Finite Group Databasebeta

← #28339923

Subgroups — #28339923

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×#16812078C211yesmaximal, characteristic
3C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
4#20292563C211yesmaximal, characteristic, complemented
5#20292555C211yesmaximal, characteristic, complemented
6#20292543C211yesmaximal, characteristic, complemented
7#20292542C211yesmaximal, characteristic, complemented
8#20292521C211yesmaximal, characteristic, complemented
9#20292520C211yesmaximal, characteristic, complemented
10#20292506C211yesmaximal, characteristic, complemented
11#20292503C211yesmaximal, characteristic, complemented
12#20292500C211yesmaximal, characteristic, complemented
13#20292487C211yesmaximal, characteristic, complemented
14#20292483C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic, complemented
20#16812091—24no—
21#16812078—24no—
22C4×C22⋊D107—24no—
23C2×C4⋊Dic107C2211yescharacteristic, complemented
24C2×C4⋊Dic107C2211yescharacteristic, complemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×Dic107⋊C4C2211yescharacteristic, complemented
27C2×Dic107⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
40C2×C4×Dic107C2211yescharacteristic, complemented
41C107⋊#95C2211yescharacteristic, complemented
42C107⋊#95C2211yescharacteristic, complemented
43C107⋊#95C2211yescharacteristic, complemented
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C25×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49C23×C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107—24no—
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C2×C22⋊D107—22no—
57C2×C22⋊D107—22no—
58C2×C22⋊D107—22no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—41no—
63C2×C22⋊D107—41no—
64C2×C22⋊D107D414yes—
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C2×C4×D107—22no—
73C2×C4×D107—22no—
74C2×C4×D107—41no—
75C4⋊Dic107—22no—
76C4⋊Dic107—41no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—41no—
80C22⋊Dic107—22no—
81C22⋊Dic107—22no—
82C22⋊Dic107—22no—
83C22⋊Dic107—22no—
84C22⋊Dic107—22no—
85C22⋊Dic107—22no—
86C22⋊Dic107—22no—
87C22⋊Dic107—41no—
88C22⋊Dic107—41no—
89(C2×D107)⋊C4—22no—
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—41no—
92C4×Dic107—22no—
93C24×C107—22no—
94C24×C107—22no—
95C24×C107—22no—
96C24×C107—22no—
97C24×C107—22no—
98C24×C107—41no—
99C24×C107—41no—
100C24×C107C2311yescharacteristic
101C24×C107C2311yescharacteristic
102C24×C107C2311yescharacteristic
103C22×C4×C107—22no—
104C22×C4×C107—22no—
105C22×C4×C107—22no—
106C22×C4×C107—22no—
107C22×C4×C107—41no—
108C22×C4×C107—41no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C22⋊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—28no—
122C23×C107—21no—
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—21no—
128C23×C107—21no—
129C23×C107—21no—
130C23×C107—21no—
131C23×C107—21no—
132C23×C107—21no—
133C23×C107—21no—
134C23×C107—44no—
135C23×C107—44no—
136C23×C107—44no—
137C23×C107—44no—
138C23×C107—44no—
139C23×C107—44no—
140C23×C107C2×D412yes—
141C23×C107C2×D412yes—
142C23×C107#4112yes—
143C23×C107#4112yes—
144C23×C107#4112yes—
145C23×C107C2411yescharacteristic
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—21no—
149C22×D107—21no—
150C22×D107—21no—
151C22×D107—21no—
152C22×D107C2×D412yes—
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—44no—
164C2×C4×C107—44no—
165C2×C4×C107C2×D412yes—
166C2×C4×C107C2×D412yes—
167C22⋊D107—28no—
168C22⋊D107—44no—
169C22⋊D107—44no—
170C2×Dic107—21no—
171C2×Dic107—21no—
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×Dic107—21no—
190C2×Dic107C2×D412yes—
191C4×D107—44no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—22no—
197C22×C107—22no—
198C22×C107—22no—
199C22×C107—22no—
200C22×C107—22no—
201C22×C107—22no—
202C22×C107—22no—
203C22×C107—22no—
204C22×C107—416no—
205C22×C107—41no—
206C22×C107—41no—
207C22×C107—41no—
208C22×C107—41no—
209C22×C107—41no—
210C22×C107—41no—
211C22×C107—41no—
212C22×C107—41no—
213C22×C107#12314yescomplemented
214C22×C107C22×D411yescharacteristic
215C22×C107C22×D411yescharacteristic
216C22×C107C2×#4111yescharacteristic
217C22×C107C2×#4111yescharacteristic
218C22×C107C2×#4111yescharacteristic
219C22×C107H(5,𝔽2)11yescharacteristic
220C22×C107H(5,𝔽2)11yescharacteristic
221C2×D107—22no—
222C2×D107—22no—
223C2×D107—22no—
224C2×D107—41no—
225C2×D107—41no—
226C2×D107—41no—
227C2×D107—41no—
228C4×C107—22no—
229C4×C107—22no—
230C4×C107—41no—
231C4×C107—41no—
232Dic107—22no—
233Dic107—41no—
234Dic107—41no—
235Dic107—41no—
236Dic107—41no—
237Dic107—41no—
238Dic107—41no—
239C2×C107—24no—
240C2×C107—42no—
