Finite Group Databasebeta

← #28346456

Subgroups — #28346456

Order 13696.

492 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293172C211yesmaximal, characteristic, complemented
2C2×D4×Dic107C211yesmaximal, characteristic, complemented
3#20292941C211yesmaximal, characteristic, complemented
4#20292938C211yesmaximal, characteristic, complemented
5#20292905C212yesmaximal, complemented
6#20292820C212yesmaximal, complemented
7#20292761C211yesmaximal, characteristic, complemented
8D8×Dic107C212yesmaximal, complemented
9#20292707C212yesmaximal, complemented
10#20292677C211yesmaximal, characteristic, complemented
11#447×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C23×Dic107—21no—
14C2×#13133803C2212yes—
15#16812108C2211yescharacteristic, complemented
16#16812105C2211yescharacteristic, complemented
17#16812095—21no—
18#16812083C2211yescharacteristic
19#16812081—21no—
20D4×Dic107—21no—
21D4×Dic107—21no—
22D4×Dic107C2212yes—
23D4×Dic107C2212yes—
24D4×Dic107C2211yescharacteristic, complemented
25#16812071C2211yescharacteristic, complemented
26(C22⋊D107)⋊C4—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33#16812031C2212yescomplemented
34#16812030C2212yes—
35#16812030C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37C8×Dic107C2212yescomplemented
38C22×D4×C107C2211yescharacteristic
39C2×D8×C107C2212yes—
40#121×C107C2211yescharacteristic
41#102×C107C2212yes—
42C22⋊C8×C107C2211yescharacteristic
43#13133817—44no—
44C2×Dic214—21no—
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C2×C22⋊D107—21no—
49C22×Dic107—21no—
50C22×Dic107—21no—
51C22×Dic107—22no—
52C22×Dic107—22no—
53C22×Dic107—44no—
54C22×Dic107D411yescharacteristic
55C22×Dic107D411yescharacteristic
56C2×C4×D107—21no—
57#13133803—24no—
58C2×C107⋊C8C2312yes—
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yes—
61C4⋊Dic107C2311yescharacteristic
62Dic107⋊C4—21no—
63C22⋊Dic107—22no—
64C22⋊Dic107—22no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68C4×Dic107D411yescharacteristic
69C4×Dic107D411yescharacteristic
70C4×Dic107C2311yescharacteristic
71C4×Dic107C2311yescharacteristic
72C24×C107—21no—
73C2×D4×C107—24no—
74C2×D4×C107—21no—
75C2×D4×C107—21no—
76C2×D4×C107C2311yescharacteristic
77C2×D4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79D8×C107—24no—
80C2×C8×C107C2312yes—
81C4⋊C4×C107C2311yescharacteristic
82C22⋊C4×C107—21no—
83C23×C107—24no—
84C23×C107—22no—
85C23×C107—21no—
86C23×C107—42no—
87C23×C107—42no—
88C23×C107C2×D411yescharacteristic
89C22×D107—21no—
90D4×C107—24no—
91D4×C107—22no—
92D4×C107—22no—
93D4×C107—42no—
94D4×C107#4114yescomplemented
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—42no—
100C22⋊D107—42no—
101C22⋊D107—42no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—41no—
105C2×Dic107—42no—
106C2×Dic107—44no—
107C2×Dic107—44no—
108C2×Dic107—42no—
109C2×Dic107D814yescomplemented
110C2×Dic107C2×D411yescharacteristic
111C2×Dic107C2×D411yescharacteristic
112C2×Dic107C2×D411yescharacteristic
113C2×Dic107C2×D411yescharacteristic
114D428—22no—
115C4×D107—42no—
116Dic214—42no—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—24no—
122C22×C107—48no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—41no—
126C22×C107—41no—
127C22×C107—41no—
128C22×C107C2×D812yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—41no—
133C4×C107—21no—
134C4×C107—41no—
135C4×C107C2×#4112yes—
136Dic107—21no—
137Dic107—21no—
138Dic107—41no—
139Dic107—42no—
140Dic107C2×D812yes—
141Dic107C2×D812yes—
142C2×C107—22no—
143C2×C107—44no—
144C2×C107—81no—
145C2×C107#52511yescharacteristic
146C2×C107C22×D811yescharacteristic
147C2×C107#57511yescharacteristic
148D107—81no—
149#3100—1071nomaximal, Sylow
150C107#310011yescharacteristic, complemented, Sylow
