Finite Group Databasebeta

← #28346613

Subgroups — #28346613

Order 13696.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293177C211yesmaximal, characteristic, complemented
2C4×#13133818C211yesmaximal, characteristic, complemented
3#20292952C212yesmaximal, complemented
4#20292913C211yesmaximal, characteristic, complemented
5#20292908C211yesmaximal, characteristic, complemented
6#20292828C211yesmaximal, characteristic, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292774C212yesmaximal, complemented
9#20292740C212yesmaximal, complemented
10#20292729C212yesmaximal, complemented
11#432×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×#13133818C2211yescharacteristic
14C2×#13133807C2212yes—
15C2×D856C2212yes—
16#16812109C2211yescharacteristic
17#16812106C2211yescharacteristic
18#16812097—21no—
19#16812083—21no—
20#16812083C2212yes—
21C4×Dic214C2211yescharacteristic
22C4×Dic214C2211yescharacteristic
23C4×D428—21no—
24C4×D428C2211yescharacteristic
25C4×D428C2212yes—
26C4×D428C2211yescharacteristic
27C4×C22⋊D107—21no—
28C4×C22⋊D107—21no—
29#16812068C2211yescharacteristic
30#16812066—21no—
31C42×D107—21no—
32#16812043C2212yes—
33#16812041C2212yes—
34#16812034C2212yes—
35#16812031C2212yes—
36#16812031C2212yes—
37#16812031C2212yes—
38C2×C42×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic
41C22⋊C8×C107C2212yes—
42#97×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45#13133818—44no—
46C2×Dic214—21no—
47C2×Dic214C2311yescharacteristic
48C2×D428—21no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107—22no—
54C2×C4×D107—21no—
55C2×C4×D107—21no—
56C2×C4×D107—21no—
57C2×C4×D107—22no—
58#13133807—24no—
59D856—24no—
60C4⋊Dic107—21no—
61C4⋊Dic107C2312yes—
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2311yescharacteristic
64Dic107⋊C4—21no—
65Dic107⋊C4—21no—
66C22⋊Dic107—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—21no—
72C4×Dic107—21no—
73C4×Dic107—21no—
74C22×C4×C107D411yescharacteristic
75C22×C4×C107D411yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C2×C8×C107C2312yes—
78C2×C8×C107C2312yes—
79C42×C107D411yescharacteristic
80C42×C107D411yescharacteristic
81C42×C107C2311yescharacteristic
82C42×C107C2311yescharacteristic
83C23×C107C2×D411yescharacteristic
84C22×D107—21no—
85C22×D107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—41no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—24no—
97C22⋊D107—44no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102D428—21no—
103D428—22no—
104D428—22no—
105D428—42no—
106D428—42no—
107D428#4112yescomplemented
108C4×D107—24no—
109C4×D107—42no—
110C4×D107—44no—
111C4×D107—42no—
112Dic214—21no—
113Dic214—42no—
114Dic214#4112yescomplemented
115C8×C107—22no—
116C8×C107—22no—
117C22×C107—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107C22×D411yescharacteristic
121C2×D107—22no—
122C2×D107—41no—
123C2×D107—41no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—42no—
127C4×C107—21no—
128C4×C107—21no—
129C4×C107—21no—
130C4×C107—21no—
131C4×C107AGL(1,ℤ/8)12yes—
132C4×C107C2×#4111yescharacteristic
133C4×C107C2×#4111yescharacteristic
134Dic107—22no—
135Dic107—42no—
136Dic107—41no—
137Dic107—41no—
138C2×C107—41no—
139C2×C107#52511yescharacteristic
140C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C107#57511yescharacteristic
142D107—42no—
143D107—82no—
144#3115—1071nomaximal, Sylow
145C107#311511yescharacteristic, complemented, Sylow
146#532—1071no—
