Finite Group Databasebeta

← #28346071

Subgroups — #28346071

Order 13696.

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

normal subgroups only

This group has 500 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×#16812131C211yesmaximal, characteristic, complemented
2C22×#13133801C211yesmaximal, characteristic, complemented
3#20293139C212yesmaximal, complemented
4#20292886C212yesmaximal, complemented
5#20292881C212yesmaximal, complemented
6#20292874C214yesmaximal, complemented
7#20292621C212yesmaximal, complemented
8#427×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2212yes—
10C22×C4×D107C2211yescharacteristic
11#16812131—24no—
12C2×Dic428C2211yescharacteristic, complemented
13C2×#13133807C2212yescomplemented
14C2×D856C2211yescharacteristic, complemented
15C2×#13133801—24no—
16C2×#13133801C2212yes—
17C2×#13133801C2212yes—
18C22×C107⋊C8C2211yescharacteristic
19#16812093—22no—
20#16812089C2214yes—
21#16812084C2214yes—
22#16812079—22no—
23C4×D428—24no—
24#16812032C2214yescomplemented
25#16812029C2214yescomplemented
26C22×C8×C107C2211yescharacteristic
27#117×C107C2212yes—
28#107×C107C2212yescomplemented
29#106×C107C2212yescomplemented
30C23×D107D411yescharacteristic
31#13133818—24no—
32#13133818—24no—
33C2×Dic214C2312yes—
34C2×D428—24no—
35C2×D428C2312yes—
36C2×C22⋊D107—22no—
37C22×Dic107D411yescharacteristic
38C2×C4×D107—22no—
39C2×C4×D107—22no—
40C2×C4×D107—22no—
41C2×C4×D107D412yes—
42C2×C4×D107C2312yes—
43Dic428—22no—
44#13133807—24no—
45D856—22no—
46#13133801—24no—
47#13133801—24no—
48C2×C107⋊C8—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—24no—
51(C2×D107)⋊C4—24no—
52(C2×D107)⋊C4—24no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107D414yescomplemented
55C2×C8×C107C2311yescharacteristic
56C2×C8×C107C2311yescharacteristic
57C4⋊C4×C107C2314yes—
58C22⋊C4×C107—22no—
59C42×C107—22no—
60C23×C107C2×D411yescharacteristic
61C22×D107—22no—
62C22×D107—22no—
63C22×D107—42no—
64C22×D107C2×D412yes—
65C2×C4×C107—24no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D412yes—
68C2×C4×C107#4112yes—
69C2×C4×C107C2411yescharacteristic
70C22⋊D107—44no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107C2×D412yes—
74D428—22no—
75D428—22no—
76D428—48no—
77C4×D107—24no—
78C4×D107—22no—
79C4×D107—22no—
80C4×D107—44no—
81Dic214—22no—
82Dic214—22no—
83C8×C107C2×D412yes—
84C8×C107C2×D412yes—
85C107⋊C8—22no—
86C22×C107—22no—
87C22×C107#12112yes—
88C22×C107C22×D411yescharacteristic
89C2×D107—22no—
90C2×D107—24no—
91C2×D107—44no—
92C2×D107—42no—
93C2×D107—44no—
94C4×C107—44no—
95C4×C107#12112yes—
96C4×C107C22×D411yescharacteristic
97C4×C107C2×#4111yescharacteristic
98Dic107—22no—
99Dic107—42no—
100C2×C107—22no—
101C2×C107C2×#12111yescharacteristic
102C2×C107#57512yes—
103D107—42no—
104D107—82no—
105#3054—1071nomaximal, Sylow
106C107#305411yescharacteristic, complemented, Sylow
107C2×#135—1071no—
108C22×M16—1071no—
109#525—1072no—
110#470—1072no—
111#469—1072no—
112#468—1074no—
113#427D10711yescharacteristic, complemented
114#416—1072no—
115C2×#41—1072no—
116C23×C4—1071no—
117#135—2144no—
118C2×Q16—1071no—
119C2×SD16—1072no—
120C2×D8—1071no—
121C2×M16—1072no—
122C2×M16—1072no—
123C2×M16—2144no—
124C22×C8—1071no—
125C22×C8C2×D10711yescharacteristic
126#123—2142no—
127C22⋊Q8—1074no—
128#121—1074no—
129#120—2142no—
130C4×D4—2144no—
131#117C2×D10712yes—
132#107C2×D10712yescomplemented
133#106C2×D10712yescomplemented
134#103—1074no—
135#102—1074no—
136C24—1071no—
137#41—2144no—
138#41—2144no—
139C2×Q8—1072no—
140C2×D4—1072no—
141C2×D4—2144no—
142C2×D4—2142no—
143C22×C4—1071no—
144C22×C4—1072no—
145C22×C4—1072no—
146C22×C4—2142no—
147C22×C4—2142no—
148C22×C4—2142no—
149C22×C4C22×D10711yescharacteristic
150Q16—2142no—
151SD16—2144no—
152D8—2142no—
153M16—2144no—
154M16—2144no—
155C2×C8—1072no—
156C2×C8—2142no—
157C2×C8D42814yescomplemented
158C2×C8C22×D10711yescharacteristic
159C2×C8C22×D10711yescharacteristic
160C4⋊C4—2144no—
161C4⋊C4C22×D10714yes—
162C22⋊C4—22no—
163C22⋊C4—2144no—
164C22⋊C4—2144no—
165C42—22no—
166C23—1072no—
167C23—4282no—
168C23—2142no—
169C23—2142no—
170C23D4×D10711yescharacteristic
171Q8—2142no—
172Q8—2142no—
173D4—4288no—
174D4—4284no—
175D4—2142no—
176D4—2142no—
177C2×C4—24no—
178C2×C4—1072no—
179C2×C4—4284no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4—2144no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4D4×D10711yescharacteristic
186C2×C4C2×D42812yes—
187C2×C4#1313381912yes—
188C2×C4C23×D10711yescharacteristic
189C8—2142no—
190C8C2×D42812yes—
191C8C2×D42812yes—
192C22—22no—
193C22—4284no—
194C22—4282no—
195C22—4284no—
196C22—2142no—
197C22—2144no—
198C22#1681208412yes—
199C22C2×D4×D10711yescharacteristic
200C4—44no—
201C4—4282no—
202C4—2142no—
203C4#1681208412yes—
204C4C22×D42811yescharacteristic
205C4C2×#1313381911yescharacteristic
206C2—22no—
207C2—8562no—
208C2—4282no—
209C2C2×#1681208411yescharacteristic
210C2#2029342412yes—
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#427×C107C2
socleC22×C107C22×D4
hypercenter#427D107
hypocenterC107#3054
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#16812131C2semidirect
2. C22×#13133801C2semidirect
3. #20293139C2semidirect
4. #20292886C2semidirect
5. #20292881C2semidirect
6. #20292874C2semidirect
7. #20292621C2semidirect
8. #427×C107C2semidirect
12. C2×Dic428C22semidirect
13. C2×#13133807C22semidirect
14. C2×D856C22semidirect
24. #16812032C22semidirect
25. #16812029C22semidirect
28. #107×C107C22semidirect
29. #106×C107C22semidirect
54. C2×C8×C107D4semidirect
106. C107#3054semidirect
113. #427D107semidirect
132. #107C2×D107semidirect
133. #106C2×D107semidirect
157. C2×C8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121312
C22×#131338012
#202931392
#202928862
#202928812
#202928742
#202926212
#427×C1072
#3054107