Finite Group Databasebeta

← #28346255

Subgroups — #28346255

Order 13696.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#20293180C211yesmaximal, characteristic, complemented
2C2×D4×Dic107C211yesmaximal, characteristic
3#20292929C212yesmaximal, complemented
4#20292925C212yesmaximal, complemented
5#20292915C212yesmaximal, complemented
6#20292905C212yesmaximal, complemented
7#20292670C211yesmaximal, characteristic, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292650C211yesmaximal, characteristic, complemented
10#20292632C211yesmaximal, characteristic
11#418×C107C211yesmaximal, characteristic
12C23×Dic107—21no—
13C2×#13133802C2211yescharacteristic
14#16812108C2211yescharacteristic, complemented
15#16812108C2211yescharacteristic, complemented
16#16812100C2211yescharacteristic, complemented
17#16812100C2211yescharacteristic, complemented
18#16812086—21no—
19#16812086—21no—
20C4×Dic214—21no—
21C4×Dic214—21no—
22D4×Dic107—21no—
23D4×Dic107—21no—
24D4×Dic107C2212yes—
25D4×Dic107C2212yes—
26#16812068C2211yescharacteristic
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812041C2212yescomplemented
33#16812040C2212yescomplemented
34Dic107⋊C8C2212yescomplemented
35Dic107⋊C8C2212yescomplemented
36#16812030C2212yes—
37#16812030C2212yes—
38C22×D4×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41#102×C107C2212yes—
42#102×C107C2212yes—
43C2×Dic214—21no—
44C2×Dic214—21no—
45C22×Dic107—21no—
46C22×Dic107—22no—
47C22×Dic107—22no—
48C22×Dic107—44no—
49C22×Dic107D411yescharacteristic
50C22×Dic107D411yescharacteristic
51#13133802—22no—
52C2×C107⋊C8C2312yes—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107—21no—
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62Dic107⋊C4—21no—
63C22⋊Dic107—21no—
64C22⋊Dic107—22no—
65C22⋊Dic107—22no—
66C4×Dic107—21no—
67C4×Dic107D411yescharacteristic
68C4×Dic107D411yescharacteristic
69C4×Dic107C2311yescharacteristic
70C4×Dic107C2311yescharacteristic
71C24×C107—21no—
72C2×D4×C107—21no—
73C2×D4×C107—22no—
74C2×D4×C107Q814yescomplemented
75C2×D4×C107C2312yes—
76C22×C4×C107C2311yescharacteristic
77M16×C107—22no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107—24no—
84C23×C107—22no—
85C23×C107—42no—
86C23×C107—42no—
87C23×C107C2×D411yescharacteristic
88D4×C107—24no—
89D4×C107—22no—
90D4×C107C2×Q814yes—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×Q812yes—
95C2×C4×C107#4112yes—
96C2×C4×C107C2411yescharacteristic
97C2×Dic107—22no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—22no—
101C2×Dic107—41no—
102C2×Dic107—44no—
103C2×Dic107—42no—
104C2×Dic107—44no—
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108C2×Dic107C2×D411yescharacteristic
109Dic214—42no—
110Dic214—42no—
111C8×C107—22no—
112C107⋊C8—22no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107—24no—
116C22×C107—48no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107C22⋊Q812yes—
120C22×C107C22×D411yescharacteristic
121C4×C107—41no—
122C4×C107—41no—
123C4×C107C22⋊Q812yes—
124C4×C107C22×Q811yescharacteristic
125C4×C107C2×#4111yescharacteristic
126Dic107—21no—
127Dic107—21no—
128Dic107—21no—
129Dic107—41no—
130Dic107—42no—
131Dic107—41no—
132Dic107AGL(1,ℤ/8)12yes—
133C2×C107—22no—
134C2×C107—44no—
135C2×C107C2×C22⋊Q811yescharacteristic
136C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C107#57511yescharacteristic
