Finite Group Databasebeta

← #28346852

Subgroups — #28346852

Order 13696.

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

normal subgroups only

This group has 424 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#20293173C211yesmaximal, characteristic
2C2×#16812108C211yesmaximal, characteristic, complemented
3#20292988C212yesmaximal, complemented
4#20292985C212yesmaximal, complemented
5#20292976C212yesmaximal, complemented
6#20292972C212yesmaximal, complemented
7#20292629C211yesmaximal, characteristic, complemented
8#20292628C211yesmaximal, characteristic, complemented
9#20292626C211yesmaximal, characteristic, complemented
10M16×Dic107C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214—21no—
14C2×#13133802C2211yescharacteristic
15#16812108—21no—
16#16812108—21no—
17#16812108C2212yes—
18#16812108C2212yes—
19#16812107C2211yescharacteristic
20#16812105C2211yescharacteristic
21#16812098C2211yescharacteristic
22#16812096C2211yescharacteristic
23#16812081—21no—
24#16812081—21no—
25C4×C22⋊D107—21no—
26C4×C22⋊D107—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×Dic107⋊C4—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812034C2212yes—
31#16812033C2212yes—
32#16812032C2212yes—
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812019C2212yes—
37C8×Dic107C2212yes—
38C2×#41×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yes—
42#102×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—44no—
48C2×Dic214C2311yescharacteristic
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—21no—
51C2×C22⋊D107—21no—
52C2×C22⋊D107—21no—
53C22×Dic107—21no—
54C22×Dic107D411yescharacteristic
55C22×Dic107D411yescharacteristic
56C2×C4×D107—21no—
57#13133802—22no—
58C2×C107⋊C8C2312yes—
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yes—
61Dic107⋊C4—22no—
62Dic107⋊C4—22no—
63Dic107⋊C4—21no—
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66C22⋊Dic107—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C4×Dic107C2311yescharacteristic
73#41×C107—22no—
74#41×C107—22no—
75C2×Q8×C107C2311yescharacteristic
76C2×D4×C107—21no—
77C2×D4×C107C2311yescharacteristic
78C22×C4×C107—21no—
79C22×C4×C107—21no—
80C22×C4×C107C2311yescharacteristic
81M16×C107—22no—
82C2×C8×C107C2312yes—
83C4⋊C4×C107—21no—
84C4⋊C4×C107C2312yes—
85C23×C107—21no—
86C23×C107C2×D411yescharacteristic
87C22×D107—21no—
88Q8×C107—21no—
89Q8×C107—21no—
90D4×C107—21no—
91D4×C107—21no—
92D4×C107—42no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—42no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107#4112yes—
99C2×C4×C107#4112yes—
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—42no—
102C22⋊D107—42no—
103C22⋊D107—42no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—22no—
108C2×Dic107—42no—
109C2×Dic107C2×D411yescharacteristic
110C2×Dic107C2×D411yescharacteristic
111C2×Dic107C2×D411yescharacteristic
112C2×Dic107C2×D411yescharacteristic
113D428—21no—
114D428—21no—
115C4×D107—42no—
116Dic214—21no—
117Dic214—21no—
118Dic214—44no—
119Dic214—44no—
120C8×C107—22no—
121C107⋊C8—22no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107—41no—
127C22×C107#12412yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—41no—
130C2×D107—41no—
131C2×D107—41no—
132C4×C107—41no—
133C4×C107—42no—
134C4×C107#12412yes—
135C4×C107C2×#4111yescharacteristic
136C4×C107C2×#4111yescharacteristic
137Dic107—21no—
138Dic107—21no—
139Dic107—41no—
140Dic107—42no—
141Dic107AGL(1,ℤ/8)12yes—
142Dic107#13712yes—
143C2×C107—22no—
144C2×C107—81no—
145C2×C107C2×#12411yescharacteristic
