Finite Group Databasebeta

← #28339928

Subgroups — #28339928

Order 13696.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812091C211yesmaximal, characteristic, complemented
2C2×#16812079C211yesmaximal, characteristic
3C2×C4×D428C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292564C211yesmaximal, characteristic, complemented
6#20292552C211yesmaximal, characteristic, complemented
7#20292544C211yesmaximal, characteristic, complemented
8#20292543C211yesmaximal, characteristic, complemented
9#20292518C211yesmaximal, characteristic, complemented
10#20292505C211yesmaximal, characteristic, complemented
11#20292505C211yesmaximal, characteristic, complemented
12#20292501C211yesmaximal, characteristic, complemented
13#20292484C211yesmaximal, characteristic, complemented
14#20292482C211yesmaximal, characteristic, complemented
15#388×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic, complemented
23#16812091—24no—
24#16812079—24no—
25C4×D428—24no—
26C2×C4⋊Dic107C2211yescharacteristic, complemented
27C2×C4⋊Dic107C2211yescharacteristic, complemented
28C2×C4⋊Dic107C2211yescharacteristic, complemented
29C2×Dic107⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D107—24no—
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C107C2211yescharacteristic, complemented
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C42×C107C2211yescharacteristic, complemented
53#95×C107C2211yescharacteristic, complemented
54#95×C107C2211yescharacteristic, complemented
55C23×D107—22no—
56C23×D107—22no—
57C23×D107—22no—
58C23×D107—22no—
59C23×D107—22no—
60C23×D107—41no—
61C23×D107—41no—
62C23×D107C2311yescharacteristic
63C23×D107C2311yescharacteristic
64C23×D107C2311yescharacteristic
65C2×D428—22no—
66C2×D428—22no—
67C2×D428—22no—
68C2×D428—22no—
69C2×D428—41no—
70C2×D428D414yes—
71C2×C22⋊D107—22no—
72C2×C22⋊D107—22no—
73C2×C22⋊D107—41no—
74C22×Dic107C2311yescharacteristic
75C22×Dic107C2311yescharacteristic
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C2×C4×D107—22no—
80C2×C4×D107—22no—
81C2×C4×D107—22no—
82C2×C4×D107—22no—
83C2×C4×D107—22no—
84C2×C4×D107—22no—
85C2×C4×D107—41no—
86C2×C4×D107—41no—
87C2×C4×D107—41no—
88C4⋊Dic107—22no—
89C4⋊Dic107—22no—
90C4⋊Dic107—41no—
91Dic107⋊C4—22no—
92C22⋊Dic107—22no—
93C22⋊Dic107—22no—
94(C2×D107)⋊C4—22no—
95(C2×D107)⋊C4—22no—
96(C2×D107)⋊C4—22no—
97(C2×D107)⋊C4—22no—
98(C2×D107)⋊C4—22no—
99(C2×D107)⋊C4—22no—
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4—41no—
104(C2×D107)⋊C4—41no—
105C24×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C4⋊C4×C107—41no—
113C22⋊C4×C107—22no—
114C22⋊C4×C107—41no—
115C22⋊C4×C107D414yescomplemented
116C42×C107—22no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C23×C107C2×D412yes—
124C23×C107C2411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—28no—
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—21no—
135C22×D107—21no—
136C22×D107—21no—
137C22×D107—21no—
138C22×D107—21no—
139C22×D107—21no—
140C22×D107—21no—
141C22×D107—21no—
142C22×D107—44no—
143C22×D107—44no—
144C22×D107—44no—
145C22×D107—44no—
146C22×D107—44no—
147C22×D107—44no—
148C22×D107C2×D412yes—
149C22×D107C2×D412yes—
150C22×D107#4112yes—
151C22×D107#4112yes—
152C22×D107#4112yes—
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—21no—
159C2×C4×C107—21no—
160C2×C4×C107—21no—
161C2×C4×C107—21no—
162C2×C4×C107—21no—
163C2×C4×C107—21no—
164C2×C4×C107—21no—
165C2×C4×C107—21no—
166C2×C4×C107—21no—
167C2×C4×C107—21no—
168C2×C4×C107C2×D412yes—
169C2×C4×C107C2×D412yes—
170C2×C4×C107C2×D412yes—
171C22⋊D107—44no—
172C2×Dic107—21no—
173C2×Dic107—21no—
174C2×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—21no—
