Finite Group Databasebeta

← #28345741

Subgroups — #28345741

Order 13696.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812155C211yesmaximal, characteristic
2C2×#16812139C211yesmaximal, characteristic, complemented
3C22×#13133807C211yesmaximal, characteristic, complemented
4#20293203C211yesmaximal, characteristic, complemented
5#20292792C214yesmaximal, complemented
6#20292779C214yesmaximal, complemented
7#20292631C211yesmaximal, characteristic, complemented
8#20292593C211yesmaximal, characteristic, complemented
9#416×C107C211yesmaximal, characteristic, complemented
10#16812155—24no—
11#16812155—24no—
12C2×#13133819C2212yes—
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C22×Dic214C2211yescharacteristic, complemented
16C22×D428—21no—
17C22×D428C2211yescharacteristic
18C2×D4×D107—22no—
19C2×D4×D107C2212yes—
20#16812139—24no—
21C2×#13133808C2212yescomplemented
22C2×#13133807—24no—
23C2×#13133807C2214yescomplemented
24C2×#13133806C2212yescomplemented
25C2×#13133802C2211yescharacteristic, complemented
26#16812109—21no—
27#16812097—21no—
28#16812093—22no—
29#16812089C2214yescomplemented
30#16812029C2214yescomplemented
31(C2×D107)⋊C8C2214yescomplemented
32C2×#41×C107C2211yescharacteristic
33C22×C8×C107C2211yescharacteristic, complemented
34#117×C107C2211yescharacteristic, complemented
35#103×C107C2212yescomplemented
36#102×C107C2212yescomplemented
37C23×D107—21no—
38C23×D107—22no—
39#13133819—24no—
40#13133819—24no—
41#13133818—28no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214—22no—
45C2×Dic214—22no—
46C2×Dic214—21no—
47C2×Dic214D411yescharacteristic
48C2×Dic214C2312yes—
49C2×D428—28no—
50C2×D428—22no—
51C2×D428—21no—
52C2×D428—22no—
53C2×D428D411yescharacteristic
54C2×D428D414yes—
55C2×D428C2312yes—
56C2×C22⋊D107—22no—
57C2×C22⋊D107D412yes—
58D4×D107—216no—
59D4×D107—24no—
60D4×D107—24no—
61C22×Dic107—21no—
62C2×C4×D107—22no—
63C2×C4×D107C2312yes—
64#13133808—24no—
65#13133807—28no—
66#13133806—24no—
67#13133802—22no—
68C2×C107⋊C8C2312yes—
69C4⋊Dic107—24no—
70(C2×D107)⋊C4—24no—
71#41×C107—22no—
72#41×C107D414yes—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107—21no—
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78C2×C8×C107—22no—
79C2×C8×C107C2312yes—
80C4⋊C4×C107C2312yes—
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107—22no—
86C22×D107—28no—
87C22×D107—24no—
88C22×D107—84no—
89C22×D107—82no—
90C22×D107C2×D412yes—
91Q8×C107—21no—
92Q8×C107C2×D412yes—
93D4×C107—24no—
94D4×C107—21no—
95D4×C107C2×D412yes—
96C2×C4×C107—21no—
97C2×C4×C107—24no—
98C2×C4×C107—22no—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—28no—
104C22⋊D107—24no—
105C2×Dic107—22no—
106C2×Dic107—42no—
107C2×Dic107C2×D412yes—
108D428—28no—
109D428—22no—
110D428—24no—
111D428—22no—
112D428C2×D414yes—
113C4×D107—28no—
114C4×D107—22no—
115C4×D107—22no—
116Dic214—22no—
117Dic214—22no—
118Dic214—22no—
119Dic214—42no—
120C8×C107—22no—
121C107⋊C8—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—42no—
125C22×C107#12012yes—
126C22×C107C22×D411yescharacteristic
127C2×D107—22no—
128C2×D107—24no—
129C2×D107—24no—
130C2×D107—44no—
131C2×D107—84no—
132C2×D107—88no—
133C4×C107—22no—
134C4×C107—42no—
135C4×C107#12012yes—
136C4×C107C22×D411yescharacteristic
137C4×C107C22×D411yescharacteristic
138Dic107—22no—
139Dic107—42no—
140C2×C107—22no—
141C2×C107—42no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57712yes—
144D107—42no—
145D107—44no—
146#3009—1071nomaximal, Sylow
147C107#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1071no—
149C2×#137—1071no—
150C22×SD16—1071no—
151#536—1071no—
152#452—1074no—
