Finite Group Databasebeta

← #28346153

Subgroups — #28346153

Order 13696.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic, complemented
3#20293208C211yesmaximal, characteristic
4#20293203C211yesmaximal, characteristic, complemented
5#20292886C212yesmaximal, complemented
6#20292874C212yesmaximal, complemented
7#20292869C212yesmaximal, complemented
8#20292857C212yesmaximal, complemented
9#20292621C211yesmaximal, characteristic, complemented
10C2×#16812032C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14C22×Dic214C2211yescharacteristic
15#16812142—24no—
16C2×Dic428C2212yes—
17C2×#13133807C2212yes—
18#16812117—24no—
19C2×#13133801C2212yes—
20C22×C107⋊C8C2211yescharacteristic
21C2×C8×D107C2212yes—
22#16812109—21no—
23#16812097—21no—
24#16812093—22no—
25#16812089—21no—
26#16812089C2212yes—
27#16812089C2212yes—
28#16812089C2212yes—
29#16812084—21no—
30#16812084C2212yes—
31#16812083—22no—
32#16812032C2211yescharacteristic
33#16812032C2212yes—
34#16812032C2212yes—
35#16812032C2211yescharacteristic
36#16812029C2211yescharacteristic
37#16812029C2211yescharacteristic
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yes—
42#106×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45#13133818—22no—
46#13133818—22no—
47C2×Dic214—22no—
48C2×Dic214—22no—
49C2×Dic214—21no—
50C2×Dic214D411yescharacteristic
51C2×Dic214C2312yes—
52C2×Dic214C2311yescharacteristic
53C2×D428—22no—
54C2×D428D411yescharacteristic
55C2×D428C2311yescharacteristic
56C2×C22⋊D107—21no—
57C2×C22⋊D107D412yes—
58C22×Dic107—21no—
59C2×C4×D107—21no—
60C2×C4×D107C2312yes—
61Dic428—24no—
62#13133807—24no—
63#13133801—24no—
64C2×C107⋊C8—22no—
65C2×C107⋊C8C2311yescharacteristic
66C2×C107⋊C8C2311yescharacteristic
67C8×D107—24no—
68C4⋊Dic107—22no—
69C4⋊Dic107—22no—
70C4⋊Dic107—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73(C2×D107)⋊C4—22no—
74(C2×D107)⋊C4—22no—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107D414yescomplemented
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2312yes—
81C4⋊C4×C107C2311yescharacteristic
82C4⋊C4×C107C2311yescharacteristic
83C22⋊C4×C107—21no—
84C42×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—21no—
87C22×D107C2×D412yes—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—81no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D412yes—
94C2×C4×C107#4112yes—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—24no—
97C22⋊D107—42no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—42no—
101C2×Dic107C2×D412yes—
102D428—22no—
103D428—21no—
104D428—21no—
105D428—82no—
106C4×D107—22no—
107C4×D107—22no—
108C4×D107—42no—
109Dic214—22no—
110Dic214—22no—
111Dic214—21no—
112Dic214—22no—
113Dic214—21no—
114Dic214—42no—
115C8×C107C2×D414yes—
116C107⋊C8—21no—
117C107⋊C8—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107#12112yes—
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—41no—
128C4×C107—42no—
129C4×C107—41no—
130C4×C107—41no—
131C4×C107#12112yes—
132C4×C107C22×D411yescharacteristic
133C4×C107C2×#4111yescharacteristic
134Dic107—22no—
135Dic107—42no—
136Dic107—41no—
137C2×C107—22no—
138C2×C107C2×#12111yescharacteristic
139C2×C107#57711yescharacteristic
140C2×C107#57811yescharacteristic
141D107—42no—
142D107—81no—
143#3063—1071nomaximal, Sylow
