Finite Group Databasebeta

← #13355181

Subgroups — #13355181

Order 1792.

2046 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic7C211yesmaximal, characteristic, complemented
2C22×#6512C211yesmaximal, characteristic
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C2×#89903C211yesmaximal, characteristic, complemented
5C2×#89760C211yesmaximal, characteristic, complemented
6#11729733C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11727994C214yesmaximal, complemented
9#11727926C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×D28C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14#41×Dic7—24no—
15#41×Dic7—24no—
16C2×#6512—24no—
17C2×#6512—24no—
18C2×#6512C418yescomplemented
19C2×Q8×Dic7C2212yescomplemented
20C22×#1143C2212yes—
21C2×D4×Dic7—22no—
22C2×D4×Dic7C2212yescomplemented
23C22×#1141C2212yes—
24C22×(C2×D7)⋊C4—22no—
25C2×#6489—21no—
26C2×#6489C2211yescharacteristic
27C22×C4×Dic7—21no—
28C22×C4×Dic7C2211yescharacteristic
29C22×#1136C2211yescharacteristic
30C7×#417—24no—
31C2×C7×#103C2212yescomplemented
32C2×C7×#102C2212yescomplemented
33#89903—24no—
34C2×#6395C2214yes—
35#89760—21no—
36#89760—21no—
37#89760C2214yes—
38#89345C2214yescomplemented
39#89324C2214yescomplemented
40#20928—71nomaximal, Sylow
41C24×D7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41C2×C414yes—
46C2×C7×#41D414yes—
47C2×C7×#41D414yes—
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—22no—
55C23×Dic7D412yes—
56C22×D28—22no—
57C22×D28—22no—
58C22×D28—22no—
59C22×D28—22no—
60C22×D28—21no—
61C22×D28—22no—
62C22×D28C2×C414yescomplemented
63C22×D28C2312yes—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8C2312yes—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2312yes—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—22no—
74#6512—232no—
75Q8×Dic7—24no—
76Q8×Dic7—24no—
77C2×#1143—24no—
78C2×#1143—24no—
79C2×#1143C2×C418yescomplemented
80D4×Dic7—216no—
81D4×Dic7—24no—
82D4×Dic7—24no—
83C2×#1141—24no—
84C2×#1141—24no—
85C2×#1141C2×C418yescomplemented
86C2×(C2×D7)⋊C4—24no—
87C2×(C2×D7)⋊C4—24no—
88C2×(C2×D7)⋊C4—48no—
89#6489—28no—
90#6489—22no—
91#6489D414yes—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7C2312yes—
94C2×C4×Dic7—28no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7—22no—
98C2×C4×Dic7—22no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2312yes—
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136C2×C414yescomplemented
105C22×C7⋊C8C2312yes—
106C7×#103—24no—
107C7×#102—24no—
108#6395—28no—
109C22×AGL(1,ℤ/8)—71no—
110C2×C4×#41—71no—
111C2×#417D711yescharacteristic, complemented
112C2×#418—71no—
113C2×#390—71no—
114#1880—74no—
115#1879—74no—
116#1736—71no—
117#1728—71no—
118C24×C7—21no—
119C24×C7C2×D411yescharacteristic
120C23×D7—24no—
121C23×D7—22no—
122C23×D7—44no—
123C23×D7—44no—
124C23×D7—42no—
125C23×D7—42no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41—28no—
129C7×#41C22⋊C4116yescomplemented
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8C22×C412yes—
134C2×C7×Q8C2×D412yes—
135C2×C7×Q8C2×D412yes—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—24no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4C22×C412yes—
143C2×C7×D4C2×D412yes—
144C2×C7×D4C2×D412yes—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—22no—
150C22×C4×C7—44no—
151C22×C4×C7—42no—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C22×C4×C7#4112yes—
158C22×C4×C7#4112yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—28no—
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—24no—
164C22×Dic7—42no—
165C22×Dic7—44no—
166C22×Dic7—44no—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×D412yes—
169C2×D28—28no—
170C2×D28—28no—
171C2×D28—22no—
172C2×D28—22no—
173C2×D28—22no—
174C2×D28—22no—
175C2×D28—24no—
176C2×D28—22no—
177C2×D28—42no—
178C2×D28—44no—
179C2×D28—42no—
180C2×D28—44no—
181C2×D28C22×C414yes—
182C7×M16—44no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—42no—
189C22⋊Dic7—28no—
190C22⋊Dic7—24no—
191#1143—216no—
192#1141—216no—
193(C2×D7)⋊C4—48no—
194(C2×D7)⋊C4—48no—
195C4⋊Dic7—28no—
196C4⋊Dic7—22no—
197C4⋊Dic7C2×D414yes—
198C4×Dic7—28no—
199C4×Dic7—22no—
