Finite Group Databasebeta

← #28346385

Subgroups — #28346385

Order 13696.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#20293140C211yesmaximal, characteristic
2C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
3#20292961C212yesmaximal, complemented
4#20292955C211yesmaximal, characteristic, complemented
5#20292947C212yesmaximal, complemented
6#20292941C211yesmaximal, characteristic, complemented
7#20292940C211yesmaximal, characteristic, complemented
8#20292666C212yesmaximal, complemented
9#20292627C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#483×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C22×C4×D107—21no—
14C22×C4×D107C2211yescharacteristic
15C2×#13133801C2212yes—
16#16812094—21no—
17#16812085C2211yescharacteristic
18#16812084C2211yescharacteristic
19#16812083C2211yescharacteristic
20#16812079—21no—
21#16812071—21no—
22(C22⋊D107)⋊C4—21no—
23#16812067C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4—21no—
26C4⋊C4×D107—21no—
27C4⋊C4×D107—21no—
28C4⋊C4×D107C2212yes—
29C4⋊C4×D107C2212yes—
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812040C2212yescomplemented
33#16812033C2212yescomplemented
34#16812031C2212yescomplemented
35#16812029C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37(C2×D107)⋊C8C2212yes—
38C22⋊Q8×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic
40#106×C107C2212yescomplemented
41#103×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic
43C23×D107D411yescharacteristic
44#13133819—24no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428C2311yescharacteristic
48C2×C22⋊D107—21no—
49C22×Dic107—21no—
50C22×Dic107D411yescharacteristic
51C2×C4×D107—24no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—21no—
55C2×C4×D107—44no—
56C2×C4×D107D411yescharacteristic
57C2×C4×D107D411yescharacteristic
58C2×C4×D107C2311yescharacteristic
59C2×C4×D107C2311yescharacteristic
60#13133801—24no—
61C2×C107⋊C8C2312yes—
62C4⋊Dic107—21no—
63C4⋊Dic107C2312yes—
64C4⋊Dic107C2311yescharacteristic
65Dic107⋊C4—22no—
66Dic107⋊C4—22no—
67Dic107⋊C4—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73C2×Q8×C107C2311yescharacteristic
74C22×C4×C107—21no—
75C22×C4×C107C2311yescharacteristic
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2312yes—
81C22⋊C4×C107—21no—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—21no—
87C22×D107C2×D411yescharacteristic
88C22×D107C2×D411yescharacteristic
89Q8×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—42no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—42no—
98C2×Dic107—22no—
99C2×Dic107—21no—
100C2×Dic107—22no—
101C2×Dic107—42no—
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104D428—22no—
105D428—42no—
106D428—42no—
107C4×D107—22no—
108C4×D107—22no—
109C4×D107—22no—
110C4×D107—44no—
111C4×D107—44no—
112C4×D107—42no—
113C4×D107#4114yes—
114C8×C107—22no—
115C107⋊C8—22no—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107AGL(1,ℤ/8)12yes—
119C22×C107C22×D411yescharacteristic
120C2×D107—28no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—41no—
128C2×D107#12312yes—
129C2×D107#13712yes—
130C4×C107—21no—
131C4×C107—41no—
132C4×C107—42no—
133C4×C107—41no—
134C4×C107C2×#4112yes—
135Dic107—21no—
136Dic107—42no—
137Dic107—41no—
138Dic107#12312yes—
139C2×C107—22no—
140C2×C107C2×#12311yescharacteristic
141C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C107C2×#13711yescharacteristic
143D107—22no—
144D107—22no—
145D107—81no—
146#3092—1071nomaximal, Sylow
147C107#309211yescharacteristic, complemented, Sylow
