Finite Group Databasebeta

← #28345842

Subgroups — #28345842

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×#16812142C212yesmaximal, complemented
3#20293198C211yesmaximal, characteristic, complemented
4#20292812C214yesmaximal, complemented
5#20292807C214yesmaximal, complemented
6#20292643C212yesmaximal, complemented
7#413×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—
16C22×Dic214C2211yescharacteristic, complemented
17#16812142—24no—
18#16812142C2218yescomplemented
19C2×Dic428—22no—
20C2×#13133807—22no—
21#16812107C2212yescomplemented
22#16812096C2212yescomplemented
23#16812090—22no—
24#16812088—22no—
25#16812068C2211yescharacteristic, complemented
26#16812035C2218yescomplemented
27C2×#41×C107C2211yescharacteristic
28C2×M16×C107C2212yescomplemented
29#101×C107C2212yescomplemented
30#100×C107C2212yescomplemented
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×Dic214—22no—
43C2×Dic214—22no—
44C2×Dic214—21no—
45C2×Dic214D412yes—
46C2×Dic214C2312yes—
47Q8×D107—24no—
48C2×D428—22no—
49C2×D428D412yes—
50C2×C22⋊D107—24no—
51C2×C22⋊D107—22no—
52D4×D107—28no—
53D4×D107—24no—
54C22×Dic107—21no—
55C2×C4×D107—24no—
56C2×C4×D107—22no—
57Dic428—28no—
58#13133807—28no—
59C4⋊Dic107—21no—
60Dic107⋊C4—24no—
61C22⋊Dic107—21no—
62C22⋊Dic107—24no—
63C4×Dic107C2312yes—
64#41×C107—22no—
65#41×C107—22no—
66C2×Q8×C107C2311yescharacteristic
67C2×D4×C107D412yes—
68C2×D4×C107C2311yescharacteristic
69C22×C4×C107D412yes—
70C22×C4×C107C2311yescharacteristic
71M16×C107—22no—
72M16×C107C2314yes—
73C2×C8×C107—22no—
74C23×C107C2×D412yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—24no—
77C22×D107—22no—
78Q8×C107—22no—
79D4×C107—24no—
80D4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C22⋊D107—28no—
89C22⋊D107—24no—
90C22⋊D107—24no—
91C2×Dic107—24no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—42no—
96D428—24no—
97D428—22no—
98D428C2×D414yes—
99C4×D107—28no—
100C4×D107—24no—
101C4×D107—24no—
102Dic214—24no—
103Dic214—22no—
104Dic214—24no—
105Dic214—41no—
106Dic214—41no—
107Dic214C2×D414yes—
108C8×C107—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#12011yescharacteristic
119C4×C107#12011yescharacteristic
120C4×C107C22×D412yes—
121Dic107—24no—
122Dic107—42no—
123Dic107—42no—
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#536—1071no—
134#456—1074no—
135#455—1074no—
136#421—1072no—
137#413D10711yescharacteristic, complemented
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×Q8—1071no—
148#137—1078no—
149#137—2144no—
150C2×Q16—2142no—
151C2×SD16—2142no—
152C2×M16C2×D10712yescomplemented
153C4⋊Q8—1072no—
154#124—1072no—
155#123—2142no—
156C22⋊Q8—2142no—
157#117—1071no—
158#104—1078no—
159#101C2×D10712yescomplemented
160#100C2×D10712yescomplemented
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—1072no—
174C2×Q8—1072no—
175C2×Q8—2142no—
176C2×Q8—2142no—
177C2×Q8—2142no—
178C2×Q8—2144no—
179C2×Q8—2141no—
180C2×Q8C22×D10711yescharacteristic
181C2×D4—1072no—
182C2×D4—2142no—
183C2×D4—2144no—
184C2×D4—2148no—
185C2×D4—2142no—
186C2×D4—2144no—
187C2×D4D42812yes—
188C2×D4C22×D10711yescharacteristic
189C22×C4—2144no—
190C22×C4—2141no—
191C22×C4—2142no—
192C22×C4D42812yes—
193C22×C4C22×D10711yescharacteristic
194Q16—2148no—
195SD16—2148no—
196M16—22no—
197M16C22×D10714yes—
198C2×C8—22no—
199C4⋊C4—2141no—
200C4⋊C4—2144no—
201C22⋊C4—2141no—
202C22⋊C4—2144no—
203C42—1072no—
204C23—2144no—
205C23—2142no—
206C23C2×D42812yes—
207C23C2×D42811yescharacteristic
208Q8—22no—
209Q8—1074no—
210Q8—4281no—
211Q8—4281no—
212Q8—2144no—
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—4282no—
226C2×C4—2148no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2144no—
230C2×C4—2142no—
231C2×C4—2144no—
232C2×C4—2144no—
233C2×C4D4×D10712yes—
234C2×C4D4×D10712yes—
235C2×C4C2×D42812yes—
236C2×C4C2×D42811yescharacteristic
237C2×C4C23×D10711yescharacteristic
238C8—24no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8562no—
243C22—8564no—
244C22—2144no—
245C22#1681207912yes—
246C22C22×D42811yescharacteristic
247C4—22no—
248C4—4282no—
249C4—4282no—
250C4—2144no—
251C4#1681207911yescharacteristic
252C4#1681207911yescharacteristic
253C4C2×D4×D10712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—4284no—
258C2C2×#1681207911yescharacteristic
subgroupG/H
centerC2C2×#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#413×C107C2
socleC2×C107C2×#120
prime radical#20293463C2
hypercenter#413D107
hypocenterC107#3022
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. C2×#16812142C2semidirect
3. #20293198C2semidirect
4. #20292812C2semidirect
5. #20292807C2semidirect
6. #20292643C2semidirect
7. #413×C107C2semidirect
16. C22×Dic214C22semidirect
18. #16812142C22semidirect
21. #16812107C22semidirect
22. #16812096C22semidirect
25. #16812068C22semidirect
26. #16812035C22semidirect
28. C2×M16×C107C22semidirect
29. #101×C107C22semidirect
30. #100×C107C22semidirect
130. C107#3022semidirect
137. #413D107semidirect
152. C2×M16C2×D107semidirect
159. #101C2×D107semidirect
160. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
C2×#168121422
#202931982
#202928122
#202928072
#202926432
#413×C1072
#3022107