Finite Group Databasebeta

← #28345817

Subgroups — #28345817

Order 13696.

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

normal subgroups only

This group has 762 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
1C2×#16812154C211yesmaximal, characteristic, complemented
2#20293425C212yesmaximal, complemented
3#20292804C214yesmaximal, complemented
4#20292794C214yesmaximal, complemented
5#20292645C212yesmaximal, complemented
6C2×#16812026C211yesmaximal, characteristic, complemented
7#535×C107C211yesmaximal, characteristic, complemented
8#16812154—24no—
9#16812154—22no—
10#16812154C2214yes—
11C2×#13133818C2212yes—
12C2×#13133817—22no—
13C22×C22⋊D107—22no—
14C2×D4×D107—22no—
15#16812140C2214yescomplemented
16#16812139—22no—
17#16812135C2214yescomplemented
18#16812134—22no—
19#16812129—24no—
20C2×#13133802C2212yescomplemented
21#16812037C2218yescomplemented
22#16812026C2214yescomplemented
23C2×#41×C107C2211yescharacteristic, complemented
24C22×D4×C107C2211yescharacteristic
25#127×C107C2212yescomplemented
26#124×C107C2212yescomplemented
27#121×C107—22no—
28#120×C107—22no—
29#117×C107C2211yescharacteristic, complemented
30C23×D107—22no—
31#13133818—22no—
32#13133818—22no—
33#13133818D414yescomplemented
34#13133818C2314yes—
35#13133817—28no—
36#13133817—28no—
37C2×Dic214D412yescomplemented
38C2×D428D412yescomplemented
39C2×C22⋊D107—24no—
40C2×C22⋊D107—28no—
41C2×C22⋊D107—22no—
42C2×C22⋊D107—28no—
43D4×D107—28no—
44D4×D107—28no—
45C22×Dic107—22no—
46C2×C4×D107—22no—
47#13133808—24no—
48#13133806—24no—
49#13133805—24no—
50#13133803—24no—
51#13133802—22no—
52#13133802C2314yescomplemented
53C2×C107⋊C8—22no—
54C24×C107D412yes—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107C2311yescharacteristic, complemented
58C2×D4×C107—21no—
59C2×D4×C107—24no—
60C2×D4×C107—22no—
61C2×D4×C107—22no—
62C2×D4×C107D412yes—
63C2×D4×C107C2312yes—
64C2×D4×C107C2311yescharacteristic, complemented
65C22×C4×C107—21no—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—21no—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107C2312yescomplemented
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—21no—
74C23×C107—82no—
75C23×C107C2×D414yes—
76C23×C107C2×D411yescharacteristic
77C22×D107—22no—
78C22×D107—28no—
79C22×D107—84no—
80Q8×C107—22no—
81D4×C107—24no—
82D4×C107—24no—
83D4×C107—22no—
84D4×C107—41no—
85D4×C107—41no—
86D4×C107—84no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—42no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—216no—
95C22⋊D107—24no—
96C22⋊D107—24no—
97C2×Dic107—28no—
98C2×Dic107—22no—
99D428—22no—
100D428C2×D414yes—
101C4×D107—24no—
102C4×D107—24no—
103Dic214—22no—
104Dic214C2×D414yes—
105C107⋊C8—24no—
106C22×C107—24no—
107C22×C107—41no—
108C22×C107—44no—
109C22×C107—44no—
110C22×C107—42no—
111C22×C107—41no—
112C22×C107—82no—
113C22×C107#12011yescharacteristic
114C22×C107#12011yescharacteristic
115C22×C107C22×D411yescharacteristic
116C2×D107—24no—
117C2×D107—44no—
118C2×D107—88no—
119C4×C107—42no—
120C4×C107—41no—
121C4×C107#12012yes—
122C4×C107C22×D412yes—
123Dic107—24no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—44no—
128C2×C107—81no—
129C2×C107C2×#12011yescharacteristic
130D107—44no—
131#3019—1071nomaximal, Sylow
132C107#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—1071no—
134#575—1072no—
135#535D10711yescharacteristic, complemented
