Finite Group Databasebeta

← C2×#11744351 #14262265

Subgroups — C2×#11744351 #14262265

Order 1792.

11534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C22×#6554C211yesmaximal, characteristic, complemented
3#11744351C2132yesmaximal, direct product factor
4C2×#90522C214yesmaximal, complemented
5C2×#90516C213yesmaximal, complemented
6C2×#90523C213yesmaximal, complemented
7C2×#41×Dic7C211yesmaximal, characteristic, complemented
8C2×#90496C213yesmaximal, complemented
9C2×#90497C2112yesmaximal, complemented
10C22×#6499C213yesmaximal, complemented
11C2×C7×H(5,𝔽2)—216no—
12C2×C7×H(5,𝔽2)C22116yescomplemented
13C22×C7×#41C2213yes—
14C22×C7×#41C2212yes—
15C22×C7×#41C2211yescharacteristic
16C23×C7×D4C2213yes—
17C23×C7×D4C2216yes—
18C2×#6554—216no—
19C2×#6554C22116yescomplemented
20C22×Q8×D7C2212yescomplemented
21C22×#1159C2216yescomplemented
22C22×#1157C2213yescomplemented
23C23×Dic14C2213yescomplemented
24C2×#6516C22112yescomplemented
25C22×C22⋊Dic7C22112yescomplemented
26#90523C22148yescomplemented
27#90522C22164yescomplemented
28#41×Dic7C22116yescomplemented
29#90516C22148yescomplemented
30C2×#6510C2211yescharacteristic, complemented
31C2×Q8×Dic7C2212yescomplemented
32C2×#6507C2213yescomplemented
33#90497C221192yescomplemented
34#90496C22148yescomplemented
35C2×#6503C2213yescomplemented
36C2×#6502C22112yescomplemented
37C2×#6501C22112yescomplemented
38C2×#6499C2213yescomplemented
39C2×#6499C22112yescomplemented
40C2×#6499C2216yescomplemented
41C2×#6499C22148yescomplemented
42C2×#6498C22112yescomplemented
43C2×D4×Dic7C2216yescomplemented
44C2×C4×C22⋊D7C22112yescomplemented
45C2×#6489C2213yescomplemented
46C2×#6487C22112yescomplemented
47C22×C4×Dic7C2213yescomplemented
48C2×#3500—71nomaximal, Sylow
49C25×C7C2316yes—
50C7×H(5,𝔽2)—2128no—
51C2×C7×#41—216no—
52C2×C7×#41—224no—
53C2×C7×#41D4116yescomplemented
54C2×C7×#41C23124yes—
55C2×C7×#41C23116yes—
56C2×C7×#41C2318yes—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—248no—
60C22×C7×D4—224no—
61C22×C7×D4C2316yes—
62C22×C7×D4C2313yes—
63C22×C7×D4C2316yes—
64C22×C7×D4C23124yes—
65C22×C7×D4C23148yes—
66C22×C7×D4C2313yes—
67C23×C4×C7C2316yes—
68C23×C4×C7C2313yes—
69C22×C22⋊D7C2316yes—
70C22×C22⋊D7C23112yes—
71C23×Dic7C2316yes—
72C23×Dic7C2316yes—
73#6554—2128no—
74C2×Q8×D7—216no—
75C2×Q8×D7C23116yescomplemented
76C2×#1159—248no—
77C2×#1159C23148yescomplemented
78C2×#1157—224no—
79C2×#1157C23124yescomplemented
80C22×D28C2311yescharacteristic
81C22×C4×D7C2316yes—
82C22×Dic14—224no—
83C22×Dic14C2316yes—
