Finite Group Databasebeta

← #28339780

Subgroups — #28339780

Order 13696.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812078C212yesmaximal, complemented
3#20292539C212yesmaximal, complemented
4#20292536C212yesmaximal, complemented
5#20292506C212yesmaximal, complemented
6#20292503C214yesmaximal, complemented
7#20292490C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—24no—
11C22×Dic214C2211yescharacteristic, complemented
12C22×D428C2211yescharacteristic, complemented
13C22×C22⋊D107C2214yes—
14C22×C22⋊D107C2212yes—
15C22×C4×D107C2212yescomplemented
16#16812078—28no—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2214yes—
19C2×(C2×D107)⋊C4C2214yescomplemented
20C107⋊#95C2212yescomplemented
21C107⋊#95C2212yescomplemented
22C107⋊#95C2214yescomplemented
23C25×C107C2211yescharacteristic
24C23×C4×C107C2211yescharacteristic
25C23×C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107—24no—
27C2×C22⋊C4×C107C2212yes—
28C2×C22⋊C4×C107C2212yes—
29C23×D107C2312yes—
30#13133818—232no—
31C2×Dic214—22no—
32C2×Dic214—24no—
33C2×D428—24no—
34C2×D428—22no—
35C2×C22⋊D107—24no—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—28no—
38C2×C22⋊D107—28no—
39C2×C22⋊D107—44no—
40C2×C22⋊D107D418yescomplemented
41C22×Dic107C2314yes—
42C22×Dic107C2312yes—
43C2×C4×D107—24no—
44C2×C4×D107—24no—
45C2×C4×D107—24no—
46C22⋊Dic107—24no—
47C22⋊Dic107—28no—
48(C2×D107)⋊C4—44no—
49C24×C107—22no—
50C24×C107—24no—
51C24×C107—24no—
52C24×C107—42no—
53C24×C107C2312yes—
54C24×C107C2311yescharacteristic
55C22×C4×C107—24no—
56C22×C4×C107—22no—
57C22×C4×C107—42no—
58C22×C4×C107D414yescomplemented
59C22×C4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61C22×C4×C107C2311yescharacteristic
62C22⋊C4×C107—24no—
63C22⋊C4×C107—24no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—28no—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—22no—
70C23×C107—24no—
71C23×C107—416no—
72C23×C107—44no—
73C23×C107—44no—
74C23×C107C2×D414yes—
75C23×C107C2×D412yes—
76C23×C107#4114yescomplemented
77C23×C107C2411yescharacteristic
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—24no—
81C22×D107—24no—
82C22×D107C2×D414yes—
83C2×C4×C107—28no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—24no—
88C2×C4×C107—44no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C22⋊D107—216no—
92C22⋊D107—416no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—24no—
96C2×Dic107—24no—
97C2×Dic107—24no—
98C2×Dic107C2×D414yes—
99D428—28no—
100C4×D107—216no—
101Dic214—28no—
102C22×C107—22no—
103C22×C107—22no—
104C22×C107—28no—
105C22×C107—24no—
106C22×C107—28no—
107C22×C107—416no—
108C22×C107—42no—
109C22×C107—42no—
110C22×C107—44no—
111C22×C107#12014yes—
112C22×C107C22×D412yes—
113C22×C107C22×D411yescharacteristic
114C22×C107C2×#4112yes—
115C22×C107H(5,𝔽2)12yes—
116C2×D107—24no—
117C2×D107—28no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—44no—
121C4×C107—22no—
122C4×C107—22no—
123C4×C107—42no—
124Dic107—24no—
125Dic107—44no—
126C2×C107—24no—
127C2×C107—44no—
128C2×C107C2×#12011yescharacteristic
129C2×C107#52511yescharacteristic
130C2×C107#53511yescharacteristic
131C2×C107#54614yes—
132D107—44no—
133#2404—1071nomaximal, Sylow
134C107#240411yescharacteristic, complemented, Sylow
