Finite Group Databasebeta

← #28339120

Subgroups — #28339120

Order 13696.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812077C211yesmaximal, characteristic
2C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292527C211yesmaximal, characteristic, complemented
5#20292521C211yesmaximal, characteristic, complemented
6#20292521C211yesmaximal, characteristic, complemented
7#20292518C211yesmaximal, characteristic, complemented
8#20292517C211yesmaximal, characteristic, complemented
9#20292507C211yesmaximal, characteristic, complemented
10#20292504C211yesmaximal, characteristic, complemented
11#20292503C211yesmaximal, characteristic, complemented
12#20292498C211yesmaximal, characteristic, complemented
13#20292482C211yesmaximal, characteristic, complemented
14C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
15#390×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic, complemented
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812077C418yescomplemented
23(C22⋊D107)⋊C4—24no—
24C2×Dic107⋊C4C2211yescharacteristic, complemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic, complemented
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
38C22⋊C4×D107—24no—
39C2×C4×Dic107C2211yescharacteristic, complemented
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
48C2×C4⋊C4×C107C2211yescharacteristic, complemented
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic, complemented
52C2×C22⋊C4×C107C2211yescharacteristic, complemented
53C2×C42×C107C2211yescharacteristic, complemented
54C23×D107—22no—
55C23×D107—22no—
56C23×D107—41no—
57C23×D107—41no—
58C23×D107—41no—
59C23×D107—41no—
60C23×D107C2×C414yescomplemented
61C23×D107C2311yescharacteristic
62C23×D107C2311yescharacteristic
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—41no—
66C2×C22⋊D107—41no—
67C2×C22⋊D107C2×C414yes—
68C2×C22⋊D107C2×C414yescomplemented
69C22×Dic107—22no—
70C22×Dic107—41no—
71C22×Dic107—41no—
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C22×Dic107C2311yescharacteristic
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C2×C4×D107—22no—
79C2×C4×D107—41no—
80C2×C4×D107—41no—
81C2×C4×D107—41no—
82C2×C4×D107—41no—
83C2×C4×D107D414yescomplemented
84Dic107⋊C4—22no—
85Dic107⋊C4—41no—
86C22⋊Dic107—22no—
87C22⋊Dic107—41no—
88C22⋊Dic107—41no—
89C22⋊Dic107C2×C414yescomplemented
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—22no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—22no—
94(C2×D107)⋊C4—41no—
95(C2×D107)⋊C4—41no—
96(C2×D107)⋊C4C2×C414yescomplemented
97(C2×D107)⋊C4C2×C414yescomplemented
98C4×Dic107—22no—
99C4×Dic107—41no—
100C24×C107C2311yescharacteristic
101C24×C107C2311yescharacteristic
102C2×D4×C107—41no—
103C2×D4×C107—41no—
104C2×D4×C107C2×C414yescomplemented
105C22×C4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C4⋊C4×C107—41no—
111C22⋊C4×C107—22no—
112C22⋊C4×C107—22no—
113C22⋊C4×C107—41no—
114C22⋊C4×C107—41no—
115C42×C107—41no—
116C23×C107—21no—
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×C107C22×C412yes—
129C23×C107C22×C412yes—
130C23×C107C2411yescharacteristic
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—28no—
142C22×D107—44no—
143C22×D107—44no—
144C22×D107—44no—
145C22×D107—44no—
146C22×D107—44no—
147C22×D107—44no—
148C22×D107C22×C412yes—
149C22×D107C22×C412yes—
150C22×D107C2×D412yes—
151C22×D107#4112yes—
152D4×C107—44no—
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×C107C22×C412yes—
167C2×C4×C107C2×D412yes—
