Finite Group Databasebeta

← #28346148

Subgroups — #28346148

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×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812116C211yesmaximal, characteristic, complemented
3#20293205C211yesmaximal, characteristic
4#20292883C212yesmaximal, complemented
5#20292875C212yesmaximal, complemented
6#20292866C212yesmaximal, complemented
7#20292858C212yesmaximal, complemented
8#20292670C211yesmaximal, characteristic, complemented
9#20292621C211yesmaximal, characteristic, complemented
10C2×#16812033C211yesmaximal, characteristic, complemented
11#536×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812140—24no—
14C2×#13133808C2212yes—
15C2×#13133805C2212yes—
16#16812116—24no—
17C2×#13133802C2212yes—
18C2×#13133802C2211yescharacteristic
19C22×C107⋊C8C2212yes—
20C22×C107⋊C8C2211yescharacteristic
21#16812088C2212yes—
22#16812085—21no—
23#16812083C2212yes—
24#16812081—22no—
25#16812080—21no—
26C2×C4⋊Dic107C2211yescharacteristic
27#16812042C2212yes—
28#16812040C2212yes—
29#16812033C2212yes—
30#16812033C2212yes—
31#16812032C2211yescharacteristic
32#16812032C2211yescharacteristic
33#16812029C2211yescharacteristic
34#16812029C2211yescharacteristic
35C2×#41×C107C2211yescharacteristic
36C22×Q8×C107C2211yescharacteristic
37C4⋊Q8×C107—21no—
38#124×C107—21no—
39#123×C107—22no—
40C22⋊Q8×C107C2212yes—
41C22⋊Q8×C107C2212yes—
42#117×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C2×C22⋊D107—22no—
49C22×Dic107—21no—
50C2×C4×D107—21no—
51#13133808—24no—
52#13133805—24no—
53#13133802—24no—
54#13133802D414yescomplemented
55C2×C107⋊C8—22no—
56C2×C107⋊C8—24no—
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2311yescharacteristic
59C2×C107⋊C8C2312yes—
60C4⋊Dic107—21no—
61C4⋊Dic107C2312yes—
62Dic107⋊C4—22no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65#41×C107—22no—
66#41×C107—22no—
67C2×Q8×C107—21no—
68C2×Q8×C107—22no—
69C2×Q8×C107—22no—
70C2×Q8×C107D411yescharacteristic
71C2×Q8×C107C2312yes—
72C2×D4×C107D411yescharacteristic
73C2×D4×C107D412yes—
74C22×C4×C107—21no—
75C22×C4×C107C2312yes—
76C22×C4×C107C2311yescharacteristic
77C4⋊C4×C107—22no—
78C4⋊C4×C107—22no—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C22⋊C4×C107—22no—
83C22⋊C4×C107—22no—
84C42×C107—21no—
85C23×C107C2×D412yes—
86C23×C107C2×D411yescharacteristic
87C22×D107—21no—
88Q8×C107—22no—
89Q8×C107—22no—
90Q8×C107—22no—
91Q8×C107—42no—
92D4×C107—24no—
93D4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107—21no—
98C2×C4×C107—22no—
99C2×C4×C107—42no—
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107#4112yes—
104C2×C4×C107C2411yescharacteristic
105C22⋊D107—42no—
106C22⋊D107—82no—
107C2×Dic107—22no—
108C2×Dic107—21no—
109C2×Dic107—81no—
110D428—21no—
111D428—21no—
112C4×D107—42no—
113Dic214—21no—
114Dic214—21no—
115C107⋊C8—21no—
116C107⋊C8—21no—
117C107⋊C8C2×D414yes—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107#12112yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—41no—
128C4×C107—22no—
129C4×C107—42no—
130C4×C107—41no—
131C4×C107—41no—
132C4×C107#12112yes—
133C4×C107C22×D411yescharacteristic
134C4×C107C2×#4111yescharacteristic
135Dic107—41no—
136Dic107—42no—
137C2×C107—22no—
138C2×C107—42no—
139C2×C107C2×#12111yescharacteristic
140C2×C107#57711yescharacteristic
141C2×C107#57811yescharacteristic
