Finite Group Databasebeta

← #28346107

Subgroups — #28346107

Order 13696.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D107C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic, complemented
3#20293193C212yesmaximal, complemented
4#20292881C211yesmaximal, characteristic, complemented
5#20292880C211yesmaximal, characteristic, complemented
6#20292877C211yesmaximal, characteristic, complemented
7#20292862C212yesmaximal, complemented
8#20292861C212yesmaximal, complemented
9#20292662C211yesmaximal, characteristic, complemented
10#20292630C212yesmaximal, complemented
11#469×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×D4×D107C2212yes—
14C2×D856C2211yescharacteristic
15C2×#13133803C2212yes—
16D8×D107—24no—
17#16812117—24no—
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic, complemented
20C22×C107⋊C8C2211yescharacteristic, complemented
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812105—22no—
24#16812095—22no—
25#16812084C2212yescomplemented
26#16812083C2212yescomplemented
27#16812080C2212yescomplemented
28#16812079—22no—
29#16812077—22no—
30#16812067C2212yes—
31#16812042C2212yescomplemented
32#16812041C2211yescharacteristic, complemented
33#16812031C2212yescomplemented
34#16812030C2212yescomplemented
35#16812029C2212yescomplemented
36C2×D8×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic, complemented
38#121×C107C2212yescomplemented
39#106×C107C2211yescharacteristic, complemented
40#102×C107C2212yescomplemented
41C23×D107—22no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214D411yescharacteristic
45C2×D428—22no—
46C2×D428D411yescharacteristic
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107D411yescharacteristic
52C2×C22⋊D107D411yescharacteristic
53D4×D107—24no—
54D4×D107—24no—
55C2×C4×D107C2311yescharacteristic
56C2×C4×D107C2311yescharacteristic
57D856—22no—
58#13133803—24no—
59#13133801—22no—
60#13133801—22no—
61C2×C107⋊C8—22no—
62C2×C107⋊C8C2311yescharacteristic
63C2×C107⋊C8C2311yescharacteristic
64C8×D107—22no—
65C8×D107D414yescomplemented
66C4⋊Dic107C2312yes—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—22no—
72C2×D4×C107—22no—
73C2×D4×C107C2312yes—
74C22×C4×C107C2311yescharacteristic
75D8×C107—22no—
76M16×C107—22no—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107—22no—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—44no—
85C22×D107—82no—
86C22×D107—82no—
87C22×D107C2×D411yescharacteristic
88C22×D107C2×D411yescharacteristic
89D4×C107—22no—
90D4×C107—22no—
91D4×C107—82no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—22no—
98C22⋊D107—22no—
99C22⋊D107—44no—
100C22⋊D107—82no—
101C22⋊D107—82no—
102C2×Dic107—22no—
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105D428—22no—
106D428—22no—
107D428—22no—
108D428—82no—
109C4×D107—21no—
110C4×D107—21no—
111C4×D107C2×D412yes—
112C4×D107#4112yes—
113Dic214—22no—
114C8×C107—21no—
115C8×C107C2×D412yes—
116C107⋊C8—21no—
117C107⋊C8C2×D412yes—
118C22×C107—21no—
119C22×C107—22no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107C22×D411yescharacteristic
124C2×D107—22no—
125C2×D107—21no—
126C2×D107—22no—
127C2×D107—42no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—88no—
131C2×D107#12112yes—
132C4×C107—21no—
133C4×C107—42no—
134C4×C107C22×D411yescharacteristic
135C4×C107C2×#4111yescharacteristic
136Dic107—21no—
137Dic107—42no—
138Dic107#12112yes—
139C2×C107—41no—
140C2×C107—82no—
141C2×C107C2×#12111yescharacteristic
142C2×C107#57612yes—
143D107—22no—
144D107—41no—
145D107—82no—
146#3058—1071nomaximal, Sylow
147C107#305811yescharacteristic, complemented, Sylow
148C22×D8—1071no—
149C2×#131—1071no—
150#535—1072no—
151#469—1071no—
