Finite Group Databasebeta

← #28344745

Subgroups — #28344745

Order 13696.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D107C211yesmaximal, characteristic
2C2×#16812085C212yesmaximal, complemented
3#20292499C212yesmaximal, complemented
4#20292498C214yesmaximal, complemented
5#20292487C211yesmaximal, characteristic, complemented
6#20292484C212yesmaximal, complemented
7#20292482C212yesmaximal
8C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
9C24×D107C2211yescharacteristic
10C23×Dic107C2211yescharacteristic
11C22×C4×D107—28no—
12C22×C4×D107—24no—
13C22×C4×D107C2212yes—
14C22×C4×D107C2212yes—
15#16812085—28no—
16C2×C4⋊Dic107C2212yescomplemented
17C2×Dic107⋊C4C2214yescomplemented
18C2×C22⋊Dic107C2212yes—
19C2×(C2×D107)⋊C4C2214yes—
20C2×(C2×D107)⋊C4C2214yes—
21C107⋊#95C2214yescomplemented
22C107⋊#95C2212yescomplemented
23C22×Q8×C107C2211yescharacteristic, complemented
24C23×C4×C107C2211yescharacteristic
25C22⋊Q8×C107—24no—
26C2×C4⋊C4×C107C2211yescharacteristic, complemented
27C2×C4⋊C4×C107C2212yescomplemented
28C2×C22⋊C4×C107C2212yes—
29C23×D107—22no—
30C23×D107—22no—
31C23×D107—24no—
32C23×D107—24no—
33C23×D107Q814yescomplemented
34C23×D107C2312yes—
35C22×Dic107—22no—
36C22×Dic107—24no—
37C22×Dic107C2314yes—
38C22×Dic107C2312yes—
39C2×C4×D107—232no—
40C2×C4×D107—24no—
41C2×C4×D107—24no—
42C2×C4×D107—24no—
43C2×C4×D107—24no—
44C2×C4×D107—24no—
45C2×C4×D107D418yes—
46C4⋊Dic107—24no—
47Dic107⋊C4—28no—
48C22⋊Dic107—24no—
49(C2×D107)⋊C4—28no—
50(C2×D107)⋊C4—28no—
51C24×C107C2311yescharacteristic
52C2×Q8×C107—24no—
53C2×Q8×C107—22no—
54C22×C4×C107—24no—
55C22×C4×C107D414yes—
56C22×C4×C107C2312yes—
57C22×C4×C107C2312yes—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C4⋊C4×C107—22no—
61C4⋊C4×C107—24no—
62C22⋊C4×C107—24no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107C2×D412yes—
66C23×C107C2×Q812yes—
67C23×C107#4112yes—
68C23×C107C2411yescharacteristic
69C22×D107—28no—
70C22×D107—216no—
71C22×D107—28no—
72C22×D107—28no—
73C22×D107—22no—
74C22×D107—22no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—216no—
78C22×D107C2×D414yes—
79C22×D107C2×Q814yes—
80C22×D107#4114yes—
81Q8×C107—44no—
82C2×C4×C107—22no—
83C2×C4×C107—28no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—24no—
87C2×C4×C107—24no—
88C2×C4×C107C2×D414yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×Dic107—28no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—24no—
95C2×Dic107—24no—
96C2×Dic107—24no—
97C2×Dic107C2×D414yes—
98C4×D107—216no—
99C4×D107—216no—
100C22×C107—22no—
101C22×C107—24no—
102C22×C107—22no—
103C22×C107—24no—
104C22×C107C22⋊Q814yescomplemented
105C22×C107C22×D412yes—
106C22×C107C22×D411yescharacteristic
107C22×C107C22×Q811yescharacteristic
108C22×C107C2×#4112yes—
109C22×C107C2×#4111yescharacteristic
110C2×D107—232no—
111C2×D107—24no—
112C2×D107—24no—
113C2×D107—24no—
114C2×D107—24no—
115C2×D107—24no—
116C2×D107—24no—
117C2×D107C22⋊Q818yescomplemented
118C4×C107—22no—
119C4×C107—42no—
120C4×C107—42no—
121C4×C107#12014yes—
122Dic107—24no—
123Dic107—44no—
124C2×C107—24no—
125C2×C107C2×#12011yescharacteristic
126C2×C107C2×C22⋊Q812yes—
127C2×C107C2×C22⋊Q811yescharacteristic
128C2×C107#52511yescharacteristic
129C2×C107#52512yes—
130D107—28no—
131#2848—1071nomaximal, Sylow
132C107#284811yescharacteristic, complemented, Sylow
133C24×C4—1071no—
134C2×C22⋊Q8—1072no—
