Finite Group Databasebeta

← #13285770

Subgroups — #13285770

Order 1792.

1982 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6509C211yesmaximal, characteristic
2C2×C7×#395C211yesmaximal, characteristic, complemented
3C2×#89761C211yesmaximal, characteristic
4C2×#89745C212yesmaximal, complemented
5C2×#89438C211yesmaximal, characteristic
6#11729888C211yesmaximal, characteristic, complemented
7#11729324C212yesmaximal
8#11729319C211yesmaximal, characteristic, complemented
9#11727181C212yesmaximal
10#11727173C211yesmaximal, characteristic, complemented
11#11727160C212yesmaximal, complemented
12C23×C7×Q8C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2212yes—
16C2×#6509—24no—
17C2×#6509—28no—
18C22×(C2×D7)⋊C4C2212yes—
19C22×(C2×D7)⋊C4C2211yescharacteristic
20C22×(C2×D7)⋊C4C2212yes—
21C22×(C2×D7)⋊C4C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×C4⋊Dic7C2212yes—
24C22×Dic7⋊C4C2211yescharacteristic
25C22×Dic7⋊C4C2212yes—
26C22×C4×Dic7C2212yes—
27C7×#395—24no—
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2211yescharacteristic
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89761—24no—
39#89745—28no—
40#89438—24no—
41#14763—71nomaximal, Sylow
42C24×D7C2311yescharacteristic
43C22×C7×Q8—24no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2311yescharacteristic
57C23×Dic7C2311yescharacteristic
58C22×C4×D7—22no—
59C22×C4×D7—22no—
60C22×C4×D7—24no—
61C22×C4×D7—42no—
62C22×C4×D7D414yes—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—24no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—42no—
69#6509—232no—
70C2×(C2×D7)⋊C4—24no—
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—24no—
73C2×(C2×D7)⋊C4—24no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—24no—
76C2×(C2×D7)⋊C4—24no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—42no—
80C2×(C2×D7)⋊C4Q818yescomplemented
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—24no—
83C2×C4⋊Dic7—24no—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—24no—
86C2×Dic7⋊C4—24no—
87C2×Dic7⋊C4—24no—
88C2×Dic7⋊C4—24no—
89C2×Dic7⋊C4—22no—
90C2×Dic7⋊C4D418yes—
91C2×C4×Dic7—24no—
92C2×C4×Dic7—24no—
93C2×C4×Dic7—42no—
94C7×#95—24no—
95C7×#95D414yes—
96C7⋊#95—22no—
97C7⋊#95—24no—
98C7⋊#95—22no—
99C7⋊#95—24no—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—22no—
103C7⋊#95—42no—
104C22×C22⋊Q8—71no—
105C2×#395D711yescharacteristic, complemented
106C2×#389—72no—
107C2×#385—71no—
108C2×C4×C22⋊C4—71no—
109#1446—71no—
110#1444—71no—
111#1441—71no—
112#1439—72no—
113#1439—72no—
114#1436—72no—
115C24×C7C2411yescharacteristic
116C23×D7—22no—
117C23×D7—21no—
118C23×D7—22no—
119C23×D7—21no—
120C23×D7—21no—
121C23×D7—21no—
122C23×D7—22no—
123C23×D7—21no—
124C23×D7—21no—
125C23×D7C2×D412yes—
126C23×D7C2×Q814yes—
127C2×C7×Q8—28no—
128C2×C7×Q8—48no—
129C2×C7×Q8—48no—
130C2×C7×Q8—44no—
131C2×C7×Q8—44no—
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—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×Q814yes—
160C22×C4×C7#4112yes—
161C22×C4×C7#4112yes—
162C22×C4×C7#4114yes—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—21no—
180C22×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—21no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—21no—
186C22×Dic7—21no—
187C22×Dic7C2×D414yes—
188C22×Dic7C2×D412yes—
189C22×Dic7C2×D414yes—
190C22×Dic7C2×Q814yes—
191C22×Dic7#4112yes—
192C2×C4×D7—28no—
193C2×C4×D7—28no—
194C2×C4×D7—44no—
195C2×C4×D7—48no—
196C2×C4×D7—48no—
197C7×C4⋊C4—48no—
198C7×C4⋊C4—44no—
199(C2×D7)⋊C4—28no—
200(C2×D7)⋊C4—216no—
201(C2×D7)⋊C4—216no—
202(C2×D7)⋊C4—44no—
203C4⋊Dic7—216no—
204C4⋊Dic7—28no—
205Dic7⋊C4—216no—
206Dic7⋊C4—44no—
207C4×Dic7—48no—
208C23×Q8C2×D711yescharacteristic
209C24×C4—71no—
210C2×C22⋊Q8—148no—
211C2×C22⋊Q8—144no—
212C22×C4⋊C4—72no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4C2×D711yescharacteristic
217C22×C4⋊C4C2×D712yes—
218C22×C22⋊C4—72no—
219C22×C22⋊C4—72no—
220C22×C22⋊C4—71no—
221C22×C22⋊C4—71no—
222C22×C42—72no—
223#395—24no—
224#389—148no—
225#385—144no—
226C4×C22⋊C4—144no—
227C2×#95—71no—
