Finite Group Databasebeta

← #28345693

Subgroups — #28345693

Order 13696.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D107C211yesmaximal, characteristic, complemented
2C2×#16812139C211yesmaximal, characteristic, complemented
3#20293134C211yesmaximal, characteristic
4#20292781C214yesmaximal, complemented
5#20292764C214yesmaximal, complemented
6#20292637C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292590C211yesmaximal, characteristic, complemented
9C2×#137×C107C211yesmaximal, characteristic, complemented
10C22×Dic214—21no—
11C22×Dic214C2211yescharacteristic
12C2×Q8×D107—24no—
13C2×Q8×D107—22no—
14C2×Q8×D107—24no—
15C2×Q8×D107C2214yes—
16C22×C4×D107—21no—
17C22×C4×D107C2211yescharacteristic
18#16812139—24no—
19C2×#13133808C2212yescomplemented
20C2×#13133806C2212yescomplemented
21C2×#13133802C2211yescharacteristic
22#16812090—21no—
23#16812085C2212yes—
24#16812080C2212yes—
25#16812077—21no—
26C4×C22⋊D107—22no—
27#16812068C2211yescharacteristic
28#16812034C2214yescomplemented
29#16812033C2214yescomplemented
30(C2×D107)⋊C8C2214yes—
31C2×#41×C107C2211yescharacteristic
32C22×Q8×C107C2211yescharacteristic
33#137×C107—24no—
34C2×Q16×C107C2212yescomplemented
35C2×SD16×C107C2212yescomplemented
36C2×M16×C107C2211yescharacteristic
37C23×D107D411yescharacteristic
38C2×Dic214—28no—
39C2×Dic214—21no—
40C2×Dic214—22no—
41C2×Dic214—24no—
42C2×Dic214D414yescomplemented
43C2×Dic214C2312yes—
44Q8×D107—216no—
45Q8×D107—28no—
46Q8×D107—28no—
47C2×C22⋊D107—22no—
48C22×Dic107—21no—
49C22×Dic107D411yescharacteristic
50C2×C4×D107—28no—
51C2×C4×D107—24no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107D412yes—
55C2×C4×D107C2312yes—
56#13133808—24no—
57#13133806—24no—
58#13133802—22no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107C2312yes—
61Dic107⋊C4—22no—
62C22⋊Dic107—21no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66#41×C107—22no—
67#41×C107—22no—
68C2×Q8×C107—22no—
69C2×Q8×C107—21no—
70C2×Q8×C107D414yescomplemented
71C2×Q8×C107C2311yescharacteristic
72C2×Q8×C107C2312yes—
73C2×D4×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78Q16×C107—24no—
79SD16×C107—24no—
80M16×C107—22no—
81C2×C8×C107C2312yes—
82C23×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—22no—
85C22×D107—24no—
86C22×D107C2×D412yes—
87Q8×C107—24no—
88Q8×C107—22no—
89Q8×C107—21no—
90Q8×C107—21no—
91Q8×C107C2×D414yes—
92D4×C107—21no—
93D4×C107—21no—
94D4×C107—42no—
95C2×C4×C107—24no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—42no—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—44no—
104C2×Dic107—24no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107C2×D412yes—
109C4×D107—216no—
110C4×D107—24no—
111C4×D107—22no—
112C4×D107—22no—
113Dic214—216no—
114Dic214—24no—
115Dic214—22no—
116Dic214C2×D414yes—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—22no—
120C22×C107—41no—
121C22×C107—42no—
122C22×C107#12012yes—
123C22×C107C22×D411yescharacteristic
124C2×D107—28no—
125C2×D107—22no—
126C2×D107—22no—
127C2×D107#13714yescomplemented
128C4×C107—24no—
129C4×C107—41no—
130C4×C107#12012yes—
131C4×C107C22×D411yescharacteristic
132C4×C107C22×D411yescharacteristic
133Dic107—22no—
134Dic107—24no—
135Dic107—42no—
136C2×C107—22no—
137C2×C107—81no—
138C2×C107C2×#12011yescharacteristic
139C2×C107C2×#13712yes—
140D107—24no—
141#3004—1071nomaximal, Sylow
142C107#300411yescharacteristic, complemented, Sylow
143C23×Q8—1071no—
144C2×#137—1071no—
