Finite Group Databasebeta

← #28339821

Subgroups — #28339821

Order 13696.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D107C211yesmaximal, characteristic, complemented
2C2×#16812087C212yesmaximal, complemented
3C2×#16812071C211yesmaximal, characteristic
4#20292547C211yesmaximal, characteristic, complemented
5#20292540C211yesmaximal, characteristic, complemented
6#20292538C211yesmaximal, characteristic, complemented
7#20292531C212yesmaximal, complemented
8#20292505C212yesmaximal
9#20292498C212yesmaximal, complemented
10#20292486C211yesmaximal, characteristic, complemented
11#395×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×Dic214C2212yes—
14C2×Q8×D107—28no—
15C2×Q8×D107—24no—
16C22×D428C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C22×C4×D107C2212yes—
19C22×C4×D107C2211yescharacteristic
20#16812087—28no—
21#16812071—24no—
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2211yescharacteristic
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yes—
36C2×C4⋊C4×C107C2211yescharacteristic
37#95×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×Dic214—22no—
42C2×Dic214—24no—
43C2×Dic214—24no—
44C2×Dic214—24no—
45C2×Dic214D418yescomplemented
46Q8×D107—232no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428Q814yescomplemented
50C22×Dic107C2311yescharacteristic
51C22×Dic107C2312yes—
52C22×Dic107C2312yes—
53C22×Dic107C2311yescharacteristic
54C2×C4×D107—24no—
55C2×C4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107—24no—
58C2×C4×D107—22no—
59C2×C4×D107—24no—
60C2×C4×D107D414yes—
61Dic107⋊C4—24no—
62Dic107⋊C4—24no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—24no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—24no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—22no—
69C2×Q8×C107—24no—
70C2×Q8×C107—22no—
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2312yes—
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C4⋊C4×C107—24no—
78C23×C107C2411yescharacteristic
79C22×D107—22no—
80C22×D107—22no—
81C22×D107—22no—
82C22×D107—21no—
83C22×D107—21no—
84C22×D107—22no—
85C22×D107C2×D412yes—
86C22×D107C2×Q812yes—
87C22×D107C2×Q812yescomplemented
88C22×D107#4112yes—
89Q8×C107—28no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107C2×D414yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×Q812yescomplemented
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—21no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107C2×D414yes—
111C2×Dic107C2×D412yes—
112C2×Dic107C2×D414yes—
113D428—28no—
114C4×D107—216no—
115C4×D107—28no—
116C4×D107—44no—
117Dic214—28no—
118Dic214—216no—
119C22×C107C22×D411yescharacteristic
120C22×C107C22×D412yes—
121C22×C107C22×Q811yescharacteristic
122C22×C107C2×#4111yescharacteristic
123C22×C1072−1+412yes—
124C2×D107—22no—
125C2×D107—22no—
126C2×D107—24no—
127C2×D107—24no—
128C2×D107—22no—
129C2×D107—24no—
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—42no—
133C2×D107C22⋊Q814yescomplemented
134C4×C107—22no—
135C4×C107—24no—
136C4×C107—22no—
137C4×C107—41no—
138C4×C107—42no—
139Dic107—24no—
140Dic107—41no—
141Dic107—42no—
142Dic107#12014yes—
143C2×C107C2×#12011yescharacteristic
144C2×C107C2×C22⋊Q811yescharacteristic
145C2×C107#53611yescharacteristic
146C2×C107#54812yes—
147C2×C107D4×Q812yes—
148D107—24no—
149D107—42no—
150#2409—1071nomaximal, Sylow
151C107#240911yescharacteristic, complemented, Sylow