200C4×Dic7—24no—
201C4×Dic7—22no—
202C4×Dic7—22no—
203C4×Dic7—22no—
204#1136—28no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8C22×C414yes—
208C22×#41C2×D711yescharacteristic
209C23×D4—71no—
210C2×AGL(1,ℤ/8)—78no—
211C2×AGL(1,ℤ/8)—144no—
212C2×AGL(1,ℤ/8)—144no—
213C22×SD16—72no—
214C22×D8—72no—
215C22×M16—71no—
216C22×M16C2×D711yescharacteristic
217C4×#41—144no—
218C4×#41—144no—
219C2×C4×Q8—72no—
220C2×C4×D4—72no—
221C2×C4×D4—142no—
222C2×#117—71no—
223C2×#117—141no—
224C22×C4⋊C4C2×D711yescharacteristic
225C22×C22⋊C4—142no—
226C22×C42—71no—
227C22×C42—141no—
228#418—144no—
229#417—24no—
230C2×#103C2×D712yescomplemented
231C2×#102—74no—
232C2×#102C2×D712yescomplemented
233#390—74no—
234#390—141no—
235#390—141no—
236#340—74no—
237#336—74no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7C4×D412yes—
246C23×C7#12012yes—
247C23×C7#12112yes—
248C23×C7C22×D411yescharacteristic
249C22×D7—24no—
250C22×D7—416no—
251C22×D7—42no—
252C22×D7—42no—
253C22×D7—44no—
254C22×D7—44no—
255C22×D7—48no—
256C22×D7—88no—
257C22×D7—84no—
258C22×D7—84no—
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—22no—
262C7×Q8C2×C22⋊C414yescomplemented
263C7×D4—216no—
264C7×D4—22no—
265C7×D4—22no—
266C7×D4—22no—
267C7×D4C2×C22⋊C414yescomplemented
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—216no—
272C2×C4×C7—42no—
273C2×C4×C7—44no—
274C2×C4×C7—84no—
275C2×C4×C7C2×C22⋊C414yes—
276C2×C4×C7C4×D414yes—
277C2×C4×C7C4×D412yes—
278C2×C4×C7#12012yes—
279C2×C4×C7#12114yes—
280C2×C4×C7#12112yes—
281C2×C4×C7#12314yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—48no—
295C2×Dic7—416no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7AGL(1,ℤ/8)14yescomplemented
299D28—24no—
300D28—24no—
301D28—42no—
302D28—42no—
303D28—48no—
304C7×C8—42no—
305C7⋊C8—24no—
306C25—141no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41C4×D714yes—
311C2×#41D2814yes—
312C2×#41C22⋊D714yes—
313C22×Q8C22×D711yescharacteristic
314C22×D4—21no—
315C22×D4—74no—
316C22×D4—72no—
317C22×D4—142no—
318C22×D4—142no—
319C22×D4—141no—
320C22×D4—142no—
321C22×D4—142no—
322C22×D4—142no—
323C22×D4C22×D711yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4—72no—
327C23×C4—142no—
328C23×C4C22×D711yescharacteristic
329AGL(1,ℤ/8)—1432no—
330C2×SD16—78no—
331C2×SD16—144no—
332C2×SD16—144no—
333C2×D8—78no—
334C2×D8—144no—
335C2×D8—144no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16—74no—
340C2×M16—142no—
341C2×M16—142no—
342C22×C8—72no—
343C22×C8C22×D712yes—
344C4×Q8—144no—
345C4×Q8—144no—
346C4×D4—144no—
347C4×D4—1416no—
348C4×D4—144no—
349#117—74no—
350#117—148no—
351#117—142no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—72no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4C22×D712yes—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—288no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—72no—
366C2×C42—142no—
367C2×C42—148no—
368C2×C42—142no—
369C2×C42—141no—
370C2×C42—142no—
371#103—24no—
372#102—24no—
373#102—148no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—24no—
379C22×C7—44no—
380C22×C7—44no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7#38614yes—
384C22×C7C2×C4×D411yescharacteristic
385C22×C7C2×#12011yescharacteristic
386C22×C7C2×#12111yescharacteristic
387C22×C7C2×AGL(1,ℤ/8)12yes—
388C22×C7#57512yes—
389C2×D7—48no—
390C2×D7—44no—
391C2×D7—816no—
392C2×D7—84no—
393C2×D7—82no—
394C2×D7—82no—
395C4×C7—24no—
396C4×C7—82no—
397C4×C7#38614yes—
398C4×C7C22×C22⋊C411yescharacteristic
399C4×C7C2×C4×D411yescharacteristic
400C4×C7C2×#12111yescharacteristic
401C4×C7C2×#12311yescharacteristic
402Dic7—24no—
403Dic7—42no—
404Dic7—44no—
405C24—21no—
406C24—144no—
407C24—142no—
408C24—284no—
409C24—284no—
410C24—282no—
411C24—282no—
412C24D4×D711yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41(C2×D7)⋊C4116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×C4×D712yes—
421C2×Q8C2×D2812yes—
422C2×Q8C2×C22⋊D712yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—74no—
430C2×D4—148no—
431C2×D4—144no—
432C2×D4—148no—
433C2×D4—142no—
434C2×D4—142no—
435C2×D4—142no—
436C2×D4—142no—
437C2×D4—142no—
438C2×D4—282no—
439C2×D4—284no—
