Finite Group Databasebeta

← #28340124

Subgroups — #28340124

Order 13696.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812085C211yesmaximal, characteristic, complemented
2C22×(C2×D107)⋊C4C211yesmaximal, characteristic
3C2×C22⋊C4×D107C211yesmaximal, characteristic
4#20292562C211yesmaximal, characteristic, complemented
5#20292549C211yesmaximal, characteristic, complemented
6#20292549C211yesmaximal, characteristic, complemented
7#20292538C211yesmaximal, characteristic, complemented
8#20292502C211yesmaximal, characteristic, complemented
9#20292500C211yesmaximal, characteristic, complemented
10#20292498C211yesmaximal, characteristic, complemented
11#20292498C211yesmaximal, characteristic, complemented
12#20292485C211yesmaximal, characteristic, complemented
13#20292484C211yesmaximal, characteristic, complemented
14#20292482C211yesmaximal, characteristic
15C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812085—24no—
21C2×C4⋊Dic107C2211yescharacteristic, complemented
22C2×C4⋊Dic107C2211yescharacteristic, complemented
23C2×Dic107⋊C4C2211yescharacteristic, complemented
24C2×Dic107⋊C4C2211yescharacteristic, complemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×Dic107⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×(C2×D107)⋊C4—24no—
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
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C22⋊C4×D107—24no—
41C107⋊#95C2211yescharacteristic, complemented
42C107⋊#95C2211yescharacteristic, complemented
43C107⋊#95C2211yescharacteristic, complemented
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C22×Q8×C107C2211yescharacteristic, complemented
48C23×C4×C107C2211yescharacteristic
49C22⋊Q8×C107—24no—
50C2×C4⋊C4×C107C2211yescharacteristic, complemented
51C2×C4⋊C4×C107C2211yescharacteristic, complemented
52C2×C4⋊C4×C107C2211yescharacteristic, complemented
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C23×D107—22no—
56C23×D107—22no—
57C23×D107—22no—
58C23×D107—22no—
59C23×D107—41no—
60C23×D107—41no—
61C23×D107Q814yescomplemented
62C23×D107C2311yescharacteristic
63C23×D107C2311yescharacteristic
64C22×Dic107—22no—
65C22×Dic107—22no—
66C22×Dic107—41no—
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C2×C4×D107—22no—
74C2×C4×D107—22no—
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—41no—
78C2×C4×D107—41no—
79C4⋊Dic107—22no—
80C4⋊Dic107—41no—
81Dic107⋊C4—22no—
82Dic107⋊C4—22no—
83Dic107⋊C4—41no—
84Dic107⋊C4—41no—
85C22⋊Dic107—22no—
86C22⋊Dic107—22no—
87(C2×D107)⋊C4—22no—
88(C2×D107)⋊C4—22no—
89(C2×D107)⋊C4—22no—
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)⋊C4D414yes—
97C24×C107C2311yescharacteristic
98C2×Q8×C107—22no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C22×C4×C107—22no—
102C22×C4×C107—22no—
103C22×C4×C107—41no—
104C22×C4×C107C2311yescharacteristic
105C22×C4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C4⋊C4×C107—22no—
111C4⋊C4×C107—22no—
112C4⋊C4×C107—22no—
113C22⋊C4×C107—22no—
114C22⋊C4×C107D414yes—
115C23×C107—21no—
116C23×C107—21no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107C2×D412yes—
120C23×C107C2×Q812yes—
121C23×C107#4112yes—
122C23×C107C2411yescharacteristic
123C22×D107—21no—
124C22×D107—21no—
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—28no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—28no—
135C22×D107—44no—
136C22×D107—44no—
137C22×D107—44no—
138C22×D107—44no—
139C22×D107—44no—
140C22×D107C2×D412yes—
141C22×D107C2×Q812yes—
142C22×D107C2×Q812yes—
