Finite Group Databasebeta

← C2×#11736686 #14221334

Subgroups — C2×#11736686 #14221334

Order 1792.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90213C211yesmaximal, characteristic
2C22×#6453C211yesmaximal, characteristic, complemented
3C2×C7×#428C211yesmaximal, characteristic, complemented
4C2×#89905C211yesmaximal, characteristic
5C2×#89890C211yesmaximal, characteristic, complemented
6C2×#89767C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic
8#11736686C2116yesmaximal, direct product factor
9C2×#89649C212yesmaximal, complemented
10C2×#89663C212yesmaximal, complemented
11C2×#89629C212yesmaximal, complemented
12C2×#89643C212yesmaximal, complemented
13C22×#1157C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic7—21no—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4—21no—
20C22×C4×Dic7C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22#90213—24no—
23#90213C2218yes—
24C2×#6456C2211yescharacteristic
25C2×#6456C2211yescharacteristic
26C2×#6453—24no—
27C2×#6453—21no—
28C2×#6453—21no—
29C2×#6453C2212yes—
30C2×#6453C2212yes—
31C2×#6453C2218yescomplemented
32C2×#6450C2211yescharacteristic
33C2×#6450C2211yescharacteristic
34C2×(C22⋊D7)⋊C4—21no—
35C2×(C22⋊D7)⋊C4—21no—
36C2×#6435—21no—
37C2×C42×D7—21no—
38C7×#428C2218yescomplemented
39C2×C7×#107C2212yes—
40C2×C7×#106C2212yes—
41#89905C2218yes—
42#89890C2218yescomplemented
43C2×#6395C2212yes—
44C2×#6390C2212yes—
45C2×Dic7⋊C8C2212yes—
46C2×Dic7⋊C8C2212yes—
47#89767C2218yescomplemented
48C2×#6384C2212yes—
49#89738C2218yes—
50C2×#6383C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53#89663C22116yescomplemented
54#89649C22116yescomplemented
55#89643C22116yescomplemented
56#89629C22116yescomplemented
57C2×#3080—71nomaximal, Sylow
58C23×C4×C7—21no—
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—21no—
61C23×Dic7—21no—
62C23×Dic7D411yescharacteristic
63C23×Dic7D411yescharacteristic
64C2×#1157—22no—
65C2×#1157—24no—
66C2×#1157Q818yes—
67C2×#1157C2314yes—
68C22×D28C2311yescharacteristic
69C22×C4×D7—21no—
70C22×C4×D7—21no—
71C22×C4×D7—21no—
72C22×Dic14C2311yescharacteristic
73C2×C7×M16—24no—
74C2×C7×M16C2314yes—
75C22×C7×C8C2312yes—
76C7×#117—22no—
77C7×#117C2314yes—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2314yes—
84C2×C7×C22⋊C4—21no—
85C2×C42×C7—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—22no—
92C2×C4⋊Dic7—28no—
93C2×C4⋊Dic7C2312yes—
94C2×C4⋊Dic7C2314yes—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—22no—
98C2×Dic7⋊C4—22no—
99C2×Dic7⋊C4—28no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7D411yescharacteristic
103C2×C4×Dic7D411yescharacteristic
104C2×C4×Dic7D414yes—
105C2×C4×Dic7C2311yescharacteristic
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2314yes—
108C2×#1136—24no—
109C2×#1136C2314yes—
110C22×C7⋊C8C2312yes—
111#6456—22no—
112#6456—22no—
113#6456C2314yes—
114#6456C2314yes—
115#6453—24no—
116#6453—24no—
117#6453—28no—
118#6453—28no—
119#6453C2318yes—
120#6453C2318yes—
121#6450—22no—
122#6450—22no—
123#6450C2314yes—
124#6450C2314yes—
125(C22⋊D7)⋊C4—28no—
126(C22⋊D7)⋊C4—28no—