142D107—81no—
143#3063—1071nomaximal, Sylow
144C107#306311yescharacteristic, complemented, Sylow
145C2×#137—1071no—
146C2×#131—1071no—
147#537—1071no—
148#536D10711yescharacteristic, complemented
149#470—1072no—
150#468—1072no—
151#467—1072no—
152#465—1072no—
153#428—1071no—
154#416—1071no—
155C2×#103—1071no—
156C2×#41—1071no—
157C2×#41C2×D10711yescharacteristic
158C22×Q8C2×D10711yescharacteristic
159#137—2144no—
160C2×Q16—1072no—
161C2×SD16—1072no—
162#131—2144no—
163C2×M16—1071no—
164C2×M16—1072no—
165C22×C8—1071no—
166C22×C8—1072no—
167C4⋊Q8—21no—
168#124—21no—
169#123—22no—
170C22⋊Q8—1072no—
171C22⋊Q8—2141no—
172C22⋊Q8C2×D10712yes—
173C22⋊Q8C2×D10712yes—
174#121—1072no—
175#121—2141no—
176#121—2142no—
177#117C2×D10711yescharacteristic
178C2×C4⋊C4—1071no—
179#107—1072no—
180#106—1072no—
181#103—1072no—
182#103—1072no—
183#103—1071no—
184#103—1071no—
185#102—1071no—
186#102—1071no—
187#41—22no—
188#41—22no—
189#41—2142no—
190#41—2142no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1071no—
195C2×Q8C22⋊D10711yescharacteristic
196C2×Q8C22×D10712yes—
197C2×D4—1071no—
198C2×D4—2142no—
199C2×D4—2141no—
200C2×D4C22⋊D10712yes—
201C2×D4C22⋊D10711yescharacteristic
202C22×C4—21no—
203C22×C4—2141no—
204C22×C4—2141no—
205C22×C4C22×D10712yes—
206C22×C4C22×D10711yescharacteristic
207Q16—2144no—
208SD16—2144no—
209M16—1074no—
210M16—2144no—
211C2×C8—1072no—
212C2×C8—1071no—
213C2×C8—1071no—
214C2×C8—2144no—
215C2×C8—2142no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—1072no—
219C4⋊C4—2141no—
220C4⋊C4—2142no—
221C4⋊C4C22×D10711yescharacteristic
222C4⋊C4C22×D10711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—2142no—
227C22⋊C4—2142no—
228C42—21no—
229C23—2141no—
230C23C2×C22⋊D10712yes—
231C23C2×C22⋊D10711yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—2141no—
237Q8—2141no—
238D4—24no—
239D4—22no—
240D4—8562no—
241D4—4282no—
242D4—2141no—
243D4—2141no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—8561no—
251C2×C4—4282no—
252C2×C4—2142no—
253C2×C4—2141no—
254C2×C4D4×D10712yes—
255C2×C4C2×C22⋊D10712yes—
256C2×C4C2×C22⋊D10711yescharacteristic
257C2×C4#1313381712yes—
258C2×C4C23×D10711yescharacteristic
259C8—1074no—
260C8—2141no—
261C8—2141no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—4281no—
268C22—4281no—
269C22—4281no—
270C22#1681208112yes—
271C22C22×C22⋊D10711yescharacteristic
272C4—22no—
273C4—41no—
274C4—41no—
275C4—42no—
276C4—4282no—
277C4—4281no—
278C4#1681208112yes—
279C4C2×D4×D10711yescharacteristic
280C4C2×#1313381711yescharacteristic
281C2—22no—
282C2—42no—
283C2—8561no—
284C2C2×#1681208111yescharacteristic
285C2#2029343511yescharacteristic
286C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
prime radical#20293205C2
hypercenter#536D107
hypocenterC107#3063
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×#16812116C2semidirect
4. #20292883C2semidirect
5. #20292875C2semidirect
6. #20292866C2semidirect
7. #20292858C2semidirect
8. #20292670C2semidirect
9. #20292621C2semidirect
10. C2×#16812033C2semidirect
11. #536×C107C2semidirect
54. #13133802D4semidirect
144. C107#3063semidirect
148. #536D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121162
#202932052
#202928832
#202928752
#202928662
#202928582
#202926702
#202926212
C2×#168120332
#536×C1072
#3063107