Finite Group Databasebeta

← #28345737

Subgroups — #28345737

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
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293201C211yesmaximal, characteristic, complemented
4#20292788C214yesmaximal, complemented
5#20292776C214yesmaximal, complemented
6#20292633C211yesmaximal, characteristic, complemented
7#20292623C211yesmaximal, characteristic, complemented
8#20292595C211yesmaximal, characteristic, complemented
9C2×#137×C107C211yesmaximal, characteristic, complemented
10#16812155—24no—
11#16812155—24no—
12C2×#13133819C2212yes—
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C22×D428—21no—
16C22×D428C2211yescharacteristic
17C2×D4×D107—22no—
18C2×D4×D107C2212yes—
19C2×#13133805—24no—
20C2×#13133805C2214yescomplemented
21C22×C107⋊C8C2211yescharacteristic, complemented
22#16812107—21no—
23#16812098—21no—
24#16812092—22no—
25#16812085C2214yescomplemented
26#16812068C2211yescharacteristic, complemented
27#16812033C2212yescomplemented
28#16812031C2214yescomplemented
29#16812030C2212yescomplemented
30(C2×D107)⋊C8C2214yescomplemented
31C2×#41×C107C2211yescharacteristic
32C22×Q8×C107C2211yescharacteristic, complemented
33#137×C107—24no—
34C2×Q16×C107C2212yescomplemented
35C2×SD16×C107C2212yescomplemented
36C2×M16×C107C2211yescharacteristic, complemented
37C23×D107—21no—
38C23×D107—22no—
39#13133819—24no—
40#13133819—24no—
41#13133818—28no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214D411yescharacteristic
45C2×D428—28no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—22no—
49C2×D428D411yescharacteristic
50C2×D428D414yes—
51C2×D428C2312yes—
52C2×C22⋊D107—22no—
53C2×C22⋊D107D412yes—
54D4×D107—216no—
55D4×D107—24no—
56D4×D107—24no—
57C2×C4×D107—22no—
58C2×C4×D107C2312yes—
59#13133805—28no—
60C2×C107⋊C8—22no—
61C2×C107⋊C8C2312yes—
62C4⋊Dic107C2312yes—
63Dic107⋊C4—24no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—24no—
66C4×Dic107—21no—
67#41×C107—22no—
68#41×C107D414yes—
69C2×Q8×C107—21no—
70C2×Q8×C107—22no—
71C2×Q8×C107—22no—
72C2×Q8×C107C2312yes—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107—21no—
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107—21no—
78C22×C4×C107C2311yescharacteristic
79Q16×C107—24no—
80SD16×C107—24no—
81M16×C107—22no—
82C2×C8×C107C2312yes—
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×C107—22no—
93Q8×C107—22no—
94Q8×C107—42no—
95Q8×C107C2×D412yes—
96D4×C107—24no—
97D4×C107—21no—
98D4×C107C2×D412yes—
99C2×C4×C107—21no—
100C2×C4×C107—24no—
101C2×C4×C107—22no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—28no—
108C22⋊D107—24no—
109C2×Dic107—22no—
110C2×Dic107C2×D412yes—
111D428—28no—
112D428—22no—
113D428—24no—
114D428—22no—
115D428C2×D414yes—
116C4×D107—28no—
117C4×D107—22no—
118C4×D107—22no—
119Dic214—22no—
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×#137D10711yescharacteristic, complemented
150C22×SD16—1071no—
151#536—1071no—
152#452—1074no—
153#450—1074no—
154#418—1071no—
155#416—1071no—
156#408—1071no—
157H(5,𝔽2)—2144no—
158H(5,𝔽2)—2144no—
159C2×#41—1071no—
160C2×#41—1072no—
161C2×#41—2141no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8C2×D10711yescharacteristic, complemented
164C22×D4—1072no—
165C22×D4—1071no—
166C22×D4—2141no—
167C22×D4—2142no—
168#137—24no—
169C2×Q16C2×D10712yescomplemented
170C2×SD16—1074no—
171C2×SD16—2144no—
172C2×SD16C2×D10712yescomplemented
173C2×M16C2×D10711yescharacteristic, complemented
174C22×C8—1071no—
175C4⋊Q8—2141no—
176#124—2141no—
177#123—2142no—
178C22⋊Q8—1074no—
179#117—1071no—
180#103—1072no—
181#102—1072no—
182#102—1074no—
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#41C22⋊D10714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1071no—
197C2×Q8C22×D10712yes—
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—21no—
215C22×C4—1072no—
216C22×C4—2142no—
217C22×C4C22×D10711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2148no—
221M16—22no—
222C2×C8—1072no—
223C2×C8—2142no—
224C2×C8C22×D10712yes—
225C4⋊C4—1072no—
226C4⋊C4—2144no—
227C22⋊C4—2144no—
228C22⋊C4—2141no—
229C42—2141no—
230C23—21no—
231C23—8562no—
232C23—8564no—
233C23—1072no—
234C23—2148no—
235C23—2144no—
236C23—2142no—
237C23D4×D10711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2142no—
243Q8C2×C22⋊D10712yes—
244D4—24no—
245D4—21no—
246D4—1074no—
247D4—2148no—
248D4—2148no—
249D4—2144no—
250D4—2144no—
251D4—2142no—
252D4—2142no—
253D4C2×C22⋊D10712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—1072no—
259C2×C4—2148no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4D4×D10711yescharacteristic
264C2×C4D4×D10712yes—
265C2×C4C2×C22⋊D10712yes—
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#1681207712yes—
279C22C2×D4×D10711yescharacteristic
280C4—22no—
281C4—42no—
282C4—4282no—
283C4—2142no—
284C4#1681207712yes—
285C4C2×D4×D10711yescharacteristic
286C4C22×C22⋊D10711yescharacteristic
287C2—22no—
288C2—42no—
289C2—4284no—
290C2—4282no—
291C2C2×#1681207711yescharacteristic
292C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812155C2
hypercenterC2×#137D107
hypocenterC107#3009
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#13133805C2semidirect
3. #20293201C2semidirect
4. #20292788C2semidirect
5. #20292776C2semidirect
6. #20292633C2semidirect
7. #20292623C2semidirect
8. #20292595C2semidirect
9. C2×#137×C107C2semidirect
20. C2×#13133805C22semidirect
21. C22×C107⋊C8C22semidirect
25. #16812085C22semidirect
26. #16812068C22semidirect
27. #16812033C22semidirect
28. #16812031C22semidirect
29. #16812030C22semidirect
30. (C2×D107)⋊C8C22semidirect
32. C22×Q8×C107C22semidirect
34. C2×Q16×C107C22semidirect
35. C2×SD16×C107C22semidirect
36. C2×M16×C107C22semidirect
147. C107#3009semidirect
149. C2×#137D107semidirect
163. C22×Q8C2×D107semidirect
169. C2×Q16C2×D107semidirect
172. C2×SD16C2×D107semidirect
173. C2×M16C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C22×#131338052
#202932012
#202927882
#202927762
#202926332
#202926232
#202925952
C2×#137×C1072
#3009107