Finite Group Databasebeta

← #28345818

Subgroups — #28345818

Order 13696.

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

normal subgroups only

This group has 762 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
1#20293425C212yesmaximal, complemented
2#20293194C211yesmaximal, characteristic, complemented
3#20292805C214yesmaximal, complemented
4#20292795C214yesmaximal, complemented
5#20292644C212yesmaximal, complemented
6C2×#16812026C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic, complemented
9#16812140—22no—
10#16812139C2214yescomplemented
11#16812135—22no—
12#16812134C2214yescomplemented
13#16812129—24no—
14C2×#13133802C2212yescomplemented
15#16812105C2212yescomplemented
16#16812096C2212yescomplemented
17#16812080—22no—
18#16812078—22no—
19#16812068C2211yescharacteristic, complemented
20#16812036C2218yescomplemented
21#16812026C2214yescomplemented
22H(5,𝔽2)×C107—24no—
23H(5,𝔽2)×C107—22no—
24H(5,𝔽2)×C107C2214yes—
25C2×#41×C107—22no—
26C2×#41×C107C2212yes—
27C22×D4×C107—22no—
28C22×D4×C107—22no—
29C22×D4×C107C2211yescharacteristic
30#13133818—22no—
31#13133818—22no—
32C2×Dic214C2311yescharacteristic, complemented
33C2×D428C2311yescharacteristic, complemented
34C2×C22⋊D107—21no—
35C2×C22⋊D107—24no—
36C2×C4×D107—21no—
37#13133808—24no—
38#13133806—24no—
39#13133805—24no—
40#13133803—24no—
41#13133802—22no—
42#13133802C2314yescomplemented
43C2×C107⋊C8—22no—
44C4⋊Dic107—21no—
45C22⋊Dic107—21no—
46C22⋊Dic107—24no—
47C4×Dic107C2312yescomplemented
48C24×C107—22no—
49C24×C107D412yes—
50#41×C107—28no—
51#41×C107—28no—
52#41×C107—22no—
53#41×C107—22no—
54#41×C107D414yescomplemented
55#41×C107C2314yes—
56C2×Q8×C107D412yescomplemented
57C2×D4×C107—24no—
58C2×D4×C107—28no—
59C2×D4×C107—28no—
60C2×D4×C107—22no—
61C2×D4×C107—28no—
62C2×D4×C107—28no—
63C2×D4×C107—22no—
64C2×D4×C107—22no—
65C2×D4×C107D412yes—
66C2×D4×C107D412yescomplemented
67C2×D4×C107C2312yes—
68C22×C4×C107—22no—
69C22×C4×C107—22no—
70C22×C4×C107C2311yescharacteristic
71C23×C107—28no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107—82no—
76C23×C107—84no—
77C23×C107C2×D414yes—
78C23×C107C2×D411yescharacteristic
79C22×D107—21no—
80Q8×C107—22no—
81Q8×C107C2×D414yes—
82D4×C107—216no—
83D4×C107—24no—
84D4×C107—24no—
85D4×C107—24no—
86D4×C107—22no—
87D4×C107—24no—
88D4×C107C2×D414yes—
89C2×C4×C107—28no—
90C2×C4×C107—22no—
91C2×C4×C107—24no—
92C2×C4×C107—24no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—41no—
98C22⋊D107—41no—
99C22⋊D107—84no—
100C2×Dic107—22no—
101C2×Dic107—21no—
102D428—22no—
103C4×D107—42no—
104Dic214—22no—
105C107⋊C8—24no—
106C22×C107—24no—
107C22×C107—24no—
108C22×C107—41no—
109C22×C107—44no—
110C22×C107—44no—
111C22×C107—44no—
112C22×C107—82no—
113C22×C107—88no—
114C22×C107#12011yescharacteristic
115C22×C107#12011yescharacteristic
116C22×C107C22×D411yescharacteristic
117C2×D107—42no—
118C2×D107—41no—
119C4×C107—24no—
120C4×C107#12012yes—
121C4×C107C22×D412yes—
122Dic107—42no—
123Dic107—41no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—44no—
128C2×C107—44no—
129C2×C107C2×#12011yescharacteristic
130D107—81no—
131#3019—1071nomaximal, Sylow
