Finite Group Databasebeta

← C2×#11740972 #14230699

Subgroups — C2×#11740972 #14230699

Order 1792.

2374 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C2×#90242C211yesmaximal, characteristic, complemented
3#11740972C2116yesmaximal, direct product factor
4C2×#89859C211yesmaximal, characteristic, complemented
5C2×#89865C211yesmaximal, characteristic, complemented
6C2×#89809C212yesmaximal, complemented
7C2×#89814C212yesmaximal, complemented
8C2×#89791C211yesmaximal, characteristic, complemented
9C2×#89784C211yesmaximal, characteristic, complemented
10C2×C4×#1141C211yesmaximal, characteristic, complemented
11C2×#89742C212yesmaximal, complemented
12C2×#89738C212yesmaximal, complemented
13C22×C7×#41—21no—
14C22×#1157C2212yes—
15C7×#547—24no—
16C7×#547C2218yescomplemented
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×#121C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C22×C7×C4⋊C4C2212yes—
24C2×#6501—21no—
25C22×#1142C2212yes—
26C22×#1141C2211yescharacteristic
27C2×#6487C2212yes—
28C22×#1136C2212yes—
29#90242—24no—
30#90242C2218yescomplemented
31C2×D7×C4⋊C4—21no—
32C2×#6446—22no—
33C2×C4×D28C2211yescharacteristic
34C2×#6432C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2212yes—
37#89865C2218yescomplemented
38#89859C2218yescomplemented
39#89814C22116yescomplemented
40#89809C22116yescomplemented
41#89791C2218yescomplemented
42C4×#1141C2218yescomplemented
43#89784C2218yescomplemented
44#89742C22116yescomplemented
45C2×#6384C2212yes—
46C2×#6384C2212yes—
47C2×#6384C2212yes—
48#89738C22116yescomplemented
49C2×#6383C2211yescharacteristic
50C2×#6383C2212yes—
51C2×#6381C2211yescharacteristic
52C2×#6381C2212yes—
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3330—71nomaximal, Sylow
56C2×C7×#41—28no—
57C2×C7×#41—48no—
58C22×C7×Q8—21no—
59C22×C7×D4—22no—
60C22×C7×D4D411yescharacteristic
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—22no—
63C23×C4×C7—21no—
64C23×C4×C7C2312yes—
65C22×C22⋊D7—22no—
66C2×#1157—24no—
67C2×#1157—28no—
68C2×#1157—44no—
69C2×#1157C2318yes—
70C22×D28C2311yescharacteristic
71C22×C4×D7—21no—
72C22×C4×D7—22no—
73C22×Dic14C2312yes—
74C7×C4⋊Q8—22no—
75C7×C4⋊Q8C2314yes—
76C7×#123—216no—
77C7×C22⋊Q8—28no—
78C7×#121—24no—
79C7×#121C2318yes—
80C4×C7×D4—22no—
81C4×C7×D4D414yes—
82C4×C7×D4C2314yes—
83C4×C7×D4C2314yes—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—24no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2318yes—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4D412yes—
94C2×C42×C7C2311yescharacteristic
95C2×C22⋊Dic7—22no—
96#6501—28no—
97C2×#1142—44no—
98C2×#1142C2318yes—
99C2×#1141—24no—
100C2×#1141C2314yes—
101C2×(C2×D7)⋊C4—21no—
102C2×C4⋊Dic7—21no—
103C2×C4⋊Dic7—21no—
104C2×C4⋊Dic7C2311yescharacteristic
105#6487—24no—
106#6487C2318yes—
107C2×Dic7⋊C4—22no—
108C2×#1136—44no—
109C2×#1136C2318yes—
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2312yes—
113D7×C4⋊C4—28no—
114#6446—216no—
115C4×D28—22no—
116C4×D28C2314yes—
117#6432—22no—
118#6432C2314yes—
119#6406C2314yes—
120C22⋊(C7⋊C8)C2318yes—
121#6384C2318yes—
122#6384C2318yes—
123#6384C2318yes—
