Finite Group Databasebeta

← #13283358

Subgroups — #13283358

Order 1792.

2066 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2C2×#89992C212yesmaximal
3C2×#89722C212yesmaximal, complemented
4C2×#89722C212yesmaximal, complemented
5#11729319C211yesmaximal, characteristic, complemented
6#11729317C211yesmaximal, characteristic, complemented
7#11729302C212yesmaximal, complemented
8#11729299C212yesmaximal, complemented
9#11727166C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C24×Dic7C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2212yes—
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C7×#394C418yescomplemented
23C2×C7×#95C2211yescharacteristic
24#89992—28no—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33#89722—28no—
34#89722—28no—
35#14535—71nomaximal, Sylow
36C25×C7C2312yes—
37C22×C7×D4—24no—
38C22×C7×D4—24no—
39C22×C7×D4—41no—
40C22×C7×D4—41no—
41C22×C7×D4C2×C414yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—24no—
47C23×Dic7—24no—
48C23×Dic7—24no—
49C23×Dic7—42no—
50C23×Dic7—42no—
51C23×Dic7—42no—
52C23×Dic7—42no—
53C23×Dic7C2312yes—
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2312yes—
57C2×C7×C4⋊C4—41no—
58C2×C7×C4⋊C4—41no—
59C2×C7×C4⋊C4C2×C414yescomplemented
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4C2×C418yescomplemented
65C2×C7×C22⋊C4C2×C418yescomplemented
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—42no—
71C2×C22⋊Dic7—42no—
72C2×C22⋊Dic7—42no—
73C2×C22⋊Dic7—42no—
74C2×C22⋊Dic7D418yes—
75C2×C4⋊Dic7—24no—
76C2×C4⋊Dic7—24no—
77C2×C4⋊Dic7—24no—
78C2×C4⋊Dic7—41no—
79C2×C4⋊Dic7—41no—
80C2×C4⋊Dic7—42no—
81C2×C4⋊Dic7—41no—
82C2×C4⋊Dic7D414yescomplemented
83C2×C4×Dic7—22no—
84C2×C4×Dic7—41no—
85C2×C4×Dic7—41no—
86C7×#95C2×C414yescomplemented
87C7⋊#95—24no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—42no—
94C7⋊#95—41no—
95C2×#394D711yescharacteristic
96C2×#386—72no—
97C2×#385—72no—
98C2×#385—72no—
99#1442—72no—
100#1441—71no—
101#1440—72no—
102#1437—72no—
103C4⋊#95—71no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7C22×C414yes—
117C24×C7C2×D414yes—
118C24×C7#4114yes—
119C24×C7C2411yescharacteristic
120C2×C7×D4—216no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C2×C7×D4—48no—
124C2×C7×D4—44no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7C22×C414yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C22×C412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7#4112yes—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—48no—
175C22×Dic7—48no—
176C22×Dic7—48no—
177C22×Dic7—48no—
178C22×Dic7—48no—
179C22×Dic7—48no—
180C22×Dic7—48no—
181C22×Dic7C2×D414yes—
182C22×Dic7C2×D414yes—
183C22×Dic7C2×D414yes—
184C7×C4⋊C4—44no—
185C7×C22⋊C4—216no—
186C7×C22⋊C4—216no—
187C22⋊Dic7—48no—
188C22⋊Dic7—48no—
189C22⋊Dic7—48no—
190C4⋊Dic7—44no—
191C4⋊Dic7—44no—
192C4⋊Dic7—48no—
193C4×Dic7—44no—
194C23×D4C2×D711yescharacteristic
195C24×C4—72no—
196C22×C4⋊C4—71no—
197C22×C4⋊C4—71no—
198C22×C4⋊C4—72no—
199C22×C4⋊C4C2×D711yescharacteristic
200C22×C22⋊C4—72no—
201C22×C22⋊C4—72no—
202C22×C22⋊C4—72no—
203C22×C22⋊C4C2×D712yes—
204C22×C22⋊C4C2×D712yes—
205C22×C42—71no—
206#394Dic718yescomplemented
207#386—148no—
208#385—148no—
209#385—148no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95—71no—
218C2×#95C2×D711yescharacteristic
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—24no—
230C23×C7—24no—
231C23×C7—24no—
232C23×C7—24no—
233C23×C7—24no—