147C4×#41—1071no—
148#482—1072no—
149#475—1071no—
150#474—1071no—
151#460—1071no—
152#459—1071no—
153#449—1072no—
154#442—1072no—
155#440—1072no—
156#432D10711yescharacteristic, complemented
157C2×#41—1071no—
158C2×#41—2141no—
159C2×SD16—1072no—
160C2×D8—1072no—
161C4⋊Q8—1071no—
162#127—1071no—
163#124—2141no—
164#121—1072no—
165#121—2141no—
166C4×Q8—1071no—
167C4×Q8—1071no—
168C4×D4—1071no—
169C4×D4—1071no—
170C4×D4—1072no—
171C4×D4—2141no—
172C4×D4—2141no—
173C4×D4—2141no—
174#117—1071no—
175#117—2141no—
176C2×C42—2141no—
177C2×C42C2×D10711yescharacteristic
178#107—1072no—
179#106—1072no—
180C4⋊C8C2×D10711yescharacteristic
181C4⋊C8C2×D10711yescharacteristic
182#103—1072no—
183#102—1072no—
184#102—1072no—
185#102—1072no—
186C22⋊C8C2×D10712yes—
187#97C2×D10712yes—
188#41—4284no—
189#41—2142no—
190#41—2142no—
191C2×Q8—1071no—
192C2×Q8—2141no—
193C2×D4—1071no—
194C2×D4—1072no—
195C2×D4—2142no—
196C2×D4—2141no—
197C2×D4—2141no—
198C2×D4—2141no—
199C22×C4—2141no—
200C22×C4—2141no—
201C22×C4—2142no—
202C22×C4—2141no—
203C22×C4D42811yescharacteristic
204C22×C4D42811yescharacteristic
205C22×C4C22×D10711yescharacteristic
206SD16—2144no—
207D8—2144no—
208C2×C8C22×D10712yes—
209C2×C8C22×D10712yes—
210C4⋊C4—1071no—
211C4⋊C4—1071no—
212C4⋊C4—1072no—
213C4⋊C4—2141no—
214C4⋊C4—2141no—
215C4⋊C4—2141no—
216C22⋊C4—2141no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C42—2141no—
222C42—2141no—
223C42—2141no—
224C42D42811yescharacteristic
225C42D42811yescharacteristic
226C42C22×D10711yescharacteristic
227C42C22×D10711yescharacteristic
228C23—2142no—
229C23—2141no—
230C23C2×D42811yescharacteristic
231Q8—1072no—
232Q8—4282no—
233Q8—2141no—
234D4—1072no—
235D4—4282no—
236D4—4284no—
237D4—4282no—
238D4—2144no—
239D4—2142no—
240D4—2142no—
241D4—2141no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—4284no—
247C2×C4—4282no—
248C2×C4—4282no—
249C2×C4—2144no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2141no—
254C2×C4C2×D42811yescharacteristic
255C2×C4C2×D42811yescharacteristic
256C2×C4C2×D42811yescharacteristic
257C2×C4C2×D42811yescharacteristic
258C2×C4C2×D42811yescharacteristic
259C2×C4C23×D10711yescharacteristic
260C8—22no—
261C8—22no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—4282no—
266C22—4282no—
267C22—4282no—
268C22—4281no—
269C22—4281no—
270C22—2142no—
271C22C22×D42811yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—4282no—
277C4—4281no—
278C4—4281no—
279C4—2142no—
280C4#1681213712yes—
281C4#41×D10711yescharacteristic
282C4#41×D10711yescharacteristic
283C2—41no—
284C2—8562no—
285C2—4282no—
286C2#2029313911yescharacteristic
287C2C2×#1681213711yescharacteristic
288C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#432×C107C2
socleC22×C107C22×D4
hypercenter#432D107
hypocenterC107#3115
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #20293177C2semidirect
2. C4×#13133818C2semidirect
3. #20292952C2semidirect
4. #20292913C2semidirect
5. #20292908C2semidirect
6. #20292828C2semidirect
7. #20292823C2semidirect
8. #20292774C2semidirect
9. #20292740C2semidirect
10. #20292729C2semidirect
11. #432×C107C2semidirect
107. D428#41semidirect
114. Dic214#41semidirect
145. C107#3115semidirect
156. #432D107semidirect

Maximal subgroups

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

subgroupindex
#202931772
C4×#131338182
#202929522
#202929132
#202929082
#202928282
#202928232
#202927742
#202927402
#202927292
#432×C1072
#3115107