Finite Group Databasebeta

← #28345840

Subgroups — #28345840

Order 13696.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16812139C212yesmaximal, complemented
2#20293198C211yesmaximal, characteristic, complemented
3#20292810C214yesmaximal, complemented
4#20292805C214yesmaximal, complemented
5#20292644C212yesmaximal, complemented
6#20292611C211yesmaximal, characteristic, complemented
7#585×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic, complemented
9#16812139—24no—
10#16812139C2218yescomplemented
11C2×#13133808—22no—
12C2×#13133806—22no—
13C2×#13133802C2212yescomplemented
14#16812107C2212yescomplemented
15#16812096C2212yescomplemented
16#16812090—22no—
17#16812088—22no—
18#16812068C2211yescharacteristic, complemented
19#16812036C2218yescomplemented
20#16812028C2212yescomplemented
21#16812026C2212yescomplemented
222−1+4×C107—22no—
232−1+4×C107C2212yes—
24H(5,𝔽2)×C107—22no—
25H(5,𝔽2)×C107—22no—
26H(5,𝔽2)×C107C2212yes—
27C2×#41×C107—24no—
28C2×#41×C107—22no—
29C2×#41×C107C2212yes—
30C2×#41×C107C2211yescharacteristic
31C2×Dic214—22no—
32C2×Dic214—22no—
33C2×Dic214—21no—
34C2×Dic214C2312yes—
35C22×Dic107—21no—
36#13133808—28no—
37#13133806—28no—
38#13133802—22no—
39#13133802C2314yes—
40C2×C107⋊C8—22no—
41C4⋊Dic107—21no—
42Dic107⋊C4—24no—
43C22⋊Dic107—21no—
44C22⋊Dic107—24no—
45C4×Dic107C2312yes—
46#41×C107—24no—
47#41×C107—28no—
48#41×C107—24no—
49#41×C107—24no—
50#41×C107—24no—
51#41×C107—22no—
52#41×C107—22no—
53#41×C107—24no—
54#41×C107—22no—
55#41×C107—22no—
56#41×C107D414yes—
57#41×C107C2314yes—
58C2×Q8×C107—22no—
59C2×Q8×C107—24no—
60C2×Q8×C107D412yes—
61C2×Q8×C107C2311yescharacteristic
62C2×D4×C107—28no—
63C2×D4×C107—22no—
64C2×D4×C107—24no—
65C2×D4×C107—22no—
66C2×D4×C107—24no—
67C2×D4×C107D412yes—
68C2×D4×C107D412yes—
69C2×D4×C107C2311yescharacteristic
70C22×C4×C107—24no—
71C22×C4×C107—22no—
72C22×C4×C107D412yes—
73C22×C4×C107C2311yescharacteristic
74C23×C107—24no—
75C23×C107—22no—
76C23×C107C2×D412yes—
77C23×C107C2×D411yescharacteristic
78Q8×C107—24no—
79Q8×C107—22no—
80Q8×C107—22no—
81Q8×C107C2×D414yes—
82D4×C107—28no—
83D4×C107—24no—
84D4×C107—24no—
85D4×C107—24no—
86D4×C107—24no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107C2×D414yes—
90C2×C4×C107—24no—
91C2×C4×C107—28no—
92C2×C4×C107—22no—
93C2×C4×C107—24no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—24no—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2411yescharacteristic
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—42no—
105Dic214—24no—
106Dic214—41no—
107Dic214—41no—
108C107⋊C8—24no—
109C22×C107—22no—
110C22×C107—24no—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107—82no—
114C22×C107—84no—
115C22×C107#12012yes—
116C22×C107C22×D411yescharacteristic
117C4×C107—24no—
118C4×C107—22no—
119C4×C107#12011yescharacteristic
120C4×C107#12011yescharacteristic
121C4×C107C22×D412yes—
122Dic107—42no—
123Dic107—42no—
124C2×C107—21no—
125C2×C107—22no—
126C2×C107—42no—
127C2×C107—44no—
128C2×C107C2×#12011yescharacteristic
129#3022—1071nomaximal, Sylow
130C107#302211yescharacteristic, complemented, Sylow
131#585D10711yescharacteristic
132C2×#137—1072no—
133#536—1071no—
134#456—1074no—
135#455—1074no—
136#421—1072no—
137#413—1071no—
1382−1+4—22no—
1392−1+4C2×D10712yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D10712yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D10712yes—
146C2×#41C2×D10711yescharacteristic
147C22×Q8—1071no—
148#137—1078no—
149#137—2144no—
150C2×Q16—2142no—
151C2×SD16—2142no—
152C2×M16—1072no—
153C4⋊Q8—1072no—
154#124—1072no—
155#123—2142no—
156C22⋊Q8—2142no—
157#117—1071no—
158#104—1078no—
159#101—1072no—
160#100—1072no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D10714yes—
172#41C22×D10714yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—1072no—
176C2×Q8—2142no—
177C2×Q8—2142no—
178C2×Q8—2141no—
179C2×Q8C22⋊D10712yes—
180C2×Q8C22×D10711yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D10712yes—
187C2×D4C22⋊D10712yes—
188C2×D4C22×D10711yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—2141no—
192C22×C4C22⋊D10712yes—
193C22×C4C22×D10711yescharacteristic
194Q16—2148no—
195SD16—2148no—
196M16—1074no—
197M16—2142no—
198C2×C8—2142no—
199C4⋊C4—2141no—
200C4⋊C4—2144no—
201C22⋊C4—2141no—
202C22⋊C4—2144no—
203C42—1072no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D10712yes—
207C23C2×C22⋊D10711yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—4281no—
212Q8—4281no—
213Q8—2144no—
214Q8C2×C22⋊D10714yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D10714yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—4282no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4C2×C22⋊D10712yes—
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×C22⋊D10712yes—
236C2×C4C2×C22⋊D10712yes—
237C2×C4C23×D10711yescharacteristic
238C8—2144no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1681207812yes—
246C22C22×C22⋊D10711yescharacteristic
247C4—24no—
248C4—22no—
249C4—4282no—
250C4—4282no—
251C4#1681207811yescharacteristic
252C4#1681207811yescharacteristic
253C4C22×C22⋊D10712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1681207811yescharacteristic
subgroupG/H
centerC2C2×#16812078
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#585×C107C2
socleC2×C107C2×#120
hypercenter#585D107
hypocenterC107#3022
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. #20293198C2semidirect
3. #20292810C2semidirect
4. #20292805C2semidirect
5. #20292644C2semidirect
6. #20292611C2semidirect
8. C22×Dic214C22semidirect
10. #16812139C22semidirect
13. C2×#13133802C22semidirect
14. #16812107C22semidirect
15. #16812096C22semidirect
18. #16812068C22semidirect
19. #16812036C22semidirect
20. #16812028C22semidirect
21. #16812026C22semidirect
130. C107#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
#202931982
#202928102
#202928052
#202926442
#202926112
#585×C1072
#3022107