228C2×#95—72no—
229C2×#95—72no—
230C2×#95—72no—
231C2×#95—72no—
232C2×#95—71no—
233C2×#95—71no—
234C2×#95—72no—
235C2×#95C2×D712yes—
236C2×#95C2×D711yescharacteristic
237C23×C7C22×D411yescharacteristic
238C23×C7C22×D412yes—
239C23×C7C22×D411yescharacteristic
240C23×C7C22×Q812yes—
241C23×C7C2×#4111yescharacteristic
242C23×C7C2×#4112yes—
243C23×C7C2×#4111yescharacteristic
244C23×C7C2×#4111yescharacteristic
245C23×C7H(5,𝔽2)12yes—
246C23×C72−1+411yescharacteristic
247C23×C72−1+411yescharacteristic
248C22×D7—22no—
249C22×D7—22no—
250C22×D7—24no—
251C22×D7—22no—
252C22×D7—22no—
253C22×D7—24no—
254C22×D7—22no—
255C22×D7—22no—
256C22×D7—24no—
257C22×D7—24no—
258C22×D7—24no—
259C22×D7—24no—
260C22×D7—42no—
261C22×D7—42no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—42no—
265C22×D7—42no—
266C22×D7—41no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7C4⋊Q814yes—
271C7×Q8—416no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—22no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—41no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7#12114yes—
302C2×C4×C7C22⋊Q818yescomplemented
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—24no—
306C2×Dic7—24no—
307C2×Dic7—24no—
308C2×Dic7—24no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—41no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—41no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7#12014yes—
332C4×D7—416no—
333C25—71no—
334C22×Q8—24no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—42no—
339C23×C4—71no—
340C23×C4—72no—
341C23×C4—71no—
342C23×C4—72no—
343C23×C4—74no—
344C23×C4—71no—
345C23×C4—144no—
346C23×C4—142no—
347C23×C4—142no—
348C23×C4—282no—
349C23×C4C22×D711yescharacteristic
350C23×C4C22×D712yes—
351C23×C4C22×D711yescharacteristic
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D712yes—
354C22⋊Q8—1432no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—78no—
362C2×C4⋊C4—144no—
363C2×C4⋊C4—144no—
364C2×C4⋊C4—144no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—144no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4—144no—
371C2×C22⋊C4—78no—
372C2×C22⋊C4—144no—
373C2×C22⋊C4—142no—
374C2×C22⋊C4—144no—
375C2×C22⋊C4—142no—
376C2×C22⋊C4—144no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—144no—
380C2×C22⋊C4—142no—
381C2×C22⋊C4—282no—
382C2×C42—144no—
383C2×C42—144no—
384C2×C42—282no—
385#95—24no—
386#95—142no—
387#95—144no—
388#95—142no—
389#95—144no—
390#95—144no—
391#95—144no—
392#95—142no—
393#95—282no—
394#95C22⋊D714yes—
395C22×C7C2×#12011yescharacteristic
396C22×C7C2×#12111yescharacteristic
397C22×C7C2×C22⋊Q812yes—
398C22×C7#52511yescharacteristic
399C22×C7#52512yes—
400C22×C7C2×C4⋊Q811yescharacteristic
401C22×C7#53211yescharacteristic
402C22×C7#53212yes—
403C22×C7#53312yes—
404C22×C7#53411yescharacteristic
405C22×C7#53611yescharacteristic
406C22×C7#53912yes—
407C22×C7#54212yes—
408C22×C7D4212yes—
409C22×C7#54612yes—
410C22×C7#54711yescharacteristic
411C22×C7#54811yescharacteristic
412C22×C7D4×Q812yes—
413C22×C7D4×Q812yes—
414C22×C7#55211yescharacteristic
415C22×C7#55412yes—
416C22×C7#55412yes—
417C22×C7#55611yescharacteristic
418C2×D7—24no—
419C2×D7—24no—
420C2×D7—24no—
421C2×D7—42no—
422C2×D7—42no—
423C2×D7—42no—
424C2×D7—42no—
425C2×D7—44no—
426C2×D7—44no—
427C2×D7—44no—
428C2×D7—44no—
429C4×C7—24no—
430C4×C7—28no—
431C4×C7—44no—
432C4×C7—42no—
433C4×C7—81no—
434Dic7—24no—
435Dic7—44no—
436Dic7—42no—
437Dic7—44no—
438Dic7—81no—
439C24—72no—
440C24—74no—
441C24—142no—
442C24—141no—
443C24—142no—
444C24—142no—
445C24—141no—
446C24—141no—
447C24—141no—
448C24—141no—
449C24—141no—
450C24C23×D711yescharacteristic
451C2×Q8—28no—
452C2×Q8—48no—
453C2×Q8—48no—
454C2×Q8—44no—
455C2×Q8—44no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—72no—
480C22×C4—74no—
481C22×C4—74no—
482C22×C4—74no—
483C22×C4—72no—
484C22×C4—148no—
485C22×C4—141no—
486C22×C4—142no—
487C22×C4—141no—
488C22×C4—142no—
489C22×C4—141no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—141no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—148no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—141no—
