Finite Group Databasebeta

← #28339633

Subgroups — #28339633

Order 13696.

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

normal subgroups only

This group has 520 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
1C2×#16812098C211yesmaximal, characteristic
2#20292534C211yesmaximal, characteristic, complemented
3#20292518C212yesmaximal, complemented
4#20292516C212yesmaximal, complemented
5#20292505C212yesmaximal
6#20292501C212yesmaximal, complemented
7#20292498C212yesmaximal, complemented
8#20292491C212yesmaximal, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C4×D107C2212yes—
12#16812098C418yescomplemented
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4C2212yes—
15C2×Dic107⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×C4×Dic107C2211yescharacteristic
22C2×C4×Dic107C2211yescharacteristic
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C22×Q8×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2212yes—
31#95×C107C2212yes—
32C23×D107C2312yes—
33C2×D428—41no—
34C2×D428—41no—
35C2×D428C2×C414yes—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C2×C4×D107—42no—
40C2×C4×D107—42no—
41C2×C4×D107—42no—
42C4⋊Dic107—41no—
43Dic107⋊C4—42no—
44Dic107⋊C4—42no—
45(C2×D107)⋊C4—42no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4—42no—
48(C2×D107)⋊C4C2×C418yescomplemented
49(C2×D107)⋊C4C2×C418yescomplemented
50C4×Dic107—41no—
51C4×Dic107C2×C414yescomplemented
52C2×Q8×C107—41no—
53C2×Q8×C107—41no—
54C2×Q8×C107C2×C414yescomplemented
55C22×C4×C107C2312yes—
56C22×C4×C107C2312yes—
57C22×C4×C107C2311yescharacteristic
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C4⋊C4×C107—41no—
61C42×C107—41no—
62C23×C107C2411yescharacteristic
63C22×D107—22no—
64C22×D107—22no—
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—22no—
68C22×D107—22no—
69C22×D107C22×C414yes—
70Q8×C107—44no—
71C2×C4×C107—21no—
72C2×C4×C107—21no—
73C2×C4×C107—21no—
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107C22×C412yes—
84C2×C4×C107C22×C412yes—
85C2×C4×C107C22×C412yes—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107C22×C414yes—
95D428—44no—
96C4×D107—48no—
97C22×C107C23×C411yescharacteristic
98C22×C107H(5,𝔽2)11yescharacteristic
99C22×C107H(5,𝔽2)11yescharacteristic
100C22×C107H(5,𝔽2)12yes—
101C22×C1072−1+412yes—
102C2×D107—24no—
103C2×D107—24no—
104C2×D107—24no—
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—42no—
109C4×C107—22no—
110C4×C107—22no—
111C4×C107—22no—
112C4×C107—42no—
113C4×C107—42no—
114Dic107—24no—
115Dic107—42no—
116Dic107—42no—
117C2×C107#51811yescharacteristic
118C2×C107#52012yes—
119C2×C107H(3,𝔽4)11yescharacteristic
120C2×C107#56212yes—
121C2×C107#56311yescharacteristic
122D107—44no—
123#2379—1071nomaximal, Sylow
124C107#237911yescharacteristic, complemented, Sylow
125C2×#124—1071no—
126#391—1071no—
127#391D10711yescharacteristic, complemented
128#388—1072no—
129#388—1072no—
130#386—1072no—
131#385—1072no—
132#385—1072no—
133#382—1072no—
134C22×Q8C2×D10711yescharacteristic
135C22×D4—1071no—
136C23×C4—1072no—
137#124—1078no—
138C2×C4⋊C4—1072no—
139C2×C4⋊C4—1072no—
140C2×C4⋊C4—1071no—
141C2×C4⋊C4C2×D10711yescharacteristic
142C2×C22⋊C4—1072no—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1072no—
147C2×C42—1071no—
148C2×C42—1071no—
149C2×C42C2×D10711yescharacteristic
150#95—1072no—
151#95—1072no—
152#95—1072no—
153#95—1072no—
154#95C2×D10712yes—
155#95C2×D10712yes—
156C24—1072no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D10714yescomplemented
160C2×D4—1074no—
161C2×D4—4281no—
162C2×D4—4281no—
163C22×C4—1072no—
164C22×C4—1072no—
165C22×C4—1072no—
166C22×C4—4282no—
167C22×C4—4282no—
168C22×C4—4282no—
169C22×C4C22×D10712yes—
170C22×C4C22×D10712yes—
171C22×C4C22×D10711yescharacteristic
172C22×C4C22×D10711yescharacteristic
173C22×C4C22×D10711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—4282no—
176C4⋊C4—4282no—
177C4⋊C4—4281no—
178C22⋊C4—1078no—
179C22⋊C4—1078no—
180C22⋊C4—4282no—
181C22⋊C4—4282no—
182C22⋊C4—4282no—
183C42—41no—
184C42—1074no—
185C42—4281no—
186C23—1074no—
187C23—2142no—
188C23—2142no—
189C23—2142no—
190C23—2142no—
191C23—2142no—
192C23—2142no—
193C23C23×D10711yescharacteristic
194Q8—44no—
195D4—4284no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—1074no—
209C2×C4—4288no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4C2×C4×D10712yes—
219C2×C4C2×C4×D10712yes—
220C2×C4C2×C4×D10712yes—
221C22—4282no—
222C22—4282no—
223C22—4282no—
224C22—4282no—
225C22—2144no—
226C22—2144no—
227C22—2144no—
228C22C22×C4×D10711yescharacteristic
229C22#1681215411yescharacteristic
230C22#1681215411yescharacteristic
231C22#1681215512yes—
232C22#1681215712yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—4282no—
239C4—4282no—
240C4—2144no—
241C2—4284no—
242C2#2029310011yescharacteristic
243C2#2029310912yes—
244C2#2029336911yescharacteristic
245C2#2029337312yes—
246C2#2029337611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#391×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D428C22
hypercenter#391D107
hypocenterC107#2379
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20292534C2semidirect
3. #20292518C2semidirect
4. #20292516C2semidirect
6. #20292501C2semidirect
7. #20292498C2semidirect
8. #20292491C2semidirect
9. #391×C107C2semidirect
12. #16812098C4semidirect
48. (C2×D107)⋊C4C2×C4semidirect
49. (C2×D107)⋊C4C2×C4semidirect
51. C4×Dic107C2×C4semidirect
54. C2×Q8×C107C2×C4semidirect
124. C107#2379semidirect
127. #391D107semidirect
159. C2×Q8C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
#202925342
#202925182
#202925162
#202925052
#202925012
#202924982
#202924912
#391×C1072
#2379107