Finite Group Databasebeta

← #28339850

Subgroups — #28339850

Order 13696.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812094C211yesmaximal, characteristic, complemented
2C2×#16812079C211yesmaximal, characteristic
3C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292555C211yesmaximal, characteristic, complemented
6#20292543C211yesmaximal, characteristic, complemented
7#20292542C211yesmaximal, characteristic, complemented
8#20292505C211yesmaximal, characteristic, complemented
9#20292505C211yesmaximal, characteristic, complemented
10#20292505C211yesmaximal, characteristic, complemented
11#20292501C211yesmaximal, characteristic, complemented
12#20292489C211yesmaximal, characteristic, complemented
13#20292486C211yesmaximal, characteristic, complemented
14#20292484C211yesmaximal, characteristic, complemented
15#397×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic
23C22×C4×D107C2211yescharacteristic
24#16812094—24no—
25#16812079—24no—
26C2×C4⋊Dic107C2211yescharacteristic, complemented
27C2×Dic107⋊C4C2211yescharacteristic, complemented
28C2×Dic107⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
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, complemented
38C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
41C22⋊C4×D107—24no—
42C22⋊C4×D107—24no—
43C107⋊#95C2211yescharacteristic, complemented
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
52#95×C107C2211yescharacteristic, complemented
53#95×C107C2211yescharacteristic, complemented
54#95×C107C2211yescharacteristic, complemented
55C23×D107—22no—
56C23×D107—22no—
57C23×D107—22no—
58C23×D107—22no—
59C23×D107—41no—
60C23×D107—41no—
61C23×D107D414yes—
62C23×D107C2311yescharacteristic
63C23×D107C2311yescharacteristic
64C23×D107C2311yescharacteristic
65C2×D428—22no—
66C2×D428—22no—
67C2×D428—22no—
68C2×D428—22no—
69C2×D428—22no—
70C2×D428—22no—
71C2×C22⋊D107—22no—
72C2×C22⋊D107—41no—
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—22no—
86C2×C4×D107—22no—
87C2×C4×D107—41no—
88C2×C4×D107—41no—
89C2×C4×D107—41no—
90C2×C4×D107—41no—
91C4⋊Dic107—41no—
92Dic107⋊C4—22no—
93Dic107⋊C4—41no—
94C22⋊Dic107—22no—
95C22⋊Dic107—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—
105(C2×D107)⋊C4D414yescomplemented
106C24×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C4⋊C4×C107—22no—
114C22⋊C4×C107—22no—
115C22⋊C4×C107—22no—
116C22⋊C4×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—21no—
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—28no—
134C22×D107—28no—
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×D107C2×D412yes—
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×C107—21no—
169C2×C4×C107—21no—
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×Dic107C2×D412yes—
187D428—44no—
188D428—44no—
189C4×D107—44no—
190C4×D107—44no—
191C4×D107—44no—
192C4×D107—44no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—41no—
197C22×C107—41no—
198C22×C107—41no—
199C22×C107—41no—
200C22×C107C22×D411yescharacteristic
201C22×C107C22×D411yescharacteristic
202C22×C107C2×#4111yescharacteristic
203C22×C107C2×#4111yescharacteristic
204C22×C107C2×#4111yescharacteristic
205C22×C107H(5,𝔽2)11yescharacteristic
206C22×C107H(5,𝔽2)11yescharacteristic
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—22no—
219C2×D107—416no—
220C2×D107—41no—
221C2×D107—41no—
222C2×D107—41no—
223C2×D107—41no—
224C2×D107—41no—
225C2×D107—41no—
226C2×D107—41no—
227C2×D107#12114yes—
228C2×D107#12314yescomplemented
229C4×C107—22no—
230C4×C107—41no—
231C4×C107—41no—
232C4×C107—41no—
233C4×C107—41no—
234C4×C107—41no—
235Dic107—22no—
236Dic107—41no—
237Dic107—41no—
238Dic107—41no—
239Dic107—41no—
240C2×C107—42no—
241C2×C107C2×#12111yescharacteristic
242C2×C107C2×#12311yescharacteristic
243C2×C107#53411yescharacteristic
244C2×C107#54611yescharacteristic
245C2×C107#54611yescharacteristic
246C2×C107#55111yescharacteristic
247C2×C107#55111yescharacteristic
248D107—24no—
249D107—24no—
250D107—81no—
251#2412—1071nomaximal, Sylow
252C107#241211yescharacteristic, complemented, Sylow
253C2×#123—1071no—
254C2×#120—1071no—
255C22×C22⋊C4—1071no—
256C22×C22⋊C4—1071no—
257#397—1071no—
258#397D10711yescharacteristic, complemented
259#394—1071no—
260#394—1071no—
261#386—1071no—
262#386—1071no—
263#386—1071no—
264#385—1071no—
265#381—1071no—
266#380—1071no—
267#380—1071no—
268C25—1071no—
269C22×D4—1071no—
270C22×D4—1071no—
271C22×D4—1071no—
272C23×C4—1071no—
273C23×C4—1071no—
