Finite Group Databasebeta

← C2×#20293300 #28349955

Subgroups — C2×#20293300 #28349955

Order 13696.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293300C2116yesmaximal, direct product factor
2C2×#16812104C211yesmaximal, characteristic, complemented
3C2×#16812104C211yesmaximal, characteristic, complemented
4C2×#16812095C211yesmaximal, characteristic, complemented
5C2×#16812093C211yesmaximal, characteristic, complemented
6C2×#16812091C211yesmaximal, characteristic, complemented
7C2×#16812089C211yesmaximal, characteristic, complemented
8C2×#16812086C211yesmaximal, characteristic, complemented
9C2×#16812077C211yesmaximal, characteristic
10C2×D4×Dic107C211yesmaximal, characteristic, complemented
11C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
12C2×#16812068C211yesmaximal, characteristic, complemented
13C2×#16812067C211yesmaximal, characteristic, complemented
14C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
16C2×#123×C107C211yesmaximal, characteristic, complemented
17C24×D107C2211yescharacteristic
18C22×Dic214C2211yescharacteristic, complemented
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic
22C23×Dic107C2211yescharacteristic, complemented
23C22×C4×D107C2211yescharacteristic
24C22×C4×D107C2211yescharacteristic
25#16812104C2218yescomplemented
26#16812104C2218yescomplemented
27#16812095C2218yescomplemented
28#16812093C2218yescomplemented
29#16812091C2218yescomplemented
30#16812089C2218yescomplemented
31#16812086C2218yescomplemented
32#16812077C2218yes—
33D4×Dic107C2218yescomplemented
34(C22⋊D107)⋊C4C2218yescomplemented
35#16812068C2218yescomplemented
36#16812067C2218yescomplemented
37C2×C4⋊Dic107C2211yescharacteristic, complemented
38C2×C4⋊Dic107C2211yescharacteristic, complemented
39C2×C4⋊Dic107C2211yescharacteristic, complemented
40C2×C4⋊Dic107C2211yescharacteristic, complemented
41C2×Dic107⋊C4C2211yescharacteristic, complemented
42C2×Dic107⋊C4C2211yescharacteristic, complemented
43C2×C22⋊Dic107C2211yescharacteristic
44C2×C22⋊Dic107C2211yescharacteristic
45C2×C22⋊Dic107C2211yescharacteristic, complemented
46C2×(C2×D107)⋊C4C2211yescharacteristic
47C2×(C2×D107)⋊C4C2211yescharacteristic
48C2×(C2×D107)⋊C4C2211yescharacteristic
49C2×(C2×D107)⋊C4C2211yescharacteristic
50C2×(C2×D107)⋊C4C2211yescharacteristic
51C2×(C2×D107)⋊C4C2211yescharacteristic
52C2×(C2×D107)⋊C4C2211yescharacteristic
53C2×(C2×D107)⋊C4C2211yescharacteristic
54C2×(C2×D107)⋊C4C2218yescomplemented
55C22⋊C4×D107C2218yescomplemented
56C2×C4×Dic107C2211yescharacteristic, complemented
57C2×C4×Dic107C2211yescharacteristic, complemented
58C2×C4×Dic107C2211yescharacteristic, complemented
59C22×D4×C107C2211yescharacteristic
60C23×C4×C107C2211yescharacteristic