135C22×#41—1071no—
136C2×#120—1072no—
137C22×C22⋊C4D10711yescharacteristic, complemented
138#393—1072no—
139#392—1072no—
140#386—1072no—
141#386—1074no—
142#381—1071no—
143C25C2×D10711yescharacteristic
144C2×#41—2148no—
145C2×#41—2144no—
146C22×Q8—1071no—
147C22×D4—1072no—
148C22×D4—1074no—
149C22×D4—1071no—
150C23×C4—1072no—
151C23×C4C2×D10711yescharacteristic
152C23×C4C2×D10711yescharacteristic
153#120—2148no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—1074no—
156C2×C22⋊C4—1072no—
157C2×C22⋊C4—1074no—
158C2×C22⋊C4C2×D10712yes—
159C2×C22⋊C4C2×D10712yes—
160#95—1072no—
161#95—1072no—
162#95—1074no—
163C24—22no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—1072no—
168C24C22×D10712yes—
169C24C22×D10711yescharacteristic
170#41—21432no—
171C2×Q8—2144no—
172C2×Q8—2142no—
173C2×D4—1078no—
174C2×D4—4284no—
175C2×D4—2144no—
176C2×D4—2148no—
177C2×D4—2148no—
178C2×D4—2144no—
179C2×D4—2144no—
180C2×D4—2142no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—1072no—
185C22×C4—1074no—
186C22×C4—2144no—
187C22×C4—2144no—
188C22×C4—2144no—
189C22×C4C22⋊D10714yescomplemented
190C22×C4C22×D10712yes—
191C22×C4C22×D10711yescharacteristic
192C22×C4C22×D10711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—4284no—
196C22⋊C4—2148no—
197C22⋊C4—2144no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—1074no—
209C23—2142no—
210C23—2142no—
211C23—2144no—
212C23—2144no—
213C23D4×D10714yes—
214C23C2×C22⋊D10712yes—
215C23#1313381814yescomplemented
216C23C23×D10711yescharacteristic
217Q8—2148no—
218D4—42816no—
219D4—2148no—
220D4—21416no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—1074no—
228C2×C4—21416no—
229C2×C4—2144no—
230C2×C4—2144no—
231C2×C4—2144no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4C2×C22⋊D10712yes—
235C2×C4C2×C22⋊D10712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—416no—
242C22—42no—
243C22—44no—
244C22—42no—
245C22—4282no—
246C22—4282no—
247C22—4284no—
248C22—2148no—
249C22—2144no—
250C22#1681207714yes—
251C22C2×D4×D10712yes—
252C22C22×C22⋊D10711yescharacteristic
253C22C2×#1313381812yes—
254C22#1681215412yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—4284no—
259C4—2144no—
260C2—24no—
261C2—44no—
262C2—4284no—
263C2C2×#1681207711yescharacteristic
264C2#2029313811yescharacteristic
265C2#2029319411yescharacteristic
266C2#2029326714yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C22⋊C4×C107C2
hypercenterC22×C22⋊C4D107
hypocenterC107#2404
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812078C2semidirect
3. #20292539C2semidirect
4. #20292536C2semidirect
5. #20292506C2semidirect
6. #20292503C2semidirect
7. #20292490C2semidirect
8. C22×C22⋊C4×C107C2semidirect
11. C22×Dic214C22semidirect
12. C22×D428C22semidirect
15. C22×C4×D107C22semidirect
19. C2×(C2×D107)⋊C4C22semidirect
20. C107⋊#95C22semidirect
21. C107⋊#95C22semidirect
22. C107⋊#95C22semidirect
40. C2×C22⋊D107D4semidirect
58. C22×C4×C107D4semidirect
76. C23×C107#41semidirect
134. C107#2404semidirect
137. C22×C22⋊C4D107semidirect
189. C22×C4C22⋊D107semidirect
215. C23#13133818semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168120782
#202925392
#202925362
#202925062
#202925032
#202924902
C22×C22⋊C4×C1072
#2404107