84C22×Dic14C2313yes—
85C22×Dic14C23124yescomplemented
86#6516C23196yescomplemented
87C2×C22⋊Dic7C23124yes—
88C2×C22⋊Dic7C2316yes—
89C2×C22⋊Dic7C23112yes—
90C2×C22⋊Dic7C23196yescomplemented
91#6510C2318yescomplemented
92Q8×Dic7C23116yescomplemented
93#6507C23124yescomplemented
94#6503C23124yescomplemented
95#6502C23196yescomplemented
96#6501C23196yescomplemented
97#6499C23124yescomplemented
98#6499C23148yescomplemented
99#6499C23196yescomplemented
100#6498C23196yescomplemented
101D4×Dic7C23148yescomplemented
102C4×C22⋊D7C23196yescomplemented
103C2×(C2×D7)⋊C4C2314yes—
104#6489C23124yescomplemented
105C2×C4⋊Dic7C2316yes—
106#6487C23196yescomplemented
107C2×Dic7⋊C4C23112yes—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2313yes—
110C2×C4×Dic7C2316yes—
111C2×C4×Dic7C23124yescomplemented
112C22×2−1+4—71no—
113C22×H(5,𝔽2)D711yescharacteristic, complemented
114#3500—732no—
115C2×#548—74no—
116C2×#546—712no—
117C2×#536—73no—
118C2×#535—73no—
119C2×#532—73no—
120C22×#124—73no—
121C2×C4×#41—71no—
122C24×C7—224no—
123C24×C7—424no—
124C24×C7C2416yes—
125C24×C7C2413yes—
126C24×C7C24124yes—
127C23×D7C2412yes—
128C7×#41—264no—
129C7×#41—296no—
130C7×#41C2×D4164yescomplemented
131C2×C7×Q8—24no—
132C2×C7×Q8C2×D418yes—
133C2×C7×Q8C2414yes—
134C2×C7×Q8C2414yes—
135C2×C7×D4—224no—
136C2×C7×D4—2192no—
137C2×C7×D4—212no—
138C2×C7×D4—224no—
139C2×C7×D4—296no—
140C2×C7×D4C2×D4124yes—
141C2×C7×D4C24124yes—
142C2×C7×D4C24112yes—
143C2×C7×D4C24124yes—
144C2×C7×D4C24112yes—
145C22×C4×C7—224no—
146C22×C4×C7C2×D4124yes—
147C22×C4×C7C2412yes—
148C22×C4×C7C24124yes—
149C22×C4×C7C2411yescharacteristic
150C22×C4×C7C2413yes—
151C22×C4×C7C24112yes—
152C2×C22⋊D7—224no—
153C2×C22⋊D7—424no—
154C2×C22⋊D7C24124yes—
155C2×C22⋊D7C24148yes—
156C22×Dic7—224no—
157C22×Dic7—412no—
158C22×Dic7C2412yes—
159C22×Dic7C2416yes—
160C22×Dic7C2416yes—
161C22×Dic7C24124yes—
162C22×Dic7C24124yes—
163Q8×D7—264no—
164#1159—2192no—
165#1157—296no—
166C2×D28—24no—
167C2×D28C2414yes—
168C2×C4×D7—224no—
169C2×C4×D7C24124yes—
170C2×Dic14—212no—
171C2×Dic14—224no—
172C2×Dic14—296no—
173C2×Dic14C24124yes—
174C2×Dic14C24112yes—
175C22⋊Dic7C24196yes—
176C22⋊Dic7C24124yes—
177C22⋊Dic7C24148yes—
178(C2×D7)⋊C4C24116yes—
179C4⋊Dic7C24124yes—
180Dic7⋊C4C24148yes—
181C4×Dic7C2414yes—
182C4×Dic7C24112yes—
183C4×Dic7C24124yes—
184C2×2−1+4—716no—
185C2×2−1+4—1416no—
186C2×H(5,𝔽2)—216no—
187C2×H(5,𝔽2)C2×D7116yescomplemented
188C22×#41—76no—
189C22×#41—73no—
190C22×#41C2×D713yes—
191C22×#41C2×D712yes—