241C2×C107—42no—
242C2×C107C2×#12311yescharacteristic
243C2×C107#52511yescharacteristic
244C2×C107#53811yescharacteristic
245C2×C107D4211yescharacteristic
246C2×C107#54611yescharacteristic
247C2×C107#54611yescharacteristic
248C2×C107#55111yescharacteristic
249D107—42no—
250#2418—1071nomaximal, Sylow
251C107#241811yescharacteristic, complemented, Sylow
252C2×#123—1071no—
253C2×#120—1071no—
254C2×C4×D4—1071no—
255C22×C22⋊C4D10711yescharacteristic, complemented
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—
267C25C2×D10711yescharacteristic
268C22×D4—1071no—
269C22×D4—1071no—
270C22×D4—1071no—
271C23×C4—1071no—
272C23×C4C2×D10711yescharacteristic
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⋊C4—1071no—
281C2×C4⋊C4—1071no—
282C2×C22⋊C4—24no—
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⋊C4C2×D10711yescharacteristic
296C2×C22⋊C4C2×D10711yescharacteristic
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic
299C2×C42—1071no—
300#95—1071no—
301#95—1071no—
302#95—1071no—
303#95—1071no—
304#95—1071no—
305#95—1071no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—1071no—
314C24C22×D10711yescharacteristic
315C24C22×D10711yescharacteristic
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—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—1071no—
333C22×C4—1071no—
334C22×C4—1071no—
335C22×C4—1071no—
336C22×C4—1071no—
337C22×C4—1071no—
338C22×C4—1071no—
339C22×C4—4281no—
340C22×C4—2142no—
341C22×C4—2142no—
342C22×C4C22×D10711yescharacteristic
343C22×C4C22×D10711yescharacteristic
344C22×C4C22×D10711yescharacteristic
345C22×C4C22×D10711yescharacteristic
346C4⋊C4—4281no—
347C4⋊C4—4281no—
348C4⋊C4—2142no—
349C4⋊C4—2142no—
350C4⋊C4—2142no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
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⋊C4C22⋊D10714yescomplemented
367C42—2142no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—1072no—
392C23—2141no—
393C23—2141no—
394C23—2141no—
395C23—2141no—
396C23—2141no—
397C23—2141no—
398C23D4×D10712yes—
399C23C2×C22⋊D10712yes—
400C23#1313381712yes—
401C23#1313381812yes—
402C23#1313381812yes—
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—44no—
418C2×C4—44no—
419C2×C4—1072no—
420C2×C4—4284no—
421C2×C4—2141no—
422C2×C4—2141no—
423C2×C4—2141no—
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—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—4281no—
465C22—4281no—
466C22—4281no—
467C22—4281no—
468C22—2142no—
469C22—2142no—
470C22—2142no—
471C22#1681209114yescomplemented
472C22C2×D4×D10711yescharacteristic
473C22C22×C22⋊D10711yescharacteristic
474C22C2×#1313381711yescharacteristic
475C22C2×#1313381811yescharacteristic
476C22C2×#1313381811yescharacteristic
477C22#1681215411yescharacteristic
478C22#1681215411yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—4281no—
484C4—4281no—
485C4—4281no—
486C4—4281no—
487C4—4281no—
488C4—4281no—
489C4—2142no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—4282no—
494C2C2×#1681209111yescharacteristic
495C2#2029313811yescharacteristic
496C2#2029321511yescharacteristic
497C2D4×C22⋊D10711yescharacteristic
498C2#2029326711yescharacteristic
499C2#2029327111yescharacteristic
500C2#2029330311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C22⋊C4×C107C2
prime radicalC2×#16812078C2
hypercenterC22×C22⋊C4D107
hypocenterC107#2418
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16812091C2semidirect
3. C2×C4×C22⋊D107C2semidirect
4. #20292563C2semidirect
5. #20292555C2semidirect
6. #20292543C2semidirect
7. #20292542C2semidirect
8. #20292521C2semidirect
9. #20292520C2semidirect
10. #20292506C2semidirect
11. #20292503C2semidirect
12. #20292500C2semidirect
13. #20292487C2semidirect
14. #20292483C2semidirect
15. C22×C22⋊C4×C107C2semidirect
19. C22×C4×D107C22semidirect
23. C2×C4⋊Dic107C22semidirect
24. C2×C4⋊Dic107C22semidirect
25. C2×Dic107⋊C4C22semidirect
26. C2×Dic107⋊C4C22semidirect
27. C2×Dic107⋊C4C22semidirect
37. C2×(C2×D107)⋊C4C22semidirect
38. C2×(C2×D107)⋊C4C22semidirect
39. C2×(C2×D107)⋊C4C22semidirect
40. C2×C4×Dic107C22semidirect
41. C107⋊#95C22semidirect
42. C107⋊#95C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
116. C22⋊C4×C107D4semidirect
213. C22×C107#123semidirect
251. C107#2418semidirect
255. C22×C22⋊C4D107semidirect
366. C22⋊C4C22⋊D107semidirect
471. C22#16812091semidirect

Maximal subgroups

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

subgroupindex
C2×#168120912
C2×#168120782
C2×C4×C22⋊D1072
#202925632
#202925552
#202925432
#202925422
#202925212
#202925202
#202925062
#202925032
#202925002
#202924872
#202924832
C22×C22⋊C4×C1072
#2418107