Finite Group Databasebeta

← #28345983

Subgroups — #28345983

Order 13696.

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

normal subgroups only

This group has 490 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×#16812142C211yesmaximal, characteristic, complemented
2C22×#13133807C211yesmaximal, characteristic, complemented
3#20293196C211yesmaximal, characteristic
4#20293104C211yesmaximal, characteristic, complemented
5#20292850C214yesmaximal, complemented
6#20292828C214yesmaximal, complemented
7#20292633C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic, complemented
9#424×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C22×D428C2211yescharacteristic
13#16812142—24no—
14C2×Dic428C2212yescomplemented
15C2×#13133807—24no—
16C2×#13133807C2214yes—
17C2×#13133807C2212yescomplemented
18#16812106C2212yes—
19#16812097C2212yes—
20#16812084—22no—
21#16812079—22no—
22C4×Dic214C2214yescomplemented
23#16812074—22no—
24#16812068C2211yescharacteristic
25#16812065—22no—
26#16812034C2212yescomplemented
27#16812031C2214yes—
28#16812031C2212yescomplemented
29C2×M16×C107C2211yescharacteristic
30C22×C8×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32C4⋊C8×C107C2214yescomplemented
33C23×D107—21no—
34#13133818—22no—
35#13133818—22no—
36C2×Dic214—21no—
37C2×Dic214—22no—
38C2×Dic214D414yescomplemented
39C2×Dic214C2311yescharacteristic
40C2×Dic214C2312yes—
41C2×D428—22no—
42C2×D428—22no—
43C2×D428—21no—
44C2×D428—24no—
45C2×D428C2312yes—
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C2×C4×D107—21no—
50Dic428—24no—
51#13133807—28no—
52#13133807—24no—
53C4⋊Dic107C2312yes—
54Dic107⋊C4—24no—
55C22⋊Dic107—21no—
56(C2×D107)⋊C4—24no—
57C4×Dic107—24no—
58C4×Dic107—21no—
59C22×C4×C107C2311yescharacteristic
60M16×C107—22no—
61C2×C8×C107—22no—
62C2×C8×C107C2312yes—
63C2×C8×C107C2312yes—
64C4⋊C4×C107D412yes—
65C22⋊C4×C107D412yes—
66C42×C107C2312yes—
67C23×C107C2×D411yescharacteristic
68C22×D107—22no—
69C22×D107—21no—
70C22×D107—42no—
71C22×D107—82no—
72C2×C4×C107C2×D412yes—
73C2×C4×C107C2×D414yes—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107#4112yes—
76C2×C4×C107C2411yescharacteristic
77C22⋊D107—42no—
78C2×Dic107—24no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—21no—
82D428—22no—
83D428—21no—
84D428—22no—
85D428—21no—
86D428—42no—
87D428—84no—
88C4×D107—42no—
89Dic214—24no—
90Dic214—22no—
91Dic214—21no—
92Dic214—21no—
93Dic214C2×D414yes—
94C8×C107—22no—
95C8×C107—22no—
96C22×C107—22no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C2×D107—41no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—44no—
103C2×D107—84no—
104C4×C107—24no—
105C4×C107#12112yes—
106C4×C107C22×D411yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—24no—
109Dic107—41no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107C2×#12111yescharacteristic
113C2×C107#57712yes—
114D107—81no—
115D107—82no—
116#3043—1071nomaximal, Sylow
117C107#304311yescharacteristic, complemented, Sylow
118C2×#137—1071no—
119C22×SD16—1071no—
120#535—1071no—
121#519—1071no—
122#464—1074no—
123#460—1074no—
124#424D10711yescharacteristic, complemented
125#418—1071no—
126#416—1071no—
127C2×#41—1071no—
128C22×Q8—1071no—
129C22×D4—1071no—
130#137—2144no—
131C2×Q16—1072no—
132C2×SD16—1074no—
133C2×SD16—1072no—
134C2×SD16—2144no—
135C2×M16C2×D10711yescharacteristic
136C22×C8C2×D10711yescharacteristic
137#127—1072no—
138#124—1072no—
139#121—2142no—
140#120—2142no—
141C4×Q8—1074no—
142C4×Q8—2142no—
143#117—1071no—
144#117—2142no—
145#117C2×D10711yescharacteristic
146C4⋊C8C2×D10714yescomplemented
147#103—1072no—
148#102—1072no—
149#102—1074no—
150C24—2141no—
151#41—2142no—
152#41—2142no—
153C2×Q8—1074no—
154C2×Q8—1072no—
155C2×Q8—1071no—
156C2×Q8—2141no—
157C2×Q8—2142no—
158C2×D4—1072no—
159C2×D4—1071no—
160C2×D4—2141no—
161C2×D4—2142no—
162C2×D4—2144no—
163C2×D4—2141no—
164C2×D4—2142no—
165C22×C4—2141no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168Q16—2144no—
169SD16—2148no—
170SD16—2144no—
171M16—22no—
172C2×C8—22no—
173C2×C8C22×D10712yes—
174C2×C8C22×D10712yes—
175C4⋊C4—1072no—
176C4⋊C4—2144no—
177C4⋊C4D42812yes—
178C22⋊C4—2141no—
179C22⋊C4—2144no—
180C22⋊C4D42812yes—
181C42—2141no—
182C42—2144no—
183C42C22×D10712yes—
184C23—8562no—
185C23—4282no—
186C23—2142no—
187C23—2141no—
188C23C2×D42811yescharacteristic
189Q8—1074no—
190Q8—2141no—
191Q8—2141no—
192Q8—2144no—
193Q8—2142no—
194D4—8564no—
195D4—4282no—
196D4—4282no—
197D4—2142no—
198D4—2141no—
199D4—2142no—
200D4—2141no—
201C2×C4—4282no—
202C2×C4—2142no—
203C2×C4—2141no—
204C2×C4—2144no—
205C2×C4—2142no—
206C2×C4D4×D10712yes—
207C2×C4C2×D42814yes—
208C2×C4C2×D42811yescharacteristic
209C2×C4#1313381912yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—22no—
213C22—22no—
214C22—8564no—
215C22—4281no—
216C22—4282no—
217C22—4282no—
218C22—4284no—
219C22#1681208412yes—
220C22C22×D42811yescharacteristic
221C4—24no—
222C4—4281no—
223C4—4282no—
224C4—2144no—
225C4#1681208412yes—
226C4C2×D4×D10711yescharacteristic
227C4C2×#1313381911yescharacteristic
228C2—22no—
229C2—8562no—
230C2—8561no—
231C2C2×#1681208411yescharacteristic
232C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#424D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C22×#13133807C2semidirect
4. #20293104C2semidirect
5. #20292850C2semidirect
6. #20292828C2semidirect
7. #20292633C2semidirect
8. #20292624C2semidirect
9. #424×C107C2semidirect
14. C2×Dic428C22semidirect
17. C2×#13133807C22semidirect
22. C4×Dic214C22semidirect
26. #16812034C22semidirect
28. #16812031C22semidirect
32. C4⋊C8×C107C22semidirect
38. C2×Dic214D4semidirect
117. C107#3043semidirect
124. #424D107semidirect
146. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C22×#131338072
#202931962
#202931042
#202928502
#202928282
#202926332
#202926242
#424×C1072
#3043107