124#6383C2318yes—
125#6383C2314yes—
126#6381C2318yes—
127#6381C2314yes—
128C4⋊(C7⋊C8)C2314yes—
129C4×C7⋊C8C2314yes—
130C2×#547—71no—
131C2×#547D711yescharacteristic, complemented
132#3330—716no—
133C2×#487—71no—
134C2×#484—72no—
135C2×#474—71no—
136C2×#468—72no—
137C2×#461—71no—
138C2×#449—72no—
139C2×C4×D8—71no—
140C2×#436—71no—
141C2×#417—72no—
142C24×C7—21no—
143C24×C7C2×D412yes—
144C23×D7—21no—
145C7×#41—432no—
146C2×C7×Q8—24no—
147C2×C7×Q8—44no—
148C2×C7×D4—28no—
149C2×C7×D4—24no—
150C2×C7×D4—21no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—48no—
154C2×C7×D4C2×D414yes—
155C2×C7×D4C2412yes—
156C22×C4×C7—24no—
157C22×C4×C7—24no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—28no—
161C22×C4×C7—24no—
162C22×C4×C7—48no—
163C22×C4×C7—44no—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D414yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2414yes—
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—28no—
171C2×C22⋊D7—48no—
172C22×Dic7—21no—
173C22×Dic7—22no—
174#1157—216no—
175#1157—48no—
176#1157—416no—
177C2×D28—21no—
178C2×D28—22no—
179C2×D28#4114yescomplemented
180C2×D28C2412yes—
181C2×C4×D7—24no—
182C2×C4×D7—28no—
183C2×C4×D7—48no—
184C2×C4×D7—44no—
185C2×Dic14—24no—
186C2×Dic14—24no—
187C2×Dic14—22no—
188C2×Dic14C2414yes—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—28no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4C2×D414yes—
194C7×C4⋊C4C2412yes—
195C7×C4⋊C4C2414yes—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—28no—
198C7×C22⋊C4C2×D418yes—
199C42×C7C2×D412yes—
200C42×C7C2412yes—
201C22⋊Dic7—28no—
202#1142—48no—
203#1141—28no—
204(C2×D7)⋊C4—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—24no—
207C4⋊Dic7—21no—
208C4⋊Dic7C2412yes—
209Dic7⋊C4—28no—
210#1136—48no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—24no—
214C2×C7⋊C8C2412yes—
215C2×C7⋊C8C2414yes—
216C2×C7⋊C8C2412yes—
217C22×#41—21no—
218C22×#41—72no—
219C22×SD16—72no—
220C22×D8—71no—
221C22×M16—72no—
222#547—24no—
223#547—78no—
224#547—144no—
225#547C2×D718yescomplemented
226C2×C4⋊Q8—71no—
227C2×C4⋊Q8C2×D711yescharacteristic
228C2×#123—22no—
229C2×#123—142no—
230C2×C22⋊Q8—21no—
231C2×C22⋊Q8—72no—
232C2×#121—141no—
233C2×#121C2×D712yes—
234C2×C4×D4—71no—
235C2×C4×D4C2×D711yescharacteristic
236C2×C4×D4C2×D711yescharacteristic
237C22×C4⋊C4—141no—
238C22×C4⋊C4C2×D712yes—
239#487—78no—
240#484—716no—
241#474—78no—
242#468—716no—
243#461—78no—
244#449—716no—
245C4×D8—78no—
246#436—78no—
247C2×#107—71no—
248C2×#107—72no—
249C2×C4⋊C8—71no—
250#417—716no—
251C2×#103—72no—
252C2×#103—72no—
253C2×#103—72no—
254C2×#102—71no—
255C2×#102—71no—
256C2×#102—72no—
257C2×C22⋊C8—72no—
258C2×C4×C8—71no—
259C23×C7—24no—
260C23×C7—24no—
261C23×C7—24no—
262C23×C7—22no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7C22×D414yes—
267C23×C7C22×D411yescharacteristic
268C22×D7—24no—
269C22×D7—22no—
270C22×D7—42no—
271C22×D7—42no—
272C7×Q8—48no—
273C7×D4—28no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4—416no—
277C7×D4—42no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—28no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—416no—