138#3077—1071nomaximal, Sylow
139C107#307711yescharacteristic, complemented, Sylow
140#533—1071no—
141C2×C4×D4—1071no—
142#478—1072no—
143#478—1072no—
144#476—1072no—
145#474—1072no—
146#428—1071no—
147#428—1071no—
148#423—1071no—
149#418—1071no—
150#418D10711yescharacteristic
151C22×D4C2×D10711yescharacteristic
152C23×C4—2141no—
153C2×M16—1071no—
154C2×M16C2×D10711yescharacteristic
155C4⋊Q8—1071no—
156C4⋊Q8—1071no—
157#125—1071no—
158#125—1071no—
159C22⋊Q8—2141no—
160C22⋊Q8—2141no—
161C4×Q8—2141no—
162C4×Q8—2141no—
163C4×D4—1072no—
164C4×D4—1072no—
165C4×D4—2141no—
166C4×D4—2141no—
167#117—1071no—
168#117C2×D10711yescharacteristic
169C2×C4⋊C4—1071no—
170C2×C22⋊C4—2141no—
171C2×C42—1071no—
172#107—1072no—
173#107—1072no—
174#106—1072no—
175#106—1072no—
176C4⋊C8—1072no—
177C4⋊C8—1072no—
178#102—1072no—
179#102—1072no—
180#102C2×D10712yes—
181#102C2×D10712yes—
182C24—21no—
183C2×Q8—2141no—
184C2×Q8—2141no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic21414yescomplemented
188C2×D4C22×D10712yes—
189C22×C4—1071no—
190C22×C4—1071no—
191C22×C4—4284no—
192C22×C4—2142no—
193C22×C4—2142no—
194C22×C4—2141no—
195C22×C4C22×D10711yescharacteristic
196M16—22no—
197M16—2142no—
198C2×C8—1072no—
199C2×C8C22×D10712yes—
200C4⋊C4—1071no—
201C4⋊C4—1071no—
202C4⋊C4—1072no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4—2141no—
209C4⋊C4—2141no—
210C4⋊C4C22×D10711yescharacteristic
211C4⋊C4C22×D10711yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C22⋊C4—2141no—
216C42—21no—
217C42—1071no—
218C42—1071no—
219C42—1071no—
220C42—1071no—
221C42—2141no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D10711yescharacteristic
227Q8—4282no—
228Q8—4282no—
229D4—24no—
230D4—22no—
231D4C2×Dic21414yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—1071no—
235C2×C4—1071no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—4284no—
239C2×C4—4282no—
240C2×C4—4284no—
241C2×C4—4281no—
242C2×C4—2142no—
243C2×C4—2141no—
244C2×C4—2141no—
245C2×C4—2142no—
246C2×C4D4×D10711yescharacteristic
247C2×C4C2×Dic21412yes—
248C2×C4#1313381712yes—
249C2×C4C23×D10711yescharacteristic
250C8—22no—
251C8—2142no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1681208612yes—
259C22C2×D4×D10711yescharacteristic
260C4—41no—
261C4—41no—
262C4—1072no—
263C4—4282no—
264C4—4281no—
265C4—4281no—
266C4—2141no—
267C4—2141no—
268C4—2141no—
269C4#1681208612yes—
270C4C22×Dic21411yescharacteristic
271C4C2×#1313381711yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1681208611yescharacteristic
275C2AGL(1,ℤ/8)×D10711yescharacteristic
276C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenter#418D107
hypocenterC107#3077
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #20293180C2semidirect
3. #20292929C2semidirect
4. #20292925C2semidirect
5. #20292915C2semidirect
6. #20292905C2semidirect
7. #20292670C2semidirect
8. #20292669C2semidirect
9. #20292650C2semidirect
14. #16812108C22semidirect
15. #16812108C22semidirect
16. #16812100C22semidirect
17. #16812100C22semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812040C22semidirect
34. Dic107⋊C8C22semidirect
35. Dic107⋊C8C22semidirect
74. C2×D4×C107Q8semidirect
139. C107#3077semidirect
187. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×D4×Dic1072
#202929292
#202929252
#202929152
#202929052
#202926702
#202926692
#202926502
#202926322
#418×C1072
#3077107