Finite Group Databasebeta

← #28339150

Subgroups — #28339150

Order 13696.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812083C211yesmaximal, characteristic
2C2×#16812071C211yesmaximal, characteristic, complemented
3C2×#16812068C211yesmaximal, characteristic, complemented
4#20292528C211yesmaximal, characteristic, complemented
5#20292527C212yesmaximal, complemented
6#20292523C211yesmaximal, characteristic, complemented
7#20292519C212yesmaximal, complemented
8#20292503C212yesmaximal, complemented
9#20292501C212yesmaximal, complemented
10C4⋊C4×Dic107C211yesmaximal, characteristic, complemented
11#380×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C4×D107C2212yes—
15#16812083C418yescomplemented
16#16812071—24no—
17#16812068—24no—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C22⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C23×C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37#95×C107C2212yes—
38C23×D107C2312yes—
39C2×D428—22no—
40C2×D428—22no—
41C2×D428C2×C414yes—
42C2×C22⋊D107—42no—
43C2×C22⋊D107—42no—
44C2×C22⋊D107C2×C418yescomplemented
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C2×C4×D107—24no—
51C2×C4×D107—42no—
52C2×C4×D107—42no—
53C4⋊Dic107—22no—
54C4⋊Dic107—22no—
55C4⋊Dic107C2×C414yescomplemented
56Dic107⋊C4—42no—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—42no—
61(C2×D107)⋊C4—42no—
62(C2×D107)⋊C4C2×C418yescomplemented
63C4×Dic107—22no—
64C4×Dic107—42no—
65C4×Dic107D414yescomplemented
66C24×C107C2311yescharacteristic
67C22×C4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107C2×C414yescomplemented
70C22×C4×C107C2312yes—
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—22no—
75C4⋊C4×C107—22no—
76C22⋊C4×C107—42no—
77C23×C107—21no—
78C23×C107—21no—
79C23×C107—21no—
80C23×C107—21no—
81C23×C107—21no—
82C23×C107—21no—
83C23×C107C22×C412yes—
84C23×C107C2411yescharacteristic
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—22no—
90C22×D107—22no—
91C22×D107C22×C414yes—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—28no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107C22×C412yes—
102C2×C4×C107C22×C412yes—
103C2×C4×C107C2×D412yes—
104C2×C4×C107#4112yes—
105C22⋊D107—48no—
106C2×Dic107—21no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107—21no—
111C2×Dic107—21no—
112C2×Dic107—21no—
113C2×Dic107—21no—
114C2×Dic107—21no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107C22×C414yes—
119C2×Dic107C2×D412yes—
120C2×Dic107C2×D412yes—
121D428—28no—
122C4×D107—48no—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—41no—
127C22×C107—41no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107C23×C411yescharacteristic
131C22×C107C22×D411yescharacteristic
132C22×C107C2×#4111yescharacteristic
133C22×C107H(5,𝔽2)11yescharacteristic
134C22×C107H(5,𝔽2)11yescharacteristic
135C22×C107H(5,𝔽2)11yescharacteristic
136C22×C1072−1+411yescharacteristic
137C2×D107—24no—
138C2×D107—24no—
139C2×D107—24no—
140C2×D107—42no—
141C2×D107—42no—
142C2×D107—42no—
143C2×D107—42no—
144C4×C107—22no—
145C4×C107—42no—
146C4×C107—42no—
147C4×C107C4×D414yes—
148Dic107—24no—
149Dic107—22no—
150Dic107—22no—
151Dic107—41no—
152Dic107—41no—
153C2×C107—42no—
154C2×C107C2×C4×D411yescharacteristic
155C2×C107#51811yescharacteristic
156C2×C107#52011yescharacteristic
157C2×C107#53511yescharacteristic
158C2×C107#53711yescharacteristic
159C2×C107#54011yescharacteristic
160C2×C107#54211yescharacteristic
161D107—44no—
162#2324—1071nomaximal, Sylow
163C107#232411yescharacteristic, complemented, Sylow
164C2×#121—1071no—
165C2×C4×D4—1071no—
166C2×#117—1071no—
167#390—1071no—
168#390—1072no—
169#389—1071no—
170#388—1072no—
171#386—1072no—
172#385—1072no—
