Finite Group Databasebeta

← #28339925

Subgroups — #28339925

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×#16812094C211yesmaximal, characteristic, complemented
2C2×#16812079C211yesmaximal, characteristic
3C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
4C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
5#20292562C211yesmaximal, characteristic, complemented
6#20292556C211yesmaximal, characteristic, complemented
7#20292545C211yesmaximal, characteristic, complemented
8#20292544C211yesmaximal, characteristic, complemented
9#20292517C211yesmaximal, characteristic, complemented
10#20292516C211yesmaximal, characteristic, complemented
11#20292505C211yesmaximal, characteristic, complemented
12#20292504C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic, complemented
14#20292482C211yesmaximal, characteristic, complemented
15#385×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C23×Dic107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic, complemented
22#16812094—24no—
23#16812079—24no—
24C4×C22⋊D107—24no—
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×Dic107⋊C4C2211yescharacteristic, complemented
27C2×Dic107⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic107C2211yescharacteristic
29C2×(C2×D107)⋊C4—24no—
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
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
43C2×C4×Dic107C2211yescharacteristic, complemented
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C23×C4×C107C2211yescharacteristic
49C2×C4⋊C4×C107C2211yescharacteristic, complemented
50C2×C4⋊C4×C107C2211yescharacteristic, complemented
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
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×D428—41no—
71C2×C22⋊D107—22no—
72C2×C22⋊D107—22no—
73C2×C22⋊D107D414yes—
74C22×Dic107—22no—
75C22×Dic107—22no—
76C22×Dic107—41no—
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C22×Dic107C2311yescharacteristic
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2311yescharacteristic
82C2×C4×D107—22no—
83C2×C4×D107—22no—
84C2×C4×D107—41no—
85Dic107⋊C4—22no—
86Dic107⋊C4—22no—
87Dic107⋊C4—41no—
88C22⋊Dic107—22no—
89(C2×D107)⋊C4—22no—
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—22no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—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—41no—
99(C2×D107)⋊C4—41no—
100(C2×D107)⋊C4—41no—
101(C2×D107)⋊C4D414yescomplemented
102C4×Dic107—22no—
103C24×C107C2311yescharacteristic
104C22×C4×C107—22no—
105C22×C4×C107—22no—
106C22×C4×C107—41no—
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C4⋊C4×C107—22no—
114C4⋊C4×C107—41no—
115C22⋊C4×C107—22no—
116C22⋊C4×C107—22no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107C2×D412yes—
122C23×C107#4112yes—
123C23×C107#4112yes—
124C23×C107C2411yescharacteristic
125C22×D107—28no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
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—21no—
143C22×D107—21no—
144C22×D107—44no—
145C22×D107—44no—
146C22×D107—44no—
147C22×D107—44no—
148C22×D107—44no—
149C22×D107—44no—
150C22×D107C2×D412yes—
151C22×D107C2×D412yes—
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×C107—21no—
169C2×C4×C107—21no—
170C2×C4×C107—44no—
171C2×C4×C107C2×D412yes—
172C22⋊D107—28no—
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—44no—
187C2×Dic107C2×D412yes—
188C2×Dic107C2×D412yes—
189D428—44no—
190D428—44no—
191C4×D107—44no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—22no—
197C22×C107—22no—
198C22×C107#12314yescomplemented
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—416no—
216C2×D107—41no—
217C2×D107—41no—
218C2×D107—41no—
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—41no—
228C4×C107—22no—
229C4×C107—41no—
230C4×C107—41no—
231C4×C107—41no—
232C4×C107—41no—
233C4×C107—41no—
234Dic107—22no—
235Dic107—22no—
236Dic107—41no—
237Dic107—41no—
238Dic107—41no—
239C2×C107—24no—
240C2×C107C2×#12311yescharacteristic
241C2×C107#52511yescharacteristic
242C2×C107#53811yescharacteristic
243C2×C107D4211yescharacteristic
244C2×C107#54611yescharacteristic
245C2×C107#54611yescharacteristic
246C2×C107#55111yescharacteristic
247D107—42no—
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#388—1071no—
