Finite Group Databasebeta

← #28346406

Subgroups — #28346406

Order 13696.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D107C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic
3#20292970C212yesmaximal, complemented
4#20292966C212yesmaximal, complemented
5#20292962C211yesmaximal, characteristic, complemented
6#20292958C211yesmaximal, characteristic, complemented
7#20292955C211yesmaximal, characteristic, complemented
8#20292668C212yesmaximal, complemented
9#20292634C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic
11#484×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812089C2211yescharacteristic
16#16812088C2211yescharacteristic
17#16812087—21no—
18#16812086—21no—
19#16812085C2211yescharacteristic
20#16812067—21no—
21#16812067—21no—
22#16812067C2212yes—
23#16812067C2212yes—
24C2×C4⋊Dic107C2211yescharacteristic
25C4⋊C4×D107—21no—
26C4⋊C4×D107C2211yescharacteristic
27C2×(C2×D107)⋊C4—21no—
28C22⋊C4×D107—21no—
29C2×C4×Dic107—21no—
30#16812042C2212yescomplemented
31#16812041C2212yescomplemented
32#16812040C2212yescomplemented
33#16812034C2212yescomplemented
34#16812033C2212yescomplemented
35#16812032C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37(C2×D107)⋊C8C2212yes—
38C22⋊Q8×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic
40#107×C107C2212yescomplemented
41#103×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic
43C23×D107D411yescharacteristic
44C2×Dic214—21no—
45C2×Dic214C2311yescharacteristic
46Q8×D107—24no—
47C22×Dic107—21no—
48C22×Dic107D411yescharacteristic
49C2×C4×D107—24no—
50C2×C4×D107—21no—
51C2×C4×D107—21no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55C2×C4×D107C2311yescharacteristic
56#13133801—24no—
57C2×C107⋊C8C2312yes—
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yes—
61C4⋊Dic107C2311yescharacteristic
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107—22no—
71C4×Dic107—22no—
72C2×Q8×C107C2311yescharacteristic
73C22×C4×C107—21no—
74C22×C4×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C23×C107C2×D411yescharacteristic
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—41no—
85C22×D107C2×D411yescharacteristic
86C22×D107C2×D411yescharacteristic
87Q8×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—42no—
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102C4×D107—22no—
103C4×D107—22no—
104C4×D107—22no—
105C4×D107—42no—
106C4×D107—42no—
107C4×D107#4114yes—
108Dic214—22no—
109Dic214—42no—
110C8×C107—22no—
111C107⋊C8—22no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107#13712yes—
115C22×C107C22×D411yescharacteristic
116C2×D107—21no—
117C2×D107—21no—
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C2×D107—44no—
122C2×D107#12312yes—
123C4×C107—21no—
124C4×C107—41no—
125C4×C107—41no—
126C4×C107—42no—
127C4×C107C2×#4112yes—
128Dic107—21no—
129Dic107—41no—
130Dic107—42no—
131Dic107—41no—
132Dic107#12312yes—
133C2×C107—22no—
134C2×C107C2×#12311yescharacteristic
135C2×C107C2×#13711yescharacteristic
136C2×C107#57511yescharacteristic
137D107—22no—
138D107—41no—
139#3095—1071nomaximal, Sylow
140C107#309511yescharacteristic, complemented, Sylow
141C2×C22⋊Q8—1071no—
142C2×#117—1071no—
143#485—1072no—
144#485—1072no—
145#484—1071no—
146#484D10711yescharacteristic, complemented
147#483—1071no—
148#483—1071no—
149#428—1072no—
150#419—1072no—
151#407—1071no—
152C22×Q8—1071no—
153C23×C4—1071no—
154C2×M16—1072no—
155C22⋊Q8—1071no—
156C22⋊Q8—1071no—
157C22⋊Q8—1071no—
158C22⋊Q8—2141no—
159C22⋊Q8—2141no—
160C22⋊Q8C2×D10711yescharacteristic
161#117—1072no—
162#117—1072no—
163#117—2141no—
164#117—2141no—
165C2×C4⋊C4—1071no—
166C2×C4⋊C4—1071no—
167C2×C4⋊C4—2141no—
168C2×C4⋊C4C2×D10711yescharacteristic
169C2×C22⋊C4—2141no—
170C2×C22⋊C4—2141no—
171C2×C42—2141no—
172#107—1072no—
173#107C2×D10712yescomplemented
174#106—1072no—
175#106—1072no—
176#103—1072no—
177#103—1072no—
178#103—1072no—
179#103C2×D10712yescomplemented
180C22⋊C8—1071no—
181C22⋊C8—1072no—
182C22⋊C8C2×D10711yescharacteristic
183C24—1071no—
184C2×Q8—1071no—
185C2×Q8—2141no—
186C2×Q8—2144no—
187C2×Q8C22×D10711yescharacteristic
188C22×C4—21no—
189C22×C4—1071no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—1071no—
194C22×C4—2144no—
195C22×C4—2141no—
196C22×C4—2141no—
197C22×C4—2141no—
198C22×C4C22×D10711yescharacteristic
199M16—2144no—
200C2×C8—1072no—
201C2×C8C22×D10712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1072no—
205C4⋊C4—1071no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4—2141no—
209C4⋊C4—2141no—
210C4⋊C4—2141no—
211C4⋊C4C22×D10712yes—
212C4⋊C4C22×D10711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C22⋊C4—2141no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C42—2142no—
220C42—2142no—
221C23—1071no—
222C23—1071no—
223C23—4281no—
224C23—2142no—
225C23—2142no—
226C23D4×D10711yescharacteristic
227Q8—22no—
228Q8—4282no—
229Q8—2142no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1071no—
236C2×C4—1074no—
237C2×C4—1071no—
238C2×C4—4282no—
239C2×C4—4282no—
240C2×C4—4282no—
241C2×C4—2142no—
242C2×C4—2142no—
243C2×C4—2142no—
244C2×C4—2142no—
245C2×C4—2141no—
246C2×C4—2141no—
247C2×C4—2142no—
248C2×C4D4×D10711yescharacteristic
249C2×C4C23×D10711yescharacteristic
250C8—22no—
251C8—2142no—
252C22—21no—
253C22—21no—
254C22—1072no—
255C22—4284no—
256C22—2141no—
257C22—2141no—
258C22—2141no—
259C22—2141no—
260C22—2141no—
261C22#1681214112yes—
262C22C2×D4×D10711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—1072no—
268C4—4281no—
269C4—4282no—
270C4—4281no—
271C4—2141no—
272C4#41×D10712yes—
273C2—22no—
274C2—4281no—
275C2—2142no—
276C2#123×D10711yescharacteristic
277C2C2×#1681214111yescharacteristic
278C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#484×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
hypercenter#484D107
hypocenterC107#3095
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20292970C2semidirect
4. #20292966C2semidirect
5. #20292962C2semidirect
6. #20292958C2semidirect
7. #20292955C2semidirect
8. #20292668C2semidirect
9. #20292634C2semidirect
11. #484×C107C2semidirect
30. #16812042C22semidirect
31. #16812041C22semidirect
32. #16812040C22semidirect
33. #16812034C22semidirect
34. #16812033C22semidirect
35. #16812032C22semidirect
40. #107×C107C22semidirect
41. #103×C107C22semidirect
140. C107#3095semidirect
146. #484D107semidirect
173. #107C2×D107semidirect
179. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1072
C2×#168120672
#202929702
#202929662
#202929622
#202929582
#202929552
#202926682
#202926342
#202925892
#484×C1072
#3095107