168C22⋊D107—28no—
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—44no—
186C2×Dic107C22×C412yes—
187C2×Dic107C22×C412yes—
188C2×Dic107C2×D412yes—
189C4×D107—44no—
190C4×D107—44no—
191C22×C107—22no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—22no—
197C22×C107—41no—
198C22×C107—41no—
199C22×C107—41no—
200C22×C107—41no—
201C22×C107—41no—
202C22×C107—41no—
203C22×C107—41no—
204C22×C107—41no—
205C22×C107C23×C411yescharacteristic
206C22×C107C22×D411yescharacteristic
207C22×C107C2×#4111yescharacteristic
208C22×C107H(5,𝔽2)11yescharacteristic
209C22×C107H(5,𝔽2)11yescharacteristic
210C22×C107H(5,𝔽2)11yescharacteristic
211C22×C107H(5,𝔽2)11yescharacteristic
212C2×D107—22no—
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—416no—
222C2×D107—41no—
223C2×D107—41no—
224C2×D107—41no—
225C2×D107—41no—
226C2×D107C4×D414yes—
227C4×C107—22no—
228C4×C107—22no—
229C4×C107—41no—
230C4×C107—41no—
231C4×C107—41no—
232Dic107—22no—
233Dic107—22no—
234Dic107—22no—
235Dic107—41no—
236Dic107—41no—
237Dic107—41no—
238C2×C107—42no—
239C2×C107—42no—
240C2×C107C2×C4×D411yescharacteristic
241C2×C107#51811yescharacteristic
242C2×C107#51811yescharacteristic
243C2×C107#53411yescharacteristic
244C2×C107#53511yescharacteristic
245C2×C107#53811yescharacteristic
246C2×C107#53811yescharacteristic
247D107—24no—
248D107—42no—
249#2321—1071nomaximal, Sylow
250C107#232111yescharacteristic, complemented, Sylow
251C2×#120—1071no—
252C2×C4×D4—1071no—
253C22×C22⋊C4—1071no—
254#390—1071no—
255#390D10711yescharacteristic, complemented
256#388—1071no—
257#388—1071no—
258#388—1071no—
259#388—1071no—
260#386—1071no—
261#386—1071no—
262#386—1071no—
263#385—1071no—
264#379—1071no—
265C4×C22⋊C4—1071no—
266C25—1071no—
267C22×D4—1071no—
268C22×D4—1071no—
269C22×D4C2×D10711yescharacteristic
270C23×C4—1071no—
271C23×C4—1071no—
272C23×C4—1071no—
273#120—1078no—
274C4×D4—2144no—
275C2×C4⋊C4—1071no—
276C2×C4⋊C4—1071no—
277C2×C4⋊C4C2×D10711yescharacteristic, complemented
278C2×C22⋊C4—1071no—
279C2×C22⋊C4—1071no—
280C2×C22⋊C4—1071no—
281C2×C22⋊C4—1071no—
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—2144no—
291C2×C22⋊C4C2×D10711yescharacteristic
292C2×C22⋊C4C2×D10711yescharacteristic
293C2×C22⋊C4C2×D10711yescharacteristic, complemented
294C2×C22⋊C4C2×D10711yescharacteristic, complemented
295C2×C42—1071no—
296C2×C42—1071no—
297C2×C42C2×D10711yescharacteristic, complemented
298#95—1071no—
299#95—1071no—
300#95—1071no—
301#95—1071no—
302#95—1071no—
303#95—1071no—
304C24—1071no—
305C24—1074no—
306C24—1071no—
307C24—4281no—
308C24—4281no—
309C24—4281no—
310C24—4281no—
311C24—2142no—
312C24—2142no—
313C24C22×D10711yescharacteristic
314C24C22×D10711yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—1074no—
318C2×D4—1074no—
319C2×D4—4281no—
320C2×D4—4281no—
321C2×D4—2142no—
322C2×D4—2142no—
323C2×D4C4×D10714yescomplemented
324C22×C4—1071no—
325C22×C4—1071no—
326C22×C4—1071no—
327C22×C4—1074no—
328C22×C4—1071no—
329C22×C4—1071no—
330C22×C4—1071no—
331C22×C4—4281no—
332C22×C4—4281no—
333C22×C4—4281no—
334C22×C4—4281no—
335C22×C4—4281no—
336C22×C4—4281no—
337C22×C4—2142no—
338C22×C4—2142no—
339C22×C4C22×D10711yescharacteristic
340C22×C4C22×D10711yescharacteristic
341C22×C4C22×D10711yescharacteristic
342C22×C4C22×D10711yescharacteristic
343C22×C4C22×D10711yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—4281no—
346C4⋊C4—2142no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1074no—
352C22⋊C4—1074no—
353C22⋊C4—1074no—
