Finite Group Databasebeta

← #28345926

Subgroups — #28345926

Order 13696.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16812136C212yesmaximal, complemented
2#20293178C211yesmaximal, characteristic, complemented
3C4×D4×D107C211yesmaximal, characteristic, complemented
4#20292845C212yesmaximal, complemented
5#20292844C212yesmaximal, complemented
6#20292829C211yesmaximal, characteristic, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292822C211yesmaximal, characteristic, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292627C212yesmaximal, complemented
11#461×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×D4×D107C2211yescharacteristic
14C22×C4×D107—21no—
15#16812136—28no—
16C2×#13133805C2212yes—
17C2×D856C2212yes—
18C2×#13133803C2212yes—
19C2×#13133801C2212yes—
20#16812106C2211yescharacteristic, complemented
21#16812098C2212yescomplemented
22#16812084—21no—
23#16812084—21no—
24C4×D428C2211yescharacteristic
25D4×Dic107C2211yescharacteristic
26#16812071—21no—
27#16812071—21no—
28#16812071C2211yescharacteristic
29C4×C22⋊D107—21no—
30(C22⋊D107)⋊C4—21no—
31C4⋊C4×D107C2211yescharacteristic
32C22⋊C4×D107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35Dic107⋊C8C2212yescomplemented
36#16812033C2212yescomplemented
37#16812031C2212yescomplemented
38#16812029C2212yescomplemented
39C2×SD16×C107C2212yes—
40C4⋊Q8×C107C2211yescharacteristic, complemented
41C4×D4×C107C2211yescharacteristic
42C4⋊C8×C107C2211yescharacteristic, complemented
43#103×C107C2212yescomplemented
44C23×D107—21no—
45#13133819—24no—
46#13133819—24no—
47C2×D428—22no—
48C2×D428—21no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53D4×D107—22no—
54D4×D107D414yescomplemented
55C22×Dic107—21no—
56C2×C4×D107—21no—
57C2×C4×D107—21no—
58C2×C4×D107—21no—
59C2×C4×D107—22no—
60C2×C4×D107—44no—
61C2×C4×D107D411yescharacteristic
62C2×C4×D107D411yescharacteristic
63C2×C4×D107C2311yescharacteristic
64#13133805—24no—
65D856—24no—
66#13133803—24no—
67#13133801—24no—
68C2×C107⋊C8C2312yes—
69C4⋊Dic107C2311yescharacteristic
70Dic107⋊C4—21no—
71C22⋊Dic107—21no—
72(C2×D107)⋊C4—21no—
73(C2×D107)⋊C4—21no—
74(C2×D107)⋊C4—22no—
75(C2×D107)⋊C4—22no—
76C4×Dic107D411yescharacteristic
77C4×Dic107D411yescharacteristic
78C4×Dic107C2311yescharacteristic
79C2×Q8×C107C2312yes—
80C2×D4×C107C2311yescharacteristic
81C22×C4×C107—21no—
82SD16×C107—24no—
83C2×C8×C107C2312yes—
84C4⋊C4×C107—21no—
85C4⋊C4×C107—21no—
86C4⋊C4×C107C2311yescharacteristic
87C22⋊C4×C107—21no—
88C42×C107C2311yescharacteristic
89C23×C107—21no—
90C22×D107—21no—
91C22×D107—24no—
92C22×D107—22no—
93C22×D107—42no—
94C22×D107—42no—
95C22×D107C2×D411yescharacteristic
96Q8×C107—22no—
97Q8×C107—22no—
98D4×C107—21no—
99D4×C107C2×D412yes—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—24no—
107C2×Dic107—21no—
108C2×Dic107—42no—
109C2×Dic107C2×D411yescharacteristic
110C2×Dic107C2×D411yescharacteristic
111C2×Dic107C2×D411yescharacteristic
112D428—22no—
113D428—21no—
114D428—22no—
115D428—42no—
116D428—44no—
117D428—42no—
118D428C2×D412yes—
119C4×D107—22no—
120C4×D107—41no—
121C4×D107—42no—
122C4×D107—42no—
123C4×D107—42no—
124C4×D107—44no—
125C4×D107C2×D412yes—
126C4×D107#4112yes—
127C8×C107—22no—
128C107⋊C8—22no—
129C22×C107—22no—
130C22×C107—41no—
131C22×C107—41no—
132C22×C107C22×D411yescharacteristic
133C2×D107—21no—
134C2×D107—21no—
135C2×D107—22no—
136C2×D107—48no—
137C2×D107—41no—
138C2×D107—41no—
139C2×D107—42no—
140C2×D107—42no—
141C2×D107—42no—
142C2×D107#12112yes—
143C4×C107—21no—
144C4×C107—42no—
145C4×C107—41no—
146C4×C107AGL(1,ℤ/8)12yescomplemented
147C4×C107C22×D411yescharacteristic
148C4×C107C2×#4111yescharacteristic
149Dic107—21no—
150Dic107—21no—
151Dic107—41no—
152Dic107#12112yes—
153C2×C107—42no—
154C2×C107C2×#12111yescharacteristic
155C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C107#57511yescharacteristic
157D107—22no—
158D107—42no—
159D107—82no—
160#3036—1071nomaximal, Sylow
161C107#303611yescharacteristic, complemented, Sylow
162C2×AGL(1,ℤ/8)—1072no—
