Finite Group Databasebeta

← #28348232

Subgroups — #28348232

Order 13696.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293442C211yesmaximal, characteristic, complemented
2#20293435C211yesmaximal, characteristic, complemented
3#20293434C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292900C211yesmaximal, characteristic, complemented
6#20292899C211yesmaximal, characteristic, complemented
7#20292892C211yesmaximal, characteristic, complemented
8#20292890C211yesmaximal, characteristic, complemented
9#20292810C211yesmaximal, characteristic, complemented
10#20292808C211yesmaximal, characteristic, complemented
11#20292803C211yesmaximal, characteristic, complemented
12#20292800C211yesmaximal, characteristic, complemented
13#20292749C211yesmaximal, characteristic, complemented
14#20292747C211yesmaximal, characteristic, complemented
15#578×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic
17#16812155C2211yescharacteristic
18#16812141—21no—
19#16812140—21no—
20#16812140—21no—
21#16812140C2211yescharacteristic
22#16812140C2211yescharacteristic
23#16812139—21no—
24#16812139C2211yescharacteristic
25#16812139C2211yescharacteristic, complemented
26#16812134—21no—
27#16812134C2211yescharacteristic
28#16812134C2211yescharacteristic
29#16812133—21no—
30#16812130—21no—
31#16812129—21no—
32#16812129—21no—
33#16812129C2211yescharacteristic
34C2×#13133808—21no—
35C2×#13133808C2211yescharacteristic, complemented
36C2×#13133807C2211yescharacteristic
37C2×#13133805—21no—
38C2×#13133805C2211yescharacteristic
39SD16×D107—21no—
40#16812116C2211yescharacteristic
41#16812116C2211yescharacteristic, complemented
42#16812115C2211yescharacteristic
43#16812107C2211yescharacteristic, complemented
44#16812098C2211yescharacteristic
45#16812045C2211yescharacteristic, complemented
46#16812044C2211yescharacteristic, complemented
47#16812036C2211yescharacteristic
48#16812036C2211yescharacteristic, complemented
49#16812036C2211yescharacteristic, complemented
50#16812035C2211yescharacteristic
51#16812028C2211yescharacteristic, complemented
52#16812028C2211yescharacteristic, complemented
53#16812027C2211yescharacteristic, complemented
54#16812025C2211yescharacteristic
55#16812019C2211yescharacteristic, complemented
562−1+4×C107C2211yescharacteristic
572−1+4×C107C2211yescharacteristic, complemented
58#137×C107—21no—
59#137×C107—21no—
60#137×C107C2211yescharacteristic
61#137×C107C2211yescharacteristic, complemented
62#135×C107—21no—
63#135×C107C2211yescharacteristic
64C2×Q16×C107—21no—
65C2×Q16×C107C2211yescharacteristic, complemented
66#131×C107C2211yescharacteristic, complemented
67#13133819—21no—
68#13133818—21no—
69#13133818—21no—
70#13133818C2311yescharacteristic
71#13133817—21no—
72#13133817—41no—
73C2×Dic214—21no—
74C2×Dic214C2311yescharacteristic
75Q8×D107—21no—
76C2×D428—21no—
77C2×D428C2311yescharacteristic
78D4×D107—21no—
79D4×D107—41no—
80#13133808—21no—
81#13133808—21no—
82#13133808—21no—
83#13133808—21no—
84#13133808—21no—
85#13133808—21no—
86#13133808D411yescharacteristic
87#13133807—21no—
88#13133807—21no—
89#13133806—21no—
90#13133806—21no—
91#13133806D411yescharacteristic
92#13133805—21no—
93#13133805—21no—
94#13133805—21no—
95#13133805—21no—
96#13133805—21no—
97#13133805—21no—
98#13133805D411yescharacteristic
99#13133803—21no—
100#13133803—21no—
101#13133803D411yescharacteristic
102#13133802—21no—
103#13133802—21no—
104#13133802C2311yescharacteristic
105#13133802C2311yescharacteristic
106#13133802C2311yescharacteristic
107#13133801—21no—
108C2×C107⋊C8—21no—
109C2×C107⋊C8—21no—
110C2×C107⋊C8C2311yescharacteristic
111C8×D107—21no—
112Dic107⋊C4—41no—
113(C2×D107)⋊C4—41no—
114C4×Dic107C2311yescharacteristic
115#41×C107—21no—
116#41×C107—21no—
117#41×C107—21no—
118#41×C107—21no—
