Finite Group Databasebeta

← #13355152

Subgroups — #13355152

Order 1792.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6512C211yesmaximal, characteristic
2C22×#6456C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4#11741531C211yesmaximal, characteristic, complemented
5C2×#89903C211yesmaximal, characteristic, complemented
6#11729733C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11728030C214yesmaximal, complemented
9#11727962C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×D28C2211yescharacteristic
12C23×C4×D7—21no—
13C22×C7×M16C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6512—24no—
16C2×#6512—24no—
17C2×#6512C418yescomplemented
18C22×#1143C2212yes—
19C22×#1141C2212yes—
20C22×(C2×D7)⋊C4—21no—
21C2×#6489C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C22×#1136C2211yescharacteristic
24C2×#6456—24no—
25C2×#6456—24no—
26C2×#6456—22no—
27C2×#6456C2214yes—
28C7×#417—24no—
29C2×C7×#103C2212yescomplemented
30C2×C7×#102C2212yescomplemented
31#90002C2212yescomplemented
32#89993C2212yescomplemented
33#89992—21no—
34#89992—21no—
35C4×C22⋊Dic7—22no—
36#89903—24no—
37C2×#6395C2214yes—
38#89345C2214yescomplemented
39#89324C2214yescomplemented
40#20925—71nomaximal, Sylow
41C24×D7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41C2×C414yescomplemented
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7D411yescharacteristic
55C23×Dic7D411yescharacteristic
56C22×D28—21no—
57C22×D28—22no—
58C22×D28—22no—
59C22×D28—22no—
60C22×D28C2×C414yes—
61C22×D28D414yes—
62C22×D28D414yes—
63C22×D28C2312yes—
64C22×C4×D7—28no—
65C22×C4×D7—24no—
66C22×C4×D7—48no—
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C22×C7×C8C2312yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4D414yes—
74C2×C7×C4⋊C4C2312yes—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78#6512—232no—
79C2×#1143—24no—
80C2×#1143—24no—
81C2×#1143C2×C418yescomplemented
82C2×#1141—24no—
83C2×#1141—24no—
84C2×#1141C2×C418yescomplemented
85C2×(C2×D7)⋊C4—28no—
86C2×(C2×D7)⋊C4—24no—
87#6489—22no—
88#6489—22no—
89C2×C4⋊Dic7C2312yes—
90C2×C4×Dic7—22no—
91C2×C4×Dic7—22no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7D412yes—
94C2×C4×Dic7C2312yes—
95C2×#1136—22no—
96C2×#1136—22no—
97C2×#1136C2×C414yescomplemented
98C22×C7⋊C8C2312yes—
99#6456—216no—
100#6456—28no—
101#6456—28no—
102C7×#103—24no—
103C7×#102—24no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106#6395—28no—
107C22×AGL(1,ℤ/8)—71no—
108C22×C4×D4—71no—
109C2×#417D711yescharacteristic, complemented
110C2×#418—71no—
111#2297—71no—
112#1879—74no—
113#1877—74no—
114#1736—71no—
115#1728—71no—
116C24×C7—21no—
117C24×C7C2×D411yescharacteristic
118C23×D7—24no—
119C23×D7—24no—
120C23×D7—24no—
121C23×D7—22no—
122C23×D7—44no—
123C23×D7—44no—
124C7×#41—28no—
125C7×#41—28no—
126C7×#41—44no—
127C7×#41—44no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8C22×C412yescomplemented
134C2×C7×D4—24no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4C22×C412yescomplemented
143C22×C4×C7—24no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—44no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7#4112yes—
156C22×C4×C7#4112yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7—24no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—42no—
163C22×Dic7C2×D412yes—
164C22×Dic7C2×D412yes—
165C2×D28—28no—
166C2×D28—24no—
167C2×D28—24no—
168C2×D28—28no—
169C2×D28—24no—
170C2×D28—28no—
171C2×D28—22no—
172C2×D28—22no—
173C2×D28—22no—
174C2×D28C22⋊C4116yescomplemented
175C2×D28C22×C414yes—
176C2×D28C2×D414yes—
177C2×D28C2×D414yes—
178C2×C4×D7—216no—
179C2×C4×D7—432no—
180C2×C4×D7—48no—
181C2×C4×D7—48no—
182C7×M16—44no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4C2×D414yes—
189C22⋊Dic7—44no—
190C22⋊Dic7—42no—
191C22⋊Dic7—44no—
192#1143—216no—
193#1141—216no—
194(C2×D7)⋊C4—216no—
195C4⋊Dic7—22no—
196C4⋊Dic7—22no—
197C4×Dic7—24no—
198C4×Dic7—22no—
199C4×Dic7—22no—
200C4×Dic7—42no—
201#1136—28no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8C22×C414yescomplemented
205C22×#41C2×D711yescharacteristic
