Finite Group Databasebeta

← #28339799

Subgroups — #28339799

Order 13696.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic, complemented
3C2×#16812081C211yesmaximal, characteristic, complemented
4C2×D4×Dic107C211yesmaximal, characteristic, complemented
5#20292543C211yesmaximal, characteristic, complemented
6#20292541C211yesmaximal, characteristic, complemented
7#20292535C211yesmaximal, characteristic, complemented
8#20292531C211yesmaximal, characteristic, complemented
9#20292527C211yesmaximal, characteristic, complemented
10#20292507C211yesmaximal, characteristic, complemented
11#20292503C211yesmaximal, characteristic, complemented
12#20292502C211yesmaximal, characteristic, complemented
13#20292501C211yesmaximal, characteristic, complemented
14#20292485C211yesmaximal, characteristic, complemented
15#392×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—24no—
17C2×#13133817—24no—
18C2×#13133817—24no—
19C22×Dic214C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C22⋊D107C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic
23C23×Dic107C2211yescharacteristic
24C23×Dic107C2211yescharacteristic
25C23×Dic107C2211yescharacteristic
26C22×C4×D107C2211yescharacteristic
27#16812086—24no—
28#16812081—24no—
29D4×Dic107—24no—
30C2×C4⋊Dic107C2211yescharacteristic
31C2×C4⋊Dic107C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×Dic107⋊C4C2211yescharacteristic
34C2×Dic107⋊C4C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C22⋊Dic107C2211yescharacteristic
38C2×C22⋊Dic107C2211yescharacteristic
39C2×C22⋊Dic107C2211yescharacteristic
40C2×C22⋊Dic107C2211yescharacteristic
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic
43C2×(C2×D107)⋊C4C2211yescharacteristic
44C2×C4×Dic107C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C107⋊#95C2211yescharacteristic
49C107⋊#95C2211yescharacteristic
50C22×D4×C107C2211yescharacteristic
51C22×D4×C107C2211yescharacteristic
52C22×D4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×C22⋊C4×C107C2211yescharacteristic
56#95×C107C2211yescharacteristic
57C23×D107C2311yescharacteristic
58#13133817—232no—
59C2×Dic214—22no—
60C2×Dic214—22no—
61C2×Dic214D414yescomplemented
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—22no—
68C2×C22⋊D107—22no—
69C2×C22⋊D107—41no—
70C2×C22⋊D107D414yescomplemented
71C22×Dic107—22no—
72C22×Dic107—22no—
73C22×Dic107—22no—
74C22×Dic107—22no—
75C22×Dic107—22no—
76C22×Dic107—22no—
77C22×Dic107—22no—
78C22×Dic107—22no—
79C22×Dic107D414yescomplemented
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2311yescharacteristic
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C22×Dic107C2311yescharacteristic
86C22×Dic107C2311yescharacteristic
87C2×C4×D107—22no—
88C2×C4×D107—22no—
89C2×C4×D107—22no—
90C4⋊Dic107—22no—
91C4⋊Dic107—22no—
92Dic107⋊C4—22no—
93Dic107⋊C4—22no—
94Dic107⋊C4—22no—
95C22⋊Dic107—22no—
96C22⋊Dic107—22no—
97C22⋊Dic107—22no—
98C22⋊Dic107—22no—
99C22⋊Dic107—41no—
100C22⋊Dic107—41no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4—41no—
104C4×Dic107—22no—
105C24×C107C2311yescharacteristic
106C24×C107C2311yescharacteristic
107C24×C107C2311yescharacteristic
108C2×D4×C107—22no—
109C2×D4×C107—22no—
110C2×D4×C107—22no—
111C2×D4×C107—22no—
112C2×D4×C107—22no—
113C2×D4×C107—22no—
114C2×D4×C107—22no—
115C2×D4×C107—22no—
116C2×D4×C107—41no—
117C22×C4×C107C2311yescharacteristic
118C22×C4×C107C2311yescharacteristic
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22⋊C4×C107—22no—
122C22⋊C4×C107—22no—
123C22⋊C4×C107—41no—
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—21no—