143C22×D107#4112yes—
144Q8×C107—44no—
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×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—21no—
159C2×C4×C107—21no—
160C2×C4×C107—44no—
161C2×C4×C107C2×D412yes—
162C2×C4×C107C2×D412yes—
163C2×C4×C107C2×D412yes—
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×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—21no—
180C2×Dic107—21no—
181C2×Dic107—44no—
182C2×Dic107C2×D412yes—
183C4×D107—44no—
184C4×D107—44no—
185C22×C107—22no—
186C22×C107—22no—
187C22×C107—22no—
188C22×C107—22no—
189C22×C107—22no—
190C22×C107—22no—
191C22×C107C22⋊Q814yescomplemented
192C22×C107C22×D411yescharacteristic
193C22×C107C22×D411yescharacteristic
194C22×C107C22×Q811yescharacteristic
195C22×C107C2×#4111yescharacteristic
196C22×C107C2×#4111yescharacteristic
197C22×C107H(5,𝔽2)11yescharacteristic
198C22×C107H(5,𝔽2)11yescharacteristic
199C2×D107—22no—
200C2×D107—22no—
201C2×D107—22no—
202C2×D107—22no—
203C2×D107—22no—
204C2×D107—22no—
205C2×D107—22no—
206C2×D107—22no—
207C2×D107—22no—
208C2×D107—416no—
209C2×D107—41no—
210C2×D107—41no—
211C2×D107—41no—
212C2×D107—41no—
213C2×D107C22⋊Q814yescomplemented
214C4×C107—22no—
215C4×C107—22no—
216C4×C107—22no—
217C4×C107—41no—
218C4×C107—41no—
219C4×C107—41no—
220Dic107—22no—
221Dic107—41no—
222Dic107—41no—
223Dic107—41no—
224Dic107—41no—
225Dic107—41no—
226C2×C107—24no—
227C2×C107C2×C22⋊Q811yescharacteristic
228C2×C107C2×C22⋊Q811yescharacteristic
229C2×C107#54311yescharacteristic
230C2×C107D4211yescharacteristic
231C2×C107#54611yescharacteristic
232C2×C107#54611yescharacteristic
233C2×C107#55111yescharacteristic
234D107—24no—
235D107—42no—
236#2437—1071nomaximal, Sylow
237C107#243711yescharacteristic, complemented, Sylow
238C2×C22⋊Q8—1071no—
239C2×C22⋊Q8D10711yescharacteristic, complemented
240C22×C22⋊C4—1071no—
241C22×C22⋊C4—1071no—
242#399—1071no—
243#396—1071no—
244#396—1071no—
245#393—1071no—
246#385—1071no—
247#385—1071no—
248#385—1071no—
249#385—1071no—
250#380—1071no—
251#380—1071no—
252#379—1071no—
253C25—1071no—
254C22×Q8C2×D10711yescharacteristic, complemented
255C23×C4—1071no—
256C23×C4—1071no—
257C23×C4—1071no—
258C23×C4C2×D10711yescharacteristic
259C22⋊Q8—24no—
260C22⋊Q8—2144no—
261C2×C4⋊C4—1071no—
262C2×C4⋊C4—1071no—
263C2×C4⋊C4—1071no—
264C2×C4⋊C4—1071no—
265C2×C4⋊C4—1071no—
266C2×C4⋊C4—1071no—
267C2×C4⋊C4C2×D10711yescharacteristic, complemented
268C2×C4⋊C4C2×D10711yescharacteristic, complemented
269C2×C4⋊C4C2×D10711yescharacteristic, complemented
270C2×C22⋊C4—1071no—
271C2×C22⋊C4—1071no—
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⋊C4—2144no—
283C2×C22⋊C4—2144no—
284C2×C22⋊C4C2×D10711yescharacteristic
285C2×C22⋊C4C2×D10711yescharacteristic
286#95—1071no—
287#95—1071no—
288#95—1071no—
289#95—1071no—
290#95—1071no—
291#95—1071no—
292C24—1071no—
293C24—1074no—
294C24—1071no—
295C24—4281no—
296C24—4281no—
297C24—2142no—
298C24—2142no—
299C24—2142no—
300C24—2142no—
301C24C22×D10711yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—41no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—1071no—
309C22×C4—1071no—
310C22×C4—1071no—
311C22×C4—1071no—
312C22×C4—1071no—
313C22×C4—1071no—
314C22×C4—4281no—
315C22×C4—4281no—
316C22×C4—4281no—
317C22×C4—2142no—
318C22×C4—2142no—
319C22×C4—2142no—
320C22×C4—2142no—
321C22×C4—2142no—
322C22×C4—2142no—
323C22×C4C22×D10711yescharacteristic
324C22×C4C22×D10711yescharacteristic
325C22×C4C22×D10711yescharacteristic
326C22×C4C22×D10711yescharacteristic