286C2×C4×C7—42no—
287C2×C4×C7—44no—
288C2×C4×C7—48no—
289C2×C4×C7AGL(1,ℤ/8)14yes—
290C2×C4×C7C22×D414yes—
291C2×C4×C7C22×D412yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D412yes—
294C2×C4×C7C2×#4112yes—
295C2×C4×C7H(5,𝔽2)12yes—
296C2×C4×C7C2511yescharacteristic
297C22⋊D7—416no—
298C2×Dic7—22no—
299C2×Dic7—24no—
300C2×Dic7—44no—
301C2×Dic7—42no—
302D28—24no—
303D28—22no—
304D28C2×#4114yescomplemented
305C4×D7—416no—
306C4×D7—48no—
307Dic14—24no—
308Dic14—24no—
309Dic14—44no—
310C7⋊C8—22no—
311C7⋊C8—22no—
312C7⋊C8—24no—
313C2×#41—28no—
314C2×#41—48no—
315C2×#41—78no—
316C2×#41—148no—
317C2×#41—144no—
318C2×#41—284no—
319C22×Q8—21no—
320C22×Q8—72no—
321C22×D4—22no—
322C22×D4—71no—
323C22×D4—142no—
324C22×D4C22⋊D711yescharacteristic
325C22×D4C22×D711yescharacteristic
326C23×C4—22no—
327C23×C4—21no—
328C23×C4—141no—
329C23×C4—142no—
330C23×C4C22×D712yes—
331C2×SD16—78no—
332C2×SD16—284no—
333C2×D8—74no—
334C2×D8—144no—
335C2×M16—78no—
336C2×M16—284no—
337C22×C8—71no—
338C22×C8—72no—
339C22×C8—71no—
340C4⋊Q8—22no—
341C4⋊Q8—74no—
342C4⋊Q8—142no—
343C4⋊Q8C22×D714yes—
344#123—216no—
345#123—1416no—
346C22⋊Q8—28no—
347C22⋊Q8—78no—
348C22⋊Q8—144no—
349#121—24no—
350#121—148no—
351#121C22×D718yes—
352C4×D4—22no—
353C4×D4—74no—
354C4×D4—142no—
355C4×D4C22⋊D714yes—
356C4×D4C22×D714yes—
357C4×D4C22×D714yes—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—148no—
366C2×C4⋊C4C22⋊D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D712yes—
369C2×C4⋊C4C22×D718yes—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—142no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4C22⋊D712yes—
375C2×C42C22×D711yescharacteristic
376#107—74no—
377#107—78no—
378C4⋊C8—74no—
379#103—78no—
380#103—78no—
381#103—78no—
382#102—74no—
383#102—74no—
384#102—78no—
385C22⋊C8—78no—
386C4×C8—74no—
387C22×C7—24no—
388C22×C7—24no—
389C22×C7—24no—
390C22×C7—48no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7—84no—
395C22×C7#54612yes—
396C22×C7C2×AGL(1,ℤ/8)12yes—
397C22×C7#57812yes—
398C22×C7C23×D411yescharacteristic
399C2×D7—24no—
400C2×D7—48no—
401C2×D7—42no—
402C2×D7—42no—
403C4×C7—24no—
404C4×C7—22no—
405C4×C7—42no—
406C4×C7—44no—
407C4×C7#54612yes—
408C4×C7C2×AGL(1,ℤ/8)14yes—
409C4×C7C22×#4111yescharacteristic
410C4×C7C2×H(5,𝔽2)11yescharacteristic
411Dic7—44no—
412Dic7—42no—
413C24—21no—
414C24—141no—
415C24C2×C22⋊D712yes—
416#41—432no—
417#41—1416no—
418#41—288no—
419#41—2816no—
420C2×Q8—24no—
421C2×Q8—44no—
422C2×Q8—74no—
423C2×Q8—142no—
424C2×Q8—144no—
425C2×Q8—144no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—48no—
432C2×D4—74no—
433C2×D4—72no—
434C2×D4—148no—
435C2×D4—142no—
436C2×D4—141no—
437C2×D4—288no—
438C2×D4C2×C22⋊D714yes—
439C2×D4C23×D712yes—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—141no—
449C22×C4—142no—
450C22×C4—144no—
451C22×C4—148no—
452C22×C4—288no—
453C22×C4—284no—
454C22×C4C2×C22⋊D714yes—
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D712yes—
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C23×D714yes—
459C22×C4C23×D711yescharacteristic
460SD16—288no—