152C23×Q8—1071no—
153C2×C22⋊Q8—1072no—
154C2×C4×D4—1071no—
155#395—1071no—
156#395D10711yescharacteristic, complemented
157#393—1071no—
158#393—1071no—
159#391—1072no—
160#386—1072no—
161#385—1072no—
162#380—1071no—
163C22×Q8—1072no—
164C22×Q8—1071no—
165C22×Q8—2144no—
166C22×Q8—2148no—
167C22×Q8C2×D10711yescharacteristic
168C22×D4—1071no—
169C23×C4—1071no—
170C23×C4—1071no—
171C23×C4—1072no—
172C22⋊Q8—2148no—
173C4×D4—2144no—
174C2×C4⋊C4—1072no—
175C2×C4⋊C4—1072no—
176C2×C4⋊C4C2×D10711yescharacteristic
177C2×C4⋊C4C2×D10712yes—
178C2×C22⋊C4—1071no—
179C2×C22⋊C4—1071no—
180C2×C22⋊C4—1072no—
181C2×C22⋊C4—1072no—
182C2×C22⋊C4—1072no—
183C2×C42—1071no—
184#95—1071no—
185#95—1072no—
186#95—1072no—
187#95—1072no—
188#95C2×D10711yescharacteristic
189#95C2×D10712yes—
190C24—1071no—
191C24—1071no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—1078no—
195C2×Q8—2144no—
196C2×Q8—2144no—
197C2×Q8—2144no—
198C2×Q8—21432no—
199C2×Q8—2142no—
200C2×D4—1074no—
201C2×D4—2142no—
202C2×D4—2142no—
203C22×C4—1071no—
204C22×C4—1072no—
205C22×C4—1074no—
206C22×C4—1071no—
207C22×C4—1072no—
208C22×C4—2142no—
209C22×C4—2144no—
210C22×C4—2144no—
211C22×C4—2144no—
212C22×C4—2144no—
213C22×C4—2144no—
214C22×C4C22×D10712yes—
215C22×C4C22×D10711yescharacteristic
216C22×C4C22×D10711yescharacteristic
217C22×C4C22×D10712yes—
218C22×C4C22×D10711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—2144no—
222C4⋊C4—2144no—
223C22⋊C4—2144no—
224C22⋊C4—2142no—
225C22⋊C4—2144no—
226C22⋊C4—2142no—
227C22⋊C4—2144no—
228C42—2142no—
229C23—1072no—
230C23—1072no—
231C23—1072no—
232C23—1072no—
233C23—2142no—
234C23—2141no—
235C23—2141no—
236C23—2142no—
237C23—2142no—
238C23—2142no—
239C23C23×D10711yescharacteristic
240Q8—28no—
241Q8—21416no—
242Q8—2148no—
243D4—2148no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—1072no—
255C2×C4—1074no—
256C2×C4—1074no—
257C2×C4—4284no—
258C2×C4—2148no—
259C2×C4—21416no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4D4×D10712yes—
268C2×C4D4×D10714yes—
269C2×C4Q8×D10712yescomplemented
270C22—1074no—
271C22—4281no—
272C22—4282no—
273C22—4281no—
274C22—2142no—
275C22—2144no—
276C22—2142no—
277C22—2144no—
278C22—2144no—
279C22—2142no—
280C22C2×D4×D10711yescharacteristic
281C22C2×D4×D10712yes—
282C22C2×Q8×D10711yescharacteristic
283C22#41×D10711yescharacteristic
284C22#1681215712yes—
285C4—22no—
286C4—22no—
287C4—24no—
288C4—42no—
289C4—41no—
290C4—1074no—
291C4—4281no—
292C4—4282no—
293C4—2144no—
294C2—4282no—
295C2—2144no—
296C2#120×D10711yescharacteristic
297C2C22⋊Q8×D10711yescharacteristic
298C2#2029320211yescharacteristic
299C2#2029328712yes—
300C2#2029329212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#395×C107C2
prime radicalC22×D428C22
hypercenter#395D107
hypocenterC107#2409
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812087C2semidirect
4. #20292547C2semidirect
5. #20292540C2semidirect
6. #20292538C2semidirect
7. #20292531C2semidirect
9. #20292498C2semidirect
10. #20292486C2semidirect
11. #395×C107C2semidirect
45. C2×Dic214D4semidirect
49. C2×D428Q8semidirect
87. C22×D107C2×Q8semidirect
102. C2×C4×C107C2×Q8semidirect
133. C2×D107C22⋊Q8semidirect
151. C107#2409semidirect
156. #395D107semidirect
269. C2×C4Q8×D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168120872
C2×#168120712
#202925472
#202925402
#202925382
#202925312
#202925052
#202924982
#202924862
#395×C1072
#2409107