Finite Group Databasebeta

← #28345821

Subgroups — #28345821

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#20293427C212yesmaximal, complemented
3#20293194C211yesmaximal, characteristic, complemented
4#20292807C214yesmaximal, complemented
5#20292797C214yesmaximal, complemented
6#20292643C212yesmaximal, complemented
7C2×#100×C107C211yesmaximal, characteristic, complemented
8#16812154—24no—
9#16812154—22no—
10#16812154C2214yes—
11C2×#13133818C2212yes—
12C2×#13133818C2211yescharacteristic, complemented
13C2×#13133817—22no—
14C22×C22⋊D107—22no—
15C2×D4×D107—22no—
16#16812142—22no—
17#16812142C2214yescomplemented
18#16812137—22no—
19#16812137C2214yescomplemented
20#16812131—24no—
21#16812105C2212yescomplemented
22#16812096C2212yescomplemented
23#16812080—22no—
24#16812078—22no—
25#16812068C2211yescharacteristic, complemented
26#16812035C2218yescomplemented
27C22×D4×C107C2211yescharacteristic
28C2×M16×C107C2212yescomplemented
29#100×C107C2214yescomplemented
30C23×D107—22no—
31#13133818—22no—
32#13133818—22no—
33#13133818—22no—
34#13133818—22no—
35#13133818D414yescomplemented
36#13133818C2314yes—
37#13133817—28no—
38#13133817—28no—
39C2×Dic214D412yescomplemented
40C2×Dic214C2311yescharacteristic, complemented
41C2×D428D412yescomplemented
42C2×D428C2311yescharacteristic, complemented
43C2×C22⋊D107—24no—
44C2×C22⋊D107—28no—
45C2×C22⋊D107—22no—
46C2×C22⋊D107—28no—
47C2×C22⋊D107—21no—
48C2×C22⋊D107—24no—
49D4×D107—28no—
50D4×D107—28no—
51C22×Dic107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—21no—
54Dic428—24no—
55#13133807—24no—
56#13133807—24no—
57D856—24no—
58C4⋊Dic107—21no—
59C22⋊Dic107—21no—
60C22⋊Dic107—24no—
61C4×Dic107C2312yescomplemented
62C24×C107D412yes—
63C2×D4×C107—22no—
64C2×D4×C107—22no—
65C2×D4×C107D412yes—
66C2×D4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68M16×C107—22no—
69M16×C107C2314yescomplemented
70C2×C8×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—82no—
74C23×C107C2×D414yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—22no—
77C22×D107—28no—
78C22×D107—21no—
79C22×D107—84no—
80D4×C107—24no—
81D4×C107—24no—
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—216no—
87C22⋊D107—24no—
88C22⋊D107—24no—
89C22⋊D107—41no—
90C22⋊D107—41no—
91C22⋊D107—84no—
92C2×Dic107—28no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—21no—
96D428—22no—
97D428—22no—
98D428C2×D414yes—
99C4×D107—24no—
100C4×D107—24no—
101C4×D107—42no—
102Dic214—22no—
103Dic214—22no—
104Dic214C2×D414yes—
105C8×C107—24no—
106C22×C107—24no—
107C22×C107—41no—
108C22×C107—44no—
109C22×C107—44no—
110C22×C107—82no—
111C22×C107#12011yescharacteristic
112C22×C107#12011yescharacteristic
113C22×C107C22×D411yescharacteristic
114C2×D107—24no—
115C2×D107—44no—
116C2×D107—42no—
117C2×D107—41no—
118C2×D107—88no—
119C4×C107#12012yes—
120C4×C107C22×D412yes—
121Dic107—24no—
122Dic107—42no—
123Dic107—41no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—44no—
128C2×C107C2×#12011yescharacteristic
129D107—44no—
130D107—81no—
131#3019—1071nomaximal, Sylow
132C107#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—1071no—
134#575—1072no—
135#535—1071no—
136#455—1074no—
137#453—1074no—
