Finite Group Databasebeta

← C2×#20293283 #28349924

Subgroups — C2×#20293283 #28349924

Order 13696.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133817C211yesmaximal, characteristic, complemented
2C23×Dic214C211yesmaximal, characteristic, complemented
3#20293283C2116yesmaximal, direct product factor
4C2×#16812096C211yesmaximal, characteristic, complemented
5C2×#16812095C212yesmaximal, complemented
6C2×#16812089C211yesmaximal, characteristic, complemented
7C2×#16812086C212yesmaximal, complemented
8C2×#16812081C212yesmaximal, complemented
9C2×#16812074C211yesmaximal, characteristic, complemented
10C2×D4×Dic107C212yesmaximal, complemented
11C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
12C2×#121×C107C211yesmaximal, characteristic, complemented
13C2×#13133817—28no—
14C2×#13133817C2218yescomplemented
15C22×Dic214—28no—
16C22×Dic214C2212yes—
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic
19C22×Dic214C2218yescomplemented
20C22×C22⋊D107C2212yes—
21C22×C22⋊D107C2211yescharacteristic
22C23×Dic107C2212yes—
23C23×Dic107C2212yes—
24C22×C4×D107C2211yescharacteristic
25#16812096C2218yescomplemented
26#16812095C22116yescomplemented
27#16812089C2218yescomplemented
28#16812086C22116yescomplemented
29#16812081C22116yescomplemented
30#16812074C2218yescomplemented
31D4×Dic107C22116yescomplemented
32C4×C22⋊D107C2218yescomplemented
33C2×C4⋊Dic107C2212yes—
34C2×Dic107⋊C4C2211yescharacteristic
35C2×Dic107⋊C4C2212yes—
36C2×C22⋊Dic107C2212yes—
37C2×C22⋊Dic107C2212yes—
38C2×C22⋊Dic107C2211yescharacteristic
39C2×(C2×D107)⋊C4C2212yes—
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×C4×Dic107C2212yes—
42C2×C4×Dic107C2211yescharacteristic
43C22×D4×C107C2211yescharacteristic
44C22×D4×C107C2212yes—
45C23×C4×C107C2211yescharacteristic
46#121×C107C2218yescomplemented
47C2×C4⋊C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2212yes—
49C23×D107C2311yescharacteristic
50#13133817—232no—
51C2×Dic214—24no—
52C2×Dic214—28no—
53C2×Dic214—232no—
54C2×Dic214D418yescomplemented
55C2×Dic214C2318yes—
56C2×Dic214C2314yes—
57C2×Dic214C2314yes—
58C2×C22⋊D107—24no—
59C2×C22⋊D107—28no—
60C2×C22⋊D107C2314yes—
61C2×C22⋊D107C2318yes—
62C22×Dic107—28no—
63C22×Dic107—28no—
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2312yes—
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2318yes—
69C22×Dic107C2318yes—
70C22×Dic107C2312yes—
71C2×C4×D107—24no—
72C2×C4×D107C2314yes—
73C4⋊Dic107C2318yes—
74Dic107⋊C4C2314yes—
75Dic107⋊C4C2318yes—
76C22⋊Dic107C2314yes—
77C22⋊Dic107C2318yes—
78C22⋊Dic107C2318yes—
79(C2×D107)⋊C4C2314yes—
80(C2×D107)⋊C4C2318yes—
81C4×Dic107C2314yes—
82C4×Dic107C2318yes—
83C24×C107C2312yes—
84C24×C107C2311yescharacteristic
85C2×D4×C107—28no—
86C2×D4×C107—24no—
87C2×D4×C107C2318yes—
88C2×D4×C107C2314yes—
89C22×C4×C107—24no—
90C22×C4×C107C2312yes—
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2314yes—
93C22×C4×C107C2311yescharacteristic
94C4⋊C4×C107C2314yes—
95C22⋊C4×C107C2318yes—
96C23×C107—22no—
97C23×C107—22no—
98C23×C107—24no—
99C23×C107—24no—
