Finite Group Databasebeta

← #28345864

Subgroups — #28345864

Order 13696.

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

normal subgroups only

This group has 772 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, complemented
2#20293196C212yesmaximal, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292815C218yesmaximal, complemented
5#20292600C212yesmaximal, complemented
6#410×C107C211yesmaximal, characteristic, complemented
7#16812157—21no—
8#16812156—22no—
9#16812155—22no—
10#16812154—21no—
11#16812154C2214yes—
12C2×#13133818—22no—
13C2×#13133818C2212yes—
14#41×D107—24no—
15C22×D428C2211yescharacteristic, complemented
16#16812106—22no—
17#16812105—21no—
18#16812098—21no—
19#16812097—22no—
20#16812084—24no—
21#16812083—22no—
22#16812079C2218yescomplemented
23#16812077C2214yescomplemented
24#16812068C2211yescharacteristic, complemented
25#16812023C2218yescomplemented
26C2×#41×C107C2211yescharacteristic
27#117×C107C2212yescomplemented
28#99×C107C2214yescomplemented
29C23×D107C2312yes—
30#13133819—24no—
31#13133818—24no—
32#13133818—24no—
33#13133818—22no—
34#13133818—22no—
35#13133818—24no—
36#13133818—22no—
37#13133818—22no—
38#13133817—24no—
39#13133817—28no—
40C2×Dic214—22no—
41C2×Dic214D412yescomplemented
42Q8×D107—24no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—28no—
49C2×D428D412yescomplemented
50C2×C22⋊D107—24no—
51C2×C22⋊D107—24no—
52C2×C22⋊D107C2314yes—
53D4×D107—24no—
54D4×D107—28no—
55C2×C4×D107—24no—
56C2×C4×D107D414yescomplemented
57C4⋊Dic107—21no—
58C22⋊Dic107C2312yes—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—28no—
61C4×Dic107—21no—
62#41×C107—22no—
63#41×C107—22no—
64C2×Q8×C107D411yescharacteristic, complemented
65C2×D4×C107D411yescharacteristic, complemented
66C2×D4×C107C2312yes—
67C22×C4×C107D412yescomplemented
68C22×C4×C107C2311yescharacteristic
69C4⋊C4×C107—22no—
70C22⋊C4×C107C2314yes—
71C42×C107—22no—
72C23×C107C2×D412yes—
73C23×C107C2411yescharacteristic
74C22×D107—24no—
75C22×D107—22no—
76C22×D107—24no—
77C22×D107—82no—
78C22×D107C2×D414yes—
79Q8×C107—22no—
80D4×C107—24no—
81D4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—24no—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C22⋊D107—28no—
91C22⋊D107—24no—
92C22⋊D107—24no—
93C22⋊D107—84no—
94C2×Dic107—24no—
95C2×Dic107—22no—
96C2×Dic107C2×D414yes—
97D428—24no—
98D428—22no—
99D428—22no—
100D428—41no—
101D428—42no—
102D428—41no—
103D428—88no—
104C4×D107—28no—
105C4×D107—24no—
106C4×D107—24no—
107Dic214—24no—
108Dic214—22no—
109Dic214—22no—
110C22×C107—22no—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107C22×D412yes—
114C22×C107C22×D411yescharacteristic
115C2×D107—24no—
116C2×D107—44no—
117C2×D107—48no—
118C2×D107—44no—
119C2×D107—42no—
120C2×D107—84no—
121C4×C107—22no—
122C4×C107—44no—
123C4×C107#12011yescharacteristic
124C4×C107#12012yes—
125C4×C107#12011yescharacteristic
126Dic107—24no—
127Dic107—42no—
128C2×C107—21no—
129C2×C107—22no—
130C2×C107—42no—
131C2×C107C2×#12011yescharacteristic
132D107—44no—
133D107—82no—
134#3027—1071nomaximal, Sylow
135C107#302711yescharacteristic, complemented, Sylow
136#585—1071no—
137#535—1071no—
138#535—1072no—
139B(4,𝔽2)—1078no—
140#410—1072no—
141#410D10711yescharacteristic, complemented
1422−1+4—2142no—
1432−1+4—2141no—
144H(5,𝔽2)—1074no—
145H(5,𝔽2)—2142no—
