Finite Group Databasebeta

← #28344455

Subgroups — #28344455

Order 13696.

712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812077C212yesmaximal, complemented
2#20292556C212yesmaximal, complemented
3#20292552C211yesmaximal, characteristic, complemented
4#20292543C212yesmaximal, complemented
5#20292542C212yesmaximal, complemented
6#20292541C212yesmaximal, complemented
7#20292535C212yesmaximal, complemented
8#20292482C211yesmaximal, characteristic, complemented
9#392×C107C211yesmaximal, characteristic, complemented
10C24×D107C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C22×C22⋊D107C2212yes—
13#16812077—28no—
14C2×C4⋊Dic107C2211yescharacteristic, complemented
15C2×Dic107⋊C4C2212yescomplemented
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2211yescharacteristic
18C2×C22⋊Dic107C2212yescomplemented
19C2×C22⋊Dic107C2212yescomplemented
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C107⋊#95C2212yescomplemented
25C107⋊#95C2212yescomplemented
26C107⋊#95C2211yescharacteristic, complemented
27C22×D4×C107C2212yes—
28C22×D4×C107C2211yescharacteristic, complemented
29C2×C22⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107C2212yescomplemented
31#95×C107C2211yescharacteristic, complemented
32C23×D107—42no—
33C23×D107—42no—
34C23×D107—41no—
35C23×D107—41no—
36C23×D107D414yescomplemented
37C23×D107C2312yes—
38C2×C22⋊D107—24no—
39C2×C22⋊D107—24no—
40C2×C22⋊D107—42no—
41C2×C22⋊D107—42no—
42C2×C22⋊D107—42no—
43C2×C22⋊D107—42no—
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2312yes—
47C4⋊Dic107—41no—
48Dic107⋊C4—42no—
49C22⋊Dic107—24no—
50C22⋊Dic107—41no—
51C22⋊Dic107—42no—
52C22⋊Dic107—42no—
53(C2×D107)⋊C4—24no—
54(C2×D107)⋊C4—24no—
55(C2×D107)⋊C4—42no—
56(C2×D107)⋊C4—42no—
57C24×C107C2312yes—
58C24×C107C2311yescharacteristic
59C2×D4×C107—24no—
60C2×D4×C107—42no—
61C2×D4×C107—42no—
62C2×D4×C107—41no—
63C2×D4×C107—42no—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2311yescharacteristic
67C22⋊C4×C107—41no—
68C22⋊C4×C107—42no—
69C23×C107—21no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—21no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107C2×D412yes—
81C23×C107C2411yescharacteristic
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—44no—
89C22×D107—48no—
90C22×D107—44no—
91C22×D107—48no—
92C22×D107—44no—
93C22×D107C2×D414yes—
94D4×C107—48no—
95D4×C107—44no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—21no—
99C2×C4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C22⋊D107—48no—
104C22⋊D107—48no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C22×C107—24no—
115C22×C107—22no—
116C22×C107—41no—
117C22×C107—42no—
118C22×C107—41no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107C22×D411yescharacteristic
127C22×C107H(5,𝔽2)12yes—
128C22×C107H(5,𝔽2)11yescharacteristic
129C22×C107H(5,𝔽2)12yes—
130C22×C107H(5,𝔽2)11yescharacteristic
131C2×D107—24no—
132C2×D107—24no—
133C2×D107—24no—
134C2×D107—416no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—42no—
139C4×C107—42no—
140C4×C107—41no—
141C4×C107—41no—
142Dic107—42no—
143Dic107—42no—
144Dic107—42no—
145C2×C107—42no—
146C2×C107—82no—
147C2×C107#53411yescharacteristic
148C2×C107#53412yes—
149C2×C107#56011yescharacteristic
150C2×C107#56012yes—
151C2×C107H(3,𝔽4)11yescharacteristic
152D107—44no—
153#2812—1071nomaximal, Sylow