145C2×#137D10711yescharacteristic, complemented
146#525—1071no—
147#451—1074no—
148#448—1074no—
149#419—1071no—
150#419—1071no—
151#407—1071no—
152C2×#41C2×D10711yescharacteristic
153C22×Q8—1074no—
154C22×Q8—1071no—
155C22×Q8—2141no—
156C22×Q8—2142no—
157C22×Q8—2144no—
158C22×Q8—2144no—
159C22×Q8C2×D10711yescharacteristic
160C23×C4—1071no—
161C23×C4—2141no—
162#137—24no—
163#137—2144no—
164C2×Q16—1072no—
165C2×Q16C2×D10712yescomplemented
166C2×SD16—1072no—
167C2×SD16C2×D10712yescomplemented
168C2×M16—1071no—
169C2×M16C2×D10711yescharacteristic
170#123—2141no—
171C22⋊Q8—1072no—
172#121—1072no—
173#120—2141no—
174C4×D4—2142no—
175#117—1071no—
176#103—1074no—
177#103—1074no—
178C22⋊C8—1074no—
179C24—1071no—
180#41—22no—
181#41—22no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—1074no—
185C2×Q8—1072no—
186C2×Q8—2144no—
187C2×Q8—2148no—
188C2×Q8—2148no—
189C2×Q8—21416no—
190C2×Q8—2148no—
191C2×Q8—2141no—
192C2×Q8—2142no—
193C2×Q8C22⋊D10714yescomplemented
194C2×Q8C22×D10711yescharacteristic
195C2×Q8C22×D10712yes—
196C2×D4—21no—
197C2×D4—2142no—
198C2×D4C22×D10711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1071no—
202C22×C4—1072no—
203C22×C4—1072no—
204C22×C4—2144no—
205C22×C4—2148no—
206C22×C4—2141no—
207C22×C4—2142no—
208C22×C4—2142no—
209C22×C4C22×D10711yescharacteristic
210Q16—24no—
211Q16—2144no—
212SD16—24no—
213SD16—2144no—
214M16—22no—
215M16—2142no—
216C2×C8—1072no—
217C2×C8C22×D10712yes—
218C4⋊C4—1072no—
219C4⋊C4—2142no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C22⋊C4—2141no—
223C42—2141no—
224C23—21no—
225C23—1072no—
226C23—2144no—
227C23—2142no—
228C23D4×D10711yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—21no—
232Q8—21no—
233Q8—1074no—
234Q8—21416no—
235Q8—2144no—
236Q8—2142no—
237Q8C2×C22⋊D10714yes—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—4284no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—1072no—
247C2×C4—21416no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2144no—
251C2×C4—2142no—
252C2×C4—2144no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4D4×D10712yes—
257C2×C4C2×C22⋊D10712yes—
258C2×C4C23×D10711yescharacteristic
259C8—22no—
260C8—2142no—
261C22—22no—
262C22—42no—
263C22—41no—
264C22—1074no—
265C22—2148no—
266C22—2142no—
267C22—2142no—
268C22#1681207712yes—
269C22C2×D4×D10711yescharacteristic
270C4—24no—
271C4—41no—
272C4—4282no—
273C4—2142no—
274C4—2144no—
275C4#1681207712yes—
276C4C2×D4×D10711yescharacteristic
277C4C22×C22⋊D10711yescharacteristic
278C2—22no—
279C2—81no—
280C2—2144no—
281C2C2×#1681207711yescharacteristic
282C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293134C2
hypercenterC2×#137D107
hypocenterC107#3004
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
1. C22×Q8×D107C2semidirect
2. C2×#16812139C2semidirect
4. #20292781C2semidirect
5. #20292764C2semidirect
6. #20292637C2semidirect
7. #20292636C2semidirect
8. #20292590C2semidirect
9. C2×#137×C107C2semidirect
19. C2×#13133808C22semidirect
20. C2×#13133806C22semidirect
28. #16812034C22semidirect
29. #16812033C22semidirect
34. C2×Q16×C107C22semidirect
35. C2×SD16×C107C22semidirect
42. C2×Dic214D4semidirect
70. C2×Q8×C107D4semidirect
127. C2×D107#137semidirect
142. C107#3004semidirect
145. C2×#137D107semidirect
165. C2×Q16C2×D107semidirect
167. C2×SD16C2×D107semidirect
193. C2×Q8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168121392
#202931342
#202927812
#202927642
#202926372
#202926362
#202925902
C2×#137×C1072
#3004107