327C22×C4C22×D10711yescharacteristic
328C22×C4C22×D10711yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—4281no—
333C4⋊C4—4281no—
334C4⋊C4—4281no—
335C4⋊C4—2142no—
336C4⋊C4—2142no—
337C4⋊C4—2142no—
338C22⋊C4—22no—
339C22⋊C4—1074no—
340C22⋊C4—4281no—
341C22⋊C4—4281no—
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—
351C22⋊C4C22⋊D10714yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—1072no—
357C23—1072no—
358C23—1072no—
359C23—1072no—
360C23—4284no—
361C23—4284no—
362C23—4284no—
363C23—4284no—
364C23—4284no—
365C23—2148no—
366C23—2148no—
367C23—2141no—
368C23—2141no—
369C23—2141no—
370C23—2141no—
371C23—2141no—
372C23—2141no—
373C23—2141no—
374C23—2141no—
375C23—2141no—
376C23—2141no—
377C23C2×C22⋊D10712yes—
378C23Q8×D10712yes—
379C23#1313381912yes—
380C23C23×D10711yescharacteristic
381Q8—44no—
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—21no—
397C2×C4—44no—
398C2×C4—1072no—
399C2×C4—4284no—
400C2×C4—4284no—
401C2×C4—4284no—
402C2×C4—2141no—
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×C4—2141no—
417C2×C4—2141no—
418C2×C4—2141no—
419C2×C4D4×D10712yes—
420C2×C4C2×C22⋊D10712yes—
421C2×C4C2×C22⋊D10712yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—1074no—
429C22—42816no—
430C22—4281no—
431C22—4281no—
432C22—4281no—
433C22—4281no—
434C22—2142no—
435C22—2142no—
436C22—2142no—
437C22—2142no—
438C22—2142no—
439C22—2142no—
440C22—2142no—
441C22—2142no—
442C22—2142no—
443C22#1681208514yescomplemented
444C22C2×D4×D10711yescharacteristic
445C22C22×C22⋊D10711yescharacteristic
446C22C2×Q8×D10711yescharacteristic
447C22#41×D10711yescharacteristic
448C22C2×#1313381911yescharacteristic
449C22#1681215411yescharacteristic
450C22#1681215511yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—4281no—
458C4—4281no—
459C4—4281no—
460C4—4281no—
461C4—4281no—
462C4—2142no—
463C2—24no—
464C2—4282no—
465C2—2144no—
466C2C22⋊Q8×D10711yescharacteristic
467C2C2×#1681208511yescharacteristic
468C2#2029325211yescharacteristic
469C2D4×C22⋊D10711yescharacteristic
470C2#2029326511yescharacteristic
471C2#2029326811yescharacteristic
472C2#2029330411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC24×D107C22
hypercenterC2×C22⋊Q8D107
hypocenterC107#2437
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812085C2semidirect
4. #20292562C2semidirect
5. #20292549C2semidirect
6. #20292549C2semidirect
7. #20292538C2semidirect
8. #20292502C2semidirect
9. #20292500C2semidirect
10. #20292498C2semidirect
11. #20292498C2semidirect
12. #20292485C2semidirect
13. #20292484C2semidirect
15. C2×C22⋊Q8×C107C2semidirect
21. C2×C4⋊Dic107C22semidirect
22. C2×C4⋊Dic107C22semidirect
23. C2×Dic107⋊C4C22semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
26. C2×Dic107⋊C4C22semidirect
41. C107⋊#95C22semidirect
42. C107⋊#95C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C22×Q8×C107C22semidirect
50. C2×C4⋊C4×C107C22semidirect
51. C2×C4⋊C4×C107C22semidirect
52. C2×C4⋊C4×C107C22semidirect
61. C23×D107Q8semidirect
191. C22×C107C22⋊Q8semidirect
213. C2×D107C22⋊Q8semidirect
237. C107#2437semidirect
239. C2×C22⋊Q8D107semidirect
254. C22×Q8C2×D107semidirect
267. C2×C4⋊C4C2×D107semidirect
268. C2×C4⋊C4C2×D107semidirect
269. C2×C4⋊C4C2×D107semidirect
443. C22#16812085semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C22×(C2×D107)⋊C42
C2×C22⋊C4×D1072
#202925622
#202925492
#202925382
#202925022
#202925002
#202924982
#202924852
#202924842
#202924822
C2×C22⋊Q8×C1072
#2437107