Finite Group Databasebeta

← C2×#20293344 #28350004

Subgroups — C2×#20293344 #28350004

Order 13696.

1112 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293344C2116yesmaximal, direct product factor
2C2×#16812108C211yesmaximal, characteristic, complemented
3C2×#16812104C211yesmaximal, characteristic, complemented
4C2×#16812104C211yesmaximal, characteristic, complemented
5C2×#16812102C211yesmaximal, characteristic, complemented
6C2×#16812095C211yesmaximal, characteristic
7C2×#16812095C211yesmaximal, characteristic
8C2×#16812087C211yesmaximal, characteristic, complemented
9C2×#16812086C211yesmaximal, characteristic, complemented
10C2×C4×Dic214C211yesmaximal, characteristic, complemented
11C2×#16812074C211yesmaximal, characteristic, complemented
12C2×#16812074C211yesmaximal, characteristic, complemented
13C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
14C2×#16812066C211yesmaximal, characteristic, complemented
15C2×#16812065C211yesmaximal, characteristic, complemented
16C2×#126×C107C211yesmaximal, characteristic, complemented
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic
19C22×Dic214C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic
23#16812108C2218yescomplemented
24#16812104C2218yescomplemented
25#16812104C2218yescomplemented
26#16812102C2218yescomplemented
27#16812095C2218yes—
28#16812095C2218yes—
29#16812087C2218yescomplemented
30#16812086C2218yescomplemented
31C4×Dic214C2218yescomplemented
32#16812074C2218yescomplemented
33#16812074C2218yescomplemented
34(C22⋊D107)⋊C4C2218yes—
35#16812066C2218yescomplemented
36#16812065C2218yescomplemented
37C2×C4⋊Dic107C2211yescharacteristic
38C2×C4⋊Dic107C2211yescharacteristic
39C2×Dic107⋊C4C2211yescharacteristic
40C2×Dic107⋊C4C2211yescharacteristic
41C2×Dic107⋊C4C2211yescharacteristic
42C2×Dic107⋊C4C2211yescharacteristic
43C2×Dic107⋊C4C2211yescharacteristic
44C2×Dic107⋊C4C2211yescharacteristic
45C2×Dic107⋊C4C2211yescharacteristic
46C2×C22⋊Dic107C2211yescharacteristic
47C2×C22⋊Dic107C2211yescharacteristic
48C2×(C2×D107)⋊C4C2211yescharacteristic
49C2×(C2×D107)⋊C4C2211yescharacteristic
50C2×(C2×D107)⋊C4C2211yescharacteristic
51C2×(C2×D107)⋊C4C2211yescharacteristic
52C2×(C2×D107)⋊C4C2211yescharacteristic
53C2×C4×Dic107C2211yescharacteristic
54C2×C4×Dic107C2211yescharacteristic
55C2×C4×Dic107C2211yescharacteristic
56C2×C4×Dic107C2211yescharacteristic
57C2×C4×Dic107C2211yescharacteristic
58C2×C4×Dic107C2211yescharacteristic
59#126×C107C2218yescomplemented
60C2×C4⋊C4×C107C2211yescharacteristic
61C2×C4⋊C4×C107C2211yescharacteristic
62C2×C4⋊C4×C107C2211yescharacteristic
63C2×C22⋊C4×C107C2211yescharacteristic
64C2×C22⋊C4×C107C2211yescharacteristic
65C2×C22⋊C4×C107C2211yescharacteristic
66C2×C42×C107C2211yescharacteristic
67C23×D107C2311yescharacteristic
68C2×Dic214—24no—
69C2×Dic214—24no—
70C2×Dic214—24no—
71C2×Dic214C2314yes—
72C2×Dic214C2314yes—
73C2×Dic214C2314yes—
74C2×C22⋊D107—24no—
75C2×C22⋊D107C2314yes—
76C22×Dic107—24no—
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C22×Dic107C2311yescharacteristic
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2314yes—
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C2×C4×D107—24no—
86C2×C4×D107C2314yes—
87C4⋊Dic107C2314yes—
88C4⋊Dic107C2314yes—
89Dic107⋊C4C2314yes—
90Dic107⋊C4C2314yes—
91Dic107⋊C4C2314yes—
92Dic107⋊C4C2314yes—
93Dic107⋊C4C2314yes—
94Dic107⋊C4C2314yes—
95Dic107⋊C4C2314yes—
96C22⋊Dic107C2314yes—
97C22⋊Dic107C2314yes—
98(C2×D107)⋊C4C2314yes—
99(C2×D107)⋊C4C2314yes—
100(C2×D107)⋊C4C2314yes—
101(C2×D107)⋊C4C2314yes—
102(C2×D107)⋊C4C2314yes—