354C22⋊C4—4281no—
355C22⋊C4—4281no—
356C22⋊C4—4281no—
357C22⋊C4—4281no—
358C22⋊C4—2142no—
359C22⋊C4—2142no—
360C22⋊C4—2142no—
361C22⋊C4—2142no—
362C22⋊C4—2142no—
363C42—41no—
364C42—4281no—
365C42—2142no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—1072no—
379C23—1072no—
380C23—1072no—
381C23—1072no—
382C23—4284no—
383C23—4284no—
384C23—4284no—
385C23—4284no—
386C23—4284no—
387C23—4284no—
388C23—2148no—
389C23—2141no—
390C23—2141no—
391C23—2141no—
392C23—2141no—
393C23—2141no—
394C23—2141no—
395C23—2141no—
396C23—2141no—
397C23—2141no—
398C23—2141no—
399C23C2×C4×D10712yes—
400C23C2×C4×D10712yes—
401C23C23×D10711yescharacteristic
402D4—44no—
403D4—4284no—
404D4—2148no—
405C2×C4—21no—
406C2×C4—21no—
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—1072no—
419C2×C4—1072no—
420C2×C4—1072no—
421C2×C4—4284no—
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×C4C2×C4×D10712yes—
440C2×C4D4×D10712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—1074no—
456C22—42816no—
457C22—4281no—
458C22—4281no—
459C22—4281no—
460C22—4281no—
461C22—2142no—
462C22—2142no—
463C22—2142no—
464C22—2142no—
465C22—2142no—
466C22—2142no—
467C22—2142no—
468C22—2142no—
469C22—2142no—
470C22C22×C4×D10711yescharacteristic
471C22C2×D4×D10711yescharacteristic
472C22#41×D10711yescharacteristic
473C22#1681215411yescharacteristic
474C22#1681215411yescharacteristic
475C22#1681215411yescharacteristic
476C22#1681215511yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—4281no—
483C4—4281no—
484C4—4281no—
485C4—2142no—
486C4—2142no—
487C4—2142no—
488C2—42no—
489C2—42no—
490C2—4282no—
491C2—2144no—
492C2C4×D4×D10711yescharacteristic
493C2#2029309911yescharacteristic
494C2#2029310011yescharacteristic
495C2#2029318511yescharacteristic
496C2#2029319011yescharacteristic
497C2#2029320911yescharacteristic
498C2#2029321111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#390×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812077C2
hypercenter#390D107
hypocenterC107#2321
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D107)⋊C4C2semidirect
3. C2×C22⋊C4×D107C2semidirect
4. #20292527C2semidirect
5. #20292521C2semidirect
6. #20292521C2semidirect
7. #20292518C2semidirect
8. #20292517C2semidirect
9. #20292507C2semidirect
10. #20292504C2semidirect
11. #20292503C2semidirect
12. #20292498C2semidirect
13. #20292482C2semidirect
14. C4×(C2×D107)⋊C4C2semidirect
15. #390×C107C2semidirect
19. C23×Dic107C22semidirect
22. #16812077C4semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
29. C2×C22⋊Dic107C22semidirect
39. C2×C4×Dic107C22semidirect
40. C2×C4×Dic107C22semidirect
41. C107⋊#95C22semidirect
42. C107⋊#95C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
48. C2×C4⋊C4×C107C22semidirect
51. C2×C22⋊C4×C107C22semidirect
52. C2×C22⋊C4×C107C22semidirect
53. C2×C42×C107C22semidirect
60. C23×D107C2×C4semidirect
68. C2×C22⋊D107C2×C4semidirect
83. C2×C4×D107D4semidirect
89. C22⋊Dic107C2×C4semidirect
96. (C2×D107)⋊C4C2×C4semidirect
97. (C2×D107)⋊C4C2×C4semidirect
104. C2×D4×C107C2×C4semidirect
250. C107#2321semidirect
255. #390D107semidirect
277. C2×C4⋊C4C2×D107semidirect
293. C2×C22⋊C4C2×D107semidirect
294. C2×C22⋊C4C2×D107semidirect
297. C2×C42C2×D107semidirect
323. C2×D4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120772
C2×(C22⋊D107)⋊C42
C2×C22⋊C4×D1072
#202925272
#202925212
#202925182
#202925172
#202925072
#202925042
#202925032
#202924982
#202924822
C4×(C2×D107)⋊C42
#390×C1072
#2321107