146C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C107C2×#13711yescharacteristic
148D107—81no—
149#3141—1071nomaximal, Sylow
150C107#314111yescharacteristic, complemented, Sylow
151#532—1071no—
152C2×C4⋊Q8—1071no—
153#489—1072no—
154#489—1072no—
155#487—1072no—
156#486—1072no—
157#417—1071no—
158#417—1071no—
159#417—1071no—
160#417D10711yescharacteristic, complemented
161C4×M16—1071no—
162C2×#41—1071no—
163C2×#41C2×D10711yescharacteristic
164C22×Q8—2141no—
165C2×M16—1071no—
166C2×M16C2×D10711yescharacteristic
167C4⋊Q8—1072no—
168C4⋊Q8—1072no—
169C4⋊Q8—1071no—
170C4⋊Q8—2141no—
171C4⋊Q8—2141no—
172#127—1071no—
173#124—1071no—
174#124—1071no—
175#121—2141no—
176#121—2141no—
177C4×D4—2141no—
178C4×D4—2141no—
179C2×C4⋊C4—1071no—
180C2×C4⋊C4—2141no—
181C2×C4⋊C4C2×D10711yescharacteristic
182C2×C42—1071no—
183#103—1072no—
184#103—1072no—
185#103—1072no—
186#103C2×D10712yes—
187#102—1072no—
188#102—1072no—
189#102—1072no—
190#102C2×D10712yes—
191#97—1072no—
192C4×C8—1072no—
193#41—22no—
194#41—22no—
195#41—2142no—
196#41—2142no—
197C2×Q8—1071no—
198C2×Q8—4284no—
199C2×Q8—2142no—
200C2×Q8—2142no—
201C2×Q8C22×D10711yescharacteristic
202C2×D4—21no—
203C2×D4—1071no—
204C2×D4—2141no—
205C2×D4—2141no—
206C2×D4—2141no—
207C2×D4C22×D10711yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—1071no—
211C22×C4—1071no—
212C22×C4—2141no—
213C22×C4—2141no—
214C22×C4C22×D10711yescharacteristic
215M16—22no—
216M16—2142no—
217C2×C8—1072no—
218C2×C8C22×D10712yes—
219C4⋊C4—21no—
220C4⋊C4—1072no—
221C4⋊C4—2142no—
222C4⋊C4—2142no—
223C4⋊C4—2141no—
224C4⋊C4—2141no—
225C4⋊C4—2141no—
226C4⋊C4C22×D10712yes—
227C22⋊C4—2141no—
228C22⋊C4—2141no—
229C22⋊C4—2141no—
230C22⋊C4—2141no—
231C42—1071no—
232C42—1071no—
233C42—1071no—
234C42—1071no—
235C23—21no—
236C23—2141no—
237C23D4×D10711yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—4284no—
241Q8—4284no—
242Q8—2141no—
243Q8—2141no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—4282no—
248D4—4282no—
249D4—4282no—
250D4—2141no—
251D4—2141no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—1071no—
257C2×C4—1071no—
258C2×C4—1071no—
259C2×C4—1071no—
260C2×C4—4282no—
261C2×C4—4282no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2141no—
265C2×C4—2142no—
266C2×C4D4×D10711yescharacteristic
267C2×C4#1313381712yes—
268C2×C4#1313381812yes—
269C2×C4C23×D10711yescharacteristic
270C8—22no—
271C8—2142no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—4281no—
278C22—4281no—
279C22—4281no—
280C22#1681209512yes—
281C22C2×D4×D10711yescharacteristic
282C4—41no—
283C4—42no—
284C4—1072no—
285C4—1072no—
286C4—4281no—
287C4—4282no—
288C4—2141no—
289C4—2141no—
290C4#1681209512yes—
291C4C2×#1313381711yescharacteristic
292C4C2×#1313381811yescharacteristic
293C2—22no—
294C2—81no—
295C2—8561no—
296C2C2×#1681209511yescharacteristic
297C2AGL(1,ℤ/8)×D10711yescharacteristic
298C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radical#20293173C2
hypercenter#417D107
hypocenterC107#3141
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16812108C2semidirect
3. #20292988C2semidirect
4. #20292985C2semidirect
5. #20292976C2semidirect
6. #20292972C2semidirect
7. #20292629C2semidirect
8. #20292628C2semidirect
9. #20292626C2semidirect
10. M16×Dic107C2semidirect
11. #417×C107C2semidirect
150. C107#3141semidirect
160. #417D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
C2×#168121082
#202929882
#202929852
#202929762
#202929722
#202926292
#202926282
#202926262
M16×Dic1072
#417×C1072
#3141107