180C2×Dic107—21no—
181C2×Dic107—21no—
182C2×Dic107—21no—
183C2×Dic107—21no—
184C2×Dic107—21no—
185C2×Dic107—21no—
186C2×Dic107—21no—
187D428—28no—
188D428—44no—
189C4×D107—44no—
190C4×D107—44no—
191C4×D107—44no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—41no—
196C22×C107—41no—
197C22×C107—41no—
198C22×C107—41no—
199C22×C107C22×D411yescharacteristic
200C22×C107C22×D411yescharacteristic
201C22×C107C2×#4111yescharacteristic
202C22×C107C2×#4111yescharacteristic
203C22×C107C2×#4111yescharacteristic
204C22×C107H(5,𝔽2)11yescharacteristic
205C22×C107H(5,𝔽2)11yescharacteristic
206C2×D107—22no—
207C2×D107—22no—
208C2×D107—22no—
209C2×D107—22no—
210C2×D107—22no—
211C2×D107—22no—
212C2×D107—22no—
213C2×D107—22no—
214C2×D107—22no—
215C2×D107—22no—
216C2×D107—22no—
217C2×D107—22no—
218C2×D107—416no—
219C2×D107—41no—
220C2×D107—41no—
221C2×D107—41no—
222C2×D107—41no—
223C2×D107—41no—
224C2×D107—41no—
225C2×D107—41no—
226C2×D107—41no—
227C2×D107#12314yescomplemented
228C4×C107—22no—
229C4×C107—22no—
230C4×C107—22no—
231C4×C107—41no—
232C4×C107—41no—
233C4×C107—41no—
234Dic107—41no—
235Dic107—41no—
236Dic107—41no—
237Dic107—41no—
238Dic107—41no—
239C2×C107—42no—
240C2×C107C2×#12311yescharacteristic
241C2×C107#52511yescharacteristic
242C2×C107#53811yescharacteristic
243C2×C107D4211yescharacteristic
244C2×C107#54611yescharacteristic
245C2×C107#54611yescharacteristic
246C2×C107#55111yescharacteristic
247D107—24no—
248D107—42no—
249D107—42no—
250#2418—1071nomaximal, Sylow
251C107#241811yescharacteristic, complemented, Sylow
252C2×#123—1071no—
253C2×#120—1071no—
254C2×C4×D4—1071no—
255C22×C22⋊C4—1071no—
256#399—1071no—
257#397—1071no—
258#394—1071no—
259#394—1071no—
260#388—1071no—
261#388D10711yescharacteristic, complemented
262#386—1071no—
263#386—1071no—
264#385—1071no—
265#380—1071no—
266#379—1071no—
267C25—1071no—
268C22×D4—1071no—
269C22×D4—1071no—
270C22×D4—1071no—
271C23×C4—1071no—
272C23×C4—1071no—
273C23×C4—1071no—
274#123—2144no—
275#120—2144no—
276C4×D4—2144no—
277C2×C4⋊C4—1071no—
278C2×C4⋊C4—1071no—
279C2×C4⋊C4—1071no—
280C2×C4⋊C4—1071no—
281C2×C4⋊C4C2×D10711yescharacteristic, complemented
282C2×C22⋊C4—1071no—
283C2×C22⋊C4—1071no—
284C2×C22⋊C4—1071no—
285C2×C22⋊C4—1071no—
286C2×C22⋊C4—1071no—
287C2×C22⋊C4—1071no—
288C2×C22⋊C4—1071no—
289C2×C22⋊C4—1071no—
290C2×C22⋊C4—1071no—
291C2×C22⋊C4—1071no—
292C2×C22⋊C4—1071no—
293C2×C22⋊C4—1071no—
294C2×C22⋊C4—1071no—
295C2×C22⋊C4—2144no—
296C2×C22⋊C4C2×D10711yescharacteristic
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic
299C2×C42C2×D10711yescharacteristic, complemented
300#95—1071no—
301#95—1071no—
302#95—1071no—
303#95—1071no—
304#95C2×D10711yescharacteristic, complemented
305#95C2×D10711yescharacteristic, complemented
306C24—1071no—
307C24—1071no—
308C24—1071no—
309C24—4281no—
310C24—4281no—
311C24—2142no—
312C24—2142no—
313C24—2142no—
314C24—2142no—
315C24—2142no—
316C24C22×D10711yescharacteristic
317C2×D4—1074no—
318C2×D4—4281no—
319C2×D4—4281no—
320C2×D4—2142no—
321C2×D4—2142no—
322C2×D4—2142no—
323C2×D4—2142no—
324C2×D4—2142no—
325C2×D4—2142no—
326C22×C4—1071no—
327C22×C4—1071no—
328C22×C4—1071no—
329C22×C4—1071no—
330C22×C4—1071no—
331C22×C4—4281no—
332C22×C4—4281no—
333C22×C4—4281no—
334C22×C4—2142no—
335C22×C4—2142no—
336C22×C4—2142no—
337C22×C4—2142no—
338C22×C4—2142no—
339C22×C4—2142no—
340C22×C4C22×D10711yescharacteristic
341C22×C4C22×D10711yescharacteristic
342C22×C4C22×D10711yescharacteristic
343C22×C4C22×D10711yescharacteristic
344C22×C4C22×D10711yescharacteristic
345C22×C4C22×D10711yescharacteristic
346C4⋊C4—41no—
347C4⋊C4—4281no—
348C4⋊C4—2142no—
349C4⋊C4—2142no—
