Finite Group Databasebeta

← #28339920

Subgroups — #28339920

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×#16812077C211yesmaximal, characteristic
2C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292562C211yesmaximal, characteristic, complemented
5#20292555C211yesmaximal, characteristic, complemented
6#20292543C211yesmaximal, characteristic, complemented
7#20292541C211yesmaximal, characteristic, complemented
8#20292520C211yesmaximal, characteristic, complemented
9#20292517C211yesmaximal, characteristic, complemented
10#20292506C211yesmaximal, characteristic, complemented
11#20292503C211yesmaximal, characteristic, complemented
12#20292498C211yesmaximal, characteristic, complemented
13#20292484C211yesmaximal, characteristic, complemented
14#20292482C211yesmaximal, characteristic, complemented
15C2×#123×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21#16812077—24no—
22C4×C22⋊D107—24no—
23C2×C4⋊Dic107C2211yescharacteristic, complemented
24C2×Dic107⋊C4C2211yescharacteristic, complemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic, complemented
30C2×C22⋊Dic107C2211yescharacteristic, complemented
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
39C22⋊C4×D107—24no—
40C2×C4×Dic107C2211yescharacteristic, complemented
41C107⋊#95C2211yescharacteristic, complemented
42C107⋊#95C2211yescharacteristic, complemented
43C107⋊#95C2211yescharacteristic, complemented
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C22×D4×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic, complemented
49#123×C107—24no—
50C2×C4⋊C4×C107C2211yescharacteristic, complemented
51C2×C4⋊C4×C107C2211yescharacteristic, complemented
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×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
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—22no—
68C2×C22⋊D107—41no—
69C2×C22⋊D107D414yes—
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C22×Dic107C2311yescharacteristic
76C2×C4×D107—22no—
77C2×C4×D107—22no—
78C2×C4×D107—22no—
79C2×C4×D107—22no—
80C2×C4×D107—41no—
81C2×C4×D107—41no—
82C4⋊Dic107—41no—
83Dic107⋊C4—22no—
84Dic107⋊C4—41no—
85C22⋊Dic107—22no—
86C22⋊Dic107—22no—
87C22⋊Dic107—22no—
88C22⋊Dic107—41no—
89C22⋊Dic107—41no—
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—41no—
98C4×Dic107—22no—
99C24×C107C2311yescharacteristic
100C24×C107C2311yescharacteristic
101C2×D4×C107—22no—
102C2×D4×C107—22no—
103C2×D4×C107—41no—
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
112C4⋊C4×C107—22no—
113C4⋊C4×C107—22no—
114C22⋊C4×C107—22no—
115C22⋊C4×C107—22no—
116C22⋊C4×C107D414yescomplemented
117C23×C107—21no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—21no—
128C23×C107—21no—
129C23×C107C2×D412yes—
130C23×C107C2×D412yes—
131C23×C107C2411yescharacteristic
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—21no—
135C22×D107—21no—
136C22×D107—28no—
137C22×D107—21no—
138C22×D107—21no—
139C22×D107—21no—
140C22×D107—21no—
141C22×D107—21no—
142C22×D107—21no—
143C22×D107—44no—
144C22×D107—44no—
145C22×D107—44no—
146C22×D107—44no—
147C22×D107—44no—
148C22×D107—44no—
149C22×D107C2×D412yes—
150C22×D107#4112yes—
151C22×D107#4112yes—
152C22×D107#4112yes—
153D4×C107—44no—
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—44no—
168C2×C4×C107C2×D412yes—
169C2×C4×C107C2×D412yes—
170C22⋊D107—28no—
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—
187C2×Dic107—21no—
188C2×Dic107—21no—
189C2×Dic107C2×D412yes—
190C4×D107—44no—
191C4×D107—44no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—22no—
197C22×C107—22no—
198C22×C107—41no—
199C22×C107—41no—
200C22×C107—41no—
201C22×C107—41no—
202C22×C107—41no—
203C22×C107—41no—
204C22×C107—41no—
205C22×C107—41no—
206C22×C107C22×D411yescharacteristic
207C22×C107C22×D411yescharacteristic
208C22×C107C2×#4111yescharacteristic
209C22×C107C2×#4111yescharacteristic
210C22×C107C2×#4111yescharacteristic
211C22×C107H(5,𝔽2)11yescharacteristic
212C22×C107H(5,𝔽2)11yescharacteristic
213C2×D107—22no—
214C2×D107—22no—
215C2×D107—22no—
216C2×D107—22no—
217C2×D107—22no—
218C2×D107—22no—
219C2×D107—22no—
220C2×D107—22no—
221C2×D107—22no—
222C2×D107—416no—
223C2×D107—41no—
224C2×D107—41no—
225C2×D107—41no—
226C2×D107—41no—
227C2×D107#12314yescomplemented
228C4×C107—22no—
229C4×C107—22no—
230C4×C107—41no—
231C4×C107—41no—
232C4×C107—41no—
233Dic107—22no—
234Dic107—41no—
235Dic107—41no—
236Dic107—41no—
237Dic107—41no—
238Dic107—41no—
239C2×C107—42no—
240C2×C107—42no—
241C2×C107C2×#12311yescharacteristic
242C2×C107#52511yescharacteristic
243C2×C107#53811yescharacteristic
244C2×C107D4211yescharacteristic
245C2×C107#54611yescharacteristic
246C2×C107#54611yescharacteristic
