Finite Group Databasebeta

← #28340122

Subgroups — #28340122

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×#16812088C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic107C211yesmaximal, characteristic
4#20292563C211yesmaximal, characteristic, complemented
5#20292550C211yesmaximal, characteristic, complemented
6#20292550C211yesmaximal, characteristic, complemented
7#20292539C211yesmaximal, characteristic, complemented
8#20292502C211yesmaximal, characteristic, complemented
9#20292500C211yesmaximal, characteristic, complemented
10#20292500C211yesmaximal, characteristic, complemented
11#20292499C211yesmaximal, characteristic, complemented
12#20292487C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic, complemented
14#20292483C211yesmaximal, characteristic
15C22×C22⋊C4×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic, complemented
17C23×Dic107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19#16812088—24no—
20#16812086—24no—
21C2×C4⋊Dic107C2211yescharacteristic, complemented
22C2×C4⋊Dic107C2211yescharacteristic, complemented
23C2×C4⋊Dic107C2211yescharacteristic, complemented
24C2×Dic107⋊C4C2211yescharacteristic, complemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×Dic107⋊C4C2211yescharacteristic, complemented
27C2×Dic107⋊C4C2211yescharacteristic, complemented
28C2×Dic107⋊C4C2211yescharacteristic, complemented
29C2×Dic107⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic107—24no—
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C22⋊Dic107C2211yescharacteristic
38C2×C22⋊Dic107C2211yescharacteristic
39C2×C22⋊Dic107C2211yescharacteristic
40C2×C22⋊Dic107C2211yescharacteristic
41C107⋊#95C2211yescharacteristic, complemented
42C107⋊#95C2211yescharacteristic, complemented
43C107⋊#95C2211yescharacteristic, complemented
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C25×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49C23×C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107—24no—
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—22no—
57C2×Dic214—41no—
58C22×Dic107—22no—
59C22×Dic107—22no—
60C22×Dic107—22no—
61C22×Dic107—22no—
62C22×Dic107—41no—
63C22×Dic107—41no—
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C4⋊Dic107—22no—
73C4⋊Dic107—22no—
74C4⋊Dic107—41no—
75Dic107⋊C4—22no—
76Dic107⋊C4—22no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—41no—
80Dic107⋊C4—41no—
81C22⋊Dic107—22no—
82C22⋊Dic107—22no—
83C22⋊Dic107—22no—
84C22⋊Dic107—22no—
85C22⋊Dic107—22no—
86C22⋊Dic107—22no—
87C22⋊Dic107—22no—
88C22⋊Dic107—41no—
89C22⋊Dic107—41no—
90C22⋊Dic107D414yes—
91C24×C107—22no—
92C24×C107—22no—
93C24×C107—22no—
94C24×C107—22no—
95C24×C107—41no—
96C24×C107—41no—
97C24×C107Q814yescomplemented
98C24×C107C2311yescharacteristic
99C24×C107C2311yescharacteristic
100C24×C107C2311yescharacteristic
101C22×C4×C107—22no—
102C22×C4×C107—22no—
103C22×C4×C107—22no—
104C22×C4×C107—22no—
105C22×C4×C107—41no—
106C22×C4×C107—41no—
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22⋊C4×C107—22no—
112C22⋊C4×C107—22no—
113C22⋊C4×C107—22no—
114C22⋊C4×C107D414yes—
115C23×C107—21no—
116C23×C107—21no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C23×C107—28no—
124C23×C107—28no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—21no—
128C23×C107—21no—
129C23×C107—21no—
130C23×C107—21no—
131C23×C107—44no—
132C23×C107—44no—
133C23×C107—44no—
134C23×C107—44no—
135C23×C107—44no—
136C23×C107C2×D412yes—
137C23×C107C2×D412yes—
138C23×C107C2×Q812yes—
139C23×C107C2×Q812yes—
140C23×C107C2×Q812yes—
141C23×C107#4112yes—
142C23×C107#4112yes—
143C23×C107C2411yescharacteristic
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—44no—
155C2×C4×C107—44no—
156C2×C4×C107C2×D412yes—
157C2×C4×C107C2×D412yes—
158C2×Dic107—21no—
159C2×Dic107—21no—
160C2×Dic107—21no—
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×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—21no—
180C2×Dic107—44no—
181C2×Dic107—44no—
182C2×Dic107C2×D412yes—
183C2×Dic107C2×D412yes—
184Dic214—44no—
185C22×C107—22no—
186C22×C107—22no—
187C22×C107—22no—
188C22×C107—22no—
189C22×C107—22no—
190C22×C107—22no—
191C22×C107—22no—
192C22×C107—22no—
193C22×C107—22no—
194C22×C107—22no—
195C22×C107—22no—
196C22×C107—22no—
197C22×C107—22no—
198C22×C107—22no—
199C22×C107—22no—
200C22×C107—416no—
201C22×C107—41no—
202C22×C107—41no—
203C22×C107—41no—
204C22×C107—41no—
205C22×C107C22⋊Q814yescomplemented
206C22×C107C22⋊Q814yescomplemented
207C22×C107C22×D411yescharacteristic
208C22×C107C22×D411yescharacteristic
209C22×C107C22×Q811yescharacteristic
210C22×C107C2×#4111yescharacteristic
211C22×C107C2×#4111yescharacteristic
212C22×C107H(5,𝔽2)11yescharacteristic
213C22×C107H(5,𝔽2)11yescharacteristic
214C4×C107—22no—
215C4×C107—22no—
216C4×C107—41no—
217C4×C107—41no—
218Dic107—22no—
219Dic107—22no—
220Dic107—41no—
221Dic107—41no—
222Dic107—41no—
223Dic107—41no—
224Dic107—41no—