103C4×Dic107C2314yes—
104C4×Dic107C2314yes—
105C4×Dic107C2314yes—
106C4×Dic107C2314yes—
107C4×Dic107C2314yes—
108C4×Dic107C2314yes—
109C24×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114C22×C4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116C4⋊C4×C107C2314yes—
117C4⋊C4×C107C2314yes—
118C4⋊C4×C107C2314yes—
119C22⋊C4×C107C2314yes—
120C22⋊C4×C107C2314yes—
121C22⋊C4×C107C2314yes—
122C42×C107C2314yes—
123C23×C107—22no—
124C23×C107—22no—
125C23×C107—22no—
126C23×C107C2412yes—
127C23×C107C2411yescharacteristic
128C22×D107—22no—
129C22×D107—22no—
130C22×D107—22no—
131C22×D107C2412yes—
132C2×C4×C107—22no—
133C2×C4×C107—22no—
134C2×C4×C107—22no—
135C2×C4×C107—22no—
136C2×C4×C107—22no—
137C2×C4×C107—22no—
138C2×C4×C107C2412yes—
139C2×C4×C107C2412yes—
140C2×C4×C107C2412yes—
141C2×C4×C107C2412yes—
142C2×C4×C107C2412yes—
143C2×C4×C107C2412yes—
144C22⋊D107—28no—
145C2×Dic107—22no—
146C2×Dic107—22no—
147C2×Dic107—22no—
148C2×Dic107—22no—
149C2×Dic107—22no—
150C2×Dic107—28no—
151C2×Dic107#4114yes—
152C2×Dic107#4114yes—
153C2×Dic107C2412yes—
154C2×Dic107C2412yes—
155C2×Dic107C2412yes—
156C2×Dic107C2412yes—
157C2×Dic107C2412yes—
158C2×Dic107C2412yes—
159C2×Dic107C2412yes—
160C4×D107—28no—
161Dic214—28no—
162Dic214—28no—
163Dic214—28no—
164C22×C107—22no—
165C22×C107—22no—
166C22×C107—22no—
167C22×C107—44no—
168C22×C107C2×#4112yes—
169C22×C107C2×#4112yes—
170C22×C1072−1+412yes—
171C22×C107C2511yescharacteristic
172C2×D107—22no—
173C2×D107—22no—
174C2×D107—22no—
175C2×D107—44no—
176C4×C107—22no—
177C4×C107—22no—
178C4×C107—22no—
179C4×C107—22no—
180C4×C107—22no—
181C4×C107—22no—
182Dic107—22no—
183Dic107—22no—
184Dic107—22no—
185Dic107—22no—
186Dic107—22no—
187Dic107C2×#4114yes—
188Dic107C2×#4114yes—
189C2×C107—42no—
190C2×C107#55614yescomplemented
191C2×C107C22×#4111yescharacteristic
192C2×C107C22×#4111yescharacteristic
193C2×C107C2×2−1+411yescharacteristic
194D107—42no—
195C2×#556—1071nomaximal, Sylow
196C107C2×#55611yescharacteristic, complemented, Sylow
197#556—10716no—
198C2×C4⋊Q8—1071no—
199C2×#126—1071no—
200C2×#126—1071no—
201C2×#126—1071no—
202C2×#126D10711yescharacteristic, complemented
203C2×#124—1071no—
204C2×#124—1071no—
205C2×C22⋊Q8—1071no—
206C2×C22⋊Q8—1071no—
207C2×C4×Q8—1071no—
208C2×C4×Q8—1071no—
209C2×C4×Q8—1071no—
210C2×C4×D4—1071no—
211C2×#117—1071no—
212C2×#117—1071no—
213C22×Q8—1071no—
214C22×Q8—1071no—
215C22×Q8—1071no—
216C22×D4—1071no—
217C23×C4—1071no—
218C23×C4—1071no—
219C4⋊Q8—1078no—
220#126—1078no—
221#126—1078no—
222#126—1078no—
223#126C2×D10718yescomplemented
224#124—1078no—
225#124—1078no—
226C22⋊Q8—1078no—
227C22⋊Q8—1078no—
228C4×Q8—1078no—
229C4×Q8—1078no—
230C4×Q8—1078no—
231C4×D4—1078no—
232#117—1078no—
233#117—1078no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C4⋊C4—1071no—
237C2×C4⋊C4—1071no—
238C2×C4⋊C4—1071no—
239C2×C4⋊C4—1071no—
240C2×C4⋊C4—1071no—
241C2×C4⋊C4—1071no—
242C2×C4⋊C4—1071no—
243C2×C4⋊C4C2×D10711yescharacteristic
244C2×C4⋊C4C2×D10711yescharacteristic
245C2×C4⋊C4C2×D10711yescharacteristic
246C2×C22⋊C4—1071no—
247C2×C22⋊C4—1071no—
248C2×C22⋊C4—1071no—
249C2×C22⋊C4—1071no—
250C2×C22⋊C4—1071no—
251C2×C22⋊C4—1071no—
252C2×C22⋊C4—1071no—
253C2×C22⋊C4C2×D10711yescharacteristic
254C2×C22⋊C4C2×D10711yescharacteristic
255C2×C22⋊C4C2×D10711yescharacteristic
256C2×C42—1071no—
257C2×C42—1071no—
258C2×C42—1071no—
259C2×C42—1071no—
260C2×C42—1071no—
261C2×C42—1071no—
262C2×C42C2×D10711yescharacteristic
263C24—1071no—
264C24C22×D10711yescharacteristic
265C2×Q8—1074no—
266C2×Q8—1074no—
267C2×Q8—1074no—
268C2×Q8—2144no—