206C23×D4—71no—
207C24×C4—141no—
208C2×AGL(1,ℤ/8)—78no—
209C2×AGL(1,ℤ/8)—144no—
210C2×AGL(1,ℤ/8)—144no—
211C22×SD16—72no—
212C22×D8—72no—
213C22×M16—71no—
214C22×M16C2×D711yescharacteristic
215C2×C4×D4—74no—
216C2×C4×D4—142no—
217C2×C4×D4—144no—
218C2×C4×D4—144no—
219C2×#117—71no—
220C22×C4⋊C4C2×D711yescharacteristic
221C22×C22⋊C4—141no—
222C22×C42—71no—
223#418—144no—
224#417—24no—
225C2×#103C2×D712yescomplemented
226C2×#102—74no—
227C2×#102C2×D712yescomplemented
228#391—72no—
229#390—72no—
230#386—141no—
231#386—141no—
232C4×C22⋊C4—142no—
233#340—74no—
234#336—74no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7C4×D412yes—
243C23×C7#12012yes—
244C23×C7#12112yes—
245C23×C7C22×D411yescharacteristic
246C22×D7—24no—
247C22×D7—24no—
248C22×D7—24no—
249C22×D7—216no—
250C22×D7—48no—
251C22×D7—416no—
252C22×D7—48no—
253C22×D7—416no—
254C22×D7—44no—
255C22×D7—44no—
256C7×Q8—22no—
257C7×Q8—22no—
258C7×Q8—41no—
259C7×Q8—41no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—48no—
263C7×D4—41no—
264C7×D4—41no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—48no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—44no—
271C2×C4×C7—48no—
272C2×C4×C7C2×C22⋊C414yes—
273C2×C4×C7C4×D412yes—
274C2×C4×C7C4×D414yes—
275C2×C4×C7#12012yes—
276C2×C4×C7#12112yes—
277C2×C4×C7#12114yes—
278C2×C4×C7#12314yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×Dic7—28no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—44no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7AGL(1,ℤ/8)14yescomplemented
296D28—24no—
297D28—24no—
298D28—24no—
299D28—216no—
300D28C2×C22⋊C418yes—
301C4×D7—432no—
302C7×C8—42no—
303C7⋊C8—24no—
304C25—141no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41C4×D714yescomplemented
311C22×Q8C22×D711yescharacteristic
312C22×D4—21no—
313C22×D4—74no—
314C22×D4—74no—
315C22×D4—74no—
316C22×D4—72no—
317C22×D4—142no—
318C22×D4—141no—
319C22×D4—142no—
320C22×D4—142no—
321C22×D4C22×D711yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—71no—
325C23×C4—71no—
326C23×C4—144no—
327C23×C4—148no—
328C23×C4—288no—
329C23×C4C22×D711yescharacteristic
330AGL(1,ℤ/8)—1432no—
331C2×SD16—78no—
332C2×SD16—144no—
333C2×SD16—144no—
334C2×D8—78no—
335C2×D8—144no—
336C2×D8—144no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—74no—
341C2×M16—142no—
342C2×M16—142no—
343C22×C8—72no—
344C22×C8C22×D712yes—
345C4×D4—148no—
346C4×D4—1416no—
347C4×D4—148no—
348#117—142no—
349#117—142no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—72no—
353C2×C4⋊C4C22⋊D714yes—
354C2×C4⋊C4C22×D712yes—
355C2×C22⋊C4—144no—
356C2×C22⋊C4—148no—
357C2×C22⋊C4—142no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—141no—
360C2×C42—72no—
361C2×C42—72no—
362C2×C42—142no—
363C2×C42—142no—
364C2×C42—141no—
365#103—24no—
366#102—24no—
367#102—148no—
368#95—142no—
369#95—142no—
370C22×C7—24no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—24no—
374C22×C7—44no—
375C22×C7—42no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—82no—
379C22×C7#38614yes—
380C22×C7C2×C4×D411yescharacteristic
381C22×C7C2×#12011yescharacteristic
382C22×C7C2×#12111yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)12yes—
384C22×C7#57512yes—
385C2×D7—28no—
386C2×D7—432no—
387C2×D7—48no—
388C2×D7—44no—
389C2×D7—44no—
390C2×D7—48no—
391C4×C7—44no—
392C4×C7—42no—
393C4×C7#38614yes—
394C4×C7C22×C22⋊C411yescharacteristic
395C4×C7C2×C4×D411yescharacteristic
396C4×C7C2×#12111yescharacteristic
397C4×C7C2×#12311yescharacteristic
398Dic7—24no—
399Dic7—42no—
400Dic7—82no—
401C24—21no—
402C24—144no—
403C24—144no—
404C24—144no—
405C24—142no—
406C24—284no—
407C24—284no—
408C24D4×D711yescharacteristic
409#41—28no—
410#41—28no—
411#41—44no—
412#41—44no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D712yescomplemented
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—716no—
428C2×D4—74no—
429C2×D4—74no—
430C2×D4—74no—
431C2×D4—144no—
432C2×D4—148no—
433C2×D4—144no—
434C2×D4—148no—
435C2×D4—144no—
436C2×D4—148no—
437C2×D4—142no—
438C2×D4—142no—
439C2×D4—142no—
440C2×D4C2×C4×D712yescomplemented