135C23×C107—21no—
136C23×C107—21no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107—21no—
140C23×C107—21no—
141C23×C107—21no—
142C23×C107C2×D412yes—
143C23×C107C2×D412yes—
144C23×C107#4112yescomplemented
145C23×C107C2411yescharacteristic
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—21no—
149C22×D107—21no—
150C22×D107—21no—
151C22×D107—21no—
152C22×D107C2×D412yes—
153D4×C107—28no—
154D4×C107—28no—
155D4×C107—44no—
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×C107C2×D412yes—
167C2×C4×C107#4112yescomplemented
168C22⋊D107—28no—
169C22⋊D107—28no—
170C22⋊D107—44no—
171C2×Dic107—28no—
172C2×Dic107—21no—
173C2×Dic107—21no—
174C2×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107—28no—
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—44no—
190C2×Dic107C2×D412yes—
191C2×Dic107C2×D412yes—
192C2×Dic107C2×D412yes—
193C2×Dic107C2×D412yes—
194C2×Dic107C2×D412yes—
195C4×D107—28no—
196Dic214—28no—
197C22×C107—22no—
198C22×C107—22no—
199C22×C107—22no—
200C22×C107—22no—
201C22×C107—22no—
202C22×C107—22no—
203C22×C107—22no—
204C22×C107—22no—
205C22×C107—22no—
206C22×C107—41no—
207C22×C107—41no—
208C22×C107—41no—
209C22×C107—41no—
210C22×C107—41no—
211C22×C107—41no—
212C22×C107—41no—
213C22×C107—41no—
214C22×C107—41no—
215C22×C107—41no—
216C22×C107—41no—
217C22×C107—41no—
218C22×C107C22×D411yescharacteristic
219C22×C107C22×D411yescharacteristic
220C22×C107C22×D411yescharacteristic
221C22×C107C2×#4111yescharacteristic
222C22×C107C2×#4111yescharacteristic
223C22×C107H(5,𝔽2)11yescharacteristic
224C22×C1072−1+411yescharacteristic
225C2×D107—22no—
226C2×D107—22no—
227C2×D107—22no—
228C2×D107—41no—
229C2×D107—41no—
230C2×D107—41no—
231C2×D107—41no—
232C4×C107—22no—
233C4×C107—22no—
234C4×C107—41no—
235C4×C107—41no—
236Dic107—22no—
237Dic107—22no—
238Dic107—41no—
239Dic107—41no—
240Dic107—41no—
241Dic107—41no—
242Dic107#12014yes—
243C2×C107—42no—
244C2×C107—42no—
245C2×C107—42no—
246C2×C107C2×#12011yescharacteristic
247C2×C107#52511yescharacteristic
248C2×C107#53711yescharacteristic
249C2×C107#54611yescharacteristic
250C2×C107#54711yescharacteristic
251C2×C107#54811yescharacteristic
252C2×C107#55011yescharacteristic
253D107—42no—
254#2407—1071nomaximal, Sylow
255C107#240711yescharacteristic, complemented, Sylow
256C22×#41—1071no—
257C2×C22⋊Q8—1071no—
258C2×#121—1071no—
259C2×C4×D4—1071no—
260#394—1071no—
261#394—1071no—
262#392—1071no—
263#392D10711yescharacteristic, complemented
264#391—1071no—
265#390—1071no—
266#386—1071no—
267#386—1071no—
268#385—1071no—
269#385—1071no—
270#380—1071no—
271C2×#41—2144no—
272C2×#41—2144no—
273C2×#41—2144no—
274C22×Q8—1071no—
275C22×D4—1071no—
276C22×D4—1071no—
277C22×D4—1071no—
278C22×D4C2×D10711yescharacteristic
279C22×D4C2×D10711yescharacteristic
280C22×D4C2×D10711yescharacteristic
281C23×C4—1071no—
282C23×C4—1071no—
283C23×C4—1071no—
284C23×C4—1071no—
285C22⋊Q8—2144no—
286#121—2144no—
287C4×D4—2144no—
288C2×C4⋊C4—1071no—
289C2×C4⋊C4—1071no—
290C2×C4⋊C4—1071no—
291C2×C4⋊C4—1071no—
292C2×C4⋊C4—1071no—
293C2×C22⋊C4—1071no—
294C2×C22⋊C4—1071no—
295C2×C22⋊C4—1071no—
296C2×C22⋊C4—1071no—
297C2×C22⋊C4—1071no—
298C2×C22⋊C4—1071no—
299C2×C22⋊C4—1071no—
300C2×C22⋊C4—1071no—
301C2×C22⋊C4—1071no—
302C2×C22⋊C4C2×D10711yescharacteristic
303C2×C22⋊C4C2×D10711yescharacteristic
304C2×C22⋊C4C2×D10711yescharacteristic
305C2×C42—1071no—
306#95—1071no—
307#95—1071no—
308#95—1071no—
309#95—1071no—
310#95—1071no—
311#95C2×D10711yescharacteristic
312C24—1071no—
313C24C22×D10711yescharacteristic
314C24C22×D10711yescharacteristic