119#41×C107—41no—
120#41×C107—41no—
121#41×C107C2311yescharacteristic
122#41×C107C2311yescharacteristic
123#41×C107C2311yescharacteristic
124C2×Q8×C107—21no—
125C2×Q8×C107—21no—
126C2×Q8×C107—21no—
127C2×Q8×C107—21no—
128C2×Q8×C107C2311yescharacteristic
129C2×Q8×C107C2311yescharacteristic
130Q16×C107—21no—
131Q16×C107—21no—
132Q16×C107—21no—
133Q16×C107—21no—
134Q16×C107D411yescharacteristic
135SD16×C107—21no—
136SD16×C107—21no—
137SD16×C107D411yescharacteristic
138SD16×C107D411yescharacteristic
139D8×C107D411yescharacteristic
140M16×C107—21no—
141M16×C107C2311yescharacteristic
142C2×C8×C107—21no—
143C2×C8×C107C2311yescharacteristic
144C22×D107—21no—
145C22×D107—41no—
146Q8×C107—21no—
147Q8×C107—21no—
148Q8×C107—21no—
149Q8×C107—21no—
150Q8×C107—41no—
151Q8×C107—41no—
152Q8×C107C2×D411yescharacteristic
153Q8×C107C2×D411yescharacteristic
154Q8×C107C2×D411yescharacteristic
155D4×C107—21no—
156D4×C107—21no—
157D4×C107—21no—
158D4×C107—41no—
159D4×C107—41no—
160D4×C107C2×D411yescharacteristic
161D4×C107C2×D411yescharacteristic
162D4×C107C2×D411yescharacteristic
163C2×C4×C107—21no—
164C2×C4×C107—21no—
165C2×C4×C107—21no—
166C2×C4×C107—21no—
167C2×C4×C107—21no—
168C2×C4×C107—41no—
169C2×C4×C107—41no—
170C2×C4×C107—41no—
171C2×C4×C107—41no—
172C2×C4×C107C2411yescharacteristic
173C22⋊D107—21no—
174C22⋊D107—41no—
175C22⋊D107—41no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—41no—
179D428—21no—
180D428—21no—
181D428—41no—
182D428C2×D411yescharacteristic
183C4×D107—21no—
184C4×D107—41no—
185C4×D107—41no—
186Dic214—21no—
187Dic214—21no—
188Dic214—41no—
189Dic214C2×D411yescharacteristic
190C8×C107—21no—
191C8×C107C2×D411yescharacteristic
192C8×C107C2×D411yescharacteristic
193C107⋊C8—21no—
194C107⋊C8—21no—
195C107⋊C8—21no—
196C107⋊C8C2×D411yescharacteristic
197C107⋊C8C2×D411yescharacteristic
198C22×C107—21no—
199C22×C107—21no—
200C22×C107—21no—
201C22×C107H(5,𝔽2)11yescharacteristic
202C2×D107—21no—
203C2×D107—41no—
204C2×D107—81no—
205C2×D107—161no—
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—21no—
209C4×C107—41no—
210C4×C107—41no—
211C4×C107C22×D411yescharacteristic
212C4×C107C22×D411yescharacteristic
213Dic107—21no—
214Dic107—41no—
215Dic107—41no—
216C2×C107—21no—
217C2×C107—41no—
218C2×C107—41no—
219C2×C107—41no—
220C2×C107D4211yescharacteristic
221D107—41no—
222D107—81no—
223#3289—1071nomaximal, Sylow
224C107#328911yescharacteristic, complemented, Sylow
225#578—1071no—
226#578D10711yescharacteristic, complemented
227#577—1071no—
228#577—1071no—
229#497—1071no—
230#473—1071no—
231#473—1071no—
232#471—1071no—
233#471—1071no—
234#456—1071no—
235#456—1071no—
236#455—1071no—
237#454—1071no—
238#444—1071no—
239#444—1071no—
2402−1+4—1071no—
2412−1+4C2×D10711yescharacteristic
2422−1+4C2×D10711yescharacteristic, complemented
243H(5,𝔽2)—1071no—
244#137—21no—
245#137—21no—
246#137—1071no—
247#137—1071no—
248#137—1071no—
249#137—1071no—
250#137—2141no—
251#137—2141no—
252#137—2141no—
253#137—2141no—
254#137C2×D10711yescharacteristic
255#137C2×D10711yescharacteristic, complemented
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—2141no—
259AGL(1,ℤ/8)—2141no—
260#135—21no—
261#135—1071no—
262#135—2141no—
263#135—2141no—
264#135—2141no—
265#135C2×D10711yescharacteristic
266C2×Q16—21no—
267C2×Q16—1071no—
268C2×Q16—2141no—
269C2×Q16C2×D10711yescharacteristic, complemented
270C2×SD16—1071no—
271C2×SD16—1071no—
272C2×SD16—2141no—
273C2×SD16—2141no—
274#131—1071no—
275#131—1071no—
276#131—1071no—
277#131C2×D10711yescharacteristic, complemented
278C4⋊Q8—1071no—
279#124—1071no—
280#108—1071no—
281#108—1071no—
282#104—1071no—
283#104—1071no—
284#104—1071no—
285#104—1071no—
286#101—1071no—
287#101—1071no—
288#101—1071no—
289#100—1071no—
290#97—1071no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—1071no—