269C2×Q8—2144no—
270C2×Q8—2144no—
271C2×D4—1074no—
272C2×D4—2144no—
273C22×C4—1071no—
274C22×C4—1071no—
275C22×C4—1074no—
276C22×C4—1071no—
277C22×C4—1071no—
278C22×C4—1071no—
279C22×C4—1071no—
280C22×C4—1071no—
281C22×C4—1074no—
282C22×C4—2144no—
283C22×C4—2144no—
284C22×C4C22×D10711yescharacteristic
285C22×C4C22×D10711yescharacteristic
286C22×C4C22×D10711yescharacteristic
287C22×C4C22×D10711yescharacteristic
288C22×C4C22×D10711yescharacteristic
289C22×C4C22×D10711yescharacteristic
290C4⋊C4—1074no—
291C4⋊C4—1074no—
292C4⋊C4—1074no—
293C4⋊C4—1074no—
294C4⋊C4—1074no—
295C4⋊C4—1074no—
296C4⋊C4—1074no—
297C4⋊C4—1074no—
298C4⋊C4—1074no—
299C4⋊C4C22×D10714yes—
300C4⋊C4C22×D10714yes—
301C4⋊C4C22×D10714yes—
302C22⋊C4—1074no—
303C22⋊C4—1074no—
304C22⋊C4—1074no—
305C22⋊C4—1074no—
306C22⋊C4—1074no—
307C22⋊C4—1074no—
308C22⋊C4—1074no—
309C22⋊C4C22×D10714yes—
310C22⋊C4C22×D10714yes—
311C22⋊C4C22×D10714yes—
312C42—1074no—
313C42—1074no—
314C42—1074no—
315C42—1074no—
316C42—1074no—
317C42—1074no—
318C42C22×D10714yes—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—1072no—
323C23—2142no—
324C23—2142no—
325C23—2142no—
326C23C23×D10712yes—
327C23C23×D10711yescharacteristic
328Q8—2148no—
329Q8—2148no—
330Q8—2148no—
331D4—2148no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—1074no—
339C2×C4—1074no—
340C2×C4—1072no—
341C2×C4—1072no—
342C2×C4—1072no—
343C2×C4—1072no—
344C2×C4—1072no—
345C2×C4—1072no—
346C2×C4—1072no—
347C2×C4—2148no—
348C2×C4—2148no—
349C2×C4—2142no—
350C2×C4—2142no—
351C2×C4—2142no—
352C2×C4—2142no—
353C2×C4—2142no—
354C2×C4C23×D10712yes—
355C2×C4C23×D10712yes—
356C2×C4C23×D10712yes—
357C2×C4C23×D10712yes—
358C2×C4C23×D10712yes—
359C2×C4C23×D10712yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—4284no—
365C22—2142no—
366C22—2142no—
367C22—2142no—
368C22#41×D10712yes—
369C22#41×D10712yes—
370C22#1681215612yes—
371C22C24×D10711yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—1074no—
379C4—1074no—
380C4—2142no—
381C4—2142no—
382C4—2142no—
383C4—2142no—
384C4—2142no—
385C2—42no—
386C2—4282no—
387C2#2029334414yesdirect product factor
388C2C2×#41×D10711yescharacteristic
389C2C2×#41×D10711yescharacteristic
390C2C2×#1681215611yescharacteristic
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#126×C107C2
socleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenterC2×#126D107
hypocenterC107C2×#556
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20293344C2direct (N × H)
2. C2×#16812108C2semidirect
3. C2×#16812104C2semidirect
4. C2×#16812104C2semidirect
5. C2×#16812102C2semidirect
8. C2×#16812087C2semidirect
9. C2×#16812086C2semidirect
10. C2×C4×Dic214C2semidirect
11. C2×#16812074C2semidirect
12. C2×#16812074C2semidirect
14. C2×#16812066C2semidirect
15. C2×#16812065C2semidirect
16. C2×#126×C107C2semidirect
23. #16812108C22semidirect
24. #16812104C22semidirect
25. #16812104C22semidirect
26. #16812102C22semidirect
29. #16812087C22semidirect
30. #16812086C22semidirect
31. C4×Dic214C22semidirect
32. #16812074C22semidirect
33. #16812074C22semidirect
35. #16812066C22semidirect
36. #16812065C22semidirect
59. #126×C107C22semidirect
190. C2×C107#556semidirect
196. C107C2×#556semidirect
202. C2×#126D107semidirect
223. #126C2×D107semidirect
387. C2#20293344direct (N × H)

Maximal subgroups

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

subgroupindex
#202933442
C2×#168121082
C2×#168121042
C2×#168121022
C2×#168120952
C2×#168120872
C2×#168120862
C2×C4×Dic2142
C2×#168120742
C2×(C22⋊D107)⋊C42
C2×#168120662
C2×#168120652
C2×#126×C1072
C2×#556107