Finite Group Databasebeta

← #13274996

Subgroups — #13274996

Order 1792.

2110 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C2×C42×Dic7C211yesmaximal, characteristic
3C2×#89418C212yesmaximal
4C2×#89421C211yesmaximal, characteristic
5C7×#1436C211yesmaximal, characteristic
6#11729318C211yesmaximal, characteristic
7#11729290C211yesmaximal, characteristic
8#11727159C211yesmaximal, characteristic
9Dic7⋊#95C212yesmaximal
10Dic7×#95C212yesmaximal
11Dic7×#95C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×Dic7⋊C4C2211yescharacteristic
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C2×#6450—24no—
26C2×#6450—24no—
27C2×#6450C418yescomplemented
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C42×Dic7—24no—
39#89421—24no—
40#89418C4116yescomplemented
41C4×#391—71nomaximal, Sylow
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2311yescharacteristic
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14C2×C418yes—
57C22×Dic14C2×C414yes—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4C2×C414yes—
61C2×C42×C7—22no—
62C2×C42×C7—24no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4C2×C418yes—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—24no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7C2×C418yes—
86C2×C4×Dic7C2×C414yes—
87C2×C4×Dic7C2×C414yes—
88C2×C4×Dic7C2×C414yes—
89C2×C4×Dic7D414yes—
90C2×C4×Dic7D414yes—
91C2×C4×Dic7Q818yescomplemented
92#6450—232no—
93C7×#95—22no—
94C7×#95—24no—
95C7×#95C2×C418yes—
96C7⋊#95—22no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95—24no—
100C7⋊#95C2×C418yes—
101C7⋊#95C2×C418yes—
102C7⋊#95C2×C418yes—
103C22×C4×Q8—71no—
104C2×#391—72no—
105C2×C4×C4⋊C4—71no—
106C2×C43—71no—
107#1436—71no—
108#1436—71no—
109#1436—71no—
110#1436D711yescharacteristic
111C4⋊#95—72no—
112C4×#95—72no—
113C4×#95—72no—
114C24×C7C2411yescharacteristic
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7C22×C414yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C22×C414yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C414yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×Q814yes—
143C22×C4×C7#4114yes—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7C22×C414yes—
162C22×Dic7C22×C412yes—
163C22×Dic7C22×C414yes—
164C22×Dic7C22×C412yes—
165C22×Dic7C22×C414yes—
166C22×Dic7C22×C412yes—
167C22×Dic7C22×C414yes—
168C22×Dic7C22×C412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×Q814yes—
174C22×Dic7C2×Q814yes—
175C22×Dic7#4114yes—
176C22×Dic7#4114yes—
177C22×Dic7#4114yes—
178C2×Dic14—28no—
179C2×Dic14—28no—
180C2×Dic14—216no—
181C2×Dic14—216no—
182C2×Dic14C42116yescomplemented
183C7×C4⋊C4—28no—
184C42×C7—28no—
185Dic7⋊C4—28no—
186Dic7⋊C4—28no—
187Dic7⋊C4—216no—
188C4×Dic7—28no—
189C4×Dic7—28no—
190C4×Dic7—28no—
191C4×Dic7—216no—
192C4×Dic7—216no—
193C4×Dic7—48no—
194C4×Dic7C22⋊C4116yes—
195C23×Q8—71no—
196C2×C4×Q8—78no—
197C2×C4×Q8—144no—
198C2×C4×Q8—144no—
199C22×C4⋊C4—71no—
200C22×C4⋊C4—71no—
201C22×C4⋊C4—72no—
202C22×C4⋊C4C2×D711yescharacteristic
203C22×C42—71no—
204C22×C42—71no—
205C22×C42—71no—
206C22×C42—72no—
207C22×C42—72no—
208C22×C42—72no—
209C22×C42—71no—
210C22×C42C2×D711yescharacteristic
211#391—716no—