61#123×C107C2218yescomplemented
62C2×C4⋊C4×C107C2211yescharacteristic, complemented
63C2×C4⋊C4×C107C2211yescharacteristic, complemented
64C2×C22⋊C4×C107C2211yescharacteristic
65C2×C22⋊C4×C107C2211yescharacteristic, complemented
66C2×C22⋊C4×C107C2211yescharacteristic, complemented
67C23×D107—24no—
68C23×D107—24no—
69C23×D107—42no—
70C23×D107C2311yescharacteristic
71C23×D107C2314yes—
72C23×D107C2311yescharacteristic
73C2×Dic214—42no—
74C2×Dic214C2314yescomplemented
75C2×C22⋊D107—24no—
76C2×C22⋊D107—42no—
77C2×C22⋊D107C2314yes—
78C2×C22⋊D107C2314yes—
79C22×Dic107—42no—
80C22×Dic107—42no—
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2311yescharacteristic
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C22×Dic107C2311yescharacteristic
86C22×Dic107C2311yescharacteristic
87C22×Dic107C2314yes—
88C22×Dic107C2314yescomplemented
89C2×C4×D107—24no—
90C2×C4×D107—42no—
91C2×C4×D107C2314yes—
92C2×C4×D107C2314yes—
93C4⋊Dic107C2314yescomplemented
94C4⋊Dic107C2314yescomplemented
95C4⋊Dic107C2314yescomplemented
96C4⋊Dic107C2314yescomplemented
97Dic107⋊C4C2314yescomplemented
98Dic107⋊C4C2314yescomplemented
99C22⋊Dic107C2314yes—
100C22⋊Dic107C2314yes—
101C22⋊Dic107C2314yescomplemented
102(C2×D107)⋊C4C2314yes—
103(C2×D107)⋊C4C2314yes—
104(C2×D107)⋊C4C2314yes—
105(C2×D107)⋊C4C2314yes—
106(C2×D107)⋊C4C2314yes—
107(C2×D107)⋊C4C2314yes—
108(C2×D107)⋊C4C2314yes—
109(C2×D107)⋊C4C2314yes—
110C4×Dic107C2314yescomplemented
111C4×Dic107C2314yescomplemented
112C4×Dic107C2314yescomplemented
113C24×C107C2311yescharacteristic
114C24×C107C2311yescharacteristic
115C2×D4×C107—24no—
116C2×D4×C107C2314yes—
117C22×C4×C107—24no—
118C22×C4×C107C2311yescharacteristic
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2314yes—
124C4⋊C4×C107C2314yescomplemented
125C4⋊C4×C107C2314yescomplemented
126C22⋊C4×C107C2314yes—
127C22⋊C4×C107C2314yescomplemented
128C22⋊C4×C107C2314yescomplemented
129C23×C107—22no—
130C23×C107—22no—
131C23×C107—22no—
132C23×C107—22no—
133C23×C107—22no—
134C23×C107#4114yes—
135C23×C107C2412yes—
136C23×C107C2412yes—
137C23×C107C2411yescharacteristic
138C22×D107—22no—
139C22×D107—22no—
140C22×D107—22no—
141C22×D107—28no—
142C22×D107—22no—
143C22×D107—22no—
144C22×D107—28no—
145C22×D107—44no—
146C22×D107—416no—
147C22×D107#4114yes—
148C22×D107C2412yes—
149C22×D107C2412yes—
150D4×C107—28no—
151C2×C4×C107—28no—
152C2×C4×C107—22no—
153C2×C4×C107—22no—
154C2×C4×C107—22no—
155C2×C4×C107—22no—
156C2×C4×C107—22no—
157C2×C4×C107C2412yes—
158C2×C4×C107C2412yes—
159C2×C4×C107C2412yes—
160C2×C4×C107C2412yes—
161C2×C4×C107C2412yes—
162C22⋊D107—28no—
163C22⋊D107—44no—
164C2×Dic107—22no—
165C2×Dic107—22no—
166C2×Dic107—22no—
167C2×Dic107—22no—
168C2×Dic107—22no—