132C107#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)D10711yescharacteristic, complemented
134#575—1072no—
135#535—1071no—
136#455—1074no—
137#453—1074no—
138#421—1072no—
139C2×#100—1071no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D10714yes—
143C2×#41—22no—
144C2×#41—1071no—
145C2×#41C2×D10712yes—
146C22×D4—22no—
147C22×D4—22no—
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#127—1072no—
156#124—1072no—
157#121—2142no—
158#120—2142no—
159#117—1071no—
160#104—1078no—
161#100—1074no—
162C24—22no—
163C24C22⋊D10712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—2142no—
169#41—2142no—
170#41C22⋊D10714yescomplemented
171#41C22×D10714yes—
172C2×Q8—1071no—
173C2×Q8C22⋊D10712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—1071no—
183C2×D4—2144no—
184C2×D4—2141no—
185C2×D4C22⋊D10712yes—
186C2×D4C22⋊D10712yescomplemented
187C2×D4C22×D10712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2141no—
191C22×C4C22×D10711yescharacteristic
192Q16—2144no—
193SD16—2144no—
194SD16—2144no—
195D8—2144no—
196M16—1074no—
197M16—2142no—
198C2×C8—2142no—
199C4⋊C4—2141no—
200C22⋊C4—2141no—
201C22⋊C4—2144no—
202C42—1072no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—2141no—
210C23C2×C22⋊D10714yes—
211C23C2×C22⋊D10711yescharacteristic
212Q8—22no—
213Q8—2142no—
214Q8C2×C22⋊D10714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—8564no—
222D4—4281no—
223D4—4281no—
224D4—2142no—
225D4C2×C22⋊D10714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—4282no—
231C2×C4—2142no—
232C2×C4—2141no—
233C2×C4C2×C22⋊D10711yescharacteristic
234C2×C4C2×C22⋊D10712yes—
235C2×C4C2×C22⋊D10712yes—
236C2×C4C23×D10711yescharacteristic
237C8—2144no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—4282no—
247C22—4281no—
248C22#1681207811yescharacteristic
249C22#1681207811yescharacteristic
250C22C22×C22⋊D10711yescharacteristic
251C4—24no—
252C4—4282no—
253C4—4281no—
254C4#1681207812yes—
255C4C22×C22⋊D10712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—8561no—
262C2C2×#1681207811yescharacteristic
subgroupG/H
centerC2C2×#16812078
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×H(5,𝔽2)×C107C2
socleC2×C107C2×#120
hypercenterC2×H(5,𝔽2)D107
hypocenterC107#3019
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. #20293194C2semidirect
3. #20292805C2semidirect
4. #20292795C2semidirect
5. #20292644C2semidirect
6. C2×#16812026C2semidirect
7. C2×H(5,𝔽2)×C107C2semidirect
8. C2×#13133818C22semidirect
10. #16812139C22semidirect
12. #16812134C22semidirect
14. C2×#13133802C22semidirect
15. #16812105C22semidirect
16. #16812096C22semidirect
19. #16812068C22semidirect
20. #16812036C22semidirect
21. #16812026C22semidirect
32. C2×Dic214C23semidirect
33. C2×D428C23semidirect
42. #13133802C23semidirect
47. C4×Dic107C23semidirect
54. #41×C107D4semidirect
56. C2×Q8×C107D4semidirect
66. C2×D4×C107D4semidirect
132. C107#3019semidirect
133. C2×H(5,𝔽2)D107semidirect
170. #41C22⋊D107semidirect
173. C2×Q8C22⋊D107semidirect
186. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202931942
#202928052
#202927952
#202926442
C2×#168120262
C2×H(5,𝔽2)×C1072
#3019107