Finite Group Databasebeta

← #451×D107 #28345652

Subgroups — #451×D107 #28345652

Order 13696.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
2C2×Q16×D107C212yesmaximal, complemented
3C22⋊Q8×D107C211yesmaximal, characteristic
4#20292785C211yesmaximal, characteristic, complemented
5#20292784C212yesmaximal, complemented
6#20292783C211yesmaximal, characteristic, complemented
7#20292782C211yesmaximal, characteristic, complemented
8#20292781C212yesmaximal, complemented
9#103×D107C212yesmaximal, complemented
10C22⋊C8×D107C211yesmaximal, characteristic
11#451×C107C211yesmaximal, characteristic, complemented
12C22×Dic214—21no—
13C22×Dic214C2211yescharacteristic
14C2×Q8×D107—28no—
15C2×Q8×D107—21no—
16C2×Q8×D107—21no—
17C2×Q8×D107C2211yescharacteristic
18C2×Q8×D107C2212yes—
19C2×Q8×D107C2212yes—
20C22×C4×D107—21no—
21C22×C4×D107C2211yescharacteristic
22C2×Dic428C2212yescomplemented
23C2×#13133808C2212yescomplemented
24C2×#13133808C2212yescomplemented
25Q16×D107—28no—
26C2×C8×D107C2212yes—
27#16812089C2211yescharacteristic
28#16812088C2211yescharacteristic
29#16812087—21no—
30#16812086—21no—
31#16812085C2211yescharacteristic
32C4⋊C4×D107—21no—
33C4⋊C4×D107C2211yescharacteristic
34C22⋊C4×D107—21no—
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812032C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic
39(C2×D107)⋊C8C2212yes—
40C22×Q8×C107C2211yescharacteristic
41C2×Q16×C107C2212yescomplemented
42C22⋊Q8×C107C2211yescharacteristic
43#103×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46C2×Dic214—28no—
47C2×Dic214—21no—
48C2×Dic214—22no—
49C2×Dic214—22no—
50C2×Dic214—24no—
51C2×Dic214—21no—
52C2×Dic214C2311yescharacteristic
53C2×Dic214C2312yes—
54Q8×D107—28no—
55Q8×D107—28no—
56Q8×D107—24no—
57Q8×D107—24no—
58Q8×D107—24no—
59Q8×D107—24no—
60Q8×D107D418yes—
61C22×Dic107—21no—
62C22×Dic107D411yescharacteristic
63C2×C4×D107—28no—
64C2×C4×D107—22no—
65C2×C4×D107—22no—
66C2×C4×D107—24no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107D411yescharacteristic
70C2×C4×D107D411yescharacteristic
71C2×C4×D107C2311yescharacteristic
72C2×C4×D107C2311yescharacteristic
73Dic428—24no—
74#13133808—24no—
75#13133808—24no—
76C2×C107⋊C8C2312yes—
77C8×D107—24no—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86C2×Q8×C107—24no—
87C2×Q8×C107—21no—
88C2×Q8×C107C2311yescharacteristic
89C2×Q8×C107C2312yes—
90C22×C4×C107—21no—
91C22×C4×C107C2311yescharacteristic
92Q16×C107—24no—
93C2×C8×C107C2312yes—
94C4⋊C4×C107—21no—
95C4⋊C4×C107C2311yescharacteristic
96C22⋊C4×C107—21no—
97C23×C107C2×D411yescharacteristic
98C22×D107—22no—
99C22×D107—22no—
100C22×D107Q1614yescomplemented
101C22×D107C2×D411yescharacteristic
102C22×D107C2×D411yescharacteristic
103Q8×C107—24no—
104Q8×C107—22no—
105Q8×C107—22no—
106Q8×C107C2×D414yes—
107C2×C4×C107—24no—
108C2×C4×C107—22no—
109C2×C4×C107—22no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107C2×D411yescharacteristic
113C2×C4×C107C2411yescharacteristic
114C2×Dic107—24no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107C2×D411yescharacteristic
120C2×Dic107C2×D411yescharacteristic
121C4×D107—28no—
122C4×D107—28no—
123C4×D107—22no—
124C4×D107—22no—
125C4×D107—22no—
126C4×D107—42no—
127C4×D107—42no—
128C4×D107C2×D414yes—
129Dic214—28no—
130Dic214—24no—
131Dic214—22no—
132Dic214—28no—
133Dic214—22no—
134Dic214—42no—
135Dic214C2×D414yes—
136C8×C107—22no—
137C107⋊C8—22no—
138C22×C107—21no—
139C22×C107—21no—
140C22×C107C2×Q1612yescomplemented
141C22×C107C22×D411yescharacteristic
142C2×D107—21no—
143C2×D107—21no—
144C2×D107—28no—
145C2×D107#12012yes—
146C2×D107C2×Q1612yes—
147C2×D107C2×Q1612yescomplemented
148C2×D107#13712yes—
149C4×C107—24no—
150C4×C107—21no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C22×D412yes—
154Dic107—21no—
155Dic107—24no—
156Dic107—41no—
157Dic107—41no—
158Dic107#12012yes—
159C2×C107—22no—
160C2×C107C2×#12011yescharacteristic
161C2×C107C22×Q1611yescharacteristic
162C2×C107C2×#13711yescharacteristic
163D107—22no—
164D107#45114yesdirect product factor
165C2×#451—1071nomaximal, Sylow