144C107#306311yescharacteristic, complemented, Sylow
145C2×#137—1071no—
146C2×#131—1071no—
147#537—1071no—
148#536—1071no—
149#470—1072no—
150#468—1072no—
151#467—1072no—
152#465—1072no—
153#428D10711yescharacteristic, complemented
154#416—1071no—
155C2×#103—1071no—
156C2×#41—1071no—
157C2×#41—1071no—
158C22×Q8—1071no—
159#137—2144no—
160C2×Q16—1072no—
161C2×SD16—1072no—
162#131—2144no—
163C2×M16—1072no—
164C2×M16C2×D10711yescharacteristic
165C22×C8—1071no—
166C22×C8—1072no—
167C4⋊Q8—2141no—
168#124—2141no—
169#123—2142no—
170C22⋊Q8—1072no—
171C22⋊Q8—1072no—
172C22⋊Q8—1072no—
173C22⋊Q8—2141no—
174#121—1072no—
175#121—2141no—
176#121—2142no—
177#117C2×D10711yescharacteristic
178C2×C4⋊C4C2×D10711yescharacteristic
179#107C2×D10712yes—
180#106C2×D10712yes—
181#103—1072no—
182#103—1072no—
183#103—1071no—
184#103—1071no—
185#102—1071no—
186#102—1071no—
187#41—2142no—
188#41—2142no—
189#41—2142no—
190#41—2142no—
191C2×Q8—1071no—
192C2×Q8—1072no—
193C2×Q8—1071no—
194C2×Q8—2142no—
195C2×Q8—2142no—
196C2×Q8—2141no—
197C2×D4—1072no—
198C2×D4—1071no—
199C2×D4—1071no—
200C2×D4—2142no—
201C2×D4—2141no—
202C22×C4—21no—
203C22×C4—1072no—
204C22×C4—2141no—
205C22×C4—2141no—
206C22×C4C22×D10711yescharacteristic
207Q16—2144no—
208SD16—2144no—
209M16—2144no—
210M16D42814yescomplemented
211C2×C8—1071no—
212C2×C8—1071no—
213C2×C8—2142no—
214C2×C8—2144no—
215C2×C8C22×D10712yes—
216C4⋊C4—21no—
217C4⋊C4—2142no—
218C4⋊C4—2142no—
219C4⋊C4—2142no—
220C4⋊C4C22×D10711yescharacteristic
221C4⋊C4C22×D10711yescharacteristic
222C4⋊C4C22×D10712yes—
223C22⋊C4—21no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C22⋊C4—2142no—
228C42—21no—
229C23—1072no—
230C23—2141no—
231C23D4×D10711yescharacteristic
232Q8—4282no—
233Q8—2142no—
234Q8—2141no—
235Q8—2142no—
236Q8—2141no—
237Q8—2142no—
238D4—8562no—
239D4—4282no—
240D4—2144no—
241D4—2142no—
242D4—2141no—
243D4—2141no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—81no—
248C2×C4—1072no—
249C2×C4—4282no—
250C2×C4—4282no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4C2×D42812yes—
257C2×C4#1313381912yes—
258C2×C4C23×D10711yescharacteristic
259C8—2141no—
260C8—2141no—
261C8C2×D42814yes—
262C22—21no—
263C22—21no—
264C22—4281no—
265C22—4281no—
266C22—4281no—
267C22—2142no—
268C22—2142no—
269C22—2142no—
270C22#1681208412yes—
271C22C2×D4×D10711yescharacteristic
272C4—42no—
273C4—41no—
274C4—41no—
275C4—4281no—
276C4—4282no—
277C4—2142no—
278C4#1681208412yes—
279C4C22×D42811yescharacteristic
280C4C2×#1313381911yescharacteristic
281C2—22no—
282C2—8561no—
283C2—4282no—
284C2C2×#1681208411yescharacteristic
285C2#2029343711yescharacteristic
286C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radical#20293208C2
hypercenter#428D107
hypocenterC107#3063
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×#16812117C2semidirect
4. #20293203C2semidirect
5. #20292886C2semidirect
6. #20292874C2semidirect
7. #20292869C2semidirect
8. #20292857C2semidirect
9. #20292621C2semidirect
10. C2×#16812032C2semidirect
11. #428×C107C2semidirect
77. M16×C107D4semidirect
144. C107#3063semidirect
153. #428D107semidirect
210. M16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121172
#202932082
#202932032
#202928862
#202928742
#202928692
#202928572
#202926212
C2×#168120322
#428×C1072
#3063107