262#386—1071no—
263#386—1071no—
264#385D10711yescharacteristic, complemented
265#380—1071no—
266#379—1071no—
267C25—1071no—
268C22×D4—1071no—
269C22×D4—1071no—
270C22×D4—1071no—
271C23×C4—1071no—
272C23×C4—1071no—
273C23×C4C2×D10711yescharacteristic
274#123—2144no—
275#120—2144no—
276C4×D4—2144no—
277C2×C4⋊C4—1071no—
278C2×C4⋊C4—1071no—
279C2×C4⋊C4—1071no—
280C2×C4⋊C4C2×D10711yescharacteristic, complemented
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—1071no—
296C2×C22⋊C4—2144no—
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic
299C2×C42—1071no—
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—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—1071no—
330C22×C4—1071no—
331C22×C4—1071no—
332C22×C4—1071no—
333C22×C4—1071no—
334C22×C4—4281no—
335C22×C4—4281no—
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—22no—
347C4⋊C4—41no—
348C4⋊C4—4281no—
349C4⋊C4—2142no—
350C4⋊C4—2142no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—1074no—
354C22⋊C4—4281no—
355C22⋊C4—4281no—
356C22⋊C4—4281no—
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⋊C4—2142no—
367C42—2142no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—1072no—
373C23—1072no—
374C23—1072no—
375C23—4284no—
376C23—4284no—
377C23—4284no—
378C23—4284no—
379C23—4284no—
380C23—4284no—
381C23—2148no—
382C23—2141no—
383C23—2141no—
384C23—2141no—
385C23—2141no—
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—
400C23D4×D10712yes—
401C23#1313381812yes—
402C23#1313381912yes—
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—21no—
423C2×C4—21no—
424C2×C4—44no—
425C2×C4—1072no—
426C2×C4—1072no—
427C2×C4—4284no—
428C2×C4—4284no—
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×C4—2141no—
441C2×C4—2141no—
442C2×C4D4×D10712yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42816no—
450C22—4281no—
451C22—4281no—
452C22—4281no—
453C22—4281no—
454C22—4281no—
455C22—4281no—
456C22—4281no—
457C22—4281no—
458C22—4281no—
459C22—4281no—
460C22—4281no—
461C22—4281no—
462C22—2142no—
463C22—2142no—
464C22—2142no—
465C22—2142no—
466C22—2142no—
467C22—2142no—
468C22—2142no—
469C22—2142no—
470C22—2142no—
471C22#1681209414yescomplemented
472C22C2×D4×D10711yescharacteristic
473C22C2×D4×D10711yescharacteristic
474C22#41×D10711yescharacteristic
475C22C2×#1313381811yescharacteristic
476C22C2×#1313381911yescharacteristic
477C22#1681215411yescharacteristic
478C22#1681215511yescharacteristic
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—4281no—
486C4—4281no—
487C4—4281no—
488C4—2142no—
489C4—2142no—
490C2—24no—
491C2—4282no—
492C2—4282no—
493C2—4282no—
494C2C2×#1681209411yescharacteristic
495C2#2029313611yescharacteristic
496C2#2029321011yescharacteristic
497C2#2029325911yescharacteristic
498C2#2029326811yescharacteristic
499C2#2029326911yescharacteristic
500C2#2029330411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC2×#16812079C2
hypercenter#385D107
hypocenterC107#2418
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
3. C2×C4×C22⋊D107C2semidirect
4. C22×(C2×D107)⋊C4C2semidirect
5. #20292562C2semidirect
6. #20292556C2semidirect
7. #20292545C2semidirect
8. #20292544C2semidirect
9. #20292517C2semidirect
10. #20292516C2semidirect
11. #20292505C2semidirect
12. #20292504C2semidirect
13. #20292485C2semidirect
14. #20292482C2semidirect
15. #385×C107C2semidirect
21. C22×C4×D107C22semidirect
25. C2×Dic107⋊C4C22semidirect
26. C2×Dic107⋊C4C22semidirect
27. C2×Dic107⋊C4C22semidirect
40. C2×(C2×D107)⋊C4C22semidirect
41. C2×(C2×D107)⋊C4C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
43. C2×C4×Dic107C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
49. C2×C4⋊C4×C107C22semidirect
50. C2×C4⋊C4×C107C22semidirect
53. #95×C107C22semidirect
54. #95×C107C22semidirect
101. (C2×D107)⋊C4D4semidirect
198. C22×C107#123semidirect
251. C107#2418semidirect
264. #385D107semidirect
280. C2×C4⋊C4C2×D107semidirect
281. C2×C4⋊C4C2×D107semidirect
304. #95C2×D107semidirect
305. #95C2×D107semidirect
471. C22#16812094semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120792
C2×C4×C22⋊D1072
C22×(C2×D107)⋊C42
#202925622
#202925562
#202925452
#202925442
#202925172
#202925162
#202925052
#202925042
#202924852
#202924822
#385×C1072
#2418107