461D8—148no—
462M16—288no—
463C2×C8—72no—
464C2×C8—72no—
465C2×C8—74no—
466C2×C8—142no—
467C2×C8—142no—
468C2×C8—144no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—72no—
474C4⋊C4—148no—
475C4⋊C4—141no—
476C4⋊C4—144no—
477C4⋊C4—144no—
478C4⋊C4C2×C22⋊D714yes—
479C4⋊C4C23×D712yes—
480C4⋊C4C23×D714yes—
481C22⋊C4—24no—
482C22⋊C4—28no—
483C22⋊C4—148no—
484C22⋊C4—144no—
485C22⋊C4C2×C22⋊D718yes—
486C42C2×C22⋊D712yes—
487C42C23×D712yes—
488C2×C7—44no—
489C2×C7—82no—
490C2×C7#333014yescomplemented
491C2×C7C2×#54611yescharacteristic
492C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
493C2×C7C2×#57811yescharacteristic
494D7—44no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—144no—
503C23—142no—
504C23—282no—
505C23—282no—
506C23C22×C22⋊D714yes—
507C23C22×C22⋊D711yescharacteristic
508Q8—48no—
509Q8—144no—
510Q8—144no—
511Q8—284no—
512D4—28no—
513D4—22no—
514D4—22no—
515D4—416no—
516D4—42no—
517D4—74no—
518D4—144no—
519D4—142no—
520D4—2816no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—142no—
533C2×C4—144no—
534C2×C4—2816no—
535C2×C4—282no—
536C2×C4—284no—
537C2×C4—288no—
538C2×C4#649514yes—
539C2×C4D7×#4112yes—
540C2×C4#655312yes—
541C2×C4C22×C22⋊D712yes—
542C2×C4C22×C22⋊D714yes—
543C2×C4C22×C22⋊D712yes—
544C2×C4C22×C22⋊D711yescharacteristic
545C2×C4C24×D711yescharacteristic
546C8—142no—
547C8—142no—
548C8—144no—
549C7C2×#333011yescharacteristic, complemented, Sylow
550C22—24no—
551C22—24no—
552C22—24no—
553C22—48no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—84no—
558C22—144no—
559C22—282no—
560C22—288no—
561C22—282no—
562C22C2×#649512yes—
563C22#9052012yes—
564C22#9052912yes—
565C22C23×C22⋊D711yescharacteristic
566C4—24no—
567C4—22no—
568C4—42no—
569C4—44no—
570C4—282no—
571C4—284no—
572C4C2×#649514yes—
573C4#9052012yes—
574C4C2×D7×#4111yescharacteristic
575C4C2×#655311yescharacteristic
576C2—44no—
577C2—82no—
578C2—284no—
579C2#1174097214yesdirect product factor
580C2C22×#649511yescharacteristic
581C2C2×#9052011yescharacteristic
582C2C2×#9052911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#547C2
socleC23×C7C22×D4
hypercenterC2×#547D7
hypocenterC7C2×#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C2×#90242C2semidirect
3. #11740972C2direct (N × H)
4. C2×#89859C2semidirect
5. C2×#89865C2semidirect
6. C2×#89809C2semidirect
7. C2×#89814C2semidirect
8. C2×#89791C2semidirect
9. C2×#89784C2semidirect
10. C2×C4×#1141C2semidirect
11. C2×#89742C2semidirect
12. C2×#89738C2semidirect
16. C7×#547C22semidirect
30. #90242C22semidirect
37. #89865C22semidirect
38. #89859C22semidirect
39. #89814C22semidirect
40. #89809C22semidirect
41. #89791C22semidirect
42. C4×#1141C22semidirect
43. #89784C22semidirect
44. #89742C22semidirect
48. #89738C22semidirect
131. C2×#547D7semidirect
179. C2×D28#41semidirect
225. #547C2×D7semidirect
304. D28C2×#41semidirect
490. C2×C7#3330semidirect
549. C7C2×#3330semidirect
579. C2#11740972direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×#902422
#117409722
C2×#898592
C2×#898652
C2×#898092
C2×#898142
C2×#897912
C2×#897842
C2×C4×#11412
C2×#897422
C2×#897382
C2×#33307