Finite Group Databasebeta

← #28348251

Subgroups — #28348251

Order 13696.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293436C212yesmaximal, complemented
2#20293435C212yesmaximal, complemented
3#20293001C211yesmaximal, characteristic, complemented
4#20292893C212yesmaximal, complemented
5#20292892C212yesmaximal, complemented
6#20292809C211yesmaximal, characteristic, complemented
7#20292796C212yesmaximal, complemented
8#20292741C212yesmaximal, complemented
9#456×C107C211yesmaximal, characteristic, complemented
10#16812157C2211yescharacteristic
11#16812155C2212yescomplemented
12#16812141—22no—
13#16812140—22no—
14#16812140C2212yescomplemented
15#16812137C2212yes—
16#16812136—22no—
17#16812134—22no—
18#16812134C2212yes—
19#16812132—22no—
20#16812129—22no—
21#16812129C2212yes—
22C2×#13133805—22no—
23C2×#13133805C2212yescomplemented
24C2×#13133805C2212yescomplemented
25SD16×D107—22no—
26#16812117C2212yescomplemented
27#16812116C2212yescomplemented
28#16812106C2211yescharacteristic
29#16812044C2212yescomplemented
30#16812037C2212yescomplemented
31#16812028C2211yescharacteristic, complemented
32#16812025C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
342−1+4×C107C2211yescharacteristic
35#137×C107C2212yescomplemented
36C4⋊Q8×C107C2211yescharacteristic, complemented
37#104×C107C2212yescomplemented
38#101×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133819—22no—
41#13133818—22no—
42#13133818—41no—
43#13133818C2312yes—
44Q8×D107—22no—
45C2×D428—22no—
46C2×D428—41no—
47C2×D428C2312yes—
48D4×D107—22no—
49D4×D107—42no—
50#13133808—22no—
51#13133808D412yescomplemented
52#13133807—22no—
53#13133806D412yescomplemented
54#13133805—22no—
55#13133805—22no—
56#13133805—22no—
57#13133805—22no—
58#13133805—22no—
59#13133805—22no—
60#13133805D412yescomplemented
61D856—22no—
62#13133803—22no—
63#13133803D412yescomplemented
64#13133802—22no—
65#13133802C2312yes—
66#13133801—22no—
67C2×C107⋊C8—22no—
68C2×C107⋊C8C2312yes—
69C8×D107—22no—
70#41×C107—22no—
71#41×C107—41no—
72#41×C107C2312yes—
73C2×Q8×C107—22no—
74C2×Q8×C107C2311yescharacteristic
75C2×Q8×C107C2311yescharacteristic
76Q16×C107—22no—
77SD16×C107—22no—
78M16×C107C2312yes—
79C4⋊C4×C107—41no—
80C42×C107C2311yescharacteristic
81C22×D107—22no—
82C22×D107—42no—
83Q8×C107—22no—
84Q8×C107—22no—
85Q8×C107—41no—
86Q8×C107C2×D412yes—
87D4×C107—22no—
88D4×C107—41no—
89D4×C107C2×D412yes—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—42no—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—22no—
97C22⋊D107—42no—
98D428—22no—
99D428—22no—
100D428—42no—
101D428—41no—
102D428—82no—
103D428C2×D412yes—
104C4×D107—22no—
105C4×D107—42no—
106C4×D107—42no—
107Dic214—41no—
108Dic214C2×D412yes—
109C8×C107—22no—
110C107⋊C8—22no—
111C107⋊C8C2×D412yes—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107H(5,𝔽2)11yescharacteristic
115C2×D107—22no—
116C2×D107—42no—
117C2×D107—82no—
118C2×D107—162no—
119C4×C107—22no—
120C4×C107—22no—
121C4×C107—41no—
122C4×C107—41no—
123C4×C107C22×D412yes—
124Dic107—22no—
125C2×C107—21no—
126C2×C107—42no—
127C2×C107D4211yescharacteristic
128D107—42no—
129D107—82no—
130#3291—1071nomaximal, Sylow
131C107#329111yescharacteristic, complemented, Sylow
132#577—1072no—
133#577—1072no—
134#492—1071no—
135#471—1072no—
136#471—1072no—
137#456—1071no—
138#456D10711yescharacteristic, complemented