136#455—1074no—
137#453—1074no—
138#421—1072no—
139C2×#100—1071no—
140H(5,𝔽2)—1074no—
141H(5,𝔽2)—2144no—
142H(5,𝔽2)—2142no—
143C2×#41—1072no—
144C2×#41—2142no—
145C2×#41C2×D10711yescharacteristic, complemented
146C22×D4—2142no—
147C22×D4—2142no—
148C22×D4C2×D10711yescharacteristic
149#137—1074no—
150#137—2142no—
151AGL(1,ℤ/8)—1074no—
152AGL(1,ℤ/8)—2142no—
153#135—2144no—
154C2×M16—1072no—
155#127C2×D10712yescomplemented
156#124C2×D10712yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D10711yescharacteristic, complemented
160#104—1078no—
161#100—1074no—
162C24—2142no—
163C24C22⋊D10712yes—
164#41—22no—
165#41—22no—
166#41—1074no—
167#41—1074no—
168#41—2148no—
169#41—2148no—
170#41—2142no—
171#41—2142no—
172C2×Q8—1072no—
173C2×Q8C22×D10711yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—1072no—
179C2×D4—2148no—
180C2×D4—2148no—
181C2×D4—2144no—
182C2×D4—2148no—
183C2×D4—2142no—
184C2×D4—2148no—
185C2×D4C22⋊D10712yes—
186C2×D4C22×D10712yes—
187C2×D4C22×D10711yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—2142no—
190C22×C4—2142no—
191C22×C4C22×D10711yescharacteristic
192Q16—2144no—
193SD16—2144no—
194SD16—2144no—
195D8—2144no—
196M16—1074no—
197M16—2142no—
198C2×C8—2142no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D10712yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—8564no—
208C23—2148no—
209C23—2142no—
210C23C2×C22⋊D10714yes—
211C23C2×C22⋊D10711yescharacteristic
212Q8—22no—
213Q8—1074no—
214Q8—2142no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—1074no—
222D4—21416no—
223D4—2144no—
224D4—2144no—
225D4—2142no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—2148no—
230C2×C4—2142no—
231C2×C4—2144no—
232C2×C4—2144no—
233C2×C4D4×D10712yes—
234C2×C4D4×D10712yes—
235C2×C4C2×C22⋊D10711yescharacteristic
236C2×C4C23×D10711yescharacteristic
237C8—2144no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—8568no—
246C22—4284no—
247C22—2144no—
248C22#1681207711yescharacteristic
249C22#1681207711yescharacteristic
250C22C22×C22⋊D10711yescharacteristic
251C4—42no—
252C4—41no—
253C4—2144no—
254C4#1681207712yes—
255C4C2×D4×D10712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—4284no—
262C2C2×#1681207711yescharacteristic
subgroupG/H
centerC2C2×#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC2×C107C2×#120
hypercenter#535D107
hypocenterC107#3019
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812154C2semidirect
2. #20293425C2semidirect
3. #20292804C2semidirect
4. #20292794C2semidirect
5. #20292645C2semidirect
6. C2×#16812026C2semidirect
7. #535×C107C2semidirect
15. #16812140C22semidirect
17. #16812135C22semidirect
20. C2×#13133802C22semidirect
21. #16812037C22semidirect
22. #16812026C22semidirect
23. C2×#41×C107C22semidirect
25. #127×C107C22semidirect
26. #124×C107C22semidirect
29. #117×C107C22semidirect
33. #13133818D4semidirect
37. C2×Dic214D4semidirect
38. C2×D428D4semidirect
52. #13133802C23semidirect
57. C2×Q8×C107C23semidirect
64. C2×D4×C107C23semidirect
70. C42×C107C23semidirect
132. C107#3019semidirect
135. #535D107semidirect
145. C2×#41C2×D107semidirect
155. #127C2×D107semidirect
156. #124C2×D107semidirect
159. #117C2×D107semidirect
173. C2×Q8C22×D107semidirect
187. C2×D4C22×D107semidirect
202. C42C22×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
#202934252
#202928042
#202927942
#202926452
C2×#168120262
#535×C1072
#3019107