Finite Group Databasebeta

← #28340087

Subgroups — #28340087

Order 13696.

836 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133818C211yesmaximal, characteristic
2C2×#16812108C211yesmaximal, characteristic, complemented
3C2×#16812095C211yesmaximal, characteristic, complemented
4C2×#16812087C211yesmaximal, characteristic, complemented
5#20292545C211yesmaximal, characteristic, complemented
6#20292544C211yesmaximal, characteristic, complemented
7#20292543C211yesmaximal, characteristic, complemented
8#20292542C211yesmaximal, characteristic, complemented
9#20292531C211yesmaximal, characteristic, complemented
10#20292527C211yesmaximal, characteristic, complemented
11#20292501C211yesmaximal, characteristic, complemented
12#20292500C211yesmaximal, characteristic, complemented
13#20292498C211yesmaximal, characteristic, complemented
14#20292474C211yesmaximal, characteristic, complemented
15#385×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—24no—
17C2×#13133818—24no—
18C2×#13133818—24no—
19C22×Dic214C2211yescharacteristic
20C22×Dic214C2211yescharacteristic
21C22×D428C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic
23C22×C22⋊D107C2211yescharacteristic
24C22×C4×D107C2211yescharacteristic
25C22×C4×D107C2211yescharacteristic
26#16812108—24no—
27#16812095—24no—
28#16812087—24no—
29C2×C4⋊Dic107C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×Dic107⋊C4C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic
43C2×(C2×D107)⋊C4C2211yescharacteristic
44C2×C4×Dic107C2211yescharacteristic
45C2×C4×Dic107C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C107⋊#95C2211yescharacteristic
49C107⋊#95C2211yescharacteristic
50C23×C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic
52C2×C4⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55#95×C107C2211yescharacteristic
56#95×C107C2211yescharacteristic
57C23×D107C2311yescharacteristic
58C23×D107C2311yescharacteristic
59#13133818—232no—
60C2×Dic214—22no—
61C2×Dic214—22no—
62C2×Dic214—22no—
63C2×Dic214—22no—
64C2×Dic214—22no—
65C2×Dic214D414yes—
66C2×D428—22no—
67C2×D428—22no—
68C2×D428—22no—
69C2×C22⋊D107—22no—
70C2×C22⋊D107—22no—
71C2×C22⋊D107—22no—
72C2×C22⋊D107—22no—
73C2×C22⋊D107—22no—
74C2×C22⋊D107D414yes—
75C22×Dic107C2311yescharacteristic
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C22×Dic107C2311yescharacteristic
80C22×Dic107C2311yescharacteristic
81C2×C4×D107—22no—
82C2×C4×D107—22no—
83C2×C4×D107—22no—
84C2×C4×D107—22no—
85C2×C4×D107—22no—
86C2×C4×D107D414yes—
87C4⋊Dic107—22no—
88Dic107⋊C4—22no—
89Dic107⋊C4—22no—
90Dic107⋊C4—22no—
91Dic107⋊C4—22no—
92C22⋊Dic107—22no—
93C22⋊Dic107—22no—
94(C2×D107)⋊C4—22no—
95(C2×D107)⋊C4—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—
102C4×Dic107—22no—
103C4×Dic107—22no—
104C24×C107C2311yescharacteristic
105C22×C4×C107—22no—
106C22×C4×C107—22no—
107C22×C4×C107—22no—
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114C4⋊C4×C107—22no—
115C4⋊C4×C107—22no—
116C22⋊C4×C107—22no—
117C22⋊C4×C107—22no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C23×C107—21no—
124C23×C107C2×D412yes—
125C23×C107C2411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—21no—
135C22×D107—21no—
136C22×D107—21no—
137C22×D107—21no—
138C22×D107C2×D412yes—
139C22×D107C2×D412yes—
140C2×C4×C107—28no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—21no—
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107C2×D412yes—