212C4×C4⋊C4—144no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95—71no—
220C2×#95C2×D712yes—
221C2×#95C2×D711yescharacteristic
222C2×#95C2×D712yes—
223C43—144no—
224C23×C7C23×C411yescharacteristic
225C23×C7C23×C412yes—
226C23×C7C22×D411yescharacteristic
227C23×C7C22×D411yescharacteristic
228C23×C7C22×Q812yes—
229C23×C7C2×#4112yes—
230C23×C7C2×#4112yes—
231C23×C7C2×#4112yes—
232C23×C7C2×#4112yes—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7C2×C4214yes—
260C2×C4×C7C2×C22⋊C414yes—
261C2×C4×C7C4×Q818yescomplemented
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—22no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—24no—
276C2×Dic7—24no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7C2×C4214yes—
285C2×Dic7C2×C4214yes—
286C2×Dic7C2×C22⋊C414yes—
287C2×Dic7C2×C22⋊C414yes—
288C2×Dic7#11714yes—
289C2×Dic7C4×D418yes—
290C2×Dic7C4×D418yes—
291C2×Dic7C4×Q818yes—
292C2×Dic7C4×Q818yescomplemented
293C2×Dic7C22⋊Q818yes—
294C2×Dic7#12414yes—
295C2×Dic7C4⋊Q814yes—
296Dic14—232no—
297C22×Q8—74no—
298C22×Q8—78no—
299C22×Q8—142no—
300C22×Q8—142no—
301C22×Q8—144no—
302C23×C4—71no—
303C23×C4—71no—
304C23×C4—71no—
305C23×C4—72no—
306C23×C4—71no—
307C23×C4—72no—
308C23×C4C22×D711yescharacteristic
309C23×C4C22×D712yes—
310C23×C4C22×D711yescharacteristic
311C23×C4C22×D711yescharacteristic
312C23×C4C22×D712yes—
313C4×Q8—1432no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—74no—
317C2×C4⋊C4—74no—
318C2×C4⋊C4—78no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—144no—
322C2×C4⋊C4—144no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—142no—
325C2×C4⋊C4C4×D714yes—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—74no—
329C2×C42—74no—
330C2×C42—74no—
331C2×C42—74no—
332C2×C42—74no—
333C2×C42—78no—
334C2×C42—78no—
335C2×C42—142no—
336C2×C42—142no—
337C2×C42—142no—
338C2×C42—142no—
339C2×C42—144no—
340C2×C42—144no—
341C2×C42—144no—
342C2×C42—144no—
343C2×C42—144no—
344C2×C42—142no—
345C2×C42—144no—
346C2×C42—144no—
347C2×C42—144no—
348#95—22no—
349#95—24no—
350#95—78no—
351#95—78no—
352#95—78no—
353#95—144no—
354#95—144no—
355#95—142no—
356#95—144no—
357#95C4×D718yes—
358C22×C7C22×C4211yescharacteristic
359C22×C7C22×C22⋊C411yescharacteristic
360C22×C7C2×#11711yescharacteristic
361C22×C7C2×C4×D412yes—
362C22×C7C2×C4×D412yes—
363C22×C7C2×C4×Q812yes—
364C22×C7C2×C4×Q812yes—
365C22×C7C2×C4×Q812yes—
366C22×C7C4×#4112yes—
367C22×C7C4×#4112yes—
368C22×C7C4×#4112yes—
369C22×C7C2×C22⋊Q812yes—
370C22×C7#52511yescharacteristic
371C22×C7#52511yescharacteristic
372C22×C7C2×#12411yescharacteristic
373C22×C7#52912yes—
374C22×C7#52912yes—
375C22×C7C2×C4⋊Q811yescharacteristic
376C22×C7#53212yes—
377C22×C7#53312yes—
378C22×C7#53312yes—
379C4×C7—24no—
380C4×C7—24no—
381C4×C7—28no—
382C4×C7—42no—
383C4×C7—44no—
384Dic7—24no—
385Dic7—28no—
386Dic7—42no—
387Dic7—44no—
388Dic7C4×C22⋊C418yes—
389Dic7#39118yescomplemented
390C24C23×D711yescharacteristic
391C2×Q8—716no—
392C2×Q8—148no—
393C2×Q8—148no—
394C2×Q8—1416no—
395C2×Q8—1416no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—72no—
416C22×C4—72no—
417C22×C4—74no—
418C22×C4—72no—
419C22×C4—72no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—74no—