166C107C2×#45111yescharacteristic, complemented, Sylow
167C23×Q8—1071no—
168C22×Q16—1072no—
169C2×C22⋊Q8—1071no—
170#451—1071no—
171#451—1072no—
172#451—1071no—
173#451—1072no—
174#451—1071no—
175#451D10711yescharacteristic, direct product factor
176C2×#103—1072no—
177C2×C22⋊C8—1071no—
178C22×Q8—1072no—
179C22×Q8—1071no—
180C22×Q8—1072no—
181C22×Q8—1071no—
182C22×Q8—2141no—
183C22×Q8—2141no—
184C22×Q8—2141no—
185C22×Q8—2148no—
186C22×Q8C2×D10711yescharacteristic
187C23×C4—1071no—
188C23×C4—2141no—
189C2×Q16—1072no—
190C2×Q16—1072no—
191C2×Q16—1072no—
192C2×Q16—2148no—
193C2×Q16C2×D10712yescomplemented
194C22×C8—1072no—
195C22⋊Q8—1071no—
196C22⋊Q8—1071no—
197C22⋊Q8—1071no—
198C22⋊Q8—2141no—
199C22⋊Q8—2141no—
200C22⋊Q8C2×D10711yescharacteristic
201C2×C4⋊C4—1071no—
202C2×C4⋊C4—2141no—
203C2×C22⋊C4—2141no—
204#103—1072no—
205#103—1072no—
206#103—1072no—
207#103C2×D10712yescomplemented
208C22⋊C8—1071no—
209C22⋊C8—1072no—
210C22⋊C8C2×D10711yescharacteristic
211C24—1071no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—1078no—
215C2×Q8—1072no—
216C2×Q8—1071no—
217C2×Q8—2141no—
218C2×Q8—2142no—
219C2×Q8—2142no—
220C2×Q8—2144no—
221C2×Q8—2148no—
222C2×Q8—2144no—
223C2×Q8—2144no—
224C2×Q8—2144no—
225C2×Q8—2148no—
226C2×Q8—2148no—
227C2×Q8—2144no—
228C2×Q8—2141no—
229C2×Q8C22×D10711yescharacteristic
230C2×Q8C22×D10712yes—
231C22×C4—21no—
232C22×C4—1071no—
233C22×C4—1071no—
234C22×C4—1071no—
235C22×C4—1071no—
236C22×C4—1071no—
237C22×C4—2144no—
238C22×C4—2141no—
239C22×C4—2141no—
240C22×C4—2142no—
241C22×C4—2142no—
242C22×C4—2148no—
243C22×C4—2141no—
244C22×C4C22×D10711yescharacteristic
245Q16—24no—
246Q16—2144no—
247Q16—2144no—
248Q16—2144no—
249C2×C8—1072no—
250C2×C8—2144no—
251C2×C8C22×D10712yes—
252C4⋊C4—21no—
253C4⋊C4—1071no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4C22×D10711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C23—1071no—
264C23—1074no—
265C23—1071no—
266C23—2142no—
267C23—2142no—
268C23D4×D10711yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—1074no—
273Q8—4282no—
274Q8—2148no—
275Q8—2148no—
276Q8—2144no—
277Q8—2142no—
278Q8—2142no—
279Q8D4×D10714yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—1071no—
286C2×C4—1074no—
287C2×C4—1071no—
288C2×C4—4282no—
289C2×C4—4282no—
290C2×C4—2142no—
291C2×C4—2142no—
292C2×C4—2148no—
293C2×C4—2148no—
294C2×C4—2142no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2144no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4D4×D10711yescharacteristic
301C2×C4C23×D10711yescharacteristic
302C8—22no—
303C8—2142no—
304C22—21no—
305C22—21no—
306C22—1072no—
307C22—1072no—
308C22—1072no—
309C22—1072no—
310C22—2148no—
311C22—2141no—
312C22—2141no—
313C22Q16×D10712yescomplemented
314C22C2×D4×D10711yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—1072no—
320C4—4281no—
321C4—4281no—
322C4—2141no—
323C4—2144no—
324C4C2×D4×D10712yes—
325C2—22no—
326C2—1074no—
327C2—2142no—
328C2#120×D10711yescharacteristic
329C2C2×Q16×D10711yescharacteristic
330C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#451×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
hypercenter#451D107
hypocenterC107C2×#451
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D107C2semidirect
4. #20292785C2semidirect
5. #20292784C2semidirect
6. #20292783C2semidirect
7. #20292782C2semidirect
8. #20292781C2semidirect
9. #103×D107C2semidirect
11. #451×C107C2semidirect
22. C2×Dic428C22semidirect
23. C2×#13133808C22semidirect
24. C2×#13133808C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812032C22semidirect
41. C2×Q16×C107C22semidirect
43. #103×C107C22semidirect
100. C22×D107Q16semidirect
140. C22×C107C2×Q16semidirect
147. C2×D107C2×Q16semidirect
164. D107#451direct (N × H)
166. C107C2×#451semidirect
175. #451D107direct (N × H)
193. C2×Q16C2×D107semidirect
207. #103C2×D107semidirect
313. C22Q16×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×Q16×D1072
C22⋊Q8×D1072
#202927852
#202927842
#202927832
#202927822
#202927812
#103×D1072
C22⋊C8×D1072
#451×C1072
C2×#451107