Finite Group Databasebeta

← #13383783

Subgroups — #13383783

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic7×#121C211yesmaximal, characteristic
2#11731309C211yesmaximal, characteristic, complemented
3C7×#2033C211yesmaximal, characteristic
4#11729433C211yesmaximal, characteristic
5#11728373C211yesmaximal, characteristic, complemented
6Dic7×#107C211yesmaximal, characteristic, complemented
7#11727937C211yesmaximal, characteristic, complemented
8#11727919C211yesmaximal, characteristic, complemented
9Dic7×#102C211yesmaximal, characteristic, complemented
10Dic7×#102C211yesmaximal, characteristic, complemented
11#11727692C211yesmaximal, characteristic, complemented
12Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
13#11727272C211yesmaximal, characteristic, complemented
14#11726596C211yesmaximal, characteristic, complemented
15#11726539C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×D4×Dic7—21no—
18C2×D4×Dic7C2211yescharacteristic
19C22×C4×Dic7C2211yescharacteristic
20C2×#6440—21no—
21C2×C7×#107C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C7×C22⋊C8C2211yescharacteristic
25C7×#380C2211yescharacteristic
26#89993C2211yescharacteristic
27#89992—21no—
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30#89984C2211yescharacteristic
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6392C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89757C2211yescharacteristic
37#89748C2211yescharacteristic
38#89748C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89723—21no—
45#89714—21no—
46C22⋊C4×Dic7—21no—
47#89422—21no—
48#89421—21no—
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#23754—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C23×Dic7D411yescharacteristic
65C23×Dic7D411yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×#121—22no—
69C7×#121—22no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80D4×Dic7—22no—
81D4×Dic7—22no—
82D4×Dic7—44no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7—22no—
93C2×C4×Dic7—22no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7D411yescharacteristic
98C2×C4×Dic7D411yescharacteristic
99C2×C4×Dic7C2311yescharacteristic
100C2×C4×Dic7C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103#6440—44no—
104C7×#107—22no—
105C7×#102—22no—
106C7×#102—22no—
107C7×C22⋊C8—22no—
108C7×#95—21no—
109#6406—22no—
110#6406—22no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C22⋊(C7⋊C8)—22no—
118#6392—22no—
119C8×Dic7—22no—
120C8×Dic7—22no—
121#6381—22no—
122#6381—22no—
123C4×#121—71no—
124#2285—71no—
125#2099—71no—
126#2093—71no—
127#2083—71no—
128#2075—71no—
129#2075—71no—
130#2071—71no—
131#2033—71no—
132#2033D711yescharacteristic
133C4×#107—71no—
134C4×#107—71no—
135C4×#102—71no—
136C4×#102—71no—
137C4×C22⋊C8—71no—
138C24×C7—21no—
139C24×C7C2×D411yescharacteristic
140C2×C7×D4—24no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—42no—
147C2×C7×D4—42no—
148C2×C7×D4#4112yes—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—41no—
169C22×Dic7—42no—
170C22×Dic7—42no—
171C22×Dic7—42no—
172C22×Dic7D814yes—
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8#4112yes—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4#4112yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—42no—
191C22⋊Dic7—42no—
192C22⋊Dic7—42no—
193C22⋊Dic7—42no—
194C22⋊Dic7—42no—
195C22⋊Dic7—42no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—22no—
212C4×Dic7—22no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C4×Dic7#4112yes—
217C4×Dic7#4112yes—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8#4112yes—
224C2×#121C2×D711yescharacteristic
225C2×C4×D4—71no—
226C2×C4×D4—141no—
227C2×#117—141no—
228C22×C42—71no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#107—71no—
232C2×#107C2×D711yescharacteristic
233C2×#102—71no—
234C2×#102—71no—
235C2×#102C2×D711yescharacteristic
236C2×#102C2×D711yescharacteristic
237C2×C22⋊C8—71no—
238C2×C22⋊C8C2×D711yescharacteristic
239C2×C4×C8—71no—
240C2×C4×C8—71no—
241#390—71no—
242#389—71no—
243#388—141no—
244#386—141no—
245#384—71no—
246#384—71no—
247#382—141no—
248#380—71no—
249#380—71no—
250#380—141no—
251#380C2×D711yescharacteristic
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—141no—
256C4×C22⋊C4—141no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340C2×D711yescharacteristic
265#336—71no—
266#336—71no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—22no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C2×D812yes—
281C23×C7C22×D411yescharacteristic
282C7×D4—22no—
283C7×D4—22no—
284C7×D4—48no—
285C7×D4—41no—
286C7×D4—41no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—28no—
316C2×Dic7—44no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—84no—
330C2×Dic7#12312yes—
331C2×Dic7#12312yes—
332C2×Dic7C2×D812yes—
333C2×Dic7C2×D812yes—
334C2×Dic7#13512yes—
335C2×Dic7#13712yes—
336C7×C8—22no—
337C7×C8—41no—
338C7⋊C8—22no—
339C7⋊C8—41no—
340C22×D4—21no—