350C4⋊C4—2142no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—4281no—
354C22⋊C4—4281no—
355C22⋊C4—2142no—
356C22⋊C4—2142no—
357C22⋊C4—2142no—
358C22⋊C4—2142no—
359C22⋊C4—2142no—
360C22⋊C4—2142no—
361C22⋊C4—2142no—
362C22⋊C4—2142no—
363C22⋊C4—2142no—
364C22⋊C4—2142no—
365C22⋊C4—2142no—
366C22⋊C4D42814yescomplemented
367C42—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—1072no—
375C23—1072no—
376C23—1072no—
377C23—1072no—
378C23—1072no—
379C23—4284no—
380C23—4284no—
381C23—4284no—
382C23—4284no—
383C23—4284no—
384C23—4284no—
385C23—2148no—
386C23—2141no—
387C23—2141no—
388C23—2141no—
389C23—2141no—
390C23—2141no—
391C23—2141no—
392C23—2141no—
393C23—2141no—
394C23—2141no—
395C23—2141no—
396C23—2141no—
397C23—2141no—
398C23—2141no—
399C23—2141no—
400C23—2141no—
401C23—2141no—
402C23C2×D42812yes—
403C23C23×D10711yescharacteristic
404D4—4284no—
405D4—4284no—
406D4—2148no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—4284no—
423C2×C4—4284no—
424C2×C4—4284no—
425C2×C4—2141no—
426C2×C4—2141no—
427C2×C4—2141no—
428C2×C4—2141no—
429C2×C4—2141no—
430C2×C4—2141no—
431C2×C4—2141no—
432C2×C4—2141no—
433C2×C4—2141no—
434C2×C4—2141no—
435C2×C4—2141no—
436C2×C4—2141no—
437C2×C4—2141no—
438C2×C4—2141no—
439C2×C4—2141no—
440C2×C4D4×D10712yes—
441C2×C4C2×D42812yes—
442C2×C4C2×D42812yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—1074no—
451C22—42816no—
452C22—4281no—
453C22—4281no—
454C22—4281no—
455C22—4281no—
456C22—4281no—
457C22—4281no—
458C22—4281no—
459C22—4281no—
460C22—2142no—
461C22—2142no—
462C22—2142no—
463C22—2142no—
464C22—2142no—
465C22—2142no—
466C22—2142no—
467C22—2142no—
468C22—2142no—
469C22—2142no—
470C22—2142no—
471C22—2142no—
472C22C2×D4×D10711yescharacteristic
473C22C22×D42811yescharacteristic
474C22#41×D10711yescharacteristic
475C22#41×D10711yescharacteristic
476C22#41×D10711yescharacteristic
477C22#1681215411yescharacteristic
478C22#1681215511yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—4281no—
486C4—4281no—
487C4—4281no—
488C4—4281no—
489C4—4281no—
490C2—42no—
491C2—4282no—
492C2—4282no—
493C2—2144no—
494C2#123×D10711yescharacteristic
495C2#2029313911yescharacteristic
496C2#2029321111yescharacteristic
497C2D4×D42811yescharacteristic
498C2#2029327211yescharacteristic
499C2#2029327411yescharacteristic
500C2#2029330211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#388×C107C2
prime radicalC2×#16812079C2
hypercenter#388D107
hypocenterC107#2418
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16812091C2semidirect
3. C2×C4×D428C2semidirect
4. C2×C22⋊C4×D107C2semidirect
5. #20292564C2semidirect
6. #20292552C2semidirect
7. #20292544C2semidirect
8. #20292543C2semidirect
9. #20292518C2semidirect
10. #20292505C2semidirect
11. #20292505C2semidirect
12. #20292501C2semidirect
13. #20292484C2semidirect
14. #20292482C2semidirect
15. #388×C107C2semidirect
22. C22×C4×D107C22semidirect
26. C2×C4⋊Dic107C22semidirect
27. C2×C4⋊Dic107C22semidirect
28. C2×C4⋊Dic107C22semidirect
29. C2×Dic107⋊C4C22semidirect
40. C2×(C2×D107)⋊C4C22semidirect
41. C2×(C2×D107)⋊C4C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
48. C2×C4⋊C4×C107C22semidirect
52. C2×C42×C107C22semidirect
53. #95×C107C22semidirect
54. #95×C107C22semidirect
115. C22⋊C4×C107D4semidirect
227. C2×D107#123semidirect
251. C107#2418semidirect
261. #388D107semidirect
281. C2×C4⋊C4C2×D107semidirect
299. C2×C42C2×D107semidirect
304. #95C2×D107semidirect
305. #95C2×D107semidirect
366. C22⋊C4D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168120912
C2×#168120792
C2×C4×D4282
C2×C22⋊C4×D1072
#202925642
#202925522
#202925442
#202925432
#202925182
#202925052
#202925012
#202924842
#202924822
#388×C1072
#2418107