169C2×Dic107—22no—
170C2×Dic107—44no—
171C2×Dic107—44no—
172C2×Dic107C2412yes—
173C2×Dic107C2412yes—
174C2×Dic107C2412yes—
175C2×Dic107C2412yes—
176C2×Dic107C2412yes—
177C2×Dic107C2412yes—
178C4×D107—28no—
179C4×D107—44no—
180Dic214—44no—
181C22×C107—28no—
182C22×C107—22no—
183C22×C107—22no—
184C22×C107—22no—
185C22×C107—22no—
186C22×C107—22no—
187C22×C107—44no—
188C22×C107C2×#4112yes—
189C22×C107C2×#4114yes—
190C22×C107C2×#4112yes—
191C22×C107H(5,𝔽2)12yes—
192C22×C107C2511yescharacteristic
193C2×D107—28no—
194C2×D107—22no—
195C2×D107—22no—
196C2×D107—22no—
197C2×D107—22no—
198C2×D107—22no—
199C2×D107—416no—
200C2×D107—44no—
201C2×D107C2×#4114yes—
202C4×C107—22no—
203C4×C107—22no—
204C4×C107—22no—
205C4×C107—22no—
206C4×C107—22no—
207Dic107—22no—
208Dic107—22no—
209Dic107—22no—
210Dic107—22no—
211Dic107—22no—
212Dic107—22no—
213C2×C107—24no—
214C2×C107—42no—
215C2×C107#55114yescomplemented
216C2×C107C22×#4111yescharacteristic
217C2×C107C22×#4111yescharacteristic
218C2×C107C2×H(5,𝔽2)11yescharacteristic
219D107—24no—
220D107—42no—
221C2×#551—1071nomaximal, Sylow
222C107C2×#55111yescharacteristic, complemented, Sylow
223#551—10716no—
224C2×#126—1071no—
225C2×#126—1071no—
226C2×#124—1071no—
227C2×#123—1071no—
228C2×#123—1071no—
229C2×#123D10711yescharacteristic, complemented
230C2×C22⋊Q8—1071no—
231C2×C22⋊Q8—1071no—
232C2×#120—1071no—
233C2×C4×D4—1071no—
234C2×C4×D4—1071no—
235C2×#117—1071no—
236C2×#117—1071no—
237C22×C22⋊C4—1071no—
238C22×C22⋊C4—1071no—
239C25—1071no—
240C22×Q8—1071no—
241C22×D4—1071no—
242C22×D4—1071no—
243C22×D4C2×D10711yescharacteristic
244C23×C4—1071no—
245C23×C4—1071no—
246C23×C4—1071no—
247C23×C4—1071no—
248C23×C4C2×D10711yescharacteristic
249#126—1078no—
250#126—1078no—
251#124—1078no—
252#123—1078no—
253#123—1078no—
254#123C2×D10718yescomplemented
255C22⋊Q8—1078no—
256C22⋊Q8—1078no—
257#120—1078no—
258C4×D4—1078no—
259C4×D4—1078no—
260#117—1078no—
261#117—1078no—
262C2×C4⋊C4—1071no—
263C2×C4⋊C4—1071no—
264C2×C4⋊C4—1071no—
265C2×C4⋊C4—1071no—
266C2×C4⋊C4—1071no—
267C2×C4⋊C4—1071no—
268C2×C4⋊C4C2×D10711yescharacteristic, complemented
269C2×C4⋊C4C2×D10711yescharacteristic, complemented
270C2×C22⋊C4—1071no—
271C2×C22⋊C4—1071no—
272C2×C22⋊C4—1071no—
273C2×C22⋊C4—1071no—
274C2×C22⋊C4—1071no—
275C2×C22⋊C4—1071no—
276C2×C22⋊C4—1071no—
277C2×C22⋊C4—1071no—
278C2×C22⋊C4—1071no—
279C2×C22⋊C4—1078no—
280C2×C22⋊C4—1071no—
281C2×C22⋊C4—1071no—
282C2×C22⋊C4—1078no—
283C2×C22⋊C4C2×D10711yescharacteristic
284C2×C22⋊C4C2×D10711yescharacteristic, complemented
285C2×C22⋊C4C2×D10711yescharacteristic, complemented
286C2×C42—1071no—
287C2×C42—1071no—
288C2×C42—1071no—
289C24—1071no—
290C24—1074no—
291C24—1071no—
292C24—4282no—
293C24—2144no—
294C24—2144no—