173#380D10711yescharacteristic, complemented
174C4×C4⋊C4—1071no—
175C22×D4—1072no—
176C22×D4—1071no—
177C23×C4—1072no—
178C23×C4C2×D10711yescharacteristic
179#121—1078no—
180C4×D4—2144no—
181#117—2144no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4—1072no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4C2×D10711yescharacteristic
187C2×C4⋊C4C2×D10711yescharacteristic
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1071no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4C2×D10712yes—
195C2×C42—1071no—
196C2×C42—1072no—
197C2×C42—1071no—
198#95—1072no—
199#95—1072no—
200#95C2×D10712yes—
201C24—1072no—
202C24C22×D10711yescharacteristic
203C2×D4—1078no—
204C2×D4—1074no—
205C2×D4—4282no—
206C2×D4—4282no—
207C2×D4—2142no—
208C2×D4—2142no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—1072no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—1071no—
215C22×C4—1071no—
216C22×C4—4282no—
217C22×C4—4282no—
218C22×C4—2144no—
219C22×C4C4×D10714yescomplemented
220C22×C4C22×D10712yes—
221C22×C4C22×D10712yes—
222C22×C4C22×D10711yescharacteristic
223C22×C4C22×D10711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—1074no—
227C4⋊C4—4282no—
228C4⋊C4—2142no—
229C4⋊C4—2142no—
230C22⋊C4—42no—
231C22⋊C4—1078no—
232C22⋊C4—4282no—
233C22⋊C4—4282no—
234C22⋊C4—2144no—
235C22⋊C4—2142no—
236C22⋊C4—2142no—
237C42—1074no—
238C42—4282no—
239C42—2142no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—1074no—
247C23—2142no—
248C23—2142no—
249C23—2142no—
250C23—2142no—
251C23—2142no—
252C23—2142no—
253C23C2×C4×D10712yes—
254C23C23×D10711yescharacteristic
255D4—4288no—
256D4—2148no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1074no—
267C2×C4—1072no—
268C2×C4—1072no—
269C2×C4—4288no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2141no—
274C2×C4—2141no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2142no—
279C2×C4—2142no—
280C2×C4—2141no—
281C2×C4—2141no—
282C2×C4C2×C4×D10712yes—
283C2×C4C2×C4×D10712yes—
284C2×C4D4×D10712yes—
285C2×C4#1313381712yes—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—4282no—
294C22—4282no—
295C22—4282no—
296C22—4282no—
297C22—2144no—
298C22—2144no—
299C22—2144no—
300C22C22×C4×D10711yescharacteristic
301C22C2×D4×D10711yescharacteristic
302C22C2×#1313381711yescharacteristic
303C22#1681215411yescharacteristic
304C22#1681215511yescharacteristic
305C22#1681215511yescharacteristic
306C22#1681215711yescharacteristic
307C4—22no—
308C4—42no—
309C4—42no—
310C4—4281no—
311C4—4281no—
312C4—2142no—
313C4—2142no—
314C4—2144no—
315C4(C22⋊D107)⋊C414yes—
316C2—42no—
317C2—4284no—
318C2C2×(C22⋊D107)⋊C411yescharacteristic
319C2#2029310111yescharacteristic
320C2#2029310811yescharacteristic
321C2#2029319311yescharacteristic
322C2#2029320711yescharacteristic
323C2#2029323011yescharacteristic
324C2#2029325011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#380×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812083C2
hypercenter#380D107
hypocenterC107#2324
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16812071C2semidirect
3. C2×#16812068C2semidirect
4. #20292528C2semidirect
5. #20292527C2semidirect
6. #20292523C2semidirect
7. #20292519C2semidirect
8. #20292503C2semidirect
9. #20292501C2semidirect
10. C4⋊C4×Dic107C2semidirect
11. #380×C107C2semidirect
15. #16812083C4semidirect
44. C2×C22⋊D107C2×C4semidirect
55. C4⋊Dic107C2×C4semidirect
62. (C2×D107)⋊C4C2×C4semidirect
65. C4×Dic107D4semidirect
69. C22×C4×C107C2×C4semidirect
163. C107#2324semidirect
173. #380D107semidirect
219. C22×C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120712
C2×#168120682
#202925282
#202925272
#202925232
#202925192
#202925032
#202925012
C4⋊C4×Dic1072
#380×C1072
#2324107