138#421—1072no—
139C2×#100D10711yescharacteristic, complemented
140H(5,𝔽2)—1074no—
141H(5,𝔽2)—2144no—
142H(5,𝔽2)—2142no—
143C2×#41—1072no—
144C2×#41—1071no—
145C2×#41—2142no—
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×M16C2×D10712yescomplemented
155#127—1072no—
156#124—1072no—
157#121—2142no—
158#120—2142no—
159#117—1071no—
160#104—1078no—
161#100C2×D10714yescomplemented
162C24—2142no—
163C24D42812yes—
164#41—1074no—
165#41—1074no—
166#41—2142no—
167#41—2142no—
168#41—2148no—
169#41—2148no—
170#41—2142no—
171#41—2142no—
172C2×Q8—1072no—
173C2×Q8—1071no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—1072no—
177C2×D4—1071no—
178C2×D4—2148no—
179C2×D4—2148no—
180C2×D4—2144no—
181C2×D4—2141no—
182C2×D4—2144no—
183C2×D4—2148no—
184C2×D4—2142no—
185C2×D4—2148no—
186C2×D4D42812yes—
187C2×D4C22×D10712yes—
188C22×C4—2141no—
189C22×C4—2142no—
190C22×C4—2142no—
191C22×C4C22×D10711yescharacteristic
192Q16—2144no—
193SD16—2144no—
194SD16—2144no—
195D8—2144no—
196M16—22no—
197M16C22×D10714yescomplemented
198C2×C8—22no—
199C4⋊C4—2141no—
200C22⋊C4—2141no—
201C22⋊C4—2144no—
202C42—1072no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—8564no—
207C23—2148no—
208C23—2141no—
209C23—2142no—
210C23C2×D42814yes—
211C23C2×D42811yescharacteristic
212Q8—1074no—
213Q8—2142no—
214Q8—2142no—
215D4—24no—
216D4—24no—
217D4—8564no—
218D4—1074no—
219D4—4281no—
220D4—4281no—
221D4—21416no—
222D4—2144no—
223D4—2144no—
224D4—2142no—
225D4—2142no—
226C2×C4—4282no—
227C2×C4—2142no—
228C2×C4—2141no—
229C2×C4—2148no—
230C2×C4—2142no—
231C2×C4—2144no—
232C2×C4—2144no—
233C2×C4D4×D10712yes—
234C2×C4D4×D10712yes—
235C2×C4C2×D42811yescharacteristic
236C2×C4C23×D10711yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—8568no—
244C22—4282no—
245C22—4284no—
246C22—4281no—
247C22—2144no—
248C22#1681207911yescharacteristic
249C22#1681207911yescharacteristic
250C22C22×D42811yescharacteristic
251C4—4282no—
252C4—4281no—
253C4—2144no—
254C4#1681207912yes—
255C4C2×D4×D10712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—8561no—
261C2—4284no—
262C2C2×#1681207911yescharacteristic
subgroupG/H
centerC2C2×#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#100×C107C2
socleC2×C107C2×#120
hypercenterC2×#100D107
hypocenterC107#3019
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16812154C2semidirect
2. #20293427C2semidirect
3. #20293194C2semidirect
4. #20292807C2semidirect
5. #20292797C2semidirect
6. #20292643C2semidirect
7. C2×#100×C107C2semidirect
12. C2×#13133818C22semidirect
17. #16812142C22semidirect
19. #16812137C22semidirect
21. #16812105C22semidirect
22. #16812096C22semidirect
25. #16812068C22semidirect
26. #16812035C22semidirect
28. C2×M16×C107C22semidirect
29. #100×C107C22semidirect
35. #13133818D4semidirect
39. C2×Dic214D4semidirect
40. C2×Dic214C23semidirect
41. C2×D428D4semidirect
42. C2×D428C23semidirect
61. C4×Dic107C23semidirect
69. M16×C107C23semidirect
132. C107#3019semidirect
139. C2×#100D107semidirect
154. C2×M16C2×D107semidirect
161. #100C2×D107semidirect
197. M16C22×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
#202934272
#202931942
#202928072
#202927972
#202926432
C2×#100×C1072
#3019107