139#453—1072no—
140#443—1072no—
1412−1+4—1071no—
1422−1+4C2×D10711yescharacteristic
143H(5,𝔽2)—1072no—
144#137—1072no—
145#137—2142no—
146#137—2142no—
147#137C2×D10712yescomplemented
148AGL(1,ℤ/8)—1072no—
149AGL(1,ℤ/8)—1072no—
150AGL(1,ℤ/8)—2142no—
151AGL(1,ℤ/8)—2142no—
152#135—1072no—
153#135—2142no—
154#135—2142no—
155C2×SD16—1072no—
156C2×SD16—1072no—
157C2×SD16—2142no—
158C2×SD16—2142no—
159#131—1072no—
160#131—1072no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#127—1071no—
163#108—1072no—
164#104—1072no—
165#104C2×D10712yescomplemented
166#101—1071no—
167#101C2×D10711yescharacteristic, complemented
168#100—1072no—
169C4×C8—1071no—
170#41—22no—
171#41—41no—
172#41—1072no—
173#41—4281no—
174#41—2142no—
175#41—2142no—
176#41—2142no—
177#41C22×D10712yes—
178C2×Q8—22no—
179C2×Q8—2142no—
180C2×Q8C22×D10711yescharacteristic
181C2×Q8C22×D10711yescharacteristic
182C2×D4—1072no—
183C2×D4—4281no—
184C2×D4—4282no—
185C2×D4—2142no—
186C2×D4—2142no—
187Q16—22no—
188Q16—1072no—
189Q16—2142no—
190SD16—22no—
191SD16—1072no—
192SD16—1072no—
193SD16—2142no—
194SD16—2142no—
195SD16—2142no—
196SD16—2142no—
197SD16—2142no—
198SD16—2142no—
199SD16—2142no—
200D8—1072no—
201D8—2142no—
202D8—2142no—
203M16—1072no—
204M16—2142no—
205M16—2142no—
206M16C22×D10712yes—
207C2×C8—1072no—
208C2×C8—2142no—
209C2×C8—2142no—
210C4⋊C4—41no—
211C42C22×D10711yescharacteristic
212C23—4282no—
213C23—2142no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—1072no—
218Q8—4281no—
219Q8D4×D10712yes—
220D4—22no—
221D4—41no—
222D4—8562no—
223D4—1072no—
224D4—4282no—
225D4—4281no—
226D4—4282no—
227D4—2142no—
228D4—2142no—
229D4—2142no—
230D4D4×D10712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—4282no—
237C2×C4—4282no—
238C2×C4—2142no—
239C2×C4C23×D10711yescharacteristic
240C8—22no—
241C8—1072no—
242C8—1072no—
243C8—2142no—
244C22—22no—
245C22—8562no—
246C22—4282no—
247C22—17122no—
248C22—2142no—
249C22#1681215411yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2142no—
255C4C2×D4×D10712yes—
256C2—21no—
257C2—42no—
258C2—8562no—
259C2—4282no—
260C2#2029325911yescharacteristic
subgroupG/H
centerC2#20293259
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#456×C107C2
socleC2×C107D42
hypercenter#456D107
hypocenterC107#3291
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20293436C2semidirect
2. #20293435C2semidirect
3. #20293001C2semidirect
4. #20292893C2semidirect
5. #20292892C2semidirect
6. #20292809C2semidirect
7. #20292796C2semidirect
8. #20292741C2semidirect
9. #456×C107C2semidirect
11. #16812155C22semidirect
14. #16812140C22semidirect
23. C2×#13133805C22semidirect
24. C2×#13133805C22semidirect
26. #16812117C22semidirect
27. #16812116C22semidirect
29. #16812044C22semidirect
30. #16812037C22semidirect
31. #16812028C22semidirect
32. #16812025C22semidirect
33. C4×C107⋊C8C22semidirect
35. #137×C107C22semidirect
36. C4⋊Q8×C107C22semidirect
37. #104×C107C22semidirect
38. #101×C107C22semidirect
51. #13133808D4semidirect
53. #13133806D4semidirect
60. #13133805D4semidirect
63. #13133803D4semidirect
131. C107#3291semidirect
138. #456D107semidirect
147. #137C2×D107semidirect
161. C4⋊Q8C2×D107semidirect
165. #104C2×D107semidirect
167. #101C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934362
#202934352
#202930012
#202928932
#202928922
#202928092
#202927962
#202927412
#456×C1072
#3291107