192C22×#41C2×D711yescharacteristic
193C23×Q8—72no—
194C23×Q8—73no—
195C23×D4C2×D713yes—
196C23×D4C2×D716yes—
197#548—764no—
198#546—7192no—
199#536—748no—
200#535—748no—
201#532—748no—
202C2×C4⋊Q8—73no—
203C2×#127—73no—
204C2×#124—71no—
205C2×#124—73no—
206C2×#124—76no—
207C2×#124—712no—
208C2×#124—748no—
209C2×#123—712no—
210C2×C22⋊Q8—712no—
211C2×#121—712no—
212C2×#121—712no—
213C2×#120—712no—
214C4×#41—716no—
215C2×C4×Q8—72no—
216C2×C4×D4—76no—
217C2×C4×D4—712no—
218C2×#117—73no—
219C22×C22⋊C4—712no—
220C22×C42—73no—
221C23×C7—212no—
222C23×C7—212no—
223C23×C7—224no—
224C23×C7—248no—
225C23×C7—448no—
226C23×C7—848no—
227C23×C7C22×D4112yes—
228C23×C7C25112yes—
229C23×C7C2516yes—
230C23×C7C2511yescharacteristic
231C22×D7—24no—
232C22×D7—28no—
233C22×D7C2514yes—
234C7×Q8—28no—
235C7×Q8C22×D4116yes—
236C7×D4—248no—
237C7×D4—224no—
238C7×D4—248no—
239C7×D4C22×D4148yes—
240C2×C4×C7—24no—
241C2×C4×C7—248no—
242C2×C4×C7C22×D414yes—
243C2×C4×C7C22×D4112yes—
244C2×C4×C7C22×D4148yes—
245C2×C4×C7C2514yes—
246C2×C4×C7C2512yes—
247C2×C4×C7C2516yes—
248C22⋊D7—248no—
249C22⋊D7—448no—
250C2×Dic7—212no—
251C2×Dic7—248no—
252C2×Dic7—24no—
253C2×Dic7—212no—
254C2×Dic7—424no—
255C2×Dic7C2514yes—
256C2×Dic7C25112yes—
257C2×Dic7C25112yes—
258D28—28no—
259C4×D7—248no—
260Dic14—224no—
261Dic14—248no—
262C25C22×D716yes—
2632−1+4—14128no—
264H(5,𝔽2)—2128no—
265C2×#41—216no—
266C2×#41—224no—
267C2×#41—748no—
268C2×#41—724no—
269C2×#41—1448no—
270C2×#41—1424no—
271C2×#41C22⋊D7116yescomplemented
272C2×#41C22×D7124yes—
273C2×#41C22×D7116yes—
274C2×#41C22×D718yes—
275C22×Q8—716no—
276C22×Q8—76no—
277C22×Q8—73no—
278C22×Q8—724no—
279C22×Q8—1416no—
280C22×Q8—1424no—
281C22×Q8C22×D711yescharacteristic
282C22×Q8C22×D711yescharacteristic
283C22×D4—248no—
284C22×D4—224no—
285C22×D4—76no—
286C22×D4—712no—
287C22×D4—71no—
288C22×D4C22×D716yes—
289C22×D4C22×D716yes—
290C22×D4C22×D713yes—
291C22×D4C22×D7124yes—
292C22×D4C22×D7148yes—
293C22×D4C22×D713yes—
294C23×C4—76no—
295C23×C4—76no—
296C23×C4—76no—
297C23×C4C22×D716yes—
298C23×C4C22×D713yes—
299C4⋊Q8—724no—
300#127—724no—
301#124—78no—
302#124—724no—
303#124—748no—
304#124—796no—
305#123—796no—
306C22⋊Q8—796no—
307#121—796no—
308#121—796no—
309#120—796no—
310C4×Q8—716no—
311C4×D4—748no—
312C4×D4—796no—
313#117—724no—
314C2×C4⋊C4—76no—
315C2×C4⋊C4—712no—
316C2×C22⋊C4—712no—
317C2×C22⋊C4—724no—
318C2×C22⋊C4—796no—
319C2×C22⋊C4—76no—
320C2×C22⋊C4—74no—
321C2×C42—73no—
322C2×C42—76no—
323C2×C42—724no—
324C2×C42—71no—
325C22×C7—212no—
326C22×C7—224no—
