Finite Group Databasebeta

← #28339129

Subgroups — #28339129

Order 13696.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812093C211yesmaximal, characteristic
2C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
3C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
4#20292523C212yesmaximal, complemented
5#20292516C212yesmaximal, complemented
6#20292510C212yesmaximal, complemented
7#20292506C211yesmaximal, characteristic
8#20292498C212yesmaximal, complemented
9#20292488C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic107C211yesmaximal, characteristic, complemented
11#385×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C23×Dic107C2211yescharacteristic
15C22×C4×D107C2211yescharacteristic
16#16812093C418yescomplemented
17(C22⋊D107)⋊C4—24no—
18C2×C4⋊Dic107C2212yes—
19C2×Dic107⋊C4—24no—
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2211yescharacteristic
23C2×C22⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C2×C4×Dic107C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C23×C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35C2×C22⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37#95×C107C2212yes—
38C23×D107C2311yescharacteristic
39C2×C22⋊D107—22no—
40C2×C22⋊D107—22no—
41C2×C22⋊D107C2×C414yes—
42C22×Dic107—22no—
43C22×Dic107—22no—
44C22×Dic107—42no—
45C22×Dic107C2×C414yescomplemented
46C22×Dic107D414yes—
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2312yes—
50C22×Dic107C2311yescharacteristic
51C22×Dic107C2311yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—42no—
54C4⋊Dic107C2×C418yescomplemented
55Dic107⋊C4—24no—
56Dic107⋊C4—24no—
57Dic107⋊C4—22no—
58C22⋊Dic107—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—42no—
61(C2×D107)⋊C4C2×C418yescomplemented
62C4×Dic107—22no—
63C4×Dic107—42no—
64C24×C107C2311yescharacteristic
65C22×C4×C107—22no—
66C22×C4×C107—42no—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2312yes—
71C4⋊C4×C107—42no—
72C22⋊C4×C107—22no—
73C22⋊C4×C107C2×C414yescomplemented
74C23×C107—22no—
75C23×C107—22no—
76C23×C107C22×C412yes—
77C23×C107C2×D412yes—
78C23×C107#4112yes—
79C23×C107C2411yescharacteristic
80C22×D107—21no—
81C22×D107—21no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107C22×C412yes—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—44no—
95C2×C4×C107C22×C414yes—
96C22⋊D107—28no—
97C2×Dic107—28no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—44no—
108C2×Dic107C22×C414yes—
109C2×Dic107C22×C412yes—
110C2×Dic107C2×D412yes—
111C2×Dic107C2×D412yes—
112C4×D107—44no—
113C22×C107—24no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—24no—
117C22×C107C4×D414yes—
118C22×C107C23×C411yescharacteristic
119C22×C107C22×D411yescharacteristic
120C22×C107C2×#4111yescharacteristic
121C22×C107H(5,𝔽2)12yes—
122C22×C1072−1+412yes—
123C2×D107—22no—
124C2×D107—24no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—42no—
128C4×C107—24no—
129C4×C107—41no—
130C4×C107—42no—
131C4×C107—41no—
132Dic107—24no—
133Dic107—22no—
134Dic107—22no—
135Dic107—22no—
136Dic107—42no—
137C2×C107—24no—
138C2×C107C2×C4×D411yescharacteristic
139C2×C107#52012yes—
140C2×C107#53411yescharacteristic
141C2×C107#53611yescharacteristic
142C2×C107#53912yes—
143D107—42no—
144#2322—1071nomaximal, Sylow
145C107#232211yescharacteristic, complemented, Sylow
146C2×#123—1071no—
147C2×C4×D4—1071no—
148C22×C4⋊C4—1071no—
149#389—1072no—
150#388—1072no—