135C2×C22⋊Q8D10711yescharacteristic, complemented
136#385—1074no—
137#385—1072no—
138#380—1071no—
139#380—1072no—
140#379—1072no—
141C25—1071no—
142C22×Q8C2×D10711yescharacteristic, complemented
143C23×C4—1072no—
144C23×C4—1071no—
145C23×C4—1072no—
146C23×C4—2148no—
147C23×C4—2144no—
148C23×C4C2×D10711yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—2148no—
151C2×C4⋊C4—1074no—
152C2×C4⋊C4—1072no—
153C2×C4⋊C4C2×D10711yescharacteristic, complemented
154C2×C4⋊C4C2×D10712yescomplemented
155C2×C22⋊C4—1072no—
156C2×C22⋊C4—1074no—
157C2×C22⋊C4—1074no—
158C2×C22⋊C4C2×D10712yes—
159#95—1074no—
160#95—1072no—
161C24—1072no—
162C24—1074no—
163C24—2144no—
164C24—2144no—
165C24—2142no—
166C24—2142no—
167C24C22×D10711yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—1072no—
172C22×C4—1078no—
173C22×C4—1074no—
174C22×C4—2144no—
175C22×C4—2144no—
176C22×C4—21432no—
177C22×C4—2144no—
178C22×C4—2142no—
179C22×C4—2144no—
180C22×C4—2144no—
181C22×C4—2144no—
182C22×C4C22⋊D10714yes—
183C22×C4C22×D10712yes—
184C22×C4C22×D10712yes—
185C22×C4C22×D10711yescharacteristic
186C22×C4C22×D10711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—2148no—
190C4⋊C4—2144no—
191C22⋊C4—24no—
192C22⋊C4—2148no—
193C22⋊C4—2148no—
194C22⋊C4—2144no—
195C23—22no—
196C23—22no—
197C23—1074no—
198C23—1074no—
199C23—1074no—
200C23—21416no—
201C23—2148no—
202C23—2148no—
203C23—21416no—
204C23—2148no—
205C23—2142no—
206C23—2142no—
207C23—2142no—
208C23—2142no—
209C23C2×C22⋊D10712yes—
210C23Q8×D10712yes—
211C23#1313381912yes—
212C23C23×D10711yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—1074no—
221C2×C4—21416no—
222C2×C4—2144no—
223C2×C4—2144no—
224C2×C4—2144no—
225C2×C4—2148no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—21416no—
229C2×C4D4×D10714yes—
230C2×C4C2×C22⋊D10712yes—
231C2×C4C2×C22⋊D10712yes—
232C22—24no—
233C22—22no—
234C22—24no—
235C22—22no—
236C22—1078no—
237C22—21432no—
238C22—2144no—
239C22—2144no—
240C22—2144no—
241C22—2144no—
242C22—2144no—
243C22—2144no—
244C22#1681208514yescomplemented
245C22C2×D4×D10712yes—
246C22C22×C22⋊D10711yescharacteristic
247C22C2×Q8×D10711yescharacteristic
248C22#41×D10712yes—
249C22C2×#1313381911yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—4284no—
254C4—2144no—
255C4#1681207714yes—
256C2—24no—
257C2—2148no—
258C2C2×#1681207711yescharacteristic
259C2C22⋊Q8×D10712yes—
260C2C2×#1681208511yescharacteristic
261C2#2029313411yescharacteristic
262C2#2029314012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC24×D107C22
hypercenterC2×C22⋊Q8D107
hypocenterC107#2848
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812085C2semidirect
3. #20292499C2semidirect
4. #20292498C2semidirect
5. #20292487C2semidirect
6. #20292484C2semidirect
8. C2×C22⋊Q8×C107C2semidirect
16. C2×C4⋊Dic107C22semidirect
17. C2×Dic107⋊C4C22semidirect
21. C107⋊#95C22semidirect
22. C107⋊#95C22semidirect
23. C22×Q8×C107C22semidirect
26. C2×C4⋊C4×C107C22semidirect
27. C2×C4⋊C4×C107C22semidirect
33. C23×D107Q8semidirect
104. C22×C107C22⋊Q8semidirect
117. C2×D107C22⋊Q8semidirect
132. C107#2848semidirect
135. C2×C22⋊Q8D107semidirect
142. C22×Q8C2×D107semidirect
153. C2×C4⋊C4C2×D107semidirect
154. C2×C4⋊C4C2×D107semidirect
244. C22#16812085semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1072
C2×#168120852
#202924992
#202924982
#202924872
#202924842
#202924822
C2×C22⋊Q8×C1072
#2848107