Finite Group Databasebeta

← #28346384

Subgroups — #28346384

Order 13696.

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

normal subgroups only

This group has 458 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
1#20293138C211yesmaximal, characteristic
2#20292962C212yesmaximal, complemented
3#20292957C212yesmaximal, complemented
4#20292945C212yesmaximal, complemented
5#20292940C212yesmaximal, complemented
6#20292667C212yesmaximal, complemented
7#20292626C212yesmaximal, complemented
8#20292591C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C2×#13133802C2212yes—
12#16812092—21no—
13#16812088C2211yescharacteristic
14#16812088C2211yescharacteristic
15#16812083C2211yescharacteristic
16#16812083C2211yescharacteristic
17#16812078—21no—
18C4×C22⋊D107—21no—
19C4×C22⋊D107—21no—
20#16812068C2211yescharacteristic
21#16812042C2214yescomplemented
22#16812040C2214yescomplemented
23#16812032C2214yescomplemented
24#16812029C2214yescomplemented
25C22⋊(C107⋊C8)C2212yes—
26C22⋊(C107⋊C8)C2212yes—
27C23×C4×C107—21no—
28C23×C4×C107C2211yescharacteristic
29C2×C4⋊C4×C107—21no—
30C2×C4⋊C4×C107—21no—
31C2×C4⋊C4×C107—21no—
32C2×C4⋊C4×C107C2212yes—
33C2×C4⋊C4×C107C2212yes—
34C2×C4⋊C4×C107C2212yes—
35#13133818—24no—
36C2×Dic214C2311yescharacteristic
37C2×D428C2311yescharacteristic
38C2×C22⋊D107—21no—
39C2×C22⋊D107—21no—
40C2×C22⋊D107—21no—
41C2×C4×D107—21no—
42#13133802—24no—
43C2×C107⋊C8C2314yes—
44C4⋊Dic107C2311yescharacteristic
45C4⋊Dic107C2311yescharacteristic
46Dic107⋊C4—21no—
47Dic107⋊C4—21no—
48C22⋊Dic107—21no—
49C22⋊Dic107—21no—
50C22⋊Dic107—21no—
51(C2×D107)⋊C4—21no—
52(C2×D107)⋊C4—21no—
53C4×Dic107—21no—
54C24×C107D411yescharacteristic
55C22×C4×C107—24no—
56C22×C4×C107—22no—
57C22×C4×C107—22no—
58C22×C4×C107—22no—
59C22×C4×C107—44no—
60C22×C4×C107D411yescharacteristic
61C22×C4×C107D411yescharacteristic
62C22×C4×C107D411yescharacteristic
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107—22no—
69C4⋊C4×C107C2314yes—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107C2×D411yescharacteristic
74C23×C107C2×D411yescharacteristic
75C23×C107C2×D411yescharacteristic
76C22×D107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—24no—
83C2×C4×C107—44no—
84C2×C4×C107—44no—
85C2×C4×C107—44no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107#4114yes—
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—42no—
92C22⋊D107—42no—
93C22⋊D107—42no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97D428—22no—
98C4×D107—42no—
99Dic214—22no—
100C107⋊C8—24no—
101C22×C107—28no—
102C22×C107—21no—
103C22×C107—21no—
104C22×C107—21no—
105C22×C107—21no—
106C22×C107—21no—
107C22×C107—21no—
108C22×C107#12312yes—
109C22×C107AGL(1,ℤ/8)12yes—
110C22×C107#13712yes—
111C22×C107C22×D411yescharacteristic
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—21no—
116C4×C107—21no—
117C4×C107—44no—
118C4×C107#12312yes—
119C4×C107C2×#4112yes—
120Dic107—41no—
121Dic107—41no—
122Dic107—41no—
123C2×C107—22no—
124C2×C107—22no—
125C2×C107—22no—
126C2×C107C2×#12311yescharacteristic
127C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C107C2×#13711yescharacteristic
129D107—81no—
130#3092—1071nomaximal, Sylow
131C107#309211yescharacteristic, complemented, Sylow
132#525—1071no—
133C22×C4⋊C4D10711yescharacteristic, complemented
134#484—1072no—
135#483—1072no—
136#481—1072no—
137#480—1072no—
138#428—1072no—
139#417—1072no—
140#407—1071no—
141C2×#41—1071no—
142C23×C4—21no—
143C23×C4C2×D10711yescharacteristic
144C2×M16—1072no—
145#123—2141no—
146C22⋊Q8—1071no—
147C22⋊Q8—1071no—
148#121—1071no—
149#121—1071no—
150#120—2141no—
151C4×D4—2141no—
152C4×D4—2141no—
153#117—1071no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D10712yes—
158C2×C4⋊C4C2×D10712yes—
159C2×C4⋊C4C2×D10712yes—
160#107—1074no—
161#106—1074no—
162#103—1074no—
163#102—1074no—
164C22⋊C8—1072no—
165C22⋊C8—1072no—
166C24C22⋊D10711yescharacteristic
167#41—2144no—
168C2×Q8—1071no—
169C2×D4—1071no—
170C2×D4—2141no—
171C2×D4—2141no—
172C2×D4—2141no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—2141no—
179C22×C4C22⋊D10711yescharacteristic
180C22×C4C22⋊D10711yescharacteristic
181C22×C4C22⋊D10711yescharacteristic
182C22×C4C22×D10711yescharacteristic
183C22×C4C22×D10711yescharacteristic
184C22×C4C22×D10711yescharacteristic
185M16—2144no—
186C2×C8—1074no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—1071no—
191C4⋊C4—1071no—
192C4⋊C4—2141no—
193C4⋊C4—2141no—
194C4⋊C4C22×D10714yes—
195C22⋊C4—2141no—
196C22⋊C4—2141no—
197C22⋊C4—2141no—
198C22⋊C4—2141no—
199C22⋊C4—2141no—
200C42—2141no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—2141no—
205C23C2×C22⋊D10711yescharacteristic
206C23C2×C22⋊D10711yescharacteristic
207C23C2×C22⋊D10711yescharacteristic
208Q8—2142no—
209D4—4282no—
210D4—4282no—
211D4—4282no—
212D4—2142no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—4282no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2141no—
226C2×C4C2×C22⋊D10711yescharacteristic
227C2×C4C2×C22⋊D10711yescharacteristic
228C2×C4C2×C22⋊D10711yescharacteristic
229C2×C4#1313381814yes—
230C2×C4C23×D10711yescharacteristic
231C8—2144no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—4281no—
240C22—4281no—
241C22—4281no—
242C22#1681209212yes—
243C22#1681213512yes—
244C22#1681214012yes—
245C22C22×C22⋊D10711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—4281no—
250C4—4281no—
251C4—4281no—
252C4#1681209212yes—
253C4C2×#1313381812yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—8561no—
258C2C2×#1681209211yescharacteristic
259C2C2×#1681213511yescharacteristic
260C2C2×#1681214011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×C4⋊C4×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenterC22×C4⋊C4D107
hypocenterC107#3092
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20292962C2semidirect
3. #20292957C2semidirect
4. #20292945C2semidirect
5. #20292940C2semidirect
6. #20292667C2semidirect
7. #20292626C2semidirect
8. #20292591C2semidirect
9. C22×C4⋊C4×C107C2semidirect
21. #16812042C22semidirect
22. #16812040C22semidirect
23. #16812032C22semidirect
24. #16812029C22semidirect
131. C107#3092semidirect
133. C22×C4⋊C4D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
#202929622
#202929572
#202929452
#202929402
#202926672
#202926262
#202925912
C22×C4⋊C4×C1072
#3092107