295C24C22×D10711yescharacteristic
296C24C22×D10711yescharacteristic
297C2×Q8—1074no—
298C2×Q8—4282no—
299C2×D4—24no—
300C2×D4—1074no—
301C2×D4—1074no—
302C2×D4—4282no—
303C2×D4—2144no—
304C2×D4C22×D10714yes—
305C22×C4—24no—
306C22×C4—1074no—
307C22×C4—1074no—
308C22×C4—1071no—
309C22×C4—1071no—
310C22×C4—1071no—
311C22×C4—1071no—
312C22×C4—1071no—
313C22×C4—1071no—
314C22×C4—1074no—
315C22×C4—1074no—
316C22×C4—4282no—
317C22×C4—4282no—
318C22×C4—4282no—
319C22×C4—2144no—
320C22×C4C22×D10714yes—
321C22×C4C22×D10711yescharacteristic
322C22×C4C22×D10711yescharacteristic
323C22×C4C22×D10711yescharacteristic
324C22×C4C22×D10711yescharacteristic
325C22×C4C22×D10711yescharacteristic
326C4⋊C4—1074no—
327C4⋊C4—1074no—
328C4⋊C4—1074no—
329C4⋊C4—1074no—
330C4⋊C4—1074no—
331C4⋊C4—1074no—
332C4⋊C4C22×D10714yescomplemented
333C4⋊C4C22×D10714yescomplemented
334C22⋊C4—1074no—
335C22⋊C4—1074no—
336C22⋊C4—1074no—
337C22⋊C4—1074no—
338C22⋊C4—1074no—
339C22⋊C4—1074no—
340C22⋊C4—1074no—
341C22⋊C4—1074no—
342C22⋊C4—1074no—
343C22⋊C4—1074no—
344C22⋊C4—1074no—
345C22⋊C4C22×D10714yes—
346C22⋊C4C22×D10714yescomplemented
347C22⋊C4C22×D10714yescomplemented
348C42—1074no—
349C42—1074no—
350C42—1074no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—1074no—
357C23—1072no—
358C23—1072no—
359C23—42816no—
360C23—4284no—
361C23—2148no—
362C23—2142no—
363C23—2148no—
364C23—2142no—
365C23—2142no—
366C23—2142no—
367C23—2142no—
368C23#1313381714yes—
369C23C23×D10712yes—
370C23C23×D10712yes—
371C23C23×D10711yescharacteristic
372Q8—4284no—
373D4—28no—
374D4—4284no—
375D4—2148no—
376C2×C4—22no—
377C2×C4—28no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—1072no—
383C2×C4—1072no—
384C2×C4—1072no—
385C2×C4—1072no—
386C2×C4—1072no—
387C2×C4—1072no—
388C2×C4—4284no—
389C2×C4—4284no—
390C2×C4—4284no—
391C2×C4—2148no—
392C2×C4—2142no—
393C2×C4—2142no—
394C2×C4—2142no—
395C2×C4—2142no—
396C2×C4—2142no—
397C2×C4—2142no—
398C2×C4C23×D10712yes—
399C2×C4C23×D10712yes—
400C2×C4C23×D10712yes—
401C2×C4C23×D10712yes—
402C2×C4C23×D10712yes—
403C22—22no—
404C22—22no—
405C22—28no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—44no—
410C22—1074no—
411C22—42816no—
412C22—4284no—
413C22—2142no—
414C22—2142no—
415C22—2142no—
416C22—2142no—
417C22—2142no—
418C22—2148no—
419C22#41×D10712yes—
420C22C2×#1313381714yes—
421C22C2×#1313381712yes—
422C22#1681215512yes—
423C22C24×D10711yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—2142no—
430C4—2142no—
431C4—2142no—
432C4—2142no—
433C4—2142no—
434C4—2142no—
435C2—24no—
436C2—42no—
437C2—4282no—
438C2—2144no—
439C2#2029330014yesdirect product factor
440C2C2×#41×D10711yescharacteristic
441C2C22×#1313381711yescharacteristic