157C2×C4×C107C2×D412yes—
158C2×C4×C107#4112yescomplemented
159C22⋊D107—28no—
160C22⋊D107—28no—
161C2×Dic107—21no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—21no—
166C2×Dic107—21no—
167C2×Dic107—21no—
168C2×Dic107—21no—
169C2×Dic107—21no—
170C2×Dic107—21no—
171C2×Dic107—21no—
172C2×Dic107—21no—
173C2×Dic107—21no—
174C2×Dic107C2×D412yes—
175C2×Dic107C2×D412yes—
176C2×Dic107C2×D412yes—
177C2×Dic107C2×D412yes—
178C2×Dic107#4112yescomplemented
179D428—28no—
180C4×D107—28no—
181C4×D107—28no—
182Dic214—28no—
183Dic214—44no—
184C22×C107—22no—
185C22×C107—22no—
186C22×C107—22no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107—41no—
191C22×C107C22×D411yescharacteristic
192C22×C107C22×D411yescharacteristic
193C22×C107C22×D411yescharacteristic
194C22×C107C2×#4111yescharacteristic
195C22×C107C2×#4111yescharacteristic
196C22×C1072−1+411yescharacteristic
197C22×C1072−1+411yescharacteristic
198C2×D107—22no—
199C2×D107—22no—
200C2×D107—22no—
201C2×D107—22no—
202C2×D107—22no—
203C2×D107—22no—
204C2×D107—41no—
205C2×D107—41no—
206C2×D107—41no—
207C2×D107—41no—
208C2×D107—41no—
209C2×D107—41no—
210C2×D107—41no—
211C2×D107—41no—
212C4×C107—22no—
213C4×C107—22no—
214C4×C107—22no—
215C4×C107—41no—
216C4×C107—41no—
217C4×C107—41no—
218Dic107—22no—
219Dic107—22no—
220Dic107—41no—
221Dic107—41no—
222Dic107—41no—
223Dic107#12014yes—
224C2×C107—42no—
225C2×C107C2×#12011yescharacteristic
226C2×C107#53211yescharacteristic
227C2×C107#53611yescharacteristic
228C2×C107#54711yescharacteristic
229C2×C107#54711yescharacteristic
230C2×C107#54811yescharacteristic
231C2×C107#54811yescharacteristic
232D107—42no—
233D107—42no—
234#2432—1071nomaximal, Sylow
235C107#243211yescharacteristic, complemented, Sylow
236C22×#41—1071no—
237C2×C4⋊Q8—1071no—
238C2×#124—1071no—
239C2×C22⋊Q8—1071no—
240#394—1071no—
241#394—1071no—
242#394—1071no—
243#394—1071no—
244#391—1071no—
245#390—1071no—
246#385—1071no—
247#385—1071no—
248#385—1071no—
249#385D10711yescharacteristic, complemented
250C4×C22⋊C4—1071no—
251C2×#41—2144no—
252C2×#41—2144no—
253C2×#41—2144no—
254C22×Q8—1071no—
255C22×Q8—1071no—
256C22×D4—1071no—
257C22×D4—1071no—
258C22×D4—1071no—
259C23×C4—1071no—
260C23×C4—1071no—
261C23×C4C2×D10711yescharacteristic
262C4⋊Q8—2144no—
263#124—2144no—
264C22⋊Q8—2144no—
265C2×C4⋊C4—1071no—
266C2×C4⋊C4—1071no—
267C2×C4⋊C4—1071no—
268C2×C4⋊C4—1071no—
269C2×C4⋊C4—1071no—
270C2×C4⋊C4C2×D10711yescharacteristic
271C2×C4⋊C4C2×D10711yescharacteristic
272C2×C22⋊C4—1071no—
273C2×C22⋊C4—1071no—
274C2×C22⋊C4—1071no—
275C2×C22⋊C4—1071no—
276C2×C22⋊C4—1071no—
277C2×C22⋊C4—1071no—
278C2×C22⋊C4—1071no—
279C2×C22⋊C4—1071no—
280C2×C22⋊C4—1071no—
281C2×C22⋊C4—1071no—
282C2×C22⋊C4C2×D10711yescharacteristic
283C2×C22⋊C4C2×D10711yescharacteristic
284C2×C42—1071no—
285C2×C42—1071no—
286#95—1071no—
287#95—1071no—
288#95—1071no—
289#95—1071no—
290#95C2×D10711yescharacteristic
291#95C2×D10711yescharacteristic
292C24—1071no—
293C24—1071no—
294C24C22×D10711yescharacteristic
295#41—21432no—
296C2×Q8—1074no—
297C2×Q8—2142no—
298C2×Q8—2142no—
299C2×Q8—2142no—
300C2×Q8—2142no—
301C2×Q8—2142no—
302C2×D4—1074no—
303C2×D4—2142no—
304C2×D4—2142no—
305C2×D4—2142no—
306C2×D4—2142no—
307C2×D4—2142no—
308C2×D4—2142no—
309C2×D4—2142no—
310C2×D4—2142no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—1071no—
315C22×C4—1071no—