327C22×C7—212no—
328C22×C7—224no—
329C22×C7—424no—
330C22×C7—848no—
331C22×C7C23×D4112yes—
332C22×C7C23×D412yes—
333C22×C7#58514yes—
334C22×C7C2611yescharacteristic
335C2×D7—24no—
336C2×D7—28no—
337C2×D7—48no—
338C4×C7—24no—
339C4×C7C23×D414yes—
340C4×C7C23×D4112yes—
341Dic7—212no—
342Dic7—24no—
343Dic7—212no—
344C24—224no—
345C24—424no—
346C24—72no—
347C24C23×D716yes—
348C24C23×D713yes—
349C24C23×D7124yes—
350#41—264no—
351#41—296no—
352#41—14192no—
353#41—1496no—
354#41C2×C22⋊D7164yescomplemented
355C2×Q8—24no—
356C2×Q8—724no—
357C2×Q8—712no—
358C2×Q8—1464no—
359C2×Q8—1424no—
360C2×Q8—1412no—
361C2×Q8—1496no—
362C2×Q8C2×C22⋊D718yes—
363C2×Q8C23×D714yes—
364C2×Q8C23×D714yes—
365C2×D4—224no—
366C2×D4—2192no—
367C2×D4—224no—
368C2×D4—212no—
369C2×D4—296no—
370C2×D4—724no—
371C2×D4—748no—
372C2×D4—74no—
373C2×D4—1424no—
374C2×D4—144no—
375C2×D4—2824no—
376C2×D4C2×C22⋊D7124yes—
377C2×D4C23×D7124yes—
378C2×D4C23×D7124yes—
379C2×D4C23×D7112yes—
380C2×D4C23×D7112yes—
381C22×C4—224no—
382C22×C4—724no—
383C22×C4—72no—
384C22×C4—76no—
385C22×C4—76no—
386C22×C4—724no—
387C22×C4—724no—
388C22×C4—1424no—
389C22×C4—1424no—
390C22×C4—2812no—
391C22×C4C2×C22⋊D7124yes—
392C22×C4C23×D7124yes—
393C22×C4C23×D712yes—
394C22×C4C23×D7112yes—
395C22×C4C23×D711yescharacteristic
396C22×C4C23×D713yes—
397C4⋊C4—748no—
398C4⋊C4—724no—
399C22⋊C4—748no—
400C22⋊C4—796no—
401C22⋊C4—724no—
402C22⋊C4—716no—
403C42—74no—
404C42—712no—
405C42—724no—
406C2×C7—212no—
407C2×C7—412no—
408C2×C7#350014yescomplemented
409C2×C7C24×D411yescharacteristic
410C2×C7C2×#58512yes—
411D7—44no—
412C23—212no—
413C23—224no—
414C23—212no—
415C23—248no—
416C23—448no—
417C23—74no—
418C23—848no—
419C23—148no—
420C23—144no—
421C23C22×C22⋊D7112yes—
422C23C24×D7112yes—
423C23C24×D716yes—
424C23C24×D711yescharacteristic
425Q8—28no—
426Q8—1448no—
427Q8—1424no—
428Q8C22×C22⋊D7116yes—
429D4—248no—
430D4—248no—
431D4—224no—
432D4—1448no—
433D4—148no—
434D4—2848no—
435D4C22×C22⋊D7148yes—
436C2×C4—24no—
437C2×C4—248no—
438C2×C4—74no—
439C2×C4—712no—
440C2×C4—712no—
441C2×C4—144no—
442C2×C4—1412no—
443C2×C4—1412no—
444C2×C4—1448no—
445C2×C4—1448no—
446C2×C4—2824no—
447C2×C4C22×C22⋊D7112yes—
448C2×C4C22×C22⋊D714yes—
449C2×C4C22×C22⋊D7148yes—
450C2×C4C24×D714yes—
451C2×C4C24×D712yes—
452C2×C4C24×D716yes—
453C7C2×#350011yescharacteristic, complemented, Sylow
454C22—212no—
455C22—224no—
456C22—224no—
457C22—212no—
458C22—424no—
459C22—848no—
460C22—148no—
461C22—144no—
462C22—288no—
463C22#9061814yes—
464C22C23×C22⋊D7112yes—
