Finite Group Databasebeta

← #28346256

Subgroups — #28346256

Order 13696.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293183C211yesmaximal, characteristic, complemented
2C4×D4×D107C211yesmaximal, characteristic
3#20292928C211yesmaximal, characteristic, complemented
4#20292927C211yesmaximal, characteristic, complemented
5#20292924C211yesmaximal, characteristic, complemented
6#20292916C211yesmaximal, characteristic, complemented
7#20292915C211yesmaximal, characteristic, complemented
8#20292906C211yesmaximal, characteristic, complemented
9#20292905C211yesmaximal, characteristic, complemented
10#20292668C211yesmaximal, characteristic, complemented
11#20292666C211yesmaximal, characteristic, complemented
12#20292649C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic
14#20292630C211yesmaximal, characteristic
15#478×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C22×C4×D107—21no—
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20#16812108C2211yescharacteristic, complemented
21#16812108C2211yescharacteristic, complemented
22#16812101C2211yescharacteristic, complemented
23#16812087—21no—
24#16812087—21no—
25#16812074—21no—
26#16812074—21no—
27C4×D428C2211yescharacteristic
28D4×Dic107C2211yescharacteristic
29#16812071C2211yescharacteristic
30C4×C22⋊D107—21no—
31(C22⋊D107)⋊C4—21no—
32#16812067C2211yescharacteristic
33#16812067C2211yescharacteristic
34C4⋊C4×D107C2211yescharacteristic
35C22⋊C4×D107—21no—
36C42×D107C2211yescharacteristic
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42#16812040C2211yescharacteristic, complemented
43C4⋊(C107⋊C8)C2211yescharacteristic, complemented
44Dic107⋊C8C2211yescharacteristic, complemented
45Dic107⋊C8C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812029C2211yescharacteristic
51#16812029C2211yescharacteristic
52#125×C107C2211yescharacteristic, complemented
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56C4⋊C8×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64D4×D107—22no—
65D4×D107Q814yescomplemented
66C22×Dic107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—44no—
71C2×C4×D107D411yescharacteristic
72C2×C4×D107D411yescharacteristic
73C2×C4×D107C2311yescharacteristic
74#13133801—22no—
75#13133801—22no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86C22⋊Dic107—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107D411yescharacteristic
94C4×Dic107D411yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107—21no—
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C42×C107C2311yescharacteristic
107C23×C107—21no—
108C22×D107—21no—
109C22×D107—24no—
110C22×D107—42no—
111C22×D107—42no—
112C22×D107C2×D411yescharacteristic
113D4×C107—21no—
114D4×C107C2×Q812yes—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—42no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C22⋊D107—24no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—42no—
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130D428—21no—
131D428C2×Q812yes—
132C4×D107—22no—
133C4×D107—41no—
134C4×D107—42no—
135C4×D107—42no—
136C4×D107—42no—
137C4×D107C2×Q812yes—
138C4×D107#4112yes—
139Dic214—42no—
140Dic214—42no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—22no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107C22×D411yescharacteristic
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—22no—
152C2×D107—48no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107C22⋊Q812yes—
156C4×C107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C22×Q811yescharacteristic
162C4×C107C2×#4111yescharacteristic
163Dic107—21no—
164Dic107—41no—
165Dic107—41no—
166Dic107—41no—
167Dic107C22⋊Q812yes—
168Dic107AGL(1,ℤ/8)12yes—
169C2×C107—42no—
170C2×C107C2×C22⋊Q811yescharacteristic
171C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C107#57511yescharacteristic
173D107—22no—
174D107—42no—
175#3077—1071nomaximal, Sylow
176C107#307711yescharacteristic, complemented, Sylow
177#533—1071no—
178C2×C4×D4—1071no—
179#478—1071no—
180#478—1071no—
181#478—1071no—
182#478D10711yescharacteristic, complemented