509C22×C4—141no—
510C22×C4—284no—
511C22×C4—288no—
512C22×C4—288no—
513C22×C4D4×D714yes—
514C22×C4D4×D712yes—
515C22×C4#115912yes—
516C22×C4Q8×D714yes—
517C22×C4#116112yes—
518C22×C4#116114yes—
519C22×C4C2×C22⋊D712yes—
520C22×C4C2×C22⋊D714yes—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—1416no—
524C4⋊C4—1416no—
525C4⋊C4—148no—
526C4⋊C4—284no—
527C22⋊C4—148no—
528C22⋊C4—1416no—
529C22⋊C4—1416no—
530C22⋊C4—284no—
531C42—288no—
532C2×C7#238811yescharacteristic
533C2×C7#241011yescharacteristic
534C2×C7#241511yescharacteristic
535C2×C7#243411yescharacteristic
536C2×C7#244912yes—
537C2×C7#245212yes—
538C2×C7#246512yes—
539C2×C7#255612yes—
540C2×C7#258311yescharacteristic
541C2×C7#266311yescharacteristic
542C2×C7#285711yescharacteristic
543D7—44no—
544C23—74no—
545C23—142no—
546C23—144no—
547C23—144no—
548C23—144no—
549C23—142no—
550C23—144no—
551C23—142no—
552C23—142no—
553C23—142no—
554C23—142no—
555C23—144no—
556C23—144no—
557C23—282no—
558C23—282no—
559C23—282no—
560C23—281no—
561C23—281no—
562C23—282no—
563C23—282no—
564C23—281no—
565C23—281no—
566C23—282no—
567C23C2×D4×D711yescharacteristic
568C23C2×D4×D712yes—
569C23C2×#115911yescharacteristic
570C23#654812yes—
571C23C2×Q8×D712yes—
572C23C2×#116112yes—
573C23C2×#116111yescharacteristic
574C23#655111yescharacteristic
575C23D7×#4111yescharacteristic
576C23#655411yescharacteristic
577C23C22×C22⋊D711yescharacteristic
578Q8—416no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—74no—
609C2×C4—142no—
610C2×C4—144no—
611C2×C4—142no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—144no—
615C2×C4—144no—
616C2×C4—144no—
617C2×C4—144no—
618C2×C4—142no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—2816no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4—282no—
625C2×C4—281no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—281no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—281no—
634C2×C4—282no—
635C2×C4—281no—
636C2×C4—281no—
637C2×C4—282no—
638C2×C4#650114yes—
639C2×C4#650918yescomplemented
640C7#1476311yescharacteristic, complemented, Sylow
641C22—144no—
642C22—144no—
643C22—144no—
644C22—284no—
645C22—284no—
646C22—282no—
647C22—284no—
648C22—282no—
649C22—284no—
650C22—282no—
651C22—282no—
652C22D7×#12011yescharacteristic
653C22#9028011yescharacteristic
654C22#9028212yes—
655C22#9031812yes—
656C22#9031811yescharacteristic
657C22#9031911yescharacteristic
658C22#9032312yes—
659C22#9032712yes—
660C22#9033211yescharacteristic
661C22#9033311yescharacteristic
662C22#9033612yes—
663C22D7×C4⋊Q811yescharacteristic
664C22#9041612yes—
665C22#9041611yescharacteristic
666C22#9041812yes—
667C22#9041912yes—
668C22#9042112yes—
669C22#9042412yes—
670C22C2×#650111yescharacteristic
671C22D4×C22⋊D712yes—
672C22C2×#650912yes—
673C22#9050811yescharacteristic
674C22#9051511yescharacteristic
675C4—28no—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—81no—
680C4—144no—
681C4—284no—
682C4—284no—
683C4—282no—
684C4—561no—
685C2—284no—
686C2#1173461411yescharacteristic
687C2#1173461811yescharacteristic
688C2#1173462512yes—
689C2#1173463312yes—
690C2#1173465511yescharacteristic
691C2#1173470011yescharacteristic
692C2#1174005011yescharacteristic
693C2#1174020912yes—
694C2#1174025811yescharacteristic
695C2#1174068112yes—
696C2#1174071411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC2×C7×#395C2
prime radicalC24×D7C23
hypercenterC2×#395D7
hypocenterC7#14763
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×C7×#395C2semidirect
4. C2×#89745C2semidirect
6. #11729888C2semidirect
8. #11729319C2semidirect
10. #11727173C2semidirect
11. #11727160C2semidirect
80. C2×(C2×D7)⋊C4Q8semidirect
105. C2×#395D7semidirect
302. C2×C4×C7C22⋊Q8semidirect
639. C2×C4#6509semidirect
640. C7#14763semidirect

Maximal subgroups

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

subgroupindex
C22×#65092
C2×C7×#3952
C2×#897612
C2×#897452
C2×#894382
#117298882
#117293242
#117293192
#117271812
#117271732
#117271602
#147637