298#41—4281no—
299#41—2141no—
300#41—2141no—
301#41—2141no—
302#41—2141no—
303#41C22×D10711yescharacteristic
304#41C22×D10711yescharacteristic
305#41C22×D10711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—1071no—
311C2×Q8—2141no—
312C2×Q8—2141no—
313C2×Q8C22×D10711yescharacteristic
314C2×Q8C22×D10711yescharacteristic
315C2×D4—1071no—
316C2×D4—4281no—
317C2×D4—2141no—
318C2×D4—2141no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—1071no—
324Q16—2141no—
325Q16—2141no—
326Q16—2141no—
327Q16—2141no—
328Q16—2141no—
329Q16—2141no—
330Q16C22⋊D10711yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—1071no—
334SD16—1071no—
335SD16—2141no—
336SD16—2141no—
337SD16—2141no—
338SD16—2141no—
339SD16—2141no—
340SD16—2141no—
341SD16—2141no—
342SD16—2141no—
343SD16—2141no—
344SD16—2141no—
345SD16C22⋊D10711yescharacteristic
346SD16C22⋊D10711yescharacteristic
347D8—1071no—
348D8—2141no—
349D8—2141no—
350D8C22⋊D10711yescharacteristic
351M16—21no—
352M16—1071no—
353M16—1071no—
354M16—1071no—
355M16—2141no—
356M16—2141no—
357M16—2141no—
358M16C22×D10711yescharacteristic
359C2×C8—21no—
360C2×C8—1071no—
361C2×C8—2141no—
362C2×C8—2141no—
363C2×C8—2141no—
364C2×C8C22×D10711yescharacteristic
365C4⋊C4—4281no—
366C22⋊C4—4281no—
367C42—1071no—
368C23—4281no—
369C23—2141no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—1071no—
377Q8—4281no—
378Q8—2141no—
379Q8—2141no—
380Q8D4×D10711yescharacteristic
381Q8C2×C22⋊D10711yescharacteristic
382Q8C2×C22⋊D10711yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—1071no—
389D4—4281no—
390D4—4281no—
391D4—4281no—
392D4—2141no—
393D4—2141no—
394D4—2141no—
395D4D4×D10711yescharacteristic
396D4C2×C22⋊D10711yescharacteristic
397D4C2×C22⋊D10711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4281no—
408C2×C4—4281no—
409C2×C4—4281no—
410C2×C4—2141no—
411C2×C4—2141no—
412C2×C4—2141no—
413C2×C4C23×D10711yescharacteristic
414C8—21no—
415C8—1071no—
416C8—1071no—
417C8—2141no—
418C8—2141no—
419C8—2141no—
420C8C2×C22⋊D10711yescharacteristic
421C8C2×C22⋊D10711yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—8561no—
426C22—4281no—
427C22—17121no—
428C22—2141no—
429C22#1681215411yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4281no—
436C4—4281no—
437C4—2141no—
438C4C2×D4×D10711yescharacteristic
439C4C22×C22⋊D10711yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—8561no—
445C2—4281no—
446C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#578×C107C2
socleC2×C107D42
hypercenter#578D107
hypocenterC107#3289
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20293442C2semidirect
2. #20293435C2semidirect
3. #20293434C2semidirect
4. #20293020C2semidirect
5. #20292900C2semidirect
6. #20292899C2semidirect
7. #20292892C2semidirect
8. #20292890C2semidirect
9. #20292810C2semidirect
10. #20292808C2semidirect
11. #20292803C2semidirect
12. #20292800C2semidirect
13. #20292749C2semidirect
14. #20292747C2semidirect
15. #578×C107C2semidirect
25. #16812139C22semidirect
35. C2×#13133808C22semidirect
41. #16812116C22semidirect
43. #16812107C22semidirect
45. #16812045C22semidirect
46. #16812044C22semidirect
48. #16812036C22semidirect
49. #16812036C22semidirect
51. #16812028C22semidirect
52. #16812028C22semidirect
53. #16812027C22semidirect
55. #16812019C22semidirect
57. 2−1+4×C107C22semidirect
61. #137×C107C22semidirect
65. C2×Q16×C107C22semidirect
66. #131×C107C22semidirect
224. C107#3289semidirect
226. #578D107semidirect
242. 2−1+4C2×D107semidirect
255. #137C2×D107semidirect
269. C2×Q16C2×D107semidirect
277. #131C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934422
#202934352
#202934342
#202930202
#202929002
#202928992
#202928922
#202928902
#202928102
#202928082
#202928032
#202928002
#202927492
#202927472
#578×C1072
#3289107