Finite Group Databasebeta

← #28345839

Subgroups — #28345839

Order 13696.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#20293463C211yesmaximal, characteristic
2C2×#16812140C212yesmaximal, complemented
3#20292809C214yesmaximal, complemented
4#20292804C214yesmaximal, complemented
5#20292645C212yesmaximal, complemented
6#20292611C211yesmaximal, characteristic, complemented
7#536×C107C211yesmaximal, characteristic, complemented
8#16812157C2212yes—
9#16812156—22no—
10#16812155—22no—
11#16812154—22no—
12#16812154C2212yes—
13C2×#13133818—22no—
14C2×#13133818C2212yes—
15#41×D107—24no—
16#16812140—24no—
17#16812140C2218yescomplemented
18C2×#13133808—22no—
19C2×#13133805—22no—
20C2×#13133802C2212yescomplemented
21#16812037C2218yescomplemented
22#16812028C2212yescomplemented
23#16812026C2212yescomplemented
24C2×#41×C107C2211yescharacteristic
25C22×Q8×C107C2211yescharacteristic, complemented
26C4⋊Q8×C107C2212yescomplemented
27#124×C107C2212yescomplemented
28#123×C107—22no—
29C22⋊Q8×C107—22no—
30#117×C107C2211yescharacteristic, complemented
31#13133819—24no—
32#13133818—24no—
33#13133818—24no—
34#13133818—22no—
35#13133818—24no—
36#13133818—22no—
37#13133818D414yes—
38#13133818C2314yes—
39#13133817—28no—
40#13133817—24no—
41C2×Dic214—22no—
42C2×Dic214D412yes—
43Q8×D107—24no—
44C2×D428—22no—
45C2×D428D412yes—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—22no—
48D4×D107—28no—
49D4×D107—24no—
50C2×C4×D107—24no—
51C2×C4×D107—22no—
52#13133808—28no—
53#13133805—28no—
54#13133802—22no—
55#13133802C2314yes—
56C2×C107⋊C8—22no—
57#41×C107—22no—
58#41×C107—22no—
59C2×Q8×C107—21no—
60C2×Q8×C107—22no—
61C2×Q8×C107—22no—
62C2×Q8×C107C2312yes—
63C2×Q8×C107C2311yescharacteristic
64C2×D4×C107D412yes—
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67C22×C4×C107D412yes—
68C22×C4×C107C2311yescharacteristic
69C4⋊C4×C107—21no—
70C4⋊C4×C107—24no—
71C22⋊C4×C107—21no—
72C22⋊C4×C107—24no—
73C42×C107C2312yes—
74C23×C107C2×D412yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—24no—
77C22×D107—22no—
78Q8×C107—22no—
79Q8×C107—24no—
80Q8×C107—41no—
81Q8×C107—41no—
82D4×C107—24no—
83D4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—42no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—28no—
95C22⋊D107—24no—
96C22⋊D107—24no—
97C2×Dic107—24no—
98C2×Dic107—22no—
99D428—24no—
100D428—22no—
101D428C2×D414yes—
102C4×D107—28no—
103C4×D107—24no—
104C4×D107—24no—
105Dic214—24no—
106Dic214—22no—
107Dic214C2×D414yes—
108C107⋊C8—24no—
109C22×C107—22no—
110C22×C107—41no—
111C22×C107—42no—
112C22×C107#12012yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—24no—
115C2×D107—82no—
116C2×D107—84no—
117C4×C107—22no—
118C4×C107—42no—
119C4×C107—42no—
120C4×C107#12011yescharacteristic
121C4×C107#12011yescharacteristic
122C4×C107C22×D412yes—
123Dic107—24no—
124C2×C107—21no—
125C2×C107—22no—
126C2×C107—42no—
127C2×C107C2×#12011yescharacteristic
128D107—44no—
129#3022—1071nomaximal, Sylow
130C107#302211yescharacteristic, complemented, Sylow
131#585—1071no—
132C2×#137—1072no—
133#536D10711yescharacteristic, complemented
134#456—1074no—
135#455—1074no—
