Finite Group Databasebeta

← #28346099

Subgroups — #28346099

Order 13696.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133806C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic, complemented
3#20293203C211yesmaximal, characteristic, complemented
4#20293194C211yesmaximal, characteristic
5#20292872C214yesmaximal, complemented
6#20292855C214yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292637C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C2×#13133806—24no—
13C2×#13133806C2212yes—
14C2×#13133806C2212yes—
15#16812117—24no—
16C2×#13133801C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107C2212yescomplemented
19#16812109—21no—
20#16812105—21no—
21#16812097—21no—
22#16812096—21no—
23#16812093—22no—
24#16812089C2214yescomplemented
25#16812080C2214yes—
26#16812078—22no—
27#16812068C2211yescharacteristic
28#16812042C2212yescomplemented
29#16812040C2212yescomplemented
30#16812034C2214yescomplemented
31C22×D4×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#102×C107C2214yes—
35#13133818—22no—
36#13133818—22no—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214D411yescharacteristic
41C2×Dic214C2312yes—
42C2×D428D411yescharacteristic
43C2×C22⋊D107—24no—
44C2×C22⋊D107D412yes—
45C22×Dic107—21no—
46C2×C4×D107C2312yes—
47#13133806—24no—
48#13133806—24no—
49#13133801—24no—
50C2×C107⋊C8D414yescomplemented
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—24no—
54C4⋊Dic107—24no—
55C4⋊Dic107C2312yes—
56C22⋊Dic107—21no—
57C22⋊Dic107—24no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—21no—
60C24×C107—21no—
61C2×D4×C107—22no—
62C2×D4×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66M16×C107—22no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107C2312yes—
69C22⋊C4×C107—21no—
70C42×C107—21no—
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—82no—
74C23×C107C2×D411yescharacteristic
75C22×D107C2×D412yes—
76D4×C107—22no—
77D4×C107—22no—
78D4×C107—42no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D412yes—
82C2×C4×C107#4112yes—
83C2×C4×C107C2411yescharacteristic
84C22⋊D107—24no—
85C22⋊D107—84no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—42no—
89C2×Dic107C2×D412yes—
90D428—22no—
91C4×D107—22no—
92C4×D107—22no—
93Dic214—22no—
94Dic214—22no—
95Dic214—22no—
96Dic214—42no—
97C8×C107—22no—
98C107⋊C8C2×D412yes—
99C107⋊C8C2×D412yes—
100C22×C107—22no—
101C22×C107—42no—
102C22×C107—44no—
103C22×C107—84no—
104C22×C107#12112yes—
105C22×C107C22×D411yescharacteristic
106C2×D107—22no—
107C2×D107—24no—
108C4×C107—42no—
109C4×C107#12112yes—
110C4×C107C22×D411yescharacteristic
111C4×C107C2×#4111yescharacteristic
112Dic107—22no—
113Dic107—42no—
114Dic107—42no—
115C2×C107—22no—
116C2×C107—82no—
117C2×C107C2×#12111yescharacteristic
118C2×C107#57712yes—
119D107—42no—
120#3057—1071nomaximal, Sylow
121C107#305711yescharacteristic, complemented, Sylow
122C22×SD16—1071no—
123C2×#131—1071no—
124#536—1071no—
125#535—1071no—
126#468—1074no—
127#465—1074no—
128#427—1071no—
129#419—1071no—
130#418D10711yescharacteristic, complemented
131C2×#41—1071no—
132C22×Q8—1071no—
133C22×D4C2×D10711yescharacteristic
134C2×SD16—1072no—
135C2×SD16—1072no—
136C2×SD16—2144no—
137#131—2144no—
138C2×M16—1072no—
139C2×M16C2×D10711yescharacteristic
140C22×C8—1071no—
141C22×C8—1072no—
142C4⋊Q8—2141no—
143#127—2141no—
144#124—2141no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1074no—
148#121—1074no—
149#120—2142no—
150#117—1071no—
151#117C2×D10711yescharacteristic
152#107—1072no—
153#106—1072no—
154#103—1074no—
155#102C2×D10714yes—
156C24—21no—
157#41—2142no—
158#41—2142no—
159C2×Q8—1071no—
160C2×Q8—1072no—
161C2×Q8—2142no—
162C2×Q8—2142no—
163C2×Q8—2141no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—1072no—
168C2×D4—1071no—
169C2×D4—2144no—
170C2×D4C22×D10712yes—
171C22×C4—1072no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175SD16—2144no—
176M16—22no—
177M16—2144no—
178C2×C8—1074no—
179C2×C8—1071no—
180C2×C8—1071no—
181C2×C8—2144no—
182C2×C8C22×D10712yes—
183C4⋊C4—1072no—
184C4⋊C4—2144no—
185C4⋊C4C22×D10712yes—
186C22⋊C4—21no—
187C22⋊C4—2144no—
188C22⋊C4—2144no—
189C22⋊C4—2141no—
190C42—21no—
191C42—2141no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1072no—
196C23D4×D10711yescharacteristic
197Q8—4282no—
198Q8—2142no—
199Q8—2142no—
200Q8—2142no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—8564no—
205D4—2144no—
206D4—2142no—
207C2×C4—22no—
208C2×C4—1072no—
209C2×C4—4282no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4D4×D10711yescharacteristic
215C2×C4D4×D10712yes—
216C2×C4#1313381812yes—
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—1072no—
220C8—1072no—
221C22—22no—
222C22—44no—
223C22—42no—
224C22—84no—
225C22—2144no—
226C22—2142no—
227C22#1681208212yes—
228C22C2×D4×D10711yescharacteristic
229C4—42no—
230C4—4282no—
231C4—4282no—
232C4—2142no—
233C4#1681208212yes—
234C4C2×D4×D10711yescharacteristic
235C4C2×#1313381811yescharacteristic
236C2—22no—
237C2—82no—
238C2—4282no—
239C2C2×#1681208211yescharacteristic
240C2#2029343712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radical#20293194C2
hypercenter#418D107
hypocenterC107#3057
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. C22×#13133806C2semidirect
2. C2×#16812117C2semidirect
3. #20293203C2semidirect
5. #20292872C2semidirect
6. #20292855C2semidirect
7. #20292663C2semidirect
8. #20292637C2semidirect
9. #418×C107C2semidirect
16. C2×#13133801C22semidirect
18. C2×C8×D107C22semidirect
24. #16812089C22semidirect
28. #16812042C22semidirect
29. #16812040C22semidirect
30. #16812034C22semidirect
50. C2×C107⋊C8D4semidirect
121. C107#3057semidirect
130. #418D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338062
C2×#168121172
#202932032
#202931942
#202928722
#202928552
#202926632
#202926372
#418×C1072
#3057107