154C107#281211yescharacteristic, complemented, Sylow
155C2×#120—1072no—
156#397—1071no—
157#397—1072no—
158#394—1072no—
159#394—1072no—
160#394—1072no—
161#392—1072no—
162#392D10711yescharacteristic, complemented
163#379—1071no—
164C25—1071no—
165C22×D4—1072no—
166C22×D4—1072no—
167C22×D4C2×D10712yes—
168C22×D4C2×D10711yescharacteristic, complemented
169#120—2148no—
170C2×C4⋊C4—1072no—
171C2×C4⋊C4—1071no—
172C2×C22⋊C4—1071no—
173C2×C22⋊C4—1072no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1072no—
177C2×C22⋊C4—1072no—
178C2×C22⋊C4—1072no—
179C2×C22⋊C4—1072no—
180C2×C22⋊C4C2×D10711yescharacteristic
181C2×C22⋊C4C2×D10712yescomplemented
182#95—1072no—
183#95—1071no—
184#95—1072no—
185#95C2×D10711yescharacteristic, complemented
186C24—1074no—
187C24—1072no—
188C24—4281no—
189C24—4282no—
190C24—4282no—
191C24—4281no—
192C24C22×D10712yes—
193C24C22×D10711yescharacteristic
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—4282no—
200C2×D4—4282no—
201C2×D4—4282no—
202C2×D4—4282no—
203C2×D4—2144no—
204C2×D4—2144no—
205C22×C4—1072no—
206C22×C4—1072no—
207C22×C4—1072no—
208C22×C4C22×D10711yescharacteristic
209C22×C4C22×D10712yes—
210C22×C4C22×D10711yescharacteristic
211C4⋊C4—4282no—
212C4⋊C4—4281no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—4282no—
216C22⋊C4—4282no—
217C22⋊C4—4282no—
218C22⋊C4—4281no—
219C22⋊C4—4282no—
220C22⋊C4—2144no—
221C22⋊C4—2144no—
222C22⋊C4—2144no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—1074no—
235C23—4284no—
236C23—4288no—
237C23—4284no—
238C23—4288no—
239C23—4284no—
240C23—2142no—
241C23—2142no—
242C23—2142no—
243C23—2142no—
244C23—2142no—
245C23—2142no—
246C23D4×D10712yes—
247C23C23×D10711yescharacteristic
248D4—48no—
249D4—44no—
250D4—4288no—
251D4—4288no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4—2142no—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42816no—
281C22—4282no—
282C22—4282no—
283C22—4282no—
284C22—4282no—
285C22—2144no—
286C22—2144no—
287C22—2144no—
288C22C2×D4×D10711yescharacteristic
289C22#1681215411yescharacteristic
290C22#1681215412yes—
291C22#1681215412yes—
292C22#1681215511yescharacteristic
293C4—42no—
294C4—41no—
295C4—41no—
296C4—4282no—
297C4—4282no—
298C4—4282no—
299C2—42no—
300C2—82no—
301C2—4284no—
302C2#2029318511yescharacteristic
303C2#2029318712yes—
304C2#2029336411yescharacteristic
305C2#2029336512yes—
306C2#2029337011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2812
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16812077C2semidirect
2. #20292556C2semidirect
3. #20292552C2semidirect
4. #20292543C2semidirect
5. #20292542C2semidirect
6. #20292541C2semidirect
7. #20292535C2semidirect
8. #20292482C2semidirect
9. #392×C107C2semidirect
14. C2×C4⋊Dic107C22semidirect
15. C2×Dic107⋊C4C22semidirect
18. C2×C22⋊Dic107C22semidirect
19. C2×C22⋊Dic107C22semidirect
24. C107⋊#95C22semidirect
25. C107⋊#95C22semidirect
26. C107⋊#95C22semidirect
28. C22×D4×C107C22semidirect
30. C2×C22⋊C4×C107C22semidirect
31. #95×C107C22semidirect
36. C23×D107D4semidirect
154. C107#2812semidirect
162. #392D107semidirect
168. C22×D4C2×D107semidirect
181. C2×C22⋊C4C2×D107semidirect
185. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120772
#202925562
#202925522
#202925432
#202925422
#202925412
#202925352
#202924822
#392×C1072
#2812107