234C23×C7—24no—
235C23×C7—24no—
236C23×C7—24no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7C4×D418yes—
254C23×C7C23×C411yescharacteristic
255C23×C7C22×D412yes—
256C23×C7C22×D411yescharacteristic
257C23×C7C2×#4111yescharacteristic
258C23×C7C2×#4112yes—
259C23×C7H(5,𝔽2)11yescharacteristic
260C23×C7H(5,𝔽2)11yescharacteristic
261C23×C7H(5,𝔽2)11yescharacteristic
262C23×C7H(5,𝔽2)12yes—
263C23×C72−1+411yescharacteristic
264C23×C72−1+412yes—
265C7×D4—416no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7C4×D414yes—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—432no—
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×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C25C22×D712yes—
320C22×D4—24no—
321C22×D4—24no—
322C22×D4—41no—
323C22×D4—41no—
324C22×D4C2×Dic714yes—
325C23×C4—72no—
326C23×C4—72no—
327C23×C4—72no—
328C23×C4—72no—
329C23×C4—144no—
330C23×C4—144no—
331C23×C4—144no—
332C23×C4—282no—
333C23×C4—282no—
334C23×C4—282no—
335C23×C4—282no—
336C23×C4C22×D711yescharacteristic
337C23×C4C22×D712yes—
338C23×C4C22×D711yescharacteristic
339C23×C4C22×D711yescharacteristic
340C2×C4⋊C4—41no—
341C2×C4⋊C4—41no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—144no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—281no—
347C2×C4⋊C4—281no—
348C2×C4⋊C4—281no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4C2×Dic714yescomplemented
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—78no—
356C2×C22⋊C4—144no—
357C2×C22⋊C4—144no—
358C2×C22⋊C4—144no—
359C2×C22⋊C4—144no—
360C2×C22⋊C4—282no—
361C2×C22⋊C4—282no—
362C2×C22⋊C4—282no—
363C2×C22⋊C4—282no—
364C2×C22⋊C4C2×Dic718yescomplemented
365C2×C22⋊C4C2×Dic718yescomplemented
366C2×C42—142no—
367C2×C42—281no—
368C2×C42—281no—
369#95—144no—
370#95—144no—
371#95—144no—
372#95—144no—
373#95—144no—
374#95—144no—
375#95—281no—
376#95—282no—
377#95C2×Dic714yescomplemented
378C22×C7—28no—
379C22×C7—28no—
380C22×C7—28no—
381C22×C7—44no—
382C22×C7—44no—
383C22×C7—44no—
384C22×C7—44no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7—44no—
391C22×C7—44no—
392C22×C7—44no—
393C22×C7C2×C4×D412yes—
394C22×C7C2×C4×D411yescharacteristic
395C22×C7#51811yescharacteristic
396C22×C7#52011yescharacteristic
397C22×C7#52012yes—
398C22×C7#53411yescharacteristic
399C22×C7#53412yes—
400C22×C7#53612yes—
401C22×C7#53711yescharacteristic
402C22×C7#53811yescharacteristic
403C22×C7#53911yescharacteristic
404C22×C7#53912yes—
405C22×C7#53912yes—
406C22×C7D4211yescharacteristic
407C22×C7D4212yes—
408C22×C7#54612yes—
409C22×C7#54712yes—
410C22×C7#54812yes—
411C22×C7#55111yescharacteristic
412C22×C7#55312yes—
413C22×C7H(3,𝔽4)11yescharacteristic
414C22×C7#56212yes—
415C22×C7#56311yescharacteristic
416C4×C7—24no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—44no—
420Dic7—44no—
421Dic7—44no—
422Dic7—44no—
423Dic7—82no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24D4×D714yes—
437C24#115914yes—
438C24C22×Dic714yes—
439C24C23×D711yescharacteristic
440C2×D4—216no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—48no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—74no—
464C22×C4—74no—
465C22×C4—74no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—288no—
492C22×C4—288no—
493C22×C4—288no—
494C22×C4—288no—
495C22×C4—288no—
496C22×C4—288no—
497C22×C4—288no—
498C22×C4D4×D712yes—
499C22×C4#115912yes—
500C22×C4C22×Dic714yes—
501C22×C4C22×Dic712yes—
502C22×C4C22×Dic712yes—