146H(5,𝔽2)—2141no—
147C2×#41—1072no—
148C2×#41—2144no—
149C2×#41—2142no—
150C2×#41C2×D10711yescharacteristic
151C22×D4—1071no—
152#127—2142no—
153#127—2141no—
154#124—2142no—
155#124—2141no—
156#121—2144no—
157#121—2142no—
158#120—1074no—
159#120—1078no—
160#117—1071no—
161#117C2×D10712yescomplemented
162#99—1078no—
163#99C2×D10714yescomplemented
164C24—1072no—
165#41—22no—
166#41—22no—
167#41—2144no—
168#41—2144no—
169#41—2148no—
170#41—2144no—
171#41—2144no—
172#41—2142no—
173#41—2142no—
174#41—2142no—
175#41—2142no—
176#41—2144no—
177C2×Q8—1072no—
178C2×Q8—2142no—
179C2×Q8—2144no—
180C2×Q8D42811yescharacteristic, complemented
181C2×D4—1074no—
182C2×D4—1072no—
183C2×D4—2148no—
184C2×D4—2142no—
185C2×D4—2148no—
186C2×D4—2144no—
187C2×D4—2144no—
188C2×D4—2144no—
189C2×D4—2142no—
190C2×D4—2142no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4D42811yescharacteristic, complemented
194C2×D4C22×D10712yes—
195C22×C4—1074no—
196C22×C4—2144no—
197C22×C4D42812yescomplemented
198C22×C4C22×D10711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—2141no—
201C22⋊C4—1072no—
202C22⋊C4—2148no—
203C22⋊C4—2144no—
204C22⋊C4C22×D10714yes—
205C42—22no—
206C42—2141no—
207C23—8562no—
208C23—1074no—
209C23—2144no—
210C23—2144no—
211C23—2142no—
212C23C2×D42812yes—
213C23C23×D10711yescharacteristic
214Q8—22no—
215Q8—2144no—
216Q8—2142no—
217Q8—2142no—
218D4—24no—
219D4—22no—
220D4—8568no—
221D4—8564no—
222D4—4281no—
223D4—4281no—
224D4—4282no—
225D4—2144no—
226D4—2148no—
227D4—2144no—
228D4—2144no—
229D4—2142no—
230D4—2142no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—1074no—
235C2×C4—2148no—
236C2×C4—2142no—
237C2×C4—2144no—
238C2×C4—2144no—
239C2×C4—2144no—
240C2×C4D4×D10712yes—
241C2×C4D4×D10712yes—
242C2×C4C2×D42812yes—
243C2×C4C2×D42811yescharacteristic
244C2×C4C2×D42811yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—8564no—
249C22—4288no—
250C22—4284no—
251C22—4284no—
252C22—4282no—
253C22—2144no—
254C22C2×D4×D10712yes—
255C22C22×D42811yescharacteristic
256C4—22no—
257C4—44no—
258C4—4282no—
259C4—2144no—
260C4#1681207911yescharacteristic
261C4#1681207911yescharacteristic
262C4#1681207912yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—8562no—
267C2—4284no—
268C2C2×#1681207911yescharacteristic
subgroupG/H
centerC2C2×#16812079
commutator, cosocleC23×C107C24
FrattiniC23C23×D107
Fitting#410×C107C2
socleC2×C107C2×#120
hypercenter#410D107
hypocenterC107#3027
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #20293463C2semidirect
2. #20293196C2semidirect
3. #20293191C2semidirect
4. #20292815C2semidirect
5. #20292600C2semidirect
6. #410×C107C2semidirect
15. C22×D428C22semidirect
22. #16812079C22semidirect
23. #16812077C22semidirect
24. #16812068C22semidirect
25. #16812023C22semidirect
27. #117×C107C22semidirect
28. #99×C107C22semidirect
41. C2×Dic214D4semidirect
49. C2×D428D4semidirect
56. C2×C4×D107D4semidirect
64. C2×Q8×C107D4semidirect
65. C2×D4×C107D4semidirect
67. C22×C4×C107D4semidirect
135. C107#3027semidirect
141. #410D107semidirect
161. #117C2×D107semidirect
163. #99C2×D107semidirect
180. C2×Q8D428semidirect
193. C2×D4D428semidirect
197. C22×C4D428semidirect

Maximal subgroups

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

subgroupindex
#202934632
#202931962
#202931912
#202928152
#202926002
#410×C1072
#3027107