148#525—1071no—
149C22×C4⋊C4—1071no—
150#484—1072no—
151#483—1071no—
152#483D10711yescharacteristic, complemented
153#481—1072no—
154#480—1071no—
155#480—1071no—
156#428—1072no—
157#417—1072no—
158#407—1071no—
159C2×#41—1071no—
160C23×C4—1071no—
161C23×C4—2141no—
162C2×M16—1072no—
163#123—2141no—
164C22⋊Q8—1071no—
165C22⋊Q8C2×D10711yescharacteristic
166#121—1071no—
167#121—1071no—
168#120—2141no—
169C4×D4—2141no—
170C4×D4—2141no—
171#117—1071no—
172C2×C4⋊C4—1071no—
173C2×C4⋊C4—1072no—
174C2×C4⋊C4—1072no—
175C2×C4⋊C4—2141no—
176C2×C4⋊C4—2141no—
177C2×C4⋊C4—2141no—
178C2×C4⋊C4C2×D10711yescharacteristic
179#107—1072no—
180#107—1072no—
181#106—1072no—
182#106C2×D10712yescomplemented
183#103—1072no—
184#103C2×D10712yescomplemented
185#102—1072no—
186#102—1072no—
187C22⋊C8—1071no—
188C22⋊C8—1072no—
189C22⋊C8C2×D10711yescharacteristic
190C24—1071no—
191#41—2144no—
192C2×Q8C22×D10711yescharacteristic
193C2×D4—1071no—
194C2×D4—2141no—
195C2×D4—2141no—
196C2×D4—2141no—
197C22×C4—21no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—1071no—
202C22×C4—1071no—
203C22×C4—4284no—
204C22×C4—2141no—
205C22×C4—2144no—
206C22×C4—2142no—
207C22×C4—2142no—
208C22×C4—2141no—
209C22×C4C22×D10711yescharacteristic
210M16—2144no—
211C2×C8—1072no—
212C2×C8C22×D10712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1072no—
216C4⋊C4—1071no—
217C4⋊C4—2142no—
218C4⋊C4—2142no—
219C4⋊C4—2141no—
220C4⋊C4—2141no—
221C4⋊C4C22×D10712yes—
222C4⋊C4C22×D10711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2141no—
225C22⋊C4—2141no—
226C22⋊C4—2141no—
227C22⋊C4—2141no—
228C42—2141no—
229C23—1071no—
230C23—1071no—
231C23—2142no—
232C23—2142no—
233C23—2141no—
234C23—2142no—
235C23D4×D10711yescharacteristic
236Q8—22no—
237D4—4282no—
238D4—4282no—
239D4—4282no—
240D4—2142no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—1071no—
247C2×C4—1074no—
248C2×C4—1071no—
249C2×C4—4282no—
250C2×C4—4284no—
251C2×C4—4284no—
252C2×C4—4282no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2141no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4D4×D10711yescharacteristic
260C2×C4C23×D10711yescharacteristic
261C8—22no—
262C8—2142no—
263C22—21no—
264C22—21no—
265C22—1072no—
266C22—1072no—
267C22—4281no—
268C22—4281no—
269C22—4281no—
270C22—2148no—
271C22—2141no—
272C22—2141no—
273C22—2141no—
274C22—2141no—
275C22#1681213612yes—
276C22C2×D4×D10711yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—1072no—
282C4—4282no—
283C4—4281no—
284C4—2141no—
285C4#41×D10712yes—
286C2—22no—
287C2—8561no—
288C2—2142no—
289C2—2142no—
290C2#123×D10711yescharacteristic
291C2C2×#1681213611yescharacteristic
292C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#483×C107C2
socleC22×C107C22×D4
prime radical#20293140C2
hypercenter#483D107
hypocenterC107#3092
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D107C2semidirect
3. #20292961C2semidirect
4. #20292955C2semidirect
5. #20292947C2semidirect
6. #20292941C2semidirect
7. #20292940C2semidirect
8. #20292666C2semidirect
9. #20292627C2semidirect
10. #20292589C2semidirect
11. #483×C107C2semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812033C22semidirect
34. #16812031C22semidirect
35. #16812029C22semidirect
40. #106×C107C22semidirect
41. #103×C107C22semidirect
147. C107#3092semidirect
152. #483D107semidirect
182. #106C2×D107semidirect
184. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931402
C2×C4⋊C4×D1072
#202929612
#202929552
#202929472
#202929412
#202929402
#202926662
#202926272
#202925892
#483×C1072
#3092107