503C22×C4C22×Dic712yes—
504C4⋊C4—44no—
505C4⋊C4—284no—
506C4⋊C4—284no—
507C4⋊C4—288no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—288no—
511C22⋊C4—288no—
512C22⋊C4—288no—
513C42—284no—
514C2×C7—48no—
515C2×C7#232212yes—
516C2×C7#232511yescharacteristic
517C2×C7#235811yescharacteristic
518C2×C7#236112yes—
519C2×C7#237911yescharacteristic
520C2×C7#267911yescharacteristic
521C2×C7#268011yescharacteristic
522C2×C7#268112yes—
523C2×C7#270812yes—
524C2×C7#272811yescharacteristic
525C2×C7#278211yescharacteristic
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23D4×Dic718yes—
561C23C2×D4×D712yes—
562C23C2×D4×D711yescharacteristic
563C23C2×#115912yes—
564C23C2×#115911yescharacteristic
565C23#654811yescharacteristic
566C23#654811yescharacteristic
567C23#655312yes—
568C23#655311yescharacteristic
569C23#655412yes—
570C23#655411yescharacteristic
571C23C23×Dic711yescharacteristic
572D4—416no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—144no—
597C2×C4—144no—
598C2×C4—144no—
599C2×C4—144no—
600C2×C4—144no—
601C2×C4—144no—
602C2×C4—144no—
603C2×C4—144no—
604C2×C4—144no—
605C2×C4—2832no—
606C2×C4—282no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4—282no—
611C2×C4—282no—
612C2×C4—282no—
613C2×C4—282no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—282no—
619C2×C4—282no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4—282no—
625C2×C4D4×Dic714yes—
626C7#1453511yescharacteristic, complemented, Sylow
627C22—28no—
628C22—28no—
629C22—28no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22#9027811yescharacteristic
643C22#9028111yescharacteristic
644C22#9028212yes—
645C22#9028811yescharacteristic
646C22#9028912yes—
647C22#9029311yescharacteristic
648C22#9029611yescharacteristic
649C22#9030012yes—
650C22#9030212yes—
651C22#9030811yescharacteristic
652C22#9030912yes—
653C22#9033912yes—
654C22#9034412yes—
655C22#9035612yes—
656C22#9035612yes—
657C22#9036311yescharacteristic
658C22#9037411yescharacteristic
659C22#9037512yes—
660C22C2×D4×Dic712yes—
661C22C2×D4×Dic711yescharacteristic
662C22#9048911yescharacteristic
663C22#9051811yescharacteristic
664C22#9051812yes—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—284no—
670C4—284no—
671C4—284no—
672C4—562no—
673C2—48no—
674C2#1173444911yescharacteristic
675C2#1173445112yes—
676C2#1173447312yes—
677C2#1173448811yescharacteristic
678C2#1173451411yescharacteristic
679C2#1173458311yescharacteristic
680C2#1174002411yescharacteristic
681C2#1174007911yescharacteristic
682C2#1174008112yes—
683C2#1174028112yes—
684C2#1174040211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#394C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#394D7
hypocenterC7#14535
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
3. C2×#89722C2semidirect
4. C2×#89722C2semidirect
5. #11729319C2semidirect
6. #11729317C2semidirect
7. #11729302C2semidirect
8. #11729299C2semidirect
9. #11727166C2semidirect
22. C7×#394C4semidirect
59. C2×C7×C4⋊C4C2×C4semidirect
64. C2×C7×C22⋊C4C2×C4semidirect
65. C2×C7×C22⋊C4C2×C4semidirect
82. C2×C4⋊Dic7D4semidirect
86. C7×#95C2×C4semidirect
206. #394Dic7semidirect
350. C2×C4⋊C4C2×Dic7semidirect
364. C2×C22⋊C4C2×Dic7semidirect
365. C2×C22⋊C4C2×Dic7semidirect
377. #95C2×Dic7semidirect
626. C7#14535semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3942
C2×#899922
C2×#897222
#117293192
#117293172
#117293022
#117292992
#117271662
#145357