Finite Group Databasebeta

← #13281017

Subgroups — #13281017

Order 1792.

1782 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89759C211yesmaximal, characteristic
2C2×#89439C211yesmaximal, characteristic
3C2×D7×#95C211yesmaximal, characteristic
4C7×#1442C211yesmaximal, characteristic, complemented
5#11729321C211yesmaximal, characteristic, complemented
6(C2×D7)⋊#95C212yesmaximal
7(C2×D7)⋊#95C212yesmaximal
8#11727169C211yesmaximal, characteristic, complemented
9#11727168C212yesmaximal, complemented
10#11727167C212yesmaximal, complemented
11#11727166C211yesmaximal, characteristic, complemented
12C23×C4×D7C2212yes—
13C23×C4×D7C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2211yescharacteristic
20C22×Dic7⋊C4C2211yescharacteristic
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7×#95C2211yescharacteristic
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35#89759—24no—
36#89439—24no—
37D7×#95C418yescomplemented
38#14347—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C22×C4×D7—24no—
51C22×C4×D7—24no—
52C22×C4×D7—42no—
53C22×C4×D7—42no—
54C22×C4×D7—42no—
55C22×C4×D7—42no—
56C22×C4×D7—42no—
57C22×C4×D7—42no—
58C22×C4×D7C2×C418yes—
59C22×C4×D7C2×C414yes—
60C22×C4×D7D414yes—
61C22×C4×D7Q814yes—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—42no—
64C2×(C2×D7)⋊C4—22no—
65C2×(C2×D7)⋊C4—22no—
66C2×(C2×D7)⋊C4—24no—
67C2×(C2×D7)⋊C4—24no—
68C2×(C2×D7)⋊C4—24no—
69C2×(C2×D7)⋊C4—24no—
70C2×(C2×D7)⋊C4—24no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—42no—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—42no—
76C2×Dic7⋊C4—42no—
77C7×#95—22no—
78C7×#95—42no—
79C7×#95—42no—
80C7×#95C2×C414yes—
81C7⋊#95—22no—
82C7⋊#95—24no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95—42no—
89C7⋊#95C2×C414yes—
90C7⋊#95C2×C418yes—
91C2×#380—71no—
92C2×#380—71no—
93C22×#95—71no—
94#1442—71no—
95#1442—72no—
96#1442—72no—
97#1442—71no—
98#1442—71no—
99#1442D711yescharacteristic, complemented
100C22⋊#95—72no—
101C22⋊#95—72no—
102C24×C7C2411yescharacteristic
103C23×D7—22no—
104C23×D7—22no—
105C23×D7—21no—
106C23×D7—21no—
107C23×D7—21no—
108C23×D7—21no—
109C23×D7C22×C412yes—
110C23×D7C2×D412yes—
111C23×D7C2×Q812yes—
112C23×D7#4114yes—
113C23×D7#4114yes—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C414yes—
140C22×C4×C7C22×C412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×Q812yes—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7C22×C414yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7C2×Q812yes—
172C2×C4×D7—216no—
173C2×C4×D7—48no—
174C2×C4×D7—48no—
175C2×C4×D7—48no—
176C2×C4×D7—44no—
177C2×C4×D7—44no—
178C2×C4×D7—48no—
179C2×C4×D7—48no—
180C2×C4×D7—48no—
181C2×C4×D7—48no—
182C2×C4×D7—48no—
183C2×C4×D7C4⋊C4116yes—
184C7×C4⋊C4—44no—
185(C2×D7)⋊C4—44no—
186(C2×D7)⋊C4—44no—
187(C2×D7)⋊C4—48no—
188C4⋊Dic7—44no—
189Dic7⋊C4—44no—
190Dic7⋊C4—44no—
191C24×C4—71no—
192C24×C4—72no—
193C22×C4⋊C4—71no—
194C22×C4⋊C4—71no—
195C22×C4⋊C4—71no—
196C22×C4⋊C4C2×D711yescharacteristic
197C22×C22⋊C4—71no—
198C22×C22⋊C4—71no—
199C22×C22⋊C4—72no—
200#380—144no—
201#380—144no—
202C2×#95—72no—
203C2×#95—71no—
204C2×#95—78no—
205C2×#95—72no—
206C2×#95—71no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95C2×D711yescharacteristic
214C2×#95C2×D712yes—
215C2×#95C2×D712yes—
216C2×#95C2×D711yescharacteristic
217C23×C7C23×C411yescharacteristic
218C23×C7C22×D411yescharacteristic
219C23×C7C22×Q811yescharacteristic
220C23×C7C2×#4112yes—
221C23×C7C2×#4112yes—
