311yescharacteristic
305C2×D4—1734no—
306C2×D4—1734no—
307C2×D4—1734no—
308C2×D4—6921no—
309C2×D4—6921no—
310C2×D4—6921no—
311C2×D4—6921no—
312C2×D4—6921no—
313C2×D4—6921no—
314C22×C4—1731no—
315C22×C4—1731no—
316C22×C4—1731no—
317C22×C4—1731no—
318C22×C4—1731no—
319C22×C4—6921no—
320C22×C4—6921no—
321C22×C4—6921no—
322C22×C4C22×D17311yescharacteristic
323C22×C4C22×D17311yescharacteristic
324C22×C4C22×D17311yescharacteristic
325C22×C4C22×D17311yescharacteristic
326C22×C4C22×D17311yescharacteristic
327C22×C4C22×D17311yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—6921no—
330C4⋊C4—6921no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1734no—
334C22⋊C4—1734no—
335C22⋊C4—6921no—
336C22⋊C4—6921no—
337C22⋊C4—6921no—
338C22⋊C4—6921no—
339C22⋊C4—6921no—
340C22⋊C4—6921no—
341C22⋊C4—6921no—
342C22⋊C4—6921no—
343C22⋊C4—6921no—
344C22⋊C4—6921no—
345C22⋊C4—6921no—
346C22⋊C4—6921no—
347C22⋊C4—6921no—
348C22⋊C4C4×D17314yescomplemented
349C42—41no—
350C42—6921no—
351C42—6921no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—3461no—
359C23—3461no—
360C23—3461no—
361C23—3461no—
362C23—3461no—
363C23—3461no—
364C23—3461no—
365C23—3461no—
366C23—3461no—
367C23—3461no—
368C23—3461no—
369C23—3461no—
370C23—3461no—
371C23—3461no—
372C23—3461no—
373C23—3461no—
374C23—3461no—
375C23—3461no—
376C23—1732no—
377C23—1732no—
378C23—1732no—
379C23—6924no—
380C23—6924no—
381C23—6924no—
382C23—6924no—
383C23—6924no—
384C23—6924no—
385C23—6924no—
386C23C2×C4×D17312yes—
387C23C23×D17311yescharacteristic
388D4—6924no—
389D4—6924no—
390D4—6924no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—3461no—
413C2×C4—3461no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—3461no—
417C2×C4—3461no—
418C2×C4—3461no—
419C2×C4—3461no—
420C2×C4—3461no—
421C2×C4—1732no—
422C2×C4—6924no—
423C2×C4C2×C4×D17312yes—
424C2×C4C2×C4×D17312yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—3462no—
433C22—3462no—
434C22—3462no—
435C22—3462no—
436C22—3462no—
437C22—3462no—
438C22—3462no—
439C22—3462no—
440C22—3462no—
441C22—69216no—
442C22—6921no—
443C22—6921no—
444C22—6921no—
445C22—6921no—
446C22—6921no—
447C22—6921no—
448C22—6921no—
449C22—6921no—
450C22—6921no—
451C22—6921no—
452C22—6921no—
453C22—6921no—
454C22C22×C4×D17311yescharacteristic
455C22#1914972411yescharacteristic
456C22#1914972411yescharacteristic
457C22#1914972511yescharacteristic
458C22#1914972511yescharacteristic
459C22#1914972511yescharacteristic
460C22#1914972511yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—3462no—
468C4—6921no—
469C4—6921no—
470C4—6921no—
471C4—6921no—
472C2—42no—
473C2—6922no—
474C2—6922no—
475C2—6922no—
476C2#2601196311yescharacteristic
477C2#2601196311yescharacteristic
478C2#2601196511yescharacteristic
479C2#2601222911yescharacteristic
480C2#2601223111yescharacteristic
481C2#2601223211yescharacteristic
482C2#2601223311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149649C2
hypercenter#388D173
hypocenterC173#2375
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #26011393C2semidirect
3. #26011392C2semidirect
4. #26011391C2semidirect
5. #26011384C2semidirect
6. #26011383C2semidirect
7. #26011382C2semidirect
8. #26011381C2semidirect
9. #26011380C2semidirect
10. #26011369C2semidirect
11. #26011369C2semidirect
12. #26011367C2semidirect
13. #26011346C2semidirect
14. #26011346C2semidirect
15. #388×C173C2semidirect
20. C22×C4×D173C22semidirect
21. #19149649C4semidirect
22. C2×C4⋊Dic173C22semidirect
23. C2×#14583805C22semidirect
36. C2×#14583803C22semidirect
37. C2×#14583803C22semidirect
38. C2×#14583803C22semidirect
39. C2×C4×Dic173C22semidirect
40. C2×C4×Dic173C22semidirect
41. #19149586C22semidirect
42. #19149586C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. C2×C4⋊C4×C173C22semidirect
49. C2×C42×C173C22semidirect
50. #95×C173C22semidirect
51. #95×C173C22semidirect
58. C23×D173C2×C4semidirect
94. #14583803C2×C4semidirect
95. #14583803C2×C4semidirect
108. C22⋊C4×C173C2×C4semidirect
241. C173#2375semidirect
252. #388D173semidirect
266. C2×C4⋊C4C2×D173semidirect
287. C2×C42C2×D173semidirect
292. #95C2×D173semidirect
293. #95C2×D173semidirect
348. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496492
#260113932
#260113922
#260113912
#260113842
#260113832
#260113822
#260113812
#260113802
#260113692
#260113672
#260113462
#388×C1732
#2375173