423C22×C4—74no—
424C22×C4—74no—
425C22×C4—74no—
426C22×C4—74no—
427C22×C4—72no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—74no—
431C22×C4—72no—
432C22×C4—142no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4C2×C4×D714yes—
450C22×C4C2×C4×D712yes—
451C22×C4C2×C4×D714yes—
452C22×C4C2×C4×D712yes—
453C22×C4C2×C4×D712yes—
454C22×C4C2×C4×D714yes—
455C22×C4D4×D712yes—
456C22×C4D4×D712yes—
457C22×C4#115914yes—
458C22×C4Q8×D714yes—
459C4⋊C4—28no—
460C4⋊C4—1416no—
461C4⋊C4—148no—
462C4⋊C4—148no—
463C42—28no—
464C42—716no—
465C42—1416no—
466C42—1416no—
467C42—148no—
468C42—148no—
469C42—148no—
470C42—288no—
471C2×C7C2×C4×C22⋊C411yescharacteristic
472C2×C7C42×Q812yes—
473C2×C7C2×#39111yescharacteristic
474C2×C7#229512yes—
475C2×C7#229711yescharacteristic
476C2×C7C4×C22⋊Q812yes—
477C2×C7C4×C22⋊Q812yes—
478C2×C7C4×#12412yes—
479C2×C7C4×C4⋊Q812yes—
480C23C22×C4×D711yescharacteristic
481C23C22×C4×D712yes—
482C23C2×D4×D711yescharacteristic
483C23C2×D4×D711yescharacteristic
484C23C2×#115912yes—
485C23C2×Q8×D712yes—
486C23D7×#4112yes—
487C23D7×#4112yes—
488C23D7×#4112yes—
489Q8—1432no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—74no—
517C2×C4—74no—
518C2×C4—74no—
519C2×C4—74no—
520C2×C4—74no—
521C2×C4—78no—
522C2×C4—78no—
523C2×C4—78no—
524C2×C4—78no—
525C2×C4—74no—
526C2×C4—74no—
527C2×C4—78no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—142no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—282no—
544C2×C4—281no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—281no—
549C2×C4—282no—
550C2×C4C42×D714yes—
551C2×C4D7×C22⋊C414yes—
552C2×C4#645018yescomplemented
553C7C4×#39111yescharacteristic, complemented, Sylow
554C22C2×C42×D711yescharacteristic
555C22C2×D7×C22⋊C411yescharacteristic
556C22C2×#645012yes—
557C22D7×#11711yescharacteristic
558C22#9021312yes—
559C22C4×#115912yes—
560C22C4×#115912yes—
561C22C4×D4×D712yes—
562C22C4×D4×D712yes—
563C22C4×Q8×D712yes—
564C22C4×Q8×D712yes—
565C22#9027711yescharacteristic
566C22#9028712yes—
567C22#9031312yes—
568C22D7×C22⋊Q812yes—
569C22#9034012yes—
570C22#9034711yescharacteristic
571C22D7×#12411yescharacteristic
572C22#9037912yes—
573C22#9039712yes—
574C22D7×C4⋊Q811yescharacteristic
575C4—24no—
576C4—28no—
577C4—24no—
578C4—42no—
579C4—44no—
580C4—78no—
581C4—78no—
582C4—144no—
583C4—148no—
584C4—282no—
585C4—284no—
586C2C4×D7×C22⋊C411yescharacteristic
587C2C4×#645012yes—
588C2#1173182612yes—
589C2#1173210311yescharacteristic
590C2#1173238312yes—
591C2#1173238612yes—
592C2#1173289912yes—
593C2#1173341212yes—
594C2D7×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC22×C7C22×C42
FittingC7×#1436C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1436D7
hypocenterC7C4×#391
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
27. C2×#6450C4semidirect
40. #89418C4semidirect
91. C2×C4×Dic7Q8semidirect
182. C2×Dic14C42semidirect
261. C2×C4×C7C4×Q8semidirect
292. C2×Dic7C4×Q8semidirect
389. Dic7#391semidirect
552. C2×C4#6450semidirect
553. C7C4×#391semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C2×C42×Dic72
C2×#894182
C2×#894212
C7×#14362
#117293182
#117292902
#117271592
Dic7⋊#952
Dic7×#952
C4×#3917