153#450—1074no—
154#418—1071no—
155#416D10711yescharacteristic, complemented
156#408—1071no—
157H(5,𝔽2)—2144no—
158H(5,𝔽2)—2144no—
159C2×#41—1071no—
160C2×#41—1072no—
161C2×#41—2141no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8—1071no—
164C22×D4—1072no—
165C22×D4—1071no—
166C22×D4—2141no—
167C22×D4—2142no—
168#137—2144no—
169C2×Q16—1072no—
170C2×SD16—1072no—
171C2×SD16—1074no—
172C2×SD16—2144no—
173C2×M16—1071no—
174C22×C8C2×D10711yescharacteristic, complemented
175C4⋊Q8—2141no—
176#124—2141no—
177#123—2142no—
178C22⋊Q8—1074no—
179#117C2×D10711yescharacteristic, complemented
180#103C2×D10712yescomplemented
181#102—1074no—
182#102C2×D10712yescomplemented
183C22⋊C8—1074no—
184C24—2142no—
185C24—2141no—
186#41—22no—
187#41—2148no—
188#41—2144no—
189#41—2144no—
190#41—2142no—
191#41—2142no—
192#41D42814yes—
193C2×Q8—1071no—
194C2×Q8—1072no—
195C2×Q8—2142no—
196C2×Q8—2142no—
197C2×Q8—2141no—
198C2×Q8C22×D10711yescharacteristic
199C2×D4—21no—
200C2×D4—1072no—
201C2×D4—1074no—
202C2×D4—1071no—
203C2×D4—1072no—
204C2×D4—2142no—
205C2×D4—2148no—
206C2×D4—2142no—
207C2×D4—2144no—
208C2×D4—21416no—
209C2×D4—2144no—
210C2×D4—2141no—
211C2×D4—2142no—
212C2×D4C22×D10711yescharacteristic
213C22×C4—21no—
214C22×C4—1072no—
215C22×C4—2142no—
216C22×C4—2141no—
217C22×C4C22×D10711yescharacteristic
218Q16—2144no—
219SD16—2148no—
220SD16—2144no—
221M16—2142no—
222C2×C8—22no—
223C2×C8—1072no—
224C2×C8C22×D10712yes—
225C4⋊C4—2144no—
226C4⋊C4C22×D10712yes—
227C22⋊C4—21no—
228C22⋊C4—2144no—
229C42—21no—
230C23—21no—
231C23—8562no—
232C23—8564no—
233C23—1072no—
234C23—2148no—
235C23—2144no—
236C23—2142no—
237C23D4×D10711yescharacteristic
238Q8—21no—
239Q8—4282no—
240Q8—2142no—
241Q8—2142no—
242Q8—2142no—
243Q8C2×D42812yes—
244D4—24no—
245D4—21no—
246D4—1074no—
247D4—2148no—
248D4—2148no—
249D4—2144no—
250D4—2144no—
251D4—2142no—
252D4—2142no—
253D4C2×D42812yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—1072no—
258C2×C4—4282no—
259C2×C4—2148no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4D4×D10711yescharacteristic
264C2×C4D4×D10712yes—
265C2×C4C2×D42812yes—
266C2×C4C23×D10711yescharacteristic
267C8—22no—
268C8—2142no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—8564no—
273C22—8568no—
274C22—4284no—
275C22—2144no—
276C22—2144no—
277C22—2142no—
278C22#1681207912yes—
279C22C2×D4×D10711yescharacteristic
280C4—22no—
281C4—42no—
282C4—4282no—
283C4—2142no—
284C4#1681207912yes—
285C4C2×D4×D10711yescharacteristic
286C4C22×D42811yescharacteristic
287C2—22no—
288C2—42no—
289C2—4284no—
290C2—4282no—
291C2C2×#1681207911yescharacteristic
292C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812155C2
hypercenter#416D107
hypocenterC107#3009
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#16812139C2semidirect
3. C22×#13133807C2semidirect
4. #20293203C2semidirect
5. #20292792C2semidirect
6. #20292779C2semidirect
7. #20292631C2semidirect
8. #20292593C2semidirect
9. #416×C107C2semidirect
15. C22×Dic214C22semidirect
21. C2×#13133808C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133806C22semidirect
25. C2×#13133802C22semidirect
29. #16812089C22semidirect
30. #16812029C22semidirect
31. (C2×D107)⋊C8C22semidirect
33. C22×C8×C107C22semidirect
34. #117×C107C22semidirect
35. #103×C107C22semidirect
36. #102×C107C22semidirect
147. C107#3009semidirect
155. #416D107semidirect
174. C22×C8C2×D107semidirect
179. #117C2×D107semidirect
180. #103C2×D107semidirect
182. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168121392
C22×#131338072
#202932032
#202927922
#202927792
#202926312
#202925932
#416×C1072
#3009107