222C23×C7H(5,𝔽2)12yes—
223C23×C7H(5,𝔽2)11yescharacteristic
224C23×C7H(5,𝔽2)12yes—
225C23×C7H(5,𝔽2)11yescharacteristic
226C23×C72−1+411yescharacteristic
227C23×C72−1+411yescharacteristic
228C22×D7—22no—
229C22×D7—24no—
230C22×D7—22no—
231C22×D7—22no—
232C22×D7—22no—
233C22×D7—22no—
234C22×D7—22no—
235C22×D7—24no—
236C22×D7—24no—
237C22×D7—24no—
238C22×D7—24no—
239C22×D7—24no—
240C22×D7—24no—
241C22×D7—24no—
242C22×D7—24no—
243C22×D7—24no—
244C22×D7—22no—
245C22×D7—22no—
246C22×D7C2×C4⋊C414yes—
247C22×D7#11718yes—
248C22×D7#12414yes—
249C22×D7#12414yes—
250C22×D7#12518yes—
251C2×C4×C7—24no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7C2×C4⋊C414yes—
280C2×Dic7—24no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—42no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—42no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7C2×C4⋊C414yes—
309C4×D7—416no—
310C4×D7—432no—
311C25—71no—
312C23×C4—71no—
313C23×C4—74no—
314C23×C4—74no—
315C23×C4—74no—
316C23×C4—72no—
317C23×C4—78no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—72no—
321C23×C4—144no—
322C23×C4—144no—
323C23×C4—282no—
324C23×C4—282no—
325C23×C4—282no—
326C23×C4—282no—
327C23×C4—282no—
328C23×C4—282no—
329C23×C4C22×D711yescharacteristic
330C23×C4C22×D711yescharacteristic
331C23×C4C22×D712yes—
332C23×C4C22×D712yes—
333C23×C4C22×D711yescharacteristic
334C2×C4⋊C4—22no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—282no—
340C2×C4⋊C4—282no—
341C2×C4⋊C4—282no—
342C2×C22⋊C4—144no—
343C2×C22⋊C4—144no—
344C2×C22⋊C4—144no—
345C2×C22⋊C4—144no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4—142no—
348C2×C22⋊C4—144no—
349#95—22no—
350#95—42no—
351#95—42no—
352#95—78no—
353#95—74no—
354#95—144no—
355#95—142no—
356#95—282no—
357#95—282no—
358#95—282no—
359#95—282no—
360#95—282no—
361#95—282no—
362#95C4×D714yes—
363C22×C7C22×C4⋊C411yescharacteristic
364C22×C7C2×#11712yes—
365C22×C7#51811yescharacteristic
366C22×C7#51812yes—
367C22×C7#51911yescharacteristic
368C22×C7C2×#12411yescharacteristic
369C22×C7C2×#12411yescharacteristic
370C22×C7C2×#12512yes—
371C22×C7#53411yescharacteristic
372C22×C7#53411yescharacteristic
373C22×C7#53711yescharacteristic
374C22×C7#53711yescharacteristic
375C22×C7#53812yes—
376C22×C7#53812yes—
377C22×C7#53812yes—
378C22×C7#53812yes—
379C22×C7#53912yes—
380C22×C7#53912yes—
381C22×C7#53912yes—
382C22×C7#53912yes—
383C22×C7#54312yes—
384C22×C7#54411yescharacteristic
385C22×C7#54411yescharacteristic
386C2×D7—28no—
387C2×D7—28no—
388C2×D7—24no—
389C2×D7—24no—
390C2×D7—24no—
391C2×D7—24no—
392C2×D7—24no—
393C2×D7—24no—
394C2×D7—28no—
395C2×D7—28no—
396C2×D7#38218yes—
397C4×C7—24no—
398C4×C7—44no—
399C4×C7—82no—
400C4×C7—81no—
401C4×C7—81no—
402Dic7—24no—
403Dic7—44no—
404Dic7—81no—
405Dic7—81no—
406Dic7—82no—
407C24—72no—
408C24—72no—
409C24—72no—
410C24—74no—
411C24—74no—
412C24—142no—
413C24—141no—
414C24—141no—
415C24—141no—
416C24—142no—
417C24—141no—
418C24C23×D711yescharacteristic
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—716no—
448C22×C4—74no—
449C22×C4—1416no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—141no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—141no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—141no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—141no—
475C22×C4—288no—
476C22×C4—284no—
477C22×C4—288no—
478C22×C4—284no—
479C22×C4—288no—
480C22×C4—288no—
481C22×C4—288no—
482C22×C4—288no—
483C22×C4—288no—
