Finite Group Databasebeta

← #28344486

Subgroups — #28344486

Order 13696.

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

normal subgroups only

This group has 644 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×#16812081C212yesmaximal, complemented
2#20292556C212yesmaximal, complemented
3#20292555C212yesmaximal, complemented
4#20292536C212yesmaximal, complemented
5#20292536C211yesmaximal, characteristic, complemented
6#20292535C212yesmaximal, complemented
7#20292535C212yesmaximal, complemented
8#20292488C211yesmaximal, characteristic, complemented
9#392×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C22×C22⋊D107C2212yes—
13C22×C22⋊D107C2212yes—
14C23×Dic107C2211yescharacteristic
15#16812081—28no—
16C2×Dic107⋊C4C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2211yescharacteristic
19C2×C22⋊Dic107C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2211yescharacteristic
27C22×D4×C107C2212yes—
28C22×D4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107C2212yes—
31#95×C107C2211yescharacteristic
32C23×D107C2312yes—
33C2×D428—41no—
34C2×D428—42no—
35C2×C22⋊D107—24no—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—42no—
38C2×C22⋊D107—42no—
39C2×C22⋊D107—42no—
40C2×C22⋊D107—42no—
41C2×C22⋊D107—42no—
42C2×C22⋊D107—42no—
43C2×C22⋊D107—42no—
44C22×Dic107—41no—
45C22×Dic107—41no—
46C22×Dic107D414yescomplemented
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—24no—
52C22⋊Dic107—41no—
53C22⋊Dic107—42no—
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—41no—
62C2×D4×C107—42no—
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—22no—
89D4×C107—48no—
90D4×C107—44no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C22⋊D107—48no—
99C22⋊D107—48no—
100C22⋊D107—48no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—44no—
110C2×Dic107C2×D414yes—
111D428—44no—
112C22×C107—22no—
113C22×C107—24no—
114C22×C107—41no—
115C22×C107—42no—
116C22×C107—41no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107C22×D411yescharacteristic
125C22×C107H(5,𝔽2)11yescharacteristic
126C22×C107H(5,𝔽2)12yes—
127C22×C107H(5,𝔽2)12yes—
128C22×C107H(5,𝔽2)11yescharacteristic
129C2×D107—42no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C4×C107—42no—
137C4×C107—41no—
138C4×C107—41no—
139Dic107—24no—
140Dic107—42no—
141Dic107—42no—
142C2×C107—42no—
143C2×C107—82no—
144C2×C107#53411yescharacteristic
145C2×C107#53512yes—
146C2×C107#56011yescharacteristic
147C2×C107#56012yes—
148C2×C107#56011yescharacteristic
149D107—82no—
150#2815—1071nomaximal, Sylow
151C107#281511yescharacteristic, complemented, Sylow
152C2×#121—1072no—
153#397—1072no—
154#397—1072no—
155#392—1072no—
156#392—1072no—
157#392—1072no—
158#392—1071no—
159#392D10711yescharacteristic, complemented
160#381—1071no—
161C22×D4—1072no—
162C22×D4—1072no—
163C22×D4—1072no—
164C22×D4—1071no—
165C22×D4C2×D10712yes—
166C22×D4C2×D10711yescharacteristic
167C23×C4—1071no—
168#121—2148no—
169C2×C4⋊C4—1072no—
170C2×C22⋊C4—1071no—
171C2×C22⋊C4—1072no—
172C2×C22⋊C4—1072no—
173C2×C22⋊C4—1072no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4C2×D10711yescharacteristic
177C2×C22⋊C4C2×D10712yes—
178#95—1072no—
179#95—1072no—
180#95—1072no—
181#95—1071no—
182#95C2×D10711yescharacteristic
183C24—1072no—
184C24C22×D10712yes—
185C24C22×D10711yescharacteristic
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—4281no—
192C2×D4—4282no—
193C2×D4—4282no—
194C2×D4—4282no—
195C2×D4—4282no—
196C2×D4—4282no—
197C2×D4—4282no—
198C2×D4—4282no—
199C2×D4—4282no—
200C2×D4—2144no—
201C2×D4—2144no—
202C22×C4—1074no—
203C22×C4—1072no—
204C22×C4—1072no—
205C22×C4—1072no—
206C22×C4—4281no—
207C22×C4—4281no—
208C22×C4C22×D10711yescharacteristic
209C22×C4C22×D10711yescharacteristic
210C22×C4C22×D10712yes—
211C4⋊C4—2144no—
212C22⋊C4—41no—
213C22⋊C4—42no—
214C22⋊C4—4282no—
215C22⋊C4—4282no—
216C22⋊C4—4282no—
217C22⋊C4—4281no—
218C22⋊C4—2144no—
219C22⋊C4—2144no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—2142no—
232C23—2142no—
233C23—2142no—
234C23—2142no—
235C23—2142no—
236C23—2142no—
237C23—2142no—
238C23D4×D10712yes—
239C23C23×D10711yescharacteristic
240D4—48no—
241D4—44no—
242D4—4284no—
243D4—4288no—
244D4—4288no—
245D4—4288no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—1074no—
254C2×C4—4284no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4—2142no—
263C22—22no—
264C22—24no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—4282no—
276C22—4282no—
277C22—4282no—
278C22—4282no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22C2×D4×D10711yescharacteristic
283C22#1681215411yescharacteristic
284C22#1681215412yes—
285C22#1681215412yes—
286C22#1681215511yescharacteristic
287C4—42no—
288C4—41no—
289C4—41no—
290C4—4282no—
291C4—4282no—
292C4—2144no—
293C2—42no—
294C2—82no—
295C2—8562no—
296C2#2029318511yescharacteristic
297C2#2029319212yes—
298C2#2029336411yescharacteristic
299C2#2029336512yes—
300C2#2029336611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2815
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16812081C2semidirect
2. #20292556C2semidirect
3. #20292555C2semidirect
4. #20292536C2semidirect
5. #20292536C2semidirect
6. #20292535C2semidirect
7. #20292535C2semidirect
8. #20292488C2semidirect
9. #392×C107C2semidirect
46. C22×Dic107D4semidirect
151. C107#2815semidirect
159. #392D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120812
#202925562
#202925552
#202925362
#202925352
#202924882
#392×C1072
#2815107