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—72no—
449C22×C4—72no—
450C22×C4—1416no—
451C22×C4—144no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—288no—
456C22×C4—2832no—
457C22×C4—288no—
458C22×C4—282no—
459C22×C4C2×C4×D712yes—
460C22×C4#115712yes—
461C22×C4D4×D712yes—
462C22×C4D4×D712yes—
463C22×C4D4×D711yescharacteristic
464C22×C4#115912yes—
465C22×C4C2×C22⋊D712yes—
466C22×C4C23×D711yescharacteristic
467SD16—1416no—
468D8—1416no—
469M16—44no—
470M16—148no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—74no—
475C2×C8—142no—
476C2×C8—142no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—142no—
480C4⋊C4—142no—
481C4⋊C4C2×C22⋊D714yes—
482C22⋊C4—1416no—
483C22⋊C4—284no—
484C22⋊C4—284no—
485C22⋊C4—282no—
486C42—144no—
487C42—142no—
488C42—142no—
489C42—282no—
490C2×C7—24no—
491C2×C7—82no—
492C2×C7C2×#38611yescharacteristic
493C2×C7C4×AGL(1,ℤ/8)12yes—
494C2×C7#300311yescharacteristic
495C2×C7#300511yescharacteristic
496C2×C7#303612yes—
497D7—48no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—144no—
506C23—144no—
507C23—144no—
508C23—1416no—
509C23—288no—
510C23—2816no—
511C23—288no—
512C23—2816no—
513C23—284no—
514C23—284no—
515C23(C22⋊D7)⋊C412yes—
516C23#644712yes—
517C23#650012yes—
518C23C2×D4×D711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523D4—22no—
524D4—22no—
525D4—48no—
526D4—41no—
527D4—41no—
528D4—78no—
529D4—144no—
530D4—144no—
531D4—144no—
532D4—1416no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—48no—
540C2×C4—74no—
541C2×C4—148no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—2832no—
548C2×C4—284no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4D7×C22⋊C414yes—
553C2×C4(C22⋊D7)⋊C412yes—
554C2×C4#644614yes—
555C2×C4#644712yes—
556C2×C4C4×C22⋊D714yes—
557C2×C4#650012yes—
558C2×C4#650114yes—
559C2×C4C22×C4×D711yescharacteristic
560C2×C4C2×#115711yescharacteristic
561C2×C4C2×D4×D711yescharacteristic
562C2×C4C2×D4×D711yescharacteristic
563C2×C4C2×#115911yescharacteristic
564C2×C4C22×C22⋊D711yescharacteristic
565C8—42no—
566C8—144no—
567C7#2092511yescharacteristic, complemented, Sylow
568C22—24no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—42no—
573C22—44no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22—148no—
578C22—2832no—
579C22—288no—
580C22—284no—
581C22—284no—
582C22—288no—
583C22#8972814yes—
584C22C2×(C22⋊D7)⋊C411yescharacteristic
585C22C2×#644711yescharacteristic
586C22D7×AGL(1,ℤ/8)12yes—
587C22#9046812yes—
588C22C2×#650011yescharacteristic
589C4—42no—
590C4—44no—
591C4—144no—
592C4—282no—
593C4—562no—
594C4#8972814yes—
595C4C2×D7×C22⋊C411yescharacteristic
596C4C2×#644611yescharacteristic
597C4C2×C4×C22⋊D711yescharacteristic
598C4C2×#650111yescharacteristic
599C2—24no—
600C2—82no—
601C2—288no—
602C2#1173631112yes—
603C2#1173635512yes—
604C2C2×#8972811yescharacteristic
605C2#1174138911yescharacteristic
606C2#1174142511yescharacteristic
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#20925
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
2. C22×#6456C2semidirect
3. C2×C7×#417C2semidirect
4. #11741531C2semidirect
5. C2×#89903C2semidirect
6. #11729733C2semidirect
7. #11729312C2semidirect
8. #11728030C2semidirect
9. #11727962C2semidirect
17. C2×#6512C4semidirect
29. C2×C7×#103C22semidirect
30. C2×C7×#102C22semidirect
31. #90002C22semidirect
32. #89993C22semidirect
38. #89345C22semidirect
39. #89324C22semidirect
47. C2×C7×#41C2×C4semidirect
81. C2×#1143C2×C4semidirect
84. C2×#1141C2×C4semidirect
97. C2×#1136C2×C4semidirect
109. C2×#417D7semidirect
133. C2×C7×Q8C22×C4semidirect
142. C2×C7×D4C22×C4semidirect
174. C2×D28C22⋊C4semidirect
204. C2×C7⋊C8C22×C4semidirect
225. C2×#103C2×D7semidirect
227. C2×#102C2×D7semidirect
295. C2×Dic7AGL(1,ℤ/8)semidirect
310. C2×#41C4×D7semidirect
418. C2×Q8C2×C4×D7semidirect
440. C2×D4C2×C4×D7semidirect
567. C7#20925semidirect

Maximal subgroups

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

subgroupindex
C22×#65122
C22×#64562
C2×C7×#4172
#117415312
C2×#899032
#117297332
#117293122
#117280302
#117279622
#209257