Finite Group Databasebeta

← #13356352

Subgroups — #13356352

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic, complemented
2C2×#89904C211yesmaximal, characteristic
3#11739658C211yesmaximal, characteristic
4#11739647C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic
6#11729748C211yesmaximal, characteristic, complemented
7C2×#89325C211yesmaximal, characteristic, complemented
8#11728427C214yesmaximal, complemented
9#11728426C214yesmaximal, complemented
10C22×#1157C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×#1136C2211yescharacteristic
15C22×#1136C2211yescharacteristic
16C22×#1136C2211yescharacteristic
17C7×#428—22no—
18C7×#428C2214yes—
19C2×C7×#107—21no—
20C2×C7×#106—21no—
21C2×#6398C2212yes—
22#89904C418yescomplemented
23C2×#6397C2212yes—
24#89776C2212yes—
25#89774—22no—
26#89773C2212yes—
27#89769—22no—
28#89769C2214yes—
29C2×#6384C2212yes—
30#89738—21no—
31#89738—21no—
32#89738—22no—
33#89738—22no—
34C2×#6383C2212yes—
35C4×#1136—22no—
36#89353C2214yes—
37#89325C2214yes—
38#21076—71nomaximal, Sylow
39C23×C4×C7—21no—
40C23×C4×C7C2311yescharacteristic
41C22×C22⋊D7—21no—
42C2×#1157—22no—
43C2×#1157—21no—
44C2×#1157—21no—
45C2×#1157—21no—
46C2×#1157—21no—
47C2×#1157—22no—
48C2×#1157—22no—
49C2×#1157C2×C414yes—
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C22×Dic14C2311yescharacteristic
53C2×C7×M16—21no—
54C2×C7×M16—22no—
55C2×C7×M16C2×C414yes—
56C2×C7×M16C2312yes—
57C22×C7×C8—21no—
58C7×#117—21no—
59C7×#117—21no—
60C7×#117D412yes—
61C7×#117C2312yes—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—21no—
71C2×C42×C7—21no—
72C2×#1136—21no—
73C2×#1136—22no—
74C2×#1136—22no—
75C2×#1136—22no—
76C2×#1136—21no—
77C2×#1136—22no—
78C2×#1136—22no—
79C2×#1136C2×C414yes—
80C2×#1136D412yes—
81C2×#1136D412yes—
82C2×#1136C2312yes—
83C2×#1136C2312yes—
84C22×C7⋊C8—21no—
85C22×C7⋊C8—21no—
86C22×C7⋊C8C2312yes—
87C7×#107—24no—
88C7×#106—24no—
89#6398C2×C418yescomplemented
90#6397C2×C418yescomplemented
91#6384—22no—
92#6384—22no—
93#6384—24no—
94#6383—22no—
95#6383—22no—
96#6383—24no—
97#6377—24no—
98C4×C7⋊C8—24no—
99C2×#428D711yescharacteristic, complemented
100#2896—71no—
101C2×#417—71no—
102C2×#413—71no—
103#2873—71no—
104#1931—74no—
105#1930—74no—
106C2×#343—71no—
107#1738—71no—
108C24×C7#4111yescharacteristic
109C23×D7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—41no—
114C22×C4×C7—41no—
115C22×C4×C7—82no—
116C22×C4×C7C22×C412yes—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7#4111yescharacteristic
122C22×C4×C7#4112yes—
123C22×C4×C7#4112yes—
124C22×C4×C7#4112yes—
125C22×C4×C7C2411yescharacteristic
126C2×C22⋊D7—24no—
127C2×C22⋊D7—41no—
128C2×C22⋊D7—41no—
129C2×C22⋊D7—42no—
130C22×Dic7—21no—
131#1157—24no—
132#1157—24no—
133#1157—24no—
134#1157—24no—
135#1157—42no—
136#1157—42no—
137#1157—42no—
138#1157—42no—
139C2×D28—21no—
140C2×D28—21no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28C22×C412yes—
145C2×C4×D7—24no—
146C2×C4×D7—41no—
147C2×C4×D7—41no—
148C2×C4×D7—42no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14C22×C412yes—
155C7×M16—24no—
156C7×M16—22no—
157C7×M16C22×C414yes—
158C2×C7×C8—24no—
159C2×C7×C8—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D412yes—
168C7×C22⋊C4—22no—
169C7×C22⋊C4—22no—
170C42×C7—22no—
171C42×C7—22no—
172#1136—24no—
173#1136—24no—
174#1136—24no—
175#1136—22no—
176#1136—22no—
177#1136—22no—
178#1136—42no—
179#1136C22×C414yes—
180C2×C7⋊C8—24no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—42no—
186C2×C7⋊C8C2×D414yes—
187C22×#41—71no—
188C22×M16—71no—