100C23×C107—24no—
101C23×C107#4114yescomplemented
102C23×C107C2414yes—
103C23×C107C2412yes—
104C23×C107C2411yescharacteristic
105C22×D107—22no—
106C22×D107—22no—
107C22×D107—22no—
108C22×D107C2412yes—
109D4×C107—216no—
110D4×C107—28no—
111C2×C4×C107—22no—
112C2×C4×C107—24no—
113C2×C4×C107—28no—
114C2×C4×C107C2×D414yes—
115C2×C4×C107C2412yes—
116C2×C4×C107C2414yes—
117C2×C4×C107C2412yes—
118C22⋊D107—28no—
119C22⋊D107—216no—
120C2×Dic107—22no—
121C2×Dic107—24no—
122C2×Dic107—22no—
123C2×Dic107—22no—
124C2×Dic107—216no—
125C2×Dic107—216no—
126C2×Dic107C2×D418yes—
127C2×Dic107C2412yes—
128C2×Dic107C2412yes—
129C2×Dic107C2414yes—
130C2×Dic107C2412yes—
131C2×Dic107C2414yes—
132C4×D107—28no—
133Dic214—28no—
134Dic214—216no—
135Dic214C2×D4116yescomplemented
136C22×C107—28no—
137C22×C107—22no—
138C22×C107—22no—
139C22×C107—24no—
140C22×C107—24no—
141C22×C107—24no—
142C22×C107—48no—
143C22×C107C22×D412yes—
144C22×C107C2×#4112yes—
145C22×C107C2×#4114yescomplemented
146C22×C1072−1+412yes—
147C22×C107C2511yescharacteristic
148C2×D107—22no—
149C2×D107—22no—
150C2×D107—22no—
151C2×D107—44no—
152C4×C107—22no—
153C4×C107—24no—
154C4×C107C22×D414yes—
155Dic107—24no—
156Dic107—22no—
157Dic107—22no—
158Dic107—22no—
159Dic107C22×D418yes—
160C2×C107—24no—
161C2×C107—44no—
162C2×C107#54814yescomplemented
163C2×C107C23×D411yescharacteristic
164C2×C107C22×#4111yescharacteristic
165C2×C107C2×2−1+411yescharacteristic
166D107—42no—
167C2×#548—1071nomaximal, Sylow
168C107C2×#54811yescharacteristic, complemented, Sylow
169C22×#41—1071no—
170C23×Q8—1071no—
171#548—10716no—
172C2×#124—1071no—
173C2×#124—1072no—
174C2×C22⋊Q8—1071no—
175C2×C22⋊Q8—1072no—
176C2×#121—1072no—
177C2×#121D10711yescharacteristic, complemented
178C2×C4×Q8—1071no—
179C2×C4×D4—1072no—
180C2×C4×D4—1071no—
181C2×#41—1078no—
182C2×#41—2148no—
183C22×Q8—1071no—
184C22×Q8—1072no—
185C22×Q8—1071no—
186C22×Q8—1078no—
187C22×Q8—2148no—
188C22×D4—1071no—
189C22×D4—1072no—
190C22×D4C2×D10712yes—
191C22×D4C2×D10711yescharacteristic
192C23×C4—1072no—
193C23×C4—1072no—
194C23×C4—1071no—
195C23×C4C2×D10711yescharacteristic
196#124—1078no—
197#124—10716no—
198C22⋊Q8—1078no—
199C22⋊Q8—10716no—
200#121—10716no—
201#121C2×D10718yescomplemented
202C4×Q8—1078no—
203C4×D4—10716no—
204C4×D4—1078no—
205C2×C4⋊C4—1071no—
206C2×C4⋊C4—1072no—
207C2×C4⋊C4—1072no—
208C2×C4⋊C4C2×D10711yescharacteristic
209C2×C22⋊C4—1071no—
210C2×C22⋊C4—1072no—
211C2×C22⋊C4—1072no—
212C2×C22⋊C4—1071no—
213C2×C22⋊C4—1072no—
214C2×C22⋊C4C2×D10712yes—
215C2×C42—1072no—
216C2×C42—1071no—
217C24—1071no—
218C24C22×D10712yes—
219C24C22×D10711yescharacteristic
220#41—21432no—
221C2×Q8—1078no—
222C2×Q8—1074no—
223C2×Q8—1078no—
224C2×Q8—1074no—
225C2×Q8—2148no—
226C2×Q8—21432no—
227C2×Q8—2144no—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—1074no—
231C2×D4—1078no—
232C2×D4—2148no—
233C2×D4—2144no—
234C2×D4C22×D10718yes—
235C2×D4C22×D10714yes—
236C22×C4—24no—
237C22×C4—1072no—
238C22×C4—1078no—
239C22×C4—1078no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—1072no—