127#6435—28no—
128C42×D7—28no—
129C7×#107C2318yes—
130C7×#106C2318yes—
131#6395C2318yes—
132#6390C2318yes—
133Dic7⋊C8C2318yes—
134Dic7⋊C8C2318yes—
135#6384C2318yes—
136#6383C2318yes—
137#6382C2318yes—
138#6381C2318yes—
139C22×#125—71no—
140C2×C4×#41—71no—
141#3080—716no—
142C2×#479—72no—
143C2×#479—72no—
144C2×#478—72no—
145C2×#478—72no—
146C2×#428—71no—
147C2×#428D711yescharacteristic, complemented
148C2×#423—71no—
149C2×#417—71no—
150C2×#417—71no—
151C24×C7C2×D411yescharacteristic
152C23×D7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—44no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×Q814yes—
161C22×C4×C7#4114yes—
162C22×C4×C7C2411yescharacteristic
163C22×C4×C7C2412yes—
164C2×C22⋊D7—28no—
165C2×C22⋊D7—24no—
166C22×Dic7—21no—
167C22×Dic7—24no—
168C22×Dic7—22no—
169C22×Dic7—42no—
170C22×Dic7—42no—
171C22×Dic7—44no—
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D414yes—
177C22×Dic7C2×D414yes—
178#1157—216no—
179#1157—28no—
180#1157C2×Q8116yes—
181C2×D28—21no—
182C2×D28—22no—
183C2×D28C2×Q814yes—
184C2×D28C2412yes—
185C2×C4×D7—28no—
186C2×C4×D7—24no—
187C2×C4×D7—24no—
188C2×C4×D7—24no—
189C2×C4×D7—44no—
190C2×C4×D7—44no—
191C2×Dic14—21no—
192C2×Dic14—22no—
193C2×Dic14C2×Q814yes—
194C2×Dic14C2412yes—
195C7×M16—28no—
196C2×C7×C8—24no—
197C2×C7×C8C2414yes—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—24no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4C2412yes—
203C7×C4⋊C4C2414yes—
204C7×C4⋊C4C2412yes—
205C7×C22⋊C4—24no—
206C42×C7—24no—
207(C2×D7)⋊C4—24no—
208(C2×D7)⋊C4—24no—
209C4⋊Dic7—24no—
210C4⋊Dic7—22no—
211C4⋊Dic7—28no—
212C4⋊Dic7—28no—
213C4⋊Dic7C2414yes—
214Dic7⋊C4—24no—
215Dic7⋊C4—24no—
216Dic7⋊C4—28no—
217Dic7⋊C4—28no—
218C4×Dic7—24no—
219C4×Dic7—24no—
220C4×Dic7C2×D414yes—
221C4×Dic7C2×D414yes—
222C4×Dic7C2×D412yes—
223C4×Dic7C2×D412yes—
224C4×Dic7C2412yes—
225C4×Dic7C2412yes—
226#1136—28no—
227C2×C7⋊C8—24no—
228C2×C7⋊C8C2414yes—
229C22×#41—71no—
230C22×M16—71no—
231C22×M16C2×D711yescharacteristic
232C2×#125—72no—
233C2×#125—72no—
234C2×#125—78no—
235C2×#125—141no—
236C2×#125—144no—
237C2×#125—141no—
238C4×#41—78no—
239C4×#41—144no—
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4—141no—
245C2×C4×D4—141no—
246C2×#117—141no—
247C2×#117C2×D711yescharacteristic
248C22×C4⋊C4—71no—
249C22×C4⋊C4—141no—
250C22×C4⋊C4—141no—
251C22×C4⋊C4C2×D711yescharacteristic
252C22×C42—71no—
253C22×C42—141no—
254#479—716no—
255#479—716no—
256#478—716no—
257#478—716no—
258#428—78no—
259#428C2×D718yescomplemented
260C2×#107—72no—
261C2×#107C2×D712yes—
262C2×#106—72no—
263C2×#106C2×D712yes—
264#423—78no—
265C2×C4⋊C8—72no—
266C2×C4⋊C8—72no—
267#417—78no—
268#417—78no—
269C2×#103—72no—
270C2×#103—72no—
271C2×#102—72no—
272C2×#102—72no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7C22⋊Q814yes—
276C23×C7C22×D412yes—
277C23×C7C22×D411yescharacteristic
278C22×D7—24no—
279C22×D7—22no—
280C22×D7—42no—
281C22×D7—42no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—48no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—44no—
289C2×C4×C7C22⋊Q814yes—
290C2×C4×C7C22⋊Q818yes—