151#532—1071no—
152C2×C4×D4—1071no—
153#480—1071no—
154#480—1071no—
155#474—1072no—
156#459—1072no—
157#447—1071no—
158#447D10711yescharacteristic, complemented
159C4×D8—1072no—
160C4×D8—1072no—
161#431—1071no—
162C2×#41—2141no—
163C22×D4C2×D10711yescharacteristic
164C23×C4—2141no—
165C2×D8—1072no—
166C2×D8C2×D10712yes—
167C4⋊Q8—1071no—
168#127—1071no—
169#124—2141no—
170#121—1071no—
171#121—2141no—
172#121C2×D10711yescharacteristic
173C4×D4—1072no—
174C4×D4—1072no—
175C4×D4—1071no—
176C4×D4—1071no—
177C4×D4—2141no—
178C4×D4—2141no—
179C4×D4—2141no—
180C2×C4⋊C4—1071no—
181C2×C22⋊C4—2141no—
182C2×C42—1071no—
183#107—1072no—
184#107—1072no—
185C4⋊C8—1072no—
186#102—1072no—
187#102—1072no—
188#102—1072no—
189#102C2×D10712yes—
190C22⋊C8—1071no—
191C22⋊C8C2×D10711yescharacteristic
192C4×C8—1072no—
193C24—21no—
194#41—4284no—
195C2×Q8—2141no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—1071no—
200C2×D4—2141no—
201C2×D4—2141no—
202C2×D4—2141no—
203C2×D4C22×D10711yescharacteristic
204C2×D4C22×D10712yes—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—4284no—
208C22×C4—2141no—
209C22×C4—2142no—
210C22×C4—2142no—
211C22×C4—2141no—
212C22×C4—2141no—
213C22×C4C22×D10711yescharacteristic
214D8—24no—
215D8—2144no—
216C2×C8—1072no—
217C2×C8C22×D10712yes—
218C4⋊C4—1072no—
219C4⋊C4—1071no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4C22×D10711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2141no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C22⋊C4—2141no—
228C22⋊C4—2141no—
229C42—1071no—
230C42—1071no—
231C42—1071no—
232C42—1071no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—2141no—
239C23D4×D10711yescharacteristic
240Q8—4282no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—4282no—
246D4—4282no—
247D4—4282no—
248D4—2142no—
249D4#1313381714yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—1074no—
253C2×C4—1071no—
254C2×C4—1071no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—4282no—
258C2×C4—4284no—
259C2×C4—4284no—
260C2×C4—4282no—
261C2×C4—4282no—
262C2×C4—4281no—
263C2×C4—2142no—
264C2×C4—2141no—
265C2×C4D4×D10711yescharacteristic
266C2×C4C23×D10711yescharacteristic
267C8—22no—
268C8—2142no—
269C22—21no—
270C22—21no—
271C22—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—4281no—
279C22—4281no—
280C22—4281no—
281C22D8×D10712yes—
282C22C2×D4×D10711yescharacteristic
283C4—21no—
284C4—41no—
285C4—1072no—
286C4—1072no—
287C4—4282no—
288C4—4281no—
289C4—2141no—
290C4—2141no—
291C4C2×#1313381712yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—8561no—
296C2#2029313311yescharacteristic
297C2C2×D8×D10711yescharacteristic
298C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#447×C107C2
socleC22×C107C22×D4
hypercenter#447D107
hypocenterC107#3100
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20293172C2semidirect
2. C2×D4×Dic107C2semidirect
3. #20292941C2semidirect
4. #20292938C2semidirect
5. #20292905C2semidirect
6. #20292820C2semidirect
7. #20292761C2semidirect
8. D8×Dic107C2semidirect
9. #20292707C2semidirect
10. #20292677C2semidirect
11. #447×C107C2semidirect
15. #16812108C22semidirect
16. #16812105C22semidirect
24. D4×Dic107C22semidirect
25. #16812071C22semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. Dic107⋊C8C22semidirect
33. #16812031C22semidirect
35. #16812030C22semidirect
37. C8×Dic107C22semidirect
94. D4×C107#41semidirect
109. C2×Dic107D8semidirect
150. C107#3100semidirect
158. #447D107semidirect
249. D4#13133817semidirect

Maximal subgroups

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

subgroupindex
#202931722
C2×D4×Dic1072
#202929412
#202929382
#202929052
#202928202
#202927612
D8×Dic1072
#202927072
#202926772
#447×C1072
#3100107