243C22×C4—1071no—
244C22×C4—1074no—
245C22×C4—2148no—
246C22×C4—2148no—
247C22×C4—2144no—
248C22×C4C22×D10714yes—
249C22×C4C22×D10712yes—
250C22×C4C22×D10711yescharacteristic
251C22×C4C22×D10711yescharacteristic
252C4⋊C4—1074no—
253C4⋊C4—1078no—
254C4⋊C4—1078no—
255C4⋊C4C22×D10714yes—
256C22⋊C4—1074no—
257C22⋊C4—1078no—
258C22⋊C4—1078no—
259C22⋊C4—1074no—
260C22⋊C4—1078no—
261C22⋊C4C22×D10718yes—
262C42—1078no—
263C42—1074no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—1072no—
270C23—2142no—
271C23—2142no—
272C23—2142no—
273C23#1313381714yescomplemented
274C23C23×D10714yes—
275C23C23×D10712yes—
276C23C23×D10711yescharacteristic
277Q8—10716no—
278Q8—21416no—
279Q8—2148no—
280D4—216no—
281D4—28no—
282D4—21416no—
283D4—2148no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—1078no—
288C2×C4—1074no—
289C2×C4—1072no—
290C2×C4—1072no—
291C2×C4—1074no—
292C2×C4—1072no—
293C2×C4—21416no—
294C2×C4—21416no—
295C2×C4—2142no—
296C2×C4—2144no—
297C2×C4—2142no—
298C2×C4—2142no—
299C2×C4—2148no—
300C2×C4D4×D10714yes—
301C2×C4C23×D10714yes—
302C2×C4C23×D10712yes—
303C2×C4C23×D10712yes—
304C22—22no—
305C22—22no—
306C22—28no—
307C22—24no—
308C22—24no—
309C22—24no—
310C22—48no—
311C22—4284no—
312C22—2142no—
313C22—2142no—
314C22—2142no—
315C22C2×D4×D10712yes—
316C22C2×#1313381712yes—
317C22C2×#1313381714yescomplemented
318C22#1681215612yes—
319C22C24×D10711yescharacteristic
320C4—22no—
321C4—24no—
322C4—1078no—
323C4—2142no—
324C4—2144no—
325C4—2142no—
326C4—2142no—
327C4C2×D4×D10714yes—
328C2—24no—
329C2—44no—
330C2—4282no—
331C2#2029328314yesdirect product factor
332C2C22×D4×D10711yescharacteristic
333C2C22×#1313381711yescharacteristic
334C2C2×#1681215611yescharacteristic
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#121×C107C2
socleC23×C107C24
hypercenterC2×#121D107
hypocenterC107C2×#548
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C23×Dic214C2semidirect
3. #20293283C2direct (N × H)
4. C2×#16812096C2semidirect
5. C2×#16812095C2semidirect
6. C2×#16812089C2semidirect
7. C2×#16812086C2semidirect
8. C2×#16812081C2semidirect
9. C2×#16812074C2semidirect
10. C2×D4×Dic107C2semidirect
11. C2×C4×C22⋊D107C2semidirect
12. C2×#121×C107C2semidirect
14. C2×#13133817C22semidirect
19. C22×Dic214C22semidirect
25. #16812096C22semidirect
26. #16812095C22semidirect
27. #16812089C22semidirect
28. #16812086C22semidirect
29. #16812081C22semidirect
30. #16812074C22semidirect
31. D4×Dic107C22semidirect
32. C4×C22⋊D107C22semidirect
46. #121×C107C22semidirect
54. C2×Dic214D4semidirect
101. C23×C107#41semidirect
135. Dic214C2×D4semidirect
145. C22×C107C2×#41semidirect
162. C2×C107#548semidirect
168. C107C2×#548semidirect
177. C2×#121D107semidirect
201. #121C2×D107semidirect
273. C23#13133817semidirect
317. C22C2×#13133817semidirect
331. C2#20293283direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338172
C23×Dic2142
#202932832
C2×#168120962
C2×#168120952
C2×#168120892
C2×#168120862
C2×#168120812
C2×#168120742
C2×D4×Dic1072
C2×C4×C22⋊D1072
C2×#121×C1072
C2×#548107