163#532—1071no—
164C2×C4×D4—1071no—
165#463—1072no—
166#463—1072no—
167#461—1071no—
168#461D10711yescharacteristic, complemented
169#459—1071no—
170#459—1071no—
171#423—1071no—
172#417—1072no—
173C2×#41—1072no—
174C22×D4—1071no—
175C23×C4—2141no—
176AGL(1,ℤ/8)—2148no—
177C2×SD16—1072no—
178C2×SD16C2×D10712yes—
179C2×D8—1072no—
180C2×D8—1072no—
181C2×M16—1072no—
182C4⋊Q8C2×D10711yescharacteristic, complemented
183#127—1071no—
184#124—1072no—
185#121—2141no—
186#121—2141no—
187C4×D4—1071no—
188C4×D4—1071no—
189C4×D4—1071no—
190C4×D4—2141no—
191C4×D4—2141no—
192C4×D4—2141no—
193C4×D4—2141no—
194C4×D4C2×D10711yescharacteristic
195C2×C4⋊C4—1071no—
196C2×C22⋊C4—2141no—
197C2×C42—1071no—
198C4⋊C8—1072no—
199C4⋊C8—1071no—
200C4⋊C8C2×D10711yescharacteristic, complemented
201#103—1072no—
202#103C2×D10712yescomplemented
203#102—1072no—
204#102—1072no—
205C24—2141no—
206#41—2144no—
207#41—2144no—
208C2×Q8C22×D10712yes—
209C2×D4—1074no—
210C2×D4—1071no—
211C2×D4—1072no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4—2141no—
215C2×D4—2141no—
216C2×D4—2141no—
217C2×D4C22×D10711yescharacteristic
218C22×C4—21no—
219C22×C4—1071no—
220C22×C4—1071no—
221C22×C4—1071no—
222C22×C4—4284no—
223C22×C4—2141no—
224C22×C4—2142no—
225C22×C4—2141no—
226C22×C4—2141no—
227C22×C4—2141no—
228SD16—24no—
229SD16—2144no—
230D8—2144no—
231D8—2144no—
232M16—2144no—
233C2×C8—1072no—
234C2×C8C22×D10712yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—1071no—
238C4⋊C4—2141no—
239C4⋊C4C22×D10711yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—2141no—
242C22⋊C4—2142no—
243C22⋊C4—2142no—
244C22⋊C4—2141no—
245C22⋊C4—2141no—
246C42—1071no—
247C42—1071no—
248C42—1071no—
249C42C22×D10711yescharacteristic
250C23—21no—
251C23—1071no—
252C23—4282no—
253C23—4282no—
254C23—2144no—
255C23—2142no—
256C23—2141no—
257Q8—22no—
258Q8—22no—
259D4—21no—
260D4—1072no—
261D4—4282no—
262D4—4284no—
263D4—4282no—
264D4—2144no—
265D4—2142no—
266D4—2142no—
267D4—2141no—
268D4D4×D10712yes—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—1071no—
273C2×C4—1072no—
274C2×C4—1072no—
275C2×C4—1071no—
276C2×C4—1071no—
277C2×C4—4281no—
278C2×C4—4284no—
279C2×C4—4282no—
280C2×C4—4282no—
281C2×C4—4282no—
282C2×C4—4282no—
283C2×C4—2141no—
284C2×C4—2142no—
285C2×C4D4×D10711yescharacteristic
286C2×C4D4×D10711yescharacteristic
287C2×C4C23×D10711yescharacteristic
288C8—22no—
289C8—2142no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—1072no—
294C22—4288no—
295C22—4282no—
296C22—4282no—
297C22—4282no—
298C22—4281no—
299C22—4281no—
300C22—2142no—
301C22—2141no—
302C22—2141no—
303C22C2×D4×D10711yescharacteristic
304C4—21no—
305C4—41no—
306C4—42no—
307C4—1072no—
308C4—4281no—
309C4—2141no—
310C4—2141no—
311C4#1681213612yescomplemented
312C4C2×D4×D10711yescharacteristic
313C4#41×D10711yescharacteristic
314C2—42no—
315C2—8562no—
316C2—4282no—
317C2—2142no—
318C2#121×D10711yescharacteristic
319C2C2×#1681213611yescharacteristic
320C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
hypercenter#461D107
hypocenterC107#3036
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. #20293178C2semidirect
3. C4×D4×D107C2semidirect
4. #20292845C2semidirect
5. #20292844C2semidirect
6. #20292829C2semidirect
7. #20292823C2semidirect
8. #20292822C2semidirect
9. #20292649C2semidirect
10. #20292627C2semidirect
11. #461×C107C2semidirect
20. #16812106C22semidirect
21. #16812098C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. Dic107⋊C8C22semidirect
36. #16812033C22semidirect
37. #16812031C22semidirect
38. #16812029C22semidirect
40. C4⋊Q8×C107C22semidirect
42. C4⋊C8×C107C22semidirect
43. #103×C107C22semidirect
54. D4×D107D4semidirect
146. C4×C107AGL(1,ℤ/8)semidirect
161. C107#3036semidirect
168. #461D107semidirect
182. C4⋊Q8C2×D107semidirect
200. C4⋊C8C2×D107semidirect
202. #103C2×D107semidirect
311. C4#16812136semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
#202931782
C4×D4×D1072
#202928452
#202928442
#202928292
#202928232
#202928222
#202926492
#202926272
#461×C1072
#3036107