Finite Group Databasebeta

← #28348152

Subgroups — #28348152

Order 13696.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D107C212yesmaximal, complemented
2Q8×D428C211yesmaximal, characteristic, complemented
3#20293262C211yesmaximal, characteristic
4#20292999C211yesmaximal, characteristic, complemented
5#20292855C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292779C212yesmaximal, complemented
8#20292764C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10#20292692C211yesmaximal, characteristic, complemented
11#460×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2212yescomplemented
13C2×D4×D107—21no—
14C2×D4×D107C2212yes—
15C2×#13133807C2212yes—
16C2×#13133806C2212yescomplemented
17C2×#13133806C2212yescomplemented
18C2×#13133805C2211yescharacteristic
19SD16×D107—28no—
20C2×C8×D107C2212yescomplemented
21#16812109—21no—
22#16812109C2211yescharacteristic, complemented
23#16812089—22no—
24#16812089C2212yescomplemented
25#16812080—21no—
26#16812080C2212yes—
27#16812077—22no—
28C4×D428—21no—
29C4×D428C2211yescharacteristic
30C4×D428C2211yescharacteristic
31#16812040C2211yescharacteristic, complemented
32#16812034C2212yescomplemented
33#16812033C2211yescharacteristic, complemented
34#16812029C2211yescharacteristic
35(C2×D107)⋊C8C2212yescomplemented
36C4×C107⋊C8C2211yescharacteristic, complemented
37C2×SD16×C107C2212yescomplemented
38#127×C107C2211yescharacteristic
39C4×Q8×C107C2211yescharacteristic, complemented
40C4⋊C8×C107C2211yescharacteristic, complemented
41#102×C107C2212yescomplemented
42C23×D107—22no—
43C2×Dic214—22no—
44C2×Dic214C2312yes—
45Q8×D107—24no—
46Q8×D107—42no—
47C2×D428D411yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51D4×D107—24no—
52D4×D107—42no—
53D4×D107—44no—
54C2×C4×D107—22no—
55C2×C4×D107—21no—
56C2×C4×D107C2312yes—
57#13133807—24no—
58#13133806—24no—
59#13133806—42no—
60#13133805D414yescomplemented
61C2×C107⋊C8C2311yescharacteristic
62C2×C107⋊C8C2311yescharacteristic
63C8×D107—24no—
64C4⋊Dic107—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107—22no—
67C4⋊Dic107D411yescharacteristic
68C4⋊Dic107C2311yescharacteristic
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4D412yes—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107—21no—
75C2×D4×C107—21no—
76C2×D4×C107C2312yes—
77SD16×C107—24no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2311yescharacteristic
81C42×C107—21no—
82C42×C107C2311yescharacteristic
83C23×C107—22no—
84C22×D107—21no—
85C22×D107—44no—
86C22×D107—44no—
87C22×D107—82no—
88C22×D107C2×D412yes—
89Q8×C107—21no—
90Q8×C107C2×D412yescomplemented
91D4×C107—22no—
92D4×C107—22no—
93D4×C107—42no—
94D4×C107—81no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—44no—
101C22⋊D107—44no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107C2×D412yes—
105D428—21no—
106D428SD1614yescomplemented
107D428C2×D412yes—
108C4×D107—22no—
109C4×D107—22no—
110C4×D107—44no—
111C4×D107—42no—
112Dic214—22no—
113Dic214—22no—
114Dic214—44no—
115C8×C107—22no—
116C107⋊C8—21no—
117C107⋊C8C2×D412yescomplemented
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—88no—
128C2×D107C2×SD1614yescomplemented
129C4×C107—22no—
130C4×C107—21no—
131C4×C107—41no—
132C4×C107C2×SD1612yescomplemented
133C4×C107C22×D411yescharacteristic
134C4×C107H(5,𝔽2)11yescharacteristic
135Dic107—22no—
136Dic107—42no—
137Dic107—41no—
138C2×C107—82no—
139C2×C107D4211yescharacteristic
140C2×C107C22×SD1611yescharacteristic
141C2×C107#57711yescharacteristic
142D107—24no—
143D107—42no—
144D4×SD16—1071nomaximal, Sylow
145C107D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—1072no—
147D4×Q8—1071no—
148D42—1071no—
149#492—1071no—
150#465—1072no—
151#461—1071no—
152#460D10711yescharacteristic, complemented
153#450—1072no—
154#448—1072no—
155C4×SD16—1071no—
156C8×D4—1071no—
157C22×Q8—1072no—
158C22×D4—1072no—