316C22×C4—1071no—
317C22×C4—1074no—
318C22×C4—1071no—
319C22×C4—1071no—
320C22×C4—1071no—
321C22×C4—2142no—
322C22×C4—2142no—
323C22×C4—2142no—
324C22×C4—2142no—
325C22×C4—2142no—
326C22×C4C22×D10711yescharacteristic
327C22×C4C22×D10711yescharacteristic
328C22×C4C22×D10711yescharacteristic
329C22×C4C22×D10711yescharacteristic
330C22×C4C22×D10711yescharacteristic
331C22×C4C22×D10711yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—2142no—
335C4⋊C4—2142no—
336C4⋊C4—2142no—
337C4⋊C4—2142no—
338C4⋊C4—2142no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—2142no—
342C22⋊C4—2142no—
343C22⋊C4—2142no—
344C22⋊C4—2142no—
345C22⋊C4—2142no—
346C22⋊C4—2142no—
347C22⋊C4—2142no—
348C22⋊C4—2142no—
349C22⋊C4—2142no—
350C22⋊C4—2142no—
351C42—2142no—
352C42—2142no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—1072no—
360C23—1072no—
361C23—2141no—
362C23—2141no—
363C23—2141no—
364C23—2141no—
365C23—2141no—
366C23—2141no—
367C23—2141no—
368C23—2141no—
369C23—2141no—
370C23—2141no—
371C23—2141no—
372C23—2141no—
373C23D4×D10712yes—
374C23C23×D10711yescharacteristic
375Q8—4284no—
376Q8—2148no—
377D4—2148no—
378D4—2148no—
379D4—2148no—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—1072no—
397C2×C4—1072no—
398C2×C4—1072no—
399C2×C4—1072no—
400C2×C4—1072no—
401C2×C4—2148no—
402C2×C4—2148no—
403C2×C4—2141no—
404C2×C4—2141no—
405C2×C4—2141no—
406C2×C4—2141no—
407C2×C4—2141no—
408C2×C4—2141no—
409C2×C4—2141no—
410C2×C4—2141no—
411C2×C4—2141no—
412C2×C4—2141no—
413C2×C4—2141no—
414C2×C4—2141no—
415C2×C4—2141no—
416C2×C4D4×D10712yes—
417C2×C4D4×D10712yes—
418C2×C4#1313381812yescomplemented
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—4281no—
427C22—4281no—
428C22—4281no—
429C22—4281no—
430C22—4281no—
431C22—4281no—
432C22—4281no—
433C22—4281no—
434C22—2142no—
435C22—2142no—
436C22—2142no—
437C22—2142no—
438C22—2142no—
439C22—2142no—
440C22C2×D4×D10711yescharacteristic
441C22C2×D4×D10711yescharacteristic
442C22C2×D4×D10711yescharacteristic
443C22#41×D10711yescharacteristic
444C22C2×#1313381811yescharacteristic
445C22#1681215611yescharacteristic
446C22#1681215711yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—1074no—
454C4—4281no—
455C4—4281no—
456C4—4281no—
457C4—2142no—
458C4—2142no—
459C2—42no—
460C2—4282no—
461C2—4282no—
462C2#120×D10711yescharacteristic
463C2#2029317411yescharacteristic
464C2#2029320011yescharacteristic
465C2#2029327511yescharacteristic
466C2#2029327811yescharacteristic
467C2#2029328611yescharacteristic
468C2#2029328711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC22×#13133818C2
hypercenter#385D107
hypocenterC107#2432
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#16812108C2semidirect
3. C2×#16812095C2semidirect
4. C2×#16812087C2semidirect
5. #20292545C2semidirect
6. #20292544C2semidirect
7. #20292543C2semidirect
8. #20292542C2semidirect
9. #20292531C2semidirect
10. #20292527C2semidirect
11. #20292501C2semidirect
12. #20292500C2semidirect
13. #20292498C2semidirect
14. #20292474C2semidirect
15. #385×C107C2semidirect
158. C2×C4×C107#41semidirect
178. C2×Dic107#41semidirect
235. C107#2432semidirect
249. #385D107semidirect
418. C2×C4#13133818semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121082
C2×#168120952
C2×#168120872
#202925452
#202925442
#202925432
#202925422
#202925312
#202925272
#202925012
#202925002
#202924982
#202924742
#385×C1072
#2432107