274C23×C4—1071no—
275C23×C4—1071no—
276#123—2144no—
277#120—2144no—
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—2144no—
295C2×C22⋊C4—2144no—
296C2×C22⋊C4C2×D10711yescharacteristic
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic
299#95—1071no—
300#95—1071no—
301#95—1071no—
302#95—1071no—
303#95—1071no—
304#95C2×D10711yescharacteristic, complemented
305#95C2×D10711yescharacteristic, complemented
306#95C2×D10711yescharacteristic, complemented
307C24—1071no—
308C24—1071no—
309C24—1074no—
310C24—1071no—
311C24—4281no—
312C24—4281no—
313C24—2142no—
314C24—2142no—
315C24—2142no—
316C24—2142no—
317C24C22×D10711yescharacteristic
318C2×D4—4281no—
319C2×D4—4281no—
320C2×D4—2142no—
321C2×D4—2142no—
322C2×D4—2142no—
323C2×D4—2142no—
324C2×D4—2142no—
325C2×D4—2142no—
326C2×D4—2142no—
327C22×C4—1071no—
328C22×C4—1071no—
329C22×C4—1071no—
330C22×C4—1071no—
331C22×C4—1071no—
332C22×C4—4281no—
333C22×C4—4281no—
334C22×C4—4281no—
335C22×C4—4281no—
336C22×C4—2142no—
337C22×C4—2142no—
338C22×C4—2142no—
339C22×C4—2142no—
340C22×C4—2142no—
341C22×C4—2142no—
342C22×C4—2142no—
343C22×C4—2142no—
344C22×C4C22×D10711yescharacteristic
345C22×C4C22×D10711yescharacteristic
346C22×C4C22×D10711yescharacteristic
347C22×C4C22×D10711yescharacteristic
348C22×C4C22×D10711yescharacteristic
349C22×C4C22×D10711yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—4281no—
352C4⋊C4—4281no—
353C4⋊C4—2142no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—1074no—
358C22⋊C4—4281no—
359C22⋊C4—4281no—
360C22⋊C4—2142no—
361C22⋊C4—2142no—
362C22⋊C4—2142no—
363C22⋊C4—2142no—
364C22⋊C4—2142no—
365C22⋊C4—2142no—
366C22⋊C4—2142no—
367C22⋊C4—2142no—
368C22⋊C4—2142no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—1072no—
376C23—1072no—
377C23—1072no—
378C23—1072no—
379C23—1072no—
380C23—1072no—
381C23—4284no—
382C23—4284no—
383C23—4284no—
384C23—4284no—
385C23—4284no—
386C23—2148no—
387C23—2148no—
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—
402C23—2141no—
403C23D4×D10712yes—
404C23C23×D10711yescharacteristic
405D4—4284no—
406D4—4284no—
407D4—4284no—
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—21no—
425C2×C4—1072no—
426C2×C4—4284no—
427C2×C4—4284no—
428C2×C4—4284no—
429C2×C4—4284no—
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×C4—2141no—
443C2×C4—2141no—
444C2×C4D4×D10712yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—1074no—
453C22—1074no—
454C22—42816no—
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—2142no—
472C22—2142no—
473C22—2142no—
474C22C2×D4×D10711yescharacteristic
475C22C2×D4×D10711yescharacteristic
476C22#41×D10711yescharacteristic
477C22#41×D10711yescharacteristic
478C22#41×D10711yescharacteristic
479C22#1681215511yescharacteristic
480C22#1681215511yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—4281no—
488C4—4281no—
489C4—4281no—
490C4—4281no—
491C4—2142no—
492C2—42no—
493C2—8561no—
494C2—2144no—
495C2—2144no—
496C2#121×D10711yescharacteristic
497C2#123×D10711yescharacteristic
498C2#2029318811yescharacteristic
499C2#2029326911yescharacteristic
500C2#2029327011yescharacteristic
501C2#2029330211yescharacteristic
502C2#2029330211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#397×C107C2
prime radicalC2×#16812079C2
hypercenter#397D107
hypocenterC107#2412
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
3. C2×C22⋊C4×D107C2semidirect
4. C2×C22⋊C4×D107C2semidirect
5. #20292555C2semidirect
6. #20292543C2semidirect
7. #20292542C2semidirect
8. #20292505C2semidirect
9. #20292505C2semidirect
10. #20292505C2semidirect
11. #20292501C2semidirect
12. #20292489C2semidirect
13. #20292486C2semidirect
14. #20292484C2semidirect
15. #397×C107C2semidirect
26. C2×C4⋊Dic107C22semidirect
27. C2×Dic107⋊C4C22semidirect
28. C2×Dic107⋊C4C22semidirect
37. C2×(C2×D107)⋊C4C22semidirect
38. C2×(C2×D107)⋊C4C22semidirect
39. C2×(C2×D107)⋊C4C22semidirect
40. C2×(C2×D107)⋊C4C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
48. C2×C4⋊C4×C107C22semidirect
52. #95×C107C22semidirect
53. #95×C107C22semidirect
54. #95×C107C22semidirect
105. (C2×D107)⋊C4D4semidirect
228. C2×D107#123semidirect
252. C107#2412semidirect
258. #397D107semidirect
281. C2×C4⋊C4C2×D107semidirect
304. #95C2×D107semidirect
305. #95C2×D107semidirect
306. #95C2×D107semidirect

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×C22⋊C4×D1072
#202925552
#202925432
#202925422
#202925052
#202925012
#202924892
#202924862
#202924842
#397×C1072
#2412107