159C22×D4—2141no—
160C2×SD16—1071no—
161C2×SD16—1072no—
162C2×SD16—1072no—
163C2×SD16—1072no—
164C2×SD16—2148no—
165C2×SD16C2×D10712yescomplemented
166C22×C8—1072no—
167C4⋊Q8—1071no—
168C4⋊Q8—2141no—
169#127C2×D10711yescharacteristic
170C22⋊Q8—1072no—
171C22⋊Q8—2142no—
172#121—1072no—
173#121—2141no—
174#120—2142no—
175C4×Q8C2×D10711yescharacteristic, complemented
176C4×D4—1071no—
177C4×D4—1071no—
178C4×D4—2141no—
179#106—1071no—
180C4⋊C8C2×D10711yescharacteristic, complemented
181#103—1071no—
182#103—1072no—
183#102—1071no—
184#102C2×D10712yescomplemented
185C22⋊C8—1072no—
186C4×C8—1071no—
187C24—2142no—
188C2×Q8—1072no—
189C2×Q8—4282no—
190C2×Q8—2142no—
191C2×Q8—2144no—
192C2×Q8C22×D10711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—1071no—
196C2×D4—1071no—
197C2×D4—4284no—
198C2×D4—4282no—
199C2×D4—2142no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4C22×D10712yes—
203C22×C4—1072no—
204C22×C4—2142no—
205C22×C4—2141no—
206SD16—24no—
207SD16—1074no—
208SD16—4282no—
209SD16—2144no—
210SD16—2144no—
211C2×C8—1071no—
212C2×C8—1071no—
213C2×C8—2144no—
214C2×C8C22×D10712yes—
215C4⋊C4—21no—
216C4⋊C4—1071no—
217C4⋊C4—1071no—
218C4⋊C4—2142no—
219C4⋊C4—2141no—
220C4⋊C4—2141no—
221C4⋊C4C22×D10711yescharacteristic
222C22⋊C4—1072no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2141no—
226C42—21no—
227C42C22×D10711yescharacteristic
228C23—22no—
229C23—8562no—
230C23—1072no—
231C23—4284no—
232C23—4284no—
233C23—2141no—
234Q8—21no—
235Q8—4284no—
236Q8—2142no—
237Q8—2142no—
238Q8D4×D10712yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—1074no—
244D4—1072no—
245D4—4284no—
246D4—4284no—
247D4—2141no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—1072no—
251C2×C4—4284no—
252C2×C4—4282no—
253C2×C4—2141no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4D4×D10711yescharacteristic
258C2×C4D4×D10711yescharacteristic
259C2×C4C23×D10711yescharacteristic
260C8—22no—
261C8—1072no—
262C8—2141no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—8568no—
267C22—1074no—
268C22—4281no—
269C22—4281no—
270C22—2142no—
271C22—2142no—
272C22—2142no—
273C22C2×D4×D10711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—4281no—
278C4—4282no—
279C4—2142no—
280C4SD16×D10712yescomplemented
281C4C2×D4×D10711yescharacteristic
282C4#1681215511yescharacteristic
283C2—82no—
284C2—4282no—
285C2—2144no—
286C2#2029326111yescharacteristic
287C2C2×SD16×D10711yescharacteristic
288C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293262C2
hypercenter#460D107
hypocenterC107D4×SD16
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D107C2semidirect
2. Q8×D428C2semidirect
4. #20292999C2semidirect
5. #20292855C2semidirect
6. #20292833C2semidirect
7. #20292779C2semidirect
8. #20292764C2semidirect
9. C4×#13133805C2semidirect
10. #20292692C2semidirect
11. #460×C107C2semidirect
12. C2×Q8×D107C22semidirect
16. C2×#13133806C22semidirect
17. C2×#13133806C22semidirect
20. C2×C8×D107C22semidirect
22. #16812109C22semidirect
24. #16812089C22semidirect
31. #16812040C22semidirect
32. #16812034C22semidirect
33. #16812033C22semidirect
35. (C2×D107)⋊C8C22semidirect
36. C4×C107⋊C8C22semidirect
37. C2×SD16×C107C22semidirect
39. C4×Q8×C107C22semidirect
40. C4⋊C8×C107C22semidirect
41. #102×C107C22semidirect
60. #13133805D4semidirect
90. Q8×C107C2×D4semidirect
106. D428SD16semidirect
117. C107⋊C8C2×D4semidirect
128. C2×D107C2×SD16semidirect
132. C4×C107C2×SD16semidirect
145. C107D4×SD16semidirect
152. #460D107semidirect
165. C2×SD16C2×D107semidirect
175. C4×Q8C2×D107semidirect
180. C4⋊C8C2×D107semidirect
184. #102C2×D107semidirect
238. Q8D4×D107semidirect
280. C4SD16×D107semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1072
Q8×D4282
#202932622
#202929992
#202928552
#202928332
#202927792
#202927642
C4×#131338052
#202926922
#460×C1072
D4×SD16107