247C2×C107#55111yescharacteristic
248D107—24no—
249D107—42no—
250#2418—1071nomaximal, Sylow
251C107#241811yescharacteristic, complemented, Sylow
252C2×#123D10711yescharacteristic, complemented
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#385—1071no—
265#380—1071no—
266#379—1071no—
267C25—1071no—
268C22×D4—1071no—
269C22×D4—1071no—
270C22×D4C2×D10711yescharacteristic
271C23×C4—1071no—
272C23×C4—1071no—
273C23×C4C2×D10711yescharacteristic, complemented
274#123—24no—
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—2144no—
296C2×C22⋊C4C2×D10711yescharacteristic
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic, complemented
299C2×C42—1071no—
300#95—1071no—
301#95—1071no—
302#95—1071no—
303#95—1071no—
304#95—1071no—
305#95—1071no—
306C24—1071no—
307C24—1071no—
308C24—4281no—
309C24—4281no—
310C24—2142no—
311C24—2142no—
312C24—2142no—
313C24—2142no—
314C24—2142no—
315C24C22×D10711yescharacteristic
316C24C22×D10711yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—1074no—
321C2×D4—4281no—
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—1071no—
335C22×C4—4281no—
336C22×C4—4281no—
337C22×C4—2142no—
338C22×C4—2142no—
339C22×C4—2142no—
340C22×C4—2142no—
341C22×C4C22×D10711yescharacteristic
342C22×C4C22×D10711yescharacteristic
343C22×C4C22×D10711yescharacteristic
344C22×C4C22×D10711yescharacteristic
345C22×C4C22×D10711yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—4281no—
349C4⋊C4—4281no—
350C4⋊C4—2142no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—4281no—
354C22⋊C4—4281no—
355C22⋊C4—4281no—
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⋊C4C22⋊D10714yescomplemented
367C42—2142no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—1072no—
381C23—1072no—
382C23—1072no—
383C23—1072no—
384C23—4284no—
385C23—4284no—
386C23—4284no—
387C23—4284no—
388C23—4284no—
389C23—4284no—
390C23—2148no—
391C23—2141no—
392C23—2141no—
393C23—2141no—
394C23—2141no—
395C23—2141no—
396C23—2141no—
397C23—2141no—
398C23—2141no—
399C23—2141no—
400C23—2141no—
401C23D4×D10712yes—
402C23C2×C22⋊D10712yes—
403C23C23×D10711yescharacteristic
404D4—44no—
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—44no—
421C2×C4—1072no—
422C2×C4—4284no—
423C2×C4—4284no—
424C2×C4—2141no—
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×C4—2141no—
441C2×C4C2×C22⋊D10712yes—
442C2×C4C2×C22⋊D10712yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—1074no—
458C22—42816no—
459C22—4281no—
460C22—4281no—
461C22—4281no—
462C22—4281no—
463C22—2142no—
464C22—2142no—
465C22—2142no—
466C22—2142no—
467C22—2142no—
468C22—2142no—
469C22—2142no—
470C22—2142no—
471C22—2142no—
472C22C2×D4×D10711yescharacteristic
473C22C22×C22⋊D10711yescharacteristic
474C22#41×D10711yescharacteristic
475C22#41×D10711yescharacteristic
476C22#41×D10711yescharacteristic
477C22#1681215411yescharacteristic
478C22#1681215511yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—4281no—
485C4—4281no—
486C4—4281no—
487C4—4281no—
488C4—4281no—
489C4—2142no—
490C2—42no—
491C2—42no—
492C2—4282no—
493C2—2144no—
494C2#123×D10711yescharacteristic
495C2#2029313411yescharacteristic
496C2#2029321311yescharacteristic
497C2D4×C22⋊D10711yescharacteristic
498C2#2029326511yescharacteristic
499C2#2029326911yescharacteristic
500C2#2029330511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
prime radicalC2×#16812077C2
hypercenterC2×#123D107
hypocenterC107#2418
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D107C2semidirect
3. C2×C22⋊C4×D107C2semidirect
4. #20292562C2semidirect
5. #20292555C2semidirect
6. #20292543C2semidirect
7. #20292541C2semidirect
8. #20292520C2semidirect
9. #20292517C2semidirect
10. #20292506C2semidirect
11. #20292503C2semidirect
12. #20292498C2semidirect
13. #20292484C2semidirect
14. #20292482C2semidirect
15. C2×#123×C107C2semidirect
23. C2×C4⋊Dic107C22semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
29. C2×C22⋊Dic107C22semidirect
30. C2×C22⋊Dic107C22semidirect
40. C2×C4×Dic107C22semidirect
41. C107⋊#95C22semidirect
42. C107⋊#95C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
48. C23×C4×C107C22semidirect
50. C2×C4⋊C4×C107C22semidirect
51. C2×C4⋊C4×C107C22semidirect
54. C2×C22⋊C4×C107C22semidirect
116. C22⋊C4×C107D4semidirect
227. C2×D107#123semidirect
251. C107#2418semidirect
252. C2×#123D107semidirect
273. C23×C4C2×D107semidirect
280. C2×C4⋊C4C2×D107semidirect
281. C2×C4⋊C4C2×D107semidirect
298. C2×C22⋊C4C2×D107semidirect
366. C22⋊C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120772
C2×C4×C22⋊D1072
C2×C22⋊C4×D1072
#202925622
#202925552
#202925432
#202925412
#202925202
#202925172
#202925062
#202925032
#202924982
#202924842
#202924822
C2×#123×C1072
#2418107