341C22×D4C22×D711yescharacteristic
342C23×C4—71no—
343C23×C4—71no—
344C23×C4—141no—
345C23×C4C22×D711yescharacteristic
346C22×C8—71no—
347C22×C8—71no—
348C22×C8C22×D711yescharacteristic
349C22×C8C22×D711yescharacteristic
350#121—22no—
351#121—22no—
352C4×D4—142no—
353C4×D4—142no—
354C4×D4—284no—
355#117—284no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4—141no—
375C2×C42—71no—
376C2×C42—71no—
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—142no—
380C2×C42—142no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42—141no—
384#107—22no—
385#107—142no—
386#107—142no—
387#107—142no—
388#102—22no—
389#102—22no—
390#102—142no—
391#102—142no—
392C22⋊C8—22no—
393C22⋊C8—142no—
394C4×C8—142no—
395C4×C8—142no—
396#95—21no—
397#95—141no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—41no—
407C22×C7—41no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7—81no—
414C22×C7—81no—
415C22×C7—81no—
416C22×C7—81no—
417C22×C7C2×#12311yescharacteristic
418C22×C7#52511yescharacteristic
419C22×C7#53211yescharacteristic
420C22×C7C22×D811yescharacteristic
421C22×C7C2×#13511yescharacteristic
422C22×C7C2×#13711yescharacteristic
423C22×C7#57511yescharacteristic
424C4×C7—42no—
425C4×C7—41no—
426C4×C7—81no—
427C4×C7—81no—
428C4×C7#52911yescharacteristic
429C4×C7#52911yescharacteristic
430C4×C7#52911yescharacteristic
431C4×C7#52911yescharacteristic
432Dic7—22no—
433Dic7—22no—
434Dic7—22no—
435Dic7—81no—
436Dic7—81no—
437Dic7—81no—
438Dic7—81no—
439Dic7#48014yes—
440C24—21no—
441C24D4×D711yescharacteristic
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4#115912yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—71no—
461C22×C4—74no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4—282no—
474C22×C4—282no—
475C22×C4—282no—
476C22×C4—281no—
477C22×C4D4×D711yescharacteristic
478C22×C4C23×D711yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—72no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8#115912yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—141no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—282no—
505C4⋊C4—282no—
506C4⋊C4—282no—
507C4⋊C4—282no—
508C4⋊C4—282no—
509C4⋊C4#115912yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—282no—
513C22⋊C4—282no—
514C22⋊C4—282no—
515C22⋊C4—282no—
516C22⋊C4—282no—
517C42—72no—
518C42—72no—
519C42—142no—
520C42—142no—
521C42—141no—
522C42—141no—
523C42—282no—
524C42—282no—
525C42—282no—
526C2×C7—42no—
527C2×C7—82no—
528C2×C7#239511yescharacteristic
529C2×C7C2×#48011yescharacteristic
530C2×C7#309111yescharacteristic
531C2×C7#310011yescharacteristic
532C2×C7#310711yescharacteristic
533C2×C7#316911yescharacteristic
534C2×C7#317211yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23D7×D812yes—
549C23C2×D4×D711yescharacteristic
550D4—22no—
551D4—22no—
552D4—48no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—72no—
574C2×C4—72no—
575C2×C4—72no—
576C2×C4—72no—
577C2×C4—148no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—284no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—564no—
598C2×C4C2×#115911yescharacteristic
599C2×C4C2×#115911yescharacteristic
600C2×C4C2×#115911yescharacteristic
601C2×C4D7×#4111yescharacteristic
602C2×C4D7×#4111yescharacteristic
603C2×C4D7×#4111yescharacteristic
604C8—22no—
605C8—41no—
606C8—142no—
607C8—281no—
608C7#2375411yescharacteristic, complemented, Sylow
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#9027711yescharacteristic
624C22#9028711yescharacteristic
625C22D7×#12311yescharacteristic
626C22C2×D7×D811yescharacteristic
627C22D7×#13511yescharacteristic
628C22#9046411yescharacteristic
629C22D7×#13711yescharacteristic
630C4—41no—
631C4—42no—
632C4—74no—
633C4—81no—
634C4—81no—
635C4—142no—
636C4—142no—
637C4—142no—
638C4—561no—
639C4—561no—
640C4—561no—
641C4—561no—
642C4#9029111yescharacteristic
643C4#9029111yescharacteristic
644C4#9034011yescharacteristic
645C4#9037111yescharacteristic
646C2—42no—
647C2—82no—
648C2#1173433711yescharacteristic
649C2#1173743311yescharacteristic
650C2#1173765811yescharacteristic
651C2#1173814211yescharacteristic
652C2#1173823011yescharacteristic
653C2D7×#48011yescharacteristic
654C2#1173881611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2033C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2033D7
hypocenterC7#23754
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #11731309C2semidirect
5. #11728373C2semidirect
6. Dic7×#107C2semidirect
7. #11727937C2semidirect
8. #11727919C2semidirect
9. Dic7×#102C2semidirect
10. Dic7×#102C2semidirect
11. #11727692C2semidirect
12. Dic7×C22⋊C8C2semidirect
13. #11727272C2semidirect
14. #11726596C2semidirect
15. #11726539C2semidirect
608. C7#23754semidirect

Maximal subgroups

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

subgroupindex
Dic7×#1212
#117313092
C7×#20332
#117294332
#117283732
Dic7×#1072
#117279372
#117279192
Dic7×#1022
#117276922
Dic7×C22⋊C82
#117272722
#117265962
#117265392
#237547