291C2×C4×C7C22×D412yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×Q812yes—
294C2×C4×C7C22×Q814yes—
295C2×C4×C7C2×#4112yes—
296C2×C4×C7C2×#4114yes—
297C2×C4×C7C2511yescharacteristic
298C22⋊D7—216no—
299C2×Dic7—24no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—24no—
306C2×Dic7—44no—
307C2×Dic7—44no—
308C2×Dic7—48no—
309C2×Dic7—44no—
310C2×Dic7C22×D412yes—
311C2×Dic7C22×D412yes—
312C2×Dic7C22×D412yes—
313C2×Dic7C22×D412yes—
314D28—24no—
315D28—22no—
316D28C22×Q814yes—
317C4×D7—216no—
318C4×D7—48no—
319C4×D7—48no—
320Dic14—24no—
321Dic14—22no—
322Dic14C22×Q814yes—
323C7×C8—24no—
324C7⋊C8—24no—
325C2×#41—78no—
326C2×#41—74no—
327C2×#41—144no—
328C2×#41—142no—
329C22×Q8—71no—
330C22×D4—71no—
331C22×D4—141no—
332C23×C4—21no—
333C23×C4—71no—
334C23×C4—71no—
335C23×C4—141no—
336C23×C4—141no—
337C23×C4—141no—
338C23×C4—141no—
339C23×C4C22×D711yescharacteristic
340C2×M16—24no—
341C2×M16—74no—
342C2×M16—144no—
343C2×M16C22×D714yes—
344C22×C8—72no—
345C22×C8C22×D712yes—
346#125—78no—
347#125—78no—
348#125—148no—
349#125—148no—
350#125—144no—
351#125—144no—
352C4×Q8—74no—
353C4×Q8—74no—
354C4×Q8—142no—
355C4×Q8—142no—
356C4×D4—74no—
357C4×D4—74no—
358C4×D4—142no—
359C4×D4—142no—
360C4×D4—148no—
361C4×D4—148no—
362#117—22no—
363#117—148no—
364#117C22×D714yes—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—74no—
369C2×C4⋊C4—148no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—148no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—142no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4C22×D712yes—
380C2×C4⋊C4C22×D714yes—
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C42—21no—
387C2×C42—71no—
388C2×C42—74no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—74no—
393C2×C42—148no—
394C2×C42—141no—
395C2×C42—141no—
396#107—78no—
397#107C22×D718yes—
398#106—78no—
399#106C22×D718yes—
400C4⋊C8—78no—
401C4⋊C8—78no—
402#103—78no—
403#103—78no—
404#102—78no—
405#102—78no—
406C22×C7—28no—
407C22×C7—22no—
408C22×C7—22no—
409C22×C7C2×C22⋊Q814yes—
410C22×C7C2×C22⋊Q812yes—
411C22×C7#57512yes—
412C22×C7#57512yes—
413C22×C7C23×D411yescharacteristic
414C2×D7—24no—
415C2×D7—48no—
416C2×D7—42no—
417C2×D7—42no—
418C4×C7—42no—
419C4×C7—42no—
420C4×C7—44no—
421C4×C7C2×C22⋊Q812yes—
422C4×C7C2×C22⋊Q814yes—
423C4×C7C23×Q811yescharacteristic
424C4×C7C22×#4111yescharacteristic
425Dic7—24no—
426Dic7—22no—
427Dic7—22no—
428Dic7—22no—
429Dic7—22no—
430Dic7—44no—
431C24—141no—
432C24D4×D711yescharacteristic
433#41—716no—
434#41—1416no—
435#41—148no—
436C2×Q8—74no—
437C2×Q8—72no—
438C2×Q8—142no—
439C2×Q8—141no—
440C2×D4—74no—
441C2×D4—72no—
442C2×D4—148no—
443C2×D4—144no—
444C2×D4—142no—
445C2×D4—141no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—74no—
456C22×C4—74no—
457C22×C4—144no—
458C22×C4—144no—
459C22×C4—141no—
460C22×C4—148no—
461C22×C4—144no—
462C22×C4—144no—
463C22×C4—142no—
464C22×C4—284no—
465C22×C4—284no—
466C22×C4—282no—
467C22×C4—282no—
468C22×C4—284no—
469C22×C4#115714yes—
470C22×C4D4×D712yes—
471C22×C4D4×D711yescharacteristic
472C22×C4Q8×D714yes—
473C22×C4C23×D712yes—