225Dic107—41no—
226C2×C107—24no—
227C2×C107—24no—
228C2×C107—42no—
229C2×C107C2×C22⋊Q811yescharacteristic
230C2×C107C2×C22⋊Q811yescharacteristic
231C2×C107#54311yescharacteristic
232C2×C107D4211yescharacteristic
233C2×C107#54611yescharacteristic
234C2×C107#54611yescharacteristic
235C2×C107#55111yescharacteristic
236#2437—1071nomaximal, Sylow
237C107#243711yescharacteristic, complemented, Sylow
238C2×C22⋊Q8—1071no—
239C2×C22⋊Q8—1071no—
240C22×C22⋊C4—1071no—
241C22×C22⋊C4D10711yescharacteristic
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—
253C25C2×D10711yescharacteristic
254C22×Q8—1071no—
255C23×C4—1071no—
256C23×C4—1071no—
257C23×C4C2×D10711yescharacteristic
258C23×C4C2×D10711yescharacteristic
259C22⋊Q8—2144no—
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⋊C4—1071no—
268C2×C4⋊C4—1071no—
269C2×C4⋊C4—1071no—
270C2×C22⋊C4—24no—
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—2144no—
282C2×C22⋊C4C2×D10711yescharacteristic
283C2×C22⋊C4C2×D10711yescharacteristic
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—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24Dic21414yescomplemented
299C24C22×D10711yescharacteristic
300C24C22×D10711yescharacteristic
301C24C22×D10711yescharacteristic
302C2×Q8—4281no—
303C2×Q8—2142no—
304C2×Q8—2142no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—1071no—
312C22×C4—1071no—
313C22×C4—1071no—
314C22×C4—1071no—
315C22×C4—1071no—
316C22×C4—1071no—
317C22×C4—1071no—
318C22×C4—1071no—
319C22×C4—4281no—
320C22×C4—4281no—
321C22×C4—2142no—
322C22×C4—2142no—
323C22×C4—2142no—
324C22×C4—2142no—
325C22×C4C22×D10711yescharacteristic
326C22×C4C22×D10711yescharacteristic
327C22×C4C22×D10711yescharacteristic
328C22×C4C22×D10711yescharacteristic
329C4⋊C4—4281no—
330C4⋊C4—4281no—
331C4⋊C4—4281no—
332C4⋊C4—2142no—
333C4⋊C4—2142no—
334C4⋊C4—2142no—
335C4⋊C4—2142no—
336C4⋊C4—2142no—
337C4⋊C4—2142no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—1074no—
342C22⋊C4—4281no—
343C22⋊C4—4281no—
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—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—28no—
361C23—28no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23D4×D10712yes—
374C23C2×C22⋊D10712yes—
375C23C2×Dic21412yes—
376C23C2×Dic21412yes—
377C23C2×Dic21412yes—
378C23#1313381712yes—
379C23#1313381812yes—
380C23C23×D10711yescharacteristic
381Q8—4284no—
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—44no—
393C2×C4—44no—
394C2×C4—1072no—
395C2×C4—1072no—
396C2×C4—4284no—
397C2×C4—4284no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4—2141no—
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×C4—2141no—
420C2×C4C2×C22⋊D10712yes—
421C2×C4C2×C22⋊D10712yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—416no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#1681208614yescomplemented
443C22#1681208814yescomplemented
444C22C2×D4×D10711yescharacteristic
445C22C22×C22⋊D10711yescharacteristic
446C22C22×Dic21411yescharacteristic
447C22C2×#1313381711yescharacteristic
448C22C2×#1313381811yescharacteristic
449C22#1681215411yescharacteristic
450C22#1681215411yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—4281no—
456C4—4281no—
457C4—4281no—
458C4—4281no—
459C4—4281no—
460C4—4281no—
461C4—2142no—
462C4—2142no—
463C2—24no—
464C2—24no—
465C2—42no—
466C2C2×#1681208611yescharacteristic
467C2C2×#1681208811yescharacteristic
468C2#2029325311yescharacteristic
469C2D4×C22⋊D10711yescharacteristic
470C2#2029326711yescharacteristic
471C2#2029327111yescharacteristic
472C2#2029330311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C22⋊C4×C107C2
prime radicalC25×C107C22
hypercenterC22×C22⋊C4D107
hypocenterC107#2437
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16812088C2semidirect
2. C2×#16812086C2semidirect
4. #20292563C2semidirect
5. #20292550C2semidirect
6. #20292550C2semidirect
7. #20292539C2semidirect
8. #20292502C2semidirect
9. #20292500C2semidirect
10. #20292500C2semidirect
11. #20292499C2semidirect
12. #20292487C2semidirect
13. #20292485C2semidirect
16. C22×Dic214C22semidirect
21. C2×C4⋊Dic107C22semidirect
22. C2×C4⋊Dic107C22semidirect
23. C2×C4⋊Dic107C22semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
26. C2×Dic107⋊C4C22semidirect
27. C2×Dic107⋊C4C22semidirect
28. C2×Dic107⋊C4C22semidirect
29. C2×Dic107⋊C4C22semidirect
41. C107⋊#95C22semidirect
42. C107⋊#95C22semidirect
43. C107⋊#95C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
97. C24×C107Q8semidirect
205. C22×C107C22⋊Q8semidirect
206. C22×C107C22⋊Q8semidirect
237. C107#2437semidirect
298. C24Dic214semidirect
442. C22#16812086semidirect
443. C22#16812088semidirect

Maximal subgroups

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

subgroupindex
C2×#168120882
C2×#168120862
C22×C22⋊Dic1072
#202925632
#202925502
#202925392
#202925022
#202925002
#202924992
#202924872
#202924852
#202924832
C22×C22⋊C4×C1072
#2437107