440C2×D4—284no—
441C2×D4—282no—
442C2×D4C2×C4×D712yes—
443C2×D4C2×D2812yes—
444C2×D4C2×C22⋊D712yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—72no—
453C22×C4—72no—
454C22×C4—144no—
455C22×C4—148no—
456C22×C4—144no—
457C22×C4—142no—
458C22×C4—284no—
459C22×C4—282no—
460C22×C4—284no—
461C22×C4C2×C4×D712yes—
462C22×C4C2×D2812yes—
463C22×C4D4×D711yescharacteristic
464C22×C4D4×D712yes—
465C22×C4#115912yes—
466C22×C4#115912yes—
467C22×C4C2×C22⋊D712yes—
468C22×C4C23×D711yescharacteristic
469SD16—1416no—
470D8—1416no—
471M16—44no—
472M16—148no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—74no—
477C2×C8—142no—
478C2×C8—142no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—74no—
483C4⋊C4—148no—
484C4⋊C4—142no—
485C22⋊C4—148no—
486C22⋊C4—144no—
487C22⋊C4—288no—
488C22⋊C4—288no—
489C42—148no—
490C42—142no—
491C42—142no—
492C42—144no—
493C42—142no—
494C42—142no—
495C2×C7—24no—
496C2×C7—44no—
497C2×C7C2×#38611yescharacteristic
498C2×C7C4×AGL(1,ℤ/8)12yes—
499C2×C7#300612yes—
500C2×C7#303811yescharacteristic
501C2×C7#304011yescharacteristic
502D7—84no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—144no—
511C23—2816no—
512C23—282no—
513C23—284no—
514C23—284no—
515C23—282no—
516C23—288no—
517C23—568no—
518C23—564no—
519C23—564no—
520C23(C22⋊D7)⋊C412yes—
521C23#644512yes—
522C23#650212yes—
523C23C2×D4×D711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8C2×(C2×D7)⋊C414yescomplemented
528D4—216no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4—144no—
533D4—144no—
534D4—282no—
535D4—282no—
536D4—288no—
537D4C2×(C2×D7)⋊C414yescomplemented
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—216no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—74no—
545C2×C4—84no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—144no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—288no—
553C2×C4—2816no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4(C22⋊D7)⋊C414yes—
557C2×C4(C22⋊D7)⋊C412yes—
558C2×C4#644512yes—
559C2×C4#644914yes—
560C2×C4C2×(C2×D7)⋊C414yes—
561C2×C4#650214yes—
562C2×C4#650212yes—
563C2×C4C22×C4×D711yescharacteristic
564C2×C4C22×D2811yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4C2×#115911yescharacteristic
568C2×C4C22×C22⋊D711yescharacteristic
569C8—42no—
570C8—144no—
571C7#2092811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—284no—
582C22—288no—
583C22—5616no—
584C22—564no—
585C22—562no—
586C22—562no—
587C22#8973014yes—
588C22C2×(C22⋊D7)⋊C411yescharacteristic
589C22C2×#644511yescharacteristic
590C22D7×AGL(1,ℤ/8)12yes—
591C22#9046812yes—
592C22C2×#650211yescharacteristic
593C4—24no—
594C4—82no—
595C4—144no—
596C4—282no—
597C4—284no—
598C4#8973014yes—
599C4C2×(C22⋊D7)⋊C411yescharacteristic
600C4C2×#644911yescharacteristic
601C4C22×(C2×D7)⋊C411yescharacteristic
602C4C2×#650211yescharacteristic
603C2—24no—
604C2—44no—
605C2—564no—
606C2#1173631112yes—
607C2#1173632912yes—
608C2C2×#8973011yescharacteristic
609C2#1174137711yescharacteristic
610C2#1174141311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×#6512C2
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#20928
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#41×Dic7C2semidirect
3. C2×C7×#417C2semidirect
4. C2×#89903C2semidirect
5. C2×#89760C2semidirect
6. #11729733C2semidirect
7. #11729312C2semidirect
8. #11727994C2semidirect
9. #11727926C2semidirect
18. C2×#6512C4semidirect
19. C2×Q8×Dic7C22semidirect
22. C2×D4×Dic7C22semidirect
31. C2×C7×#103C22semidirect
32. C2×C7×#102C22semidirect
38. #89345C22semidirect
39. #89324C22semidirect
62. C22×D28C2×C4semidirect
79. C2×#1143C2×C4semidirect
85. C2×#1141C2×C4semidirect
104. C2×#1136C2×C4semidirect
111. C2×#417D7semidirect
129. C7×#41C22⋊C4semidirect
230. C2×#103C2×D7semidirect
232. C2×#102C2×D7semidirect
262. C7×Q8C2×C22⋊C4semidirect
267. C7×D4C2×C22⋊C4semidirect
298. C2×Dic7AGL(1,ℤ/8)semidirect
416. #41(C2×D7)⋊C4semidirect
527. Q8C2×(C2×D7)⋊C4semidirect
537. D4C2×(C2×D7)⋊C4semidirect
571. C7#20928semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic72
C22×#65122
C2×C7×#4172
C2×#899032
C2×#897602
#117297332
#117293122
#117279942
#117279262
#209287