465C22C23×C22⋊D712yes—
466C22C25×D711yescharacteristic
467C4—24no—
468C4—144no—
469C4—1412no—
470C4—1412no—
471C4C23×C22⋊D7112yes—
472C4C23×C22⋊D714yes—
473C2—212no—
474C2—412no—
475C2—284no—
476C2#1174435114yesdirect product factor
477C2C2×#9061812yes—
478C2C24×C22⋊D711yescharacteristic
subgroupG/H
centerC23C24×D7
commutator, cosocleC22×C7C26
FrattiniC22C25×D7
FittingC22×C7×H(5,𝔽2)C2
socleC23×C7C25
hypercenterC22×H(5,𝔽2)D7
hypocenterC7C2×#3500
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×C7×H(5,𝔽2)C2semidirect
2. C22×#6554C2semidirect
3. #11744351C2direct (N × H)
4. C2×#90522C2semidirect
5. C2×#90516C2semidirect
6. C2×#90523C2semidirect
7. C2×#41×Dic7C2semidirect
8. C2×#90496C2semidirect
9. C2×#90497C2semidirect
10. C22×#6499C2semidirect
12. C2×C7×H(5,𝔽2)C22semidirect
19. C2×#6554C22semidirect
20. C22×Q8×D7C22semidirect
21. C22×#1159C22semidirect
22. C22×#1157C22semidirect
23. C23×Dic14C22semidirect
24. C2×#6516C22semidirect
25. C22×C22⋊Dic7C22semidirect
26. #90523C22semidirect
27. #90522C22semidirect
28. #41×Dic7C22semidirect
29. #90516C22semidirect
30. C2×#6510C22semidirect
31. C2×Q8×Dic7C22semidirect
32. C2×#6507C22semidirect
33. #90497C22semidirect
34. #90496C22semidirect
35. C2×#6503C22semidirect
36. C2×#6502C22semidirect
37. C2×#6501C22semidirect
38. C2×#6499C22semidirect
39. C2×#6499C22semidirect
40. C2×#6499C22semidirect
41. C2×#6499C22semidirect
42. C2×#6498C22semidirect
43. C2×D4×Dic7C22semidirect
44. C2×C4×C22⋊D7C22semidirect
45. C2×#6489C22semidirect
46. C2×#6487C22semidirect
47. C22×C4×Dic7C22semidirect
53. C2×C7×#41D4semidirect
75. C2×Q8×D7C23semidirect
77. C2×#1159C23semidirect
79. C2×#1157C23semidirect
85. C22×Dic14C23semidirect
86. #6516C23semidirect
90. C2×C22⋊Dic7C23semidirect
91. #6510C23semidirect
92. Q8×Dic7C23semidirect
93. #6507C23semidirect
94. #6503C23semidirect
95. #6502C23semidirect
96. #6501C23semidirect
97. #6499C23semidirect
98. #6499C23semidirect
99. #6499C23semidirect
100. #6498C23semidirect
101. D4×Dic7C23semidirect
102. C4×C22⋊D7C23semidirect
104. #6489C23semidirect
106. #6487C23semidirect
111. C2×C4×Dic7C23semidirect
113. C22×H(5,𝔽2)D7semidirect
130. C7×#41C2×D4semidirect
187. C2×H(5,𝔽2)C2×D7semidirect
271. C2×#41C22⋊D7semidirect
354. #41C2×C22⋊D7semidirect
408. C2×C7#3500semidirect
453. C7C2×#3500semidirect
476. C2#11744351direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×H(5,𝔽2)2
C22×#65542
#117443512
C2×#905222
C2×#905162
C2×#905232
C2×#41×Dic72
C2×#904962
C2×#904972
C22×#64992
C2×#35007