189C22×M16—71no—
190C22×M16—71no—
191C22×M16C2×D711yescharacteristic
192C2×#117C2×D711yescharacteristic
193C22×C4⋊C4C2×D711yescharacteristic
194#428—22no—
195#428C2×D714yes—
196C2×#107—21no—
197C2×#106—21no—
198#419—72no—
199#418—72no—
200#417—142no—
201#417—142no—
202#417—141no—
203#417—141no—
204#416—142no—
205C2×#103—72no—
206C2×#102—72no—
207#413—78no—
208C2×#101—72no—
209C2×#100—72no—
210#405—74no—
211#405—142no—
212C4×M16—142no—
213#343—74no—
214#341—74no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7#11712yes—
218C23×C7#12111yescharacteristic
219C23×C7#12111yescharacteristic
220C23×C7#12612yes—
221C23×C7C2×#4111yescharacteristic
222C22×D7—22no—
223C22×D7—41no—
224C22×D7—41no—
225C22×D7—81no—
226C22×D7—81no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—42no—
230C2×C4×C7—81no—
231C2×C4×C7—81no—
232C2×C4×C7—82no—
233C2×C4×C7—82no—
234C2×C4×C7—82no—
235C2×C4×C7#11712yes—
236C2×C4×C7C4×D412yes—
237C2×C4×C7C4×D412yes—
238C2×C4×C7C4×D412yes—
239C2×C4×C7C4×D412yes—
240C2×C4×C7#12111yescharacteristic
241C2×C4×C7#12111yescharacteristic
242C2×C4×C7#12312yes—
243C2×C4×C7#12312yes—
244C2×C4×C7#12412yes—
245C2×C4×C7#12412yes—
246C2×C4×C7#12612yes—
247C2×C4×C7C23×C411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C2×#4111yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C22⋊D7—44no—
254C22⋊D7—44no—
255C2×Dic7—22no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258D28—22no—
259D28—22no—
260D28—41no—
261D28—41no—
262C4×D7—44no—
263C4×D7—44no—
264Dic14—22no—
265Dic14—22no—
266Dic14—41no—
267Dic14—41no—
268C7×C8—24no—
269C7⋊C8—24no—
270C7⋊C8—24no—
271C7⋊C8—42no—
272C2×#41—74no—
273C2×#41—142no—
274C2×#41—142no—
275C2×#41—141no—
276C2×#41—141no—
277C2×#41—142no—
278C2×#41—141no—
279C2×#41—141no—
280C22×Q8—71no—
281C22×D4—71no—
282C22×D4—141no—
283C23×C4—21no—
284C23×C4—141no—
285C23×C4C22×D711yescharacteristic
286C2×M16—22no—
287C2×M16—21no—
288C2×M16—72no—
289C2×M16—72no—
290C2×M16—74no—
291C2×M16—72no—
292C2×M16—72no—
293C2×M16—142no—
294C2×M16—141no—
295C2×M16—142no—
296C2×M16—142no—
297C2×M16—141no—
298C2×M16—142no—
299C2×M16—142no—
300C2×M16C4×D714yes—
301C2×M16C22×D712yes—
302C22×C8—21no—
303C22×C8—72no—
304C22×C8—141no—
305C22×C8—141no—
306#117—21no—
307#117—21no—
308#117C22⋊D712yes—
309#117C22×D712yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C22⋊D711yescharacteristic
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22×D712yes—
318C2×C22⋊C4—21no—
319C2×C42—21no—
320#107—24no—
321#106—24no—
322#103—144no—
323#103—142no—
324#103—142no—
325#102—144no—
326#102—142no—
327#102—142no—
328#101—78no—
329#100—78no—
330#97—144no—
331C4×C8—144no—
332C22×C7—21no—
333C22×C7—21no—
334C22×C7—21no—
335C22×C7—21no—
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7#38812yes—
341C22×C7#38812yes—
342C22×C7C2×#11711yescharacteristic
343C22×C7C2×#12111yescharacteristic
344C22×C7C2×#12611yescharacteristic
345C2×D7—42no—
346C2×D7—82no—
347C2×D7—81no—
348C2×D7—81no—
349C2×D7—82no—
350C4×C7—42no—
351C4×C7—42no—
352C4×C7—82no—
353C4×C7#38812yes—
354C4×C7#38812yes—
355C4×C7C2×C4×D411yescharacteristic
356C4×C7C2×C4×D411yescharacteristic
357C4×C7C2×#12311yescharacteristic
358C4×C7C2×#12411yescharacteristic
359Dic7—42no—
360C24—141no—
361C24#115911yescharacteristic
362#41—144no—
363#41—144no—
364#41—144no—
365#41—144no—
366#41—282no—
367#41—282no—
368#41—282no—
369#41—282no—
370C2×Q8—72no—
371C2×Q8—141no—
372C2×Q8—141no—
373C2×Q8—141no—
374C2×Q8—141no—
375C2×Q8—141no—
376C2×D4—72no—
377C2×D4—144no—
378C2×D4—141no—
