Finite Group Databasebeta

← #28346096

Subgroups — #28346096

Order 13696.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812115C211yesmaximal, characteristic, complemented
2#20293201C211yesmaximal, characteristic, complemented
3#20293194C211yesmaximal, characteristic
4#20292870C214yesmaximal, complemented
5#20292853C214yesmaximal, complemented
6#20292665C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9C22×SD16×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812115—24no—
12C2×#13133802C2211yescharacteristic
13C2×#13133801C2212yescomplemented
14C2×C8×D107C2212yescomplemented
15#16812107—21no—
16#16812105—21no—
17#16812098—21no—
18#16812096—21no—
19#16812092—22no—
20#16812085C2214yescomplemented
21#16812080C2214yes—
22#16812078—22no—
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25#16812043C2212yescomplemented
26#16812041C2212yescomplemented
27#16812033C2214yescomplemented
28#16812030C2214yes—
29C22×Q8×C107C2211yescharacteristic
30C22×D4×C107C2211yescharacteristic
31C2×SD16×C107—24no—
32C2×SD16×C107C2212yes—
33C2×SD16×C107C2212yes—
34C22×C8×C107C2211yescharacteristic
35#13133818—22no—
36#13133818—22no—
37C2×Dic214D411yescharacteristic
38C2×D428D411yescharacteristic
39C2×C22⋊D107—24no—
40C2×C22⋊D107D412yes—
41C2×C4×D107C2312yes—
42#13133802—22no—
43#13133801—24no—
44C2×C107⋊C8C2312yes—
45C8×D107—24no—
46C4⋊Dic107C2312yes—
47C4⋊Dic107C2312yes—
48Dic107⋊C4—24no—
49C22⋊Dic107—21no—
50C22⋊Dic107—21no—
51C22⋊Dic107—24no—
52(C2×D107)⋊C4—24no—
53C4×Dic107—21no—
54C4×Dic107—21no—
55C24×C107—21no—
56C2×Q8×C107—21no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107C2312yes—
60C2×D4×C107—22no—
61C2×D4×C107—21no—
62C2×D4×C107—22no—
63C2×D4×C107C2312yes—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66SD16×C107—24no—
67SD16×C107—24no—
68C2×C8×C107D414yescomplemented
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—82no—
74C23×C107C2×D411yescharacteristic
75C22×D107C2×D412yes—
76Q8×C107—22no—
77Q8×C107—22no—
78Q8×C107—42no—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—42no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D412yes—
86C2×C4×C107#4112yes—
87C2×C4×C107C2411yescharacteristic
88C22⋊D107—24no—
89C22⋊D107—84no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107C2×D412yes—
93D428—22no—
94C4×D107—22no—
95C4×D107—22no—
96Dic214—22no—
97C8×C107C2×D412yes—
98C8×C107C2×D412yes—
99C107⋊C8—22no—
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×SD16D10711yescharacteristic, complemented
123C2×#131—1071no—
124#536—1071no—
125#535—1071no—
126#468—1074no—
127#465—1074no—
128#427—1071no—
129#419—1071no—
130#418—1071no—
131C2×#41—1071no—
132C22×Q8C2×D10711yescharacteristic
133C22×D4C2×D10711yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D10712yes—
136C2×SD16C2×D10712yes—
137#131—2144no—
138C2×M16—1071no—
139C2×M16—1072no—
140C22×C8—1072no—
141C22×C8C2×D10711yescharacteristic
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#117—1071no—
152#107—1072no—
153#106—1072no—
154#103—1074no—
155#102—1074no—
156C24—21no—
157#41—2142no—
158#41—2142no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1071no—
163C2×Q8C22×D10712yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—1072no—
168C2×D4—1071no—
169C2×D4—2144no—
170C2×D4C22×D10712yes—
171C22×C4—21no—
172C22×C4—1072no—
173C22×C4C22×D10711yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—2142no—
177M16—2144no—
178C2×C8—1072no—
179C2×C8—2144no—
180C2×C8C22⋊D10714yescomplemented
181C2×C8C22×D10711yescharacteristic
182C2×C8C22×D10711yescharacteristic
183C4⋊C4—1072no—
184C4⋊C4—1072no—
185C4⋊C4—2144no—
186C22⋊C4—2144no—
187C22⋊C4—2144no—
188C22⋊C4—2141no—
189C22⋊C4—2141no—
190C42—2141no—
191C42—2141no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1072no—
196C23D4×D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2142no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—8564no—
205D4—2144no—
206D4—2142no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1072no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4D4×D10711yescharacteristic
215C2×C4C2×C22⋊D10712yes—
216C2×C4#1313381712yes—
217C2×C4C23×D10711yescharacteristic
218C8—2142no—
219C8C2×C22⋊D10712yes—
220C8C2×C22⋊D10712yes—
221C22—22no—
222C22—44no—
223C22—42no—
224C22—84no—
225C22—2144no—
226C22—2142no—
227C22#1681208012yes—
228C22C2×D4×D10711yescharacteristic
229C4—42no—
230C4—4282no—
231C4—4282no—
232C4—2142no—
233C4#1681208012yes—
234C4C22×C22⋊D10711yescharacteristic
235C4C2×#1313381711yescharacteristic
236C2—22no—
237C2—82no—
238C2—4282no—
239C2C2×#1681208011yescharacteristic
240C2#2029343312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293194C2
hypercenterC22×SD16D107
hypocenterC107#3057
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16812115C2semidirect
2. #20293201C2semidirect
4. #20292870C2semidirect
5. #20292853C2semidirect
6. #20292665C2semidirect
7. #20292636C2semidirect
8. #20292632C2semidirect
9. C22×SD16×C107C2semidirect
13. C2×#13133801C22semidirect
14. C2×C8×D107C22semidirect
20. #16812085C22semidirect
25. #16812043C22semidirect
26. #16812041C22semidirect
27. #16812033C22semidirect
68. C2×C8×C107D4semidirect
121. C107#3057semidirect
122. C22×SD16D107semidirect
180. C2×C8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121152
#202932012
#202931942
#202928702
#202928532
#202926652
#202926362
#202926322
C22×SD16×C1072
#3057107