474C22×C4C23×D711yescharacteristic
475M16—28no—
476M16—148no—
477C2×C8—24no—
478C2×C8—74no—
479C2×C8—144no—
480C2×C8C23×D714yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—74no—
486C4⋊C4—148no—
487C4⋊C4—148no—
488C4⋊C4—144no—
489C4⋊C4—144no—
490C4⋊C4—148no—
491C4⋊C4—148no—
492C4⋊C4—144no—
493C4⋊C4—142no—
494C4⋊C4C23×D714yes—
495C4⋊C4C23×D712yes—
496C4⋊C4C23×D712yes—
497C22⋊C4—24no—
498C22⋊C4—144no—
499C22⋊C4—144no—
500C42—24no—
501C42—74no—
502C42—74no—
503C42—72no—
504C42—72no—
505C42—72no—
506C42—72no—
507C42—144no—
508C42—144no—
509C2×C7—24no—
510C2×C7#308014yescomplemented
511C2×C7C22×C22⋊Q811yescharacteristic
512C2×C7C2×#57511yescharacteristic
513C2×C7C2×#57511yescharacteristic
514D7—44no—
515C23—22no—
516C23—22no—
517C23—144no—
518C23—142no—
519C23—282no—
520C23—282no—
521C23#645914yes—
522C23C2×D4×D712yes—
523C23C2×D4×D711yescharacteristic
524Q8—74no—
525Q8—144no—
526Q8—142no—
527D4—74no—
528D4—1416no—
529D4—144no—
530D4—142no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—72no—
539C2×C4—72no—
540C2×C4—72no—
541C2×C4—72no—
542C2×C4—144no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—1416no—
549C2×C4—144no—
550C2×C4—288no—
551C2×C4—288no—
552C2×C4—284no—
553C2×C4—284no—
554C2×C4—288no—
555C2×C4—284no—
556C2×C4#645914yes—
557C2×C4#645918yes—
558C2×C4C2×#115712yes—
559C2×C4C2×#115714yes—
560C2×C4C2×D4×D712yes—
561C2×C4C2×D4×D711yescharacteristic
562C2×C4C2×Q8×D712yes—
563C2×C4C2×Q8×D714yes—
564C2×C4C24×D711yescharacteristic
565C8—24no—
566C8—144no—
567C7C2×#308011yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—28no—
571C22—144no—
572C22—282no—
573C22—282no—
574C22—288no—
575C22C2×#645914yes—
576C22C2×#645912yes—
577C22#9046412yes—
578C22#9046812yes—
579C22C22×D4×D711yescharacteristic
580C4—44no—
581C4—42no—
582C4—42no—
583C4—142no—
584C4—142no—
585C4—142no—
586C4—142no—
587C4—144no—
588C4—284no—
589C4C2×#645914yes—
590C4C2×#645912yes—
591C4C22×#115711yescharacteristic
592C4C22×Q8×D711yescharacteristic
593C2—24no—
594C2—284no—
595C2#1173668614yesdirect product factor
596C2C22×#645911yescharacteristic
597C2C2×#9046411yescharacteristic
598C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#428C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
hypercenterC2×#428D7
hypocenterC7C2×#3080
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C22×#6453C2semidirect
3. C2×C7×#428C2semidirect
5. C2×#89890C2semidirect
6. C2×#89767C2semidirect
8. #11736686C2direct (N × H)
9. C2×#89649C2semidirect
10. C2×#89663C2semidirect
11. C2×#89629C2semidirect
12. C2×#89643C2semidirect
31. C2×#6453C22semidirect
38. C7×#428C22semidirect
42. #89890C22semidirect
47. #89767C22semidirect
53. #89663C22semidirect
54. #89649C22semidirect
55. #89643C22semidirect
56. #89629C22semidirect
147. C2×#428D7semidirect
259. #428C2×D7semidirect
510. C2×C7#3080semidirect
567. C7C2×#3080semidirect
595. C2#11736686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902132
C22×#64532
C2×C7×#4282
C2×#899052
C2×#898902
C2×#897672
C2×#897382
#117366862
C2×#896492
C2×#896632
C2×#896292
C2×#896432
C2×#30807