136#421—1072no—
137#413—1071no—
1382−1+4—1072no—
1392−1+4—2142no—
140H(5,𝔽2)—1072no—
141H(5,𝔽2)—2142no—
142H(5,𝔽2)—2142no—
143C2×#41—1072no—
144C2×#41—2144no—
145C2×#41—2142no—
146C2×#41C2×D10711yescharacteristic
147C22×Q8C2×D10711yescharacteristic, complemented
148#137—1078no—
149#137—2144no—
150C2×Q16—2142no—
151C2×SD16—2142no—
152C2×M16—1072no—
153C4⋊Q8C2×D10712yescomplemented
154#124C2×D10712yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D10711yescharacteristic, complemented
158#104—1078no—
159#101—1072no—
160#100—1072no—
161#41—22no—
162#41—22no—
163#41—1074no—
164#41—1074no—
165#41—2144no—
166#41—2144no—
167#41—2144no—
168#41—2148no—
169#41—2144no—
170#41—2142no—
171#41—2142no—
172#41—2144no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—1072no—
177C2×Q8—2142no—
178C2×Q8—2144no—
179C2×Q8C22×D10712yes—
180C2×Q8C22×D10711yescharacteristic
181C2×D4—1072no—
182C2×D4—2142no—
183C2×D4—2144no—
184C2×D4—2148no—
185C2×D4—2142no—
186C2×D4—2144no—
187C2×D4C22⋊D10712yes—
188C2×D4C22×D10711yescharacteristic
189C22×C4—21no—
190C22×C4—2144no—
191C22×C4—2142no—
192C22×C4C22⋊D10712yes—
193C22×C4C22×D10711yescharacteristic
194Q16—2148no—
195SD16—2148no—
196M16—1074no—
197M16—2142no—
198C2×C8—2142no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D10712yes—
204C23—2144no—
205C23—2142no—
206C23C2×C22⋊D10712yes—
207C23C2×C22⋊D10711yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—1074no—
213Q8—2144no—
214Q8—2142no—
215D4—24no—
216D4—22no—
217D4—1074no—
218D4—2144no—
219D4—2148no—
220D4—2144no—
221D4—2144no—
222D4—2142no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—2148no—
229C2×C4—2144no—
230C2×C4—2142no—
231C2×C4—2144no—
232C2×C4—2144no—
233C2×C4D4×D10712yes—
234C2×C4D4×D10712yes—
235C2×C4C2×C22⋊D10712yes—
236C2×C4C2×C22⋊D10711yescharacteristic
237C2×C4C23×D10711yescharacteristic
238C8—2144no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8562no—
243C22—8564no—
244C22—2144no—
245C22#1681207712yes—
246C22C22×C22⋊D10711yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—2144no—
251C4#1681207711yescharacteristic
252C4#1681207711yescharacteristic
253C4C2×D4×D10712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—4284no—
258C2C2×#1681207711yescharacteristic
subgroupG/H
centerC2C2×#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC2×C107C2×#120
prime radical#20293463C2
hypercenter#536D107
hypocenterC107#3022
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20292809C2semidirect
4. #20292804C2semidirect
5. #20292645C2semidirect
6. #20292611C2semidirect
7. #536×C107C2semidirect
17. #16812140C22semidirect
20. C2×#13133802C22semidirect
21. #16812037C22semidirect
22. #16812028C22semidirect
23. #16812026C22semidirect
25. C22×Q8×C107C22semidirect
26. C4⋊Q8×C107C22semidirect
27. #124×C107C22semidirect
30. #117×C107C22semidirect
130. C107#3022semidirect
133. #536D107semidirect
147. C22×Q8C2×D107semidirect
153. C4⋊Q8C2×D107semidirect
154. #124C2×D107semidirect
157. #117C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
C2×#168121402
#202928092
#202928042
#202926452
#202926112
#536×C1072
#3022107