379C2×D4—141no—
380C2×D4—141no—
381C2×D4—141no—
382C2×D4—141no—
383C2×D4—282no—
384C2×D4—281no—
385C2×D4—281no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—82no—
392C22×C4—141no—
393C22×C4—144no—
394C22×C4—282no—
395C22×C4—281no—
396C22×C4—281no—
397C22×C4C2×C4×D712yes—
398C22×C4#115712yes—
399C22×C4D4×D711yescharacteristic
400C22×C4D4×D711yescharacteristic
401C22×C4#115911yescharacteristic
402C22×C4#116112yes—
403C22×C4#116112yes—
404C22×C4C2×C22⋊D711yescharacteristic
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407M16—24no—
408M16—22no—
409M16—74no—
410M16—144no—
411M16—144no—
412M16—144no—
413M16—142no—
414M16—142no—
415M16—142no—
416M16—282no—
417M16C2×C4×D714yes—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—74no—
421C2×C8—144no—
422C2×C8—142no—
423C2×C8—142no—
424C2×C8—142no—
425C2×C8—142no—
426C2×C8—282no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4C2×C22⋊D712yes—
434C4⋊C4C2×C22⋊D712yes—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C42—22no—
438C42—22no—
439C2×C7—21no—
440C2×C7—21no—
441C2×C7—22no—
442C2×C7—22no—
443C2×C7C2×#38811yescharacteristic
444C2×C7#306811yescharacteristic
445C2×C7#306911yescharacteristic
446D7—82no—
447C23—41no—
448C23—41no—
449C23—142no—
450C23—281no—
451C23—281no—
452C23—561no—
453C23—561no—
454C23#645512yes—
455C23#646112yes—
456C23#650211yescharacteristic
457C23#650211yescharacteristic
458C23C2×#115911yescharacteristic
459Q8—142no—
460Q8—142no—
461Q8—281no—
462Q8—281no—
463D4—142no—
464D4—142no—
465D4—284no—
466D4—284no—
467D4—281no—
468D4—281no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—142no—
478C2×C4—281no—
479C2×C4—281no—
480C2×C4—284no—
481C2×C4—284no—
482C2×C4#645512yes—
483C2×C4#645612yes—
484C2×C4#645612yes—
485C2×C4#645712yes—
486C2×C4#645712yes—
487C2×C4#646112yes—
488C2×C4C4×C22⋊D712yes—
489C2×C4C4×C22⋊D712yes—
490C2×C4#650211yescharacteristic
491C2×C4#650211yescharacteristic
492C2×C4#651012yes—
493C2×C4#651012yes—
494C2×C4C22×C4×D711yescharacteristic
495C2×C4C2×#115711yescharacteristic
496C2×C4C2×D4×D711yescharacteristic
497C2×C4C2×#116111yescharacteristic
498C2×C4C2×#116111yescharacteristic
499C2×C4C22×C22⋊D711yescharacteristic
500C8—24no—
501C8—144no—
502C8—144no—
503C8—282no—
504C7#2107611yescharacteristic, complemented, Sylow
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—282no—
514C22—562no—
515C22—562no—
516C22—561no—
517C22—561no—
518C22#8976212yes—
519C22#8976212yes—
520C22C2×#645511yescharacteristic
521C22C2×#646111yescharacteristic
522C22C2×#650211yescharacteristic
523C4—42no—
524C4—42no—
525C4—82no—
526C4—282no—
527C4#8976212yes—
528C4#8976212yes—
529C4C2×#645611yescharacteristic
530C4C2×#645711yescharacteristic
531C4C2×C4×C22⋊D711yescharacteristic
532C4C2×#651011yescharacteristic
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—562no—
538C2C2×#8976211yescharacteristic
539C2#1174138811yescharacteristic
540C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#428C2
socleC22×C7C2×#121
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenterC2×#428D7
hypocenterC7#21076
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×C7×#428C2semidirect
4. #11739647C2semidirect
6. #11729748C2semidirect
7. C2×#89325C2semidirect
8. #11728427C2semidirect
9. #11728426C2semidirect
22. #89904C4semidirect
89. #6398C2×C4semidirect
90. #6397C2×C4semidirect
99. C2×#428D7semidirect
504. C7#21076semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×#899042
#117396582
#117396472
C2×#897382
#117297482
C2×#893252
#117284272
#117284262
#210767