183#476—1071no—
184#476—1071no—
185#474—1071no—
186#474—1071no—
187#428—1071no—
188#428—1071no—
189#423—1071no—
190#418—1071no—
191#418—1071no—
192C22×D4—1071no—
193C23×C4—2141no—
194C2×M16—1071no—
195C2×M16—1071no—
196C4⋊Q8—1071no—
197C4⋊Q8—1071no—
198#125—1071no—
199#125C2×D10711yescharacteristic, complemented
200C22⋊Q8—2141no—
201C22⋊Q8—2141no—
202C4×Q8—2141no—
203C4×Q8—2141no—
204C4×D4—1071no—
205C4×D4—1071no—
206C4×D4—1071no—
207C4×D4—2141no—
208C4×D4—2141no—
209C4×D4C2×D10711yescharacteristic
210#117—1071no—
211#117—1071no—
212C2×C4⋊C4—1071no—
213C2×C22⋊C4—2141no—
214C2×C42—1071no—
215#107—1071no—
216#107—1071no—
217#107—1071no—
218#107C2×D10711yescharacteristic, complemented
219#106—1071no—
220#106—1071no—
221#106—1071no—
222#106C2×D10711yescharacteristic, complemented
223C4⋊C8—1071no—
224C4⋊C8—1071no—
225C4⋊C8—1071no—
226C4⋊C8C2×D10711yescharacteristic, complemented
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102—1071no—
232#102—1071no—
233#102C2×D10711yescharacteristic
234#102C2×D10711yescharacteristic
235C24—2141no—
236C2×Q8—2141no—
237C2×Q8—2141no—
238C2×D4—1074no—
239C2×D4—1071no—
240C2×D4—2142no—
241C2×D4—2141no—
242C2×D4C22×D10711yescharacteristic
243C22×C4—21no—
244C22×C4—1071no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—4284no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4—2141no—
252M16—2142no—
253M16—2142no—
254C2×C8—1071no—
255C2×C8—1071no—
256C2×C8C22×D10711yescharacteristic
257C2×C8C22×D10711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—1071no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C4⋊C4C22×D10711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C42—1071no—
278C42—1071no—
279C42—1071no—
280C42—2141no—
281C42—2141no—
282C42C22×D10711yescharacteristic
283C23—21no—
284C23—1071no—
285C23—4282no—
286C23—4282no—
287C23—2144no—
288C23—2141no—
289Q8—4282no—
290Q8—4282no—
291D4—21no—
292D4—1072no—
293D4—2144no—
294D4—2141no—
295D4Q8×D10712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—1071no—
301C2×C4—1071no—
302C2×C4—1072no—
303C2×C4—1072no—
304C2×C4—1071no—
305C2×C4—4282no—
306C2×C4—4281no—
307C2×C4—4282no—
308C2×C4—4282no—
309C2×C4—4282no—
310C2×C4—2142no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4D4×D10711yescharacteristic
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2141no—
320C8—2141no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—1072no—
325C22—4288no—
326C22—4281no—
327C22—4281no—
328C22—2142no—
329C22—2141no—
330C22—2141no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—1072no—
338C4—1072no—
339C4—4281no—
340C4—4281no—
341C4—4281no—
342C4—2141no—
343C4C2×Q8×D10711yescharacteristic
344C4#41×D10711yescharacteristic
345C2—42no—
346C2—4282no—
347C2—2142no—
348C2C22⋊Q8×D10711yescharacteristic
349C2AGL(1,ℤ/8)×D10711yescharacteristic
350C2#2029342411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#478D107
hypocenterC107#3077
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20293183C2semidirect
3. #20292928C2semidirect
4. #20292927C2semidirect
5. #20292924C2semidirect
6. #20292916C2semidirect
7. #20292915C2semidirect
8. #20292906C2semidirect
9. #20292905C2semidirect
10. #20292668C2semidirect
11. #20292666C2semidirect
12. #20292649C2semidirect
15. #478×C107C2semidirect
20. #16812108C22semidirect
21. #16812108C22semidirect
22. #16812101C22semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. #16812040C22semidirect
42. #16812040C22semidirect
43. C4⋊(C107⋊C8)C22semidirect
44. Dic107⋊C8C22semidirect
45. Dic107⋊C8C22semidirect
52. #125×C107C22semidirect
54. #107×C107C22semidirect
55. #106×C107C22semidirect
56. C4⋊C8×C107C22semidirect
65. D4×D107Q8semidirect
176. C107#3077semidirect
182. #478D107semidirect
199. #125C2×D107semidirect
218. #107C2×D107semidirect
222. #106C2×D107semidirect
226. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931832
C4×D4×D1072
#202929282
#202929272
#202929242
#202929162
#202929152
#202929062
#202929052
#202926682
#202926662
#202926492
#202926302
#478×C1072
#3077107