442C2C2×#1681215511yescharacteristic
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#123×C107C2
socleC23×C107C24
prime radicalC2×#16812077C2
hypercenterC2×#123D107
hypocenterC107C2×#551
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #20293300C2direct (N × H)
2. C2×#16812104C2semidirect
3. C2×#16812104C2semidirect
4. C2×#16812095C2semidirect
5. C2×#16812093C2semidirect
6. C2×#16812091C2semidirect
7. C2×#16812089C2semidirect
8. C2×#16812086C2semidirect
10. C2×D4×Dic107C2semidirect
11. C2×(C22⋊D107)⋊C4C2semidirect
12. C2×#16812068C2semidirect
13. C2×#16812067C2semidirect
14. C22×(C2×D107)⋊C4C2semidirect
15. C2×C22⋊C4×D107C2semidirect
16. C2×#123×C107C2semidirect
18. C22×Dic214C22semidirect
22. C23×Dic107C22semidirect
25. #16812104C22semidirect
26. #16812104C22semidirect
27. #16812095C22semidirect
28. #16812093C22semidirect
29. #16812091C22semidirect
30. #16812089C22semidirect
31. #16812086C22semidirect
33. D4×Dic107C22semidirect
34. (C22⋊D107)⋊C4C22semidirect
35. #16812068C22semidirect
36. #16812067C22semidirect
37. C2×C4⋊Dic107C22semidirect
38. C2×C4⋊Dic107C22semidirect
39. C2×C4⋊Dic107C22semidirect
40. C2×C4⋊Dic107C22semidirect
41. C2×Dic107⋊C4C22semidirect
42. C2×Dic107⋊C4C22semidirect
45. C2×C22⋊Dic107C22semidirect
54. C2×(C2×D107)⋊C4C22semidirect
55. C22⋊C4×D107C22semidirect
56. C2×C4×Dic107C22semidirect
57. C2×C4×Dic107C22semidirect
58. C2×C4×Dic107C22semidirect
61. #123×C107C22semidirect
62. C2×C4⋊C4×C107C22semidirect
63. C2×C4⋊C4×C107C22semidirect
65. C2×C22⋊C4×C107C22semidirect
66. C2×C22⋊C4×C107C22semidirect
74. C2×Dic214C23semidirect
88. C22×Dic107C23semidirect
93. C4⋊Dic107C23semidirect
94. C4⋊Dic107C23semidirect
95. C4⋊Dic107C23semidirect
96. C4⋊Dic107C23semidirect
97. Dic107⋊C4C23semidirect
98. Dic107⋊C4C23semidirect
101. C22⋊Dic107C23semidirect
110. C4×Dic107C23semidirect
111. C4×Dic107C23semidirect
112. C4×Dic107C23semidirect
124. C4⋊C4×C107C23semidirect
125. C4⋊C4×C107C23semidirect
127. C22⋊C4×C107C23semidirect
128. C22⋊C4×C107C23semidirect
215. C2×C107#551semidirect
222. C107C2×#551semidirect
229. C2×#123D107semidirect
254. #123C2×D107semidirect
268. C2×C4⋊C4C2×D107semidirect
269. C2×C4⋊C4C2×D107semidirect
284. C2×C22⋊C4C2×D107semidirect
285. C2×C22⋊C4C2×D107semidirect
332. C4⋊C4C22×D107semidirect
333. C4⋊C4C22×D107semidirect
346. C22⋊C4C22×D107semidirect
347. C22⋊C4C22×D107semidirect
439. C2#20293300direct (N × H)

Maximal subgroups

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

subgroupindex
#202933002
C2×#168121042
C2×#168120952
C2×#168120932
C2×#168120912
C2×#168120892
C2×#168120862
C2×#168120772
C2×D4×Dic1072
C2×(C22⋊D107)⋊C42
C2×#168120682
C2×#168120672
C22×(C2×D107)⋊C42
C2×C22⋊C4×D1072
C2×#123×C1072
C2×#551107