315C24C22×D10711yescharacteristic
316#41—21432no—
317C2×Q8—1074no—
318C2×Q8—2142no—
319C2×Q8—2142no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—1074no—
330C2×D4—4281no—
331C2×D4—2142no—
332C2×D4—2142no—
333C2×D4—2142no—
334C2×D4—2142no—
335C2×D4—2142no—
336C2×D4—2142no—
337C2×D4—2142no—
338C22×C4—1071no—
339C22×C4—1074no—
340C22×C4—1071no—
341C22×C4—1071no—
342C22×C4—1071no—
343C22×C4—1071no—
344C22×C4—1071no—
345C22×C4—1071no—
346C22×C4—2142no—
347C22×C4—2142no—
348C22×C4—2142no—
349C22×C4—2142no—
350C22×C4—2142no—
351C22×C4—2142no—
352C22×C4—2142no—
353C22×C4—2142no—
354C22×C4—2142no—
355C22×C4—2142no—
356C22×C4—2142no—
357C22×C4C22×D10711yescharacteristic
358C22×C4C22×D10711yescharacteristic
359C22×C4C22×D10711yescharacteristic
360C22×C4C22×D10711yescharacteristic
361C4⋊C4—2142no—
362C4⋊C4—2142no—
363C4⋊C4—2142no—
364C4⋊C4—2142no—
365C4⋊C4—2142no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—4281no—
370C22⋊C4—4281no—
371C22⋊C4—4281no—
372C22⋊C4—2142no—
373C22⋊C4—2142no—
374C22⋊C4—2142no—
375C22⋊C4—2142no—
376C22⋊C4—2142no—
377C22⋊C4—2142no—
378C42—2142no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—1072no—
398C23—2141no—
399C23—2141no—
400C23—2141no—
401C23—2141no—
402C23—2141no—
403C23—2141no—
404C23D4×D10712yes—
405C23D4×D10712yes—
406C23#1313381712yescomplemented
407C23C23×D10711yescharacteristic
408Q8—2148no—
409D4—28no—
410D4—28no—
411D4—44no—
412D4—4284no—
413D4—2148no—
414D4—2148no—
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—1072no—
427C2×C4—1072no—
428C2×C4—1072no—
429C2×C4—1072no—
430C2×C4—4284no—
431C2×C4—2148no—
432C2×C4—2148no—
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×C4—2141no—
445C2×C4—2148no—
446C2×C4—2141no—
447C2×C4—2141no—
448C2×C4—2141no—
449C2×C4—2141no—
450C2×C4D4×D10712yes—
451C2×C4#1313381712yescomplemented
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—4281no—
474C22—4281no—
475C22—4281no—
476C22—4281no—
477C22—2142no—
478C22—2142no—
479C22—2142no—
480C22C2×D4×D10711yescharacteristic
481C22C2×D4×D10711yescharacteristic
482C22C2×D4×D10711yescharacteristic
483C22C2×#1313381711yescharacteristic
484C22C2×#1313381711yescharacteristic
485C22#1681215411yescharacteristic
486C22#1681215611yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—1074no—
492C4—4281no—
493C4—4281no—
494C4—4281no—
495C4—4281no—
496C4—2142no—
497C4—2142no—
498C2—42no—
499C2—42no—
500C2—42no—
501C2—4282no—
502C2#120×D10711yescharacteristic
503C2#2029313311yescharacteristic
504C2#2029320411yescharacteristic
505C2#2029326411yescharacteristic
506C2#2029327711yescharacteristic
507C2#2029328311yescharacteristic
508C2#2029329711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2407
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812086C2semidirect
3. C2×#16812081C2semidirect
4. C2×D4×Dic107C2semidirect
5. #20292543C2semidirect
6. #20292541C2semidirect
7. #20292535C2semidirect
8. #20292531C2semidirect
9. #20292527C2semidirect
10. #20292507C2semidirect
11. #20292503C2semidirect
12. #20292502C2semidirect
13. #20292501C2semidirect
14. #20292485C2semidirect
15. #392×C107C2semidirect
61. C2×Dic214D4semidirect
70. C2×C22⋊D107D4semidirect
79. C22×Dic107D4semidirect
144. C23×C107#41semidirect
167. C2×C4×C107#41semidirect
255. C107#2407semidirect
263. #392D107semidirect
406. C23#13133817semidirect
451. C2×C4#13133817semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168120862
C2×#168120812
C2×D4×Dic1072
#202925432
#202925412
#202925352
#202925312
#202925272
#202925072
#202925032
#202925022
#202925012
#202924852
#392×C1072
#2407107