152#469—1071no—
153#469—1071no—
154#469D10711yescharacteristic, complemented
155#466—1072no—
156#466—1072no—
157#427—1071no—
158#418—1072no—
159C2×#41—1071no—
160C22×D4—1072no—
161C2×D8—1072no—
162C2×D8—1071no—
163C2×D8—2144no—
164C2×D8C2×D10711yescharacteristic
165#131—2144no—
166C2×M16—1071no—
167C2×M16—1071no—
168C2×M16C2×D10711yescharacteristic, complemented
169C22×C8—1071no—
170C22×C8—1071no—
171C22×C8—1071no—
172#127—2142no—
173#124—2142no—
174#121—1072no—
175#121—1072no—
176#121—1072no—
177#121C2×D10712yescomplemented
178#120—2142no—
179#120—2142no—
180#117—1072no—
181#107—1072no—
182#106—1071no—
183#106C2×D10711yescharacteristic, complemented
184#102—1072no—
185#102—1072no—
186#102—1072no—
187#102C2×D10712yescomplemented
188C24—2142no—
189#41—2142no—
190#41—2142no—
191C2×Q8—1071no—
192C2×D4—22no—
193C2×D4—1071no—
194C2×D4—1071no—
195C2×D4—1071no—
196C2×D4—1072no—
197C2×D4—2142no—
198C2×D4—2144no—
199C2×D4—2144no—
200C2×D4—2142no—
201C2×D4—2142no—
202C2×D4—2142no—
203C2×D4C22×D10712yes—
204C22×C4—1071no—
205C22×C4—1071no—
206C22×C4C22×D10711yescharacteristic
207D8—22no—
208D8—2144no—
209D8—2142no—
210M16—22no—
211M16—2142no—
212M16—2142no—
213C2×C8—1071no—
214C2×C8—1074no—
215C2×C8—1071no—
216C2×C8—2142no—
217C2×C8—2142no—
218C2×C8C22×D10711yescharacteristic
219C2×C8C22×D10711yescharacteristic
220C4⋊C4—1072no—
221C4⋊C4C22×D10712yes—
222C22⋊C4—22no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C42—2142no—
228C23—22no—
229C23—8562no—
230C23—8562no—
231C23—1071no—
232C23—1071no—
233C23—4284no—
234C23—2142no—
235C23D4×D10711yescharacteristic
236Q8—2142no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—8562no—
241D4—8562no—
242D4—8562no—
243D4—4284no—
244D4—2142no—
245D4—2142no—
246D4—2142no—
247D4—2142no—
248D4—2142no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—1071no—
253C2×C4—1072no—
254C2×C4—1072no—
255C2×C4—1071no—
256C2×C4—2142no—
257C2×C4—2141no—
258C2×C4—2141no—
259C2×C4D4×D10711yescharacteristic
260C2×C4C23×D10711yescharacteristic
261C8—21no—
262C8—1072no—
263C8—2141no—
264C8D4×D10712yes—
265C22—22no—
266C22—21no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—8568no—
271C22—1072no—
272C22—4282no—
273C22—4282no—
274C22—4282no—
275C22—2142no—
276C22—2141no—
277C22—2142no—
278C22C2×D4×D10711yescharacteristic
279C4—21no—
280C4—42no—
281C4—1072no—
282C4—4282no—
283C4—2141no—
284C4C2×D4×D10711yescharacteristic
285C4#41×D10711yescharacteristic
286C2—41no—
287C2—82no—
288C2—8562no—
289C2—4281no—
290C2—2142no—
291C2#121×D10711yescharacteristic
292C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107#3058
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D107C2semidirect
2. C2×#16812117C2semidirect
3. #20293193C2semidirect
4. #20292881C2semidirect
5. #20292880C2semidirect
6. #20292877C2semidirect
7. #20292862C2semidirect
8. #20292861C2semidirect
9. #20292662C2semidirect
10. #20292630C2semidirect
11. #469×C107C2semidirect
19. C2×#13133801C22semidirect
20. C22×C107⋊C8C22semidirect
22. C2×C8×D107C22semidirect
25. #16812084C22semidirect
26. #16812083C22semidirect
27. #16812080C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812031C22semidirect
34. #16812030C22semidirect
35. #16812029C22semidirect
37. C2×M16×C107C22semidirect
38. #121×C107C22semidirect
39. #106×C107C22semidirect
40. #102×C107C22semidirect
65. C8×D107D4semidirect
147. C107#3058semidirect
154. #469D107semidirect
168. C2×M16C2×D107semidirect
177. #121C2×D107semidirect
183. #106C2×D107semidirect
187. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
C2×#168121172
#202931932
#202928812
#202928802
#202928772
#202928622
#202928612
#202926622
#202926302
#469×C1072
#3058107