151#387—1072no—
152#386—1071no—
153#385—1072no—
154#385D10711yescharacteristic, complemented
155#381—1071no—
156C4×C22⋊C4—1071no—
157C22×D4—1071no—
158C23×C4—1071no—
159C23×C4—1071no—
160C23×C4—1071no—
161C23×C4C2×D10711yescharacteristic
162#123—1078no—
163C4×D4—2144no—
164C2×C4⋊C4—1072no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1072no—
167C2×C4⋊C4—1071no—
168C2×C4⋊C4—2144no—
169C2×C4⋊C4C2×D10712yes—
170C2×C22⋊C4—1072no—
171C2×C22⋊C4—1072no—
172C2×C22⋊C4—1071no—
173C2×C22⋊C4—1071no—
174C2×C22⋊C4C2×D10711yescharacteristic
175C2×C22⋊C4C2×D10711yescharacteristic
176C2×C42—1072no—
177C2×C42—1071no—
178#95—1072no—
179#95—1072no—
180#95—1072no—
181#95—1072no—
182#95C2×D10712yes—
183C24—1071no—
184C24C22×D10711yescharacteristic
185C2×D4—1074no—
186C2×D4—2142no—
187C2×D4—2142no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—1074no—
191C22×C4—1074no—
192C22×C4—1071no—
193C22×C4—1071no—
194C22×C4—1072no—
195C22×C4—1071no—
196C22×C4—1072no—
197C22×C4—4282no—
198C22×C4—4282no—
199C22×C4—2142no—
200C22×C4—2142no—
201C22×C4—2142no—
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C22×C4C22×D10712yes—
205C22×C4C22×D10712yes—
206C4⋊C4—42no—
207C4⋊C4—1078no—
208C4⋊C4—2144no—
209C4⋊C4—2144no—
210C4⋊C4—2142no—
211C22⋊C4—22no—
212C22⋊C4—1078no—
213C22⋊C4—4282no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C22⋊C4C4×D10714yescomplemented
217C42—4282no—
218C42—2142no—
219C23—22no—
220C23—22no—
221C23—1072no—
222C23—2141no—
223C23—2141no—
224C23—2142no—
225C23—2142no—
226C23C2×C4×D10712yes—
227C23D4×D10712yes—
228C23#1313381912yes—
229C23C23×D10711yescharacteristic
230D4—2148no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—1074no—
242C2×C4—1072no—
243C2×C4—1072no—
244C2×C4—1072no—
245C2×C4—4284no—
246C2×C4—4284no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2148no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2141no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4C2×C4×D10714yes—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—24no—
262C22—4281no—
263C22—4281no—
264C22—4282no—
265C22—2142no—
266C22—2144no—
267C22#1681207114yes—
268C22C22×C4×D10711yescharacteristic
269C22C2×D4×D10711yescharacteristic
270C22C2×#1313381911yescharacteristic
271C22#1681215412yes—
272C22#1681215612yes—
273C4—24no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—4282no—
278C4—2142no—
279C4—2142no—
280C4—2144no—
281C4—2142no—
282C2—24no—
283C2—4282no—
284C2C2×#1681207111yescharacteristic
285C2#2029310612yes—
286C2#2029318711yescharacteristic
287C2#2029319911yescharacteristic
288C2#2029322612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#385×C107C2
socle, cosocleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenter#385D107
hypocenterC107#2322
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C22×Dic107⋊C4C2semidirect
4. #20292523C2semidirect
5. #20292516C2semidirect
6. #20292510C2semidirect
8. #20292498C2semidirect
9. #20292488C2semidirect
10. C22⋊C4×Dic107C2semidirect
11. #385×C107C2semidirect
16. #16812093C4semidirect
45. C22×Dic107C2×C4semidirect
54. C4⋊Dic107C2×C4semidirect
61. (C2×D107)⋊C4C2×C4semidirect
73. C22⋊C4×C107C2×C4semidirect
145. C107#2322semidirect
154. #385D107semidirect
216. C22⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120932
C2×(C22⋊D107)⋊C42
C22×Dic107⋊C42
#202925232
#202925162
#202925102
#202925062
#202924982
#202924882
C22⋊C4×Dic1072
#385×C1072
#2322107