484C22×C4—288no—
485C22×C4C2×C4×D714yes—
486C22×C4C2×C4×D712yes—
487C22×C4D4×D712yes—
488C22×C4Q8×D712yes—
489C4⋊C4—44no—
490C4⋊C4—284no—
491C4⋊C4—284no—
492C4⋊C4—284no—
493C22⋊C4—288no—
494C22⋊C4—284no—
495C22⋊C4—284no—
496C2×C7C2×#38211yescharacteristic
497C2×C7#231212yes—
498C2×C7#231312yes—
499C2×C7#231511yescharacteristic
500C2×C7#231611yescharacteristic
501C2×C7#263911yescharacteristic
502C2×C7#263911yescharacteristic
503C2×C7#263911yescharacteristic
504C2×C7#263911yescharacteristic
505C2×C7#264612yes—
506C2×C7#264612yes—
507D7—28no—
508C23—78no—
509C23—74no—
510C23—78no—
511C23—74no—
512C23—74no—
513C23—144no—
514C23—142no—
515C23—144no—
516C23—144no—
517C23—144no—
518C23—142no—
519C23—144no—
520C23—142no—
521C23—142no—
522C23—142no—
523C23—142no—
524C23—144no—
525C23—142no—
526C23—144no—
527C23—144no—
528C23—142no—
529C23—144no—
530C23—144no—
531C23C22×C4×D711yescharacteristic
532C23C2×D4×D711yescharacteristic
533C23#654812yes—
534C23#654811yescharacteristic
535C23C2×Q8×D711yescharacteristic
536C23#655111yescharacteristic
537C23D7×#4112yes—
538C23D7×#4112yes—
539C23#655311yescharacteristic
540C23#655312yes—
541C23#655411yescharacteristic
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—74no—
571C2×C4—144no—
572C2×C4—142no—
573C2×C4—144no—
574C2×C4—142no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—144no—
578C2×C4—2816no—
579C2×C4—2832no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4—281no—
596C2×C4—282no—
597C2×C4—282no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—282no—
601C2×C4D7×C4⋊C414yes—
602C7#1434711yescharacteristic, complemented, Sylow
603C22—78no—
604C22—148no—
605C22—148no—
606C22—148no—
607C22—144no—
608C22—144no—
609C22—144no—
610C22—144no—
611C22—148no—
612C22—144no—
613C22—144no—
614C22C2×D7×C4⋊C411yescharacteristic
615C22D7×#11712yes—
616C22#9023612yes—
617C22#9023611yescharacteristic
618C22#9026311yescharacteristic
619C22#9028011yescharacteristic
620C22#9029911yescharacteristic
621C22#9030412yes—
622C22#9032011yescharacteristic
623C22#9032512yes—
624C22#9032912yes—
625C22#9034411yescharacteristic
626C22#9034812yes—
627C22#9035112yes—
628C22D7×#12411yescharacteristic
629C22D7×#12411yescharacteristic
630C22#9036912yes—
631C22D7×#12512yes—
632C22#9038011yescharacteristic
633C22#9038312yes—
634C22#9039512yes—
635C22#9039812yes—
636C22#9042011yescharacteristic
637C4—24no—
638C4—44no—
639C4—82no—
640C4—81no—
641C4—81no—
642C4—144no—
643C4—284no—
644C4—561no—
645C4—561no—
646C4—562no—
647C2—148no—
648C2D7×#38211yescharacteristic
649C2#1173203611yescharacteristic
650C2#1173274112yes—
651C2#1173327012yes—
652C2#1173364011yescharacteristic
653C2#1173431711yescharacteristic
654C2#1173457111yescharacteristic
655C2#1173513211yescharacteristic
656C2#1173541012yes—
657C2#1173580611yescharacteristic
658C2#1173580812yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1442C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1442D7
hypocenterC7#14347
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. C7×#1442C2semidirect
5. #11729321C2semidirect
8. #11727169C2semidirect
9. #11727168C2semidirect
10. #11727167C2semidirect
11. #11727166C2semidirect
37. D7×#95C4semidirect
99. #1442D7semidirect
602. C7#14347semidirect

Maximal subgroups

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

subgroupindex
C2×#897592
C2×#894392
C2×D7×#952
C7×#14422
#117293212
(C2×D7)⋊#952
#117271692
#117271682
#117271672
#117271662
#143477