Finite Group Databasebeta

← #13373142

Subgroups — #13373142

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic
2Q8×C4⋊Dic7C211yesmaximal, characteristic, complemented
3#11739783C211yesmaximal, characteristic
4#11729802C211yesmaximal, characteristic
5#11729802C211yesmaximal, characteristic
6#11729381C211yesmaximal, characteristic, complemented
7#11729358C211yesmaximal, characteristic
8#11728364C211yesmaximal, characteristic, complemented
9#11728363C211yesmaximal, characteristic, complemented
10#11727974C211yesmaximal, characteristic, complemented
11Dic7×#103C211yesmaximal, characteristic, complemented
12#11727941C211yesmaximal, characteristic
13#11727939C211yesmaximal, characteristic
14#11727569C211yesmaximal, characteristic, complemented
15#11727564C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C7×#440C418yescomplemented
22C2×C7×#107C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28#89993C2211yescharacteristic
29#89993C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89983—21no—
33C4×C22⋊Dic7—21no—
34C2×#6389C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89757—21no—
37#89757—21no—
38#89757C2211yescharacteristic
39#89757C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42#89723—21no—
43#89723—21no—
44#89708C2211yescharacteristic
45#89707—21no—
46#89707C2211yescharacteristic
47C4×C4⋊Dic7—21no—
48C4×C4⋊Dic7C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22775—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C2×C7×SD16—22no—
65C2×C7×SD16—22no—
66C2×C7×SD16C2×C414yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C4×C7×Q8—22no—
70C4×C7×Q8C2×C414yes—
71C4×C7×D4—22no—
72C4×C7×D4C2×C414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83Q8×Dic7—22no—
84Q8×Dic7—22no—
85Q8×Dic7—41no—
86Q8×Dic7—41no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7D411yescharacteristic
94C2×C4⋊Dic7D411yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#107C2×C414yes—
108C7×#103C2×C414yes—
109C7×#102C2×C414yes—
110C7×#97C2×C414yes—
111#6409—22no—
112#6409—22no—
113#6406—41no—
114#6406—41no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95D411yescharacteristic
122C7⋊#95D411yescharacteristic
123#6389—41no—
124C8×Dic7—41no—
125C4⋊(C7⋊C8)—22no—
126C2×#440D711yescharacteristic
127#2357—71no—
128Q8×C4⋊C4—71no—
129#1936—71no—
130#1934—71no—
131#1921—71no—
132#1918—71no—
133#1880—71no—
134#1880—71no—
135#1836—71no—
136#1805—71no—
137#1802—71no—
138#1771—71no—
139#1763—71no—
140C4×#103—71no—
141C24×C7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8C22×C412yes—
146C2×C7×Q8#4112yes—
147C2×C7×Q8#4112yes—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4C22×C412yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C7×SD16—28no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8C22×C412yes—
177C2×C7×C8C22×C412yes—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C22×C412yes—
182C7×C4⋊C4C22×C412yes—
183C7×C22⋊C4—24no—
184C42×C7—24no—
185C42×C7C22×C412yes—
186C42×C7#4112yes—
187C22⋊Dic7—42no—
188C22⋊Dic7—42no—
189C22⋊Dic7—81no—
190C22⋊Dic7—81no—
191C4⋊Dic7—22no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—41no—
201C4⋊Dic7—42no—
202C4⋊Dic7—42no—
203C4⋊Dic7—42no—
204C4⋊Dic7—81no—
205C4⋊Dic7—81no—
206C4⋊Dic7—81no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7—81no—
214C4×Dic7—81no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C22×SD16C2×D711yescharacteristic
222C2×C4×Q8—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8C2×D711yescharacteristic
225C2×C4×D4C2×D711yescharacteristic
226#440Dic718yescomplemented
227C2×#107C2×D711yescharacteristic
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#103C2×D711yescharacteristic
232C2×#102—71no—
233C2×#102—71no—
234C2×#102C2×D711yescharacteristic
235C2×#97—71no—
236C2×#97C2×D711yescharacteristic
237C2×C4×C8—71no—
238#390—71no—
239#390—71no—
240#389—71no—
241#389—71no—
242#389—141no—
243#389—141no—
244#388—141no—
245#388—141no—
246#384—71no—
247#384—141no—
248#382—71no—
249#381—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—141no—
254C4×C22⋊C4—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265C23×C7—22no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C22×D411yescharacteristic
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8#11714yes—
277C7×D4—22no—
278C7×D4—22no—
279C7×D4—41no—
280C7×D4—41no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—84no—
295C2×C4×C7C4×D412yes—
296C2×C4×C7C4×D412yes—
297C2×C4×C7C23×C411yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
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#13712yes—
327C7×C8—22no—
328C7×C8—22no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×Q8C22×D711yescharacteristic
332C22×D4C22×D711yescharacteristic
333C23×C4—21no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16C2×Dic714yes—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4×Q8—22no—
342C4×Q8—142no—
343C4×Q8—142no—
344C4×Q8—281no—
345C4×Q8—281no—
346C4×Q8C2×Dic714yes—
347C4×D4—22no—
348C4×D4C2×Dic714yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—141no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—21no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377#107C2×Dic714yes—
378C4⋊C8—142no—
379#103—142no—
380#103—142no—
381#103C2×Dic714yes—
382#102—281no—
383#102—281no—
384#102C2×Dic714yes—
385#97—281no—
386#97C2×Dic714yes—
387C4×C8—281no—
388#95—71no—
389#95—71no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—81no—
400C22×C7—81no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7C2×C4×D411yescharacteristic
404C22×C7#54611yescharacteristic
405C22×C7#54811yescharacteristic
406C22×C7C2×#13711yescharacteristic
407C22×C7#57511yescharacteristic
408C22×C7#57611yescharacteristic
409C22×C7#57711yescharacteristic
410C4×C7—24no—
411C4×C7—22no—
412C4×C7—22no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7C2×#11711yescharacteristic
416C4×C7#52011yescharacteristic
417C4×C7#55211yescharacteristic
418C4×C7#55311yescharacteristic
419Dic7—22no—
420Dic7—41no—
421Dic7—41no—
422Dic7—41no—
423Dic7—81no—
424Dic7—81no—
425Dic7—81no—
426Dic7—81no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8#115712yes—
432C2×Q8#115712yes—
433C2×Q8C22×Dic712yes—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C22×Dic712yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457SD16—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8C22×Dic712yes—
469C2×C8C22×Dic712yes—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—142no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—281no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—561no—
487C4⋊C4—561no—
488C4⋊C4—561no—
489C4⋊C4C22×Dic712yes—
490C4⋊C4C22×Dic712yes—
491C22⋊C4—24no—
492C22⋊C4—282no—
493C22⋊C4—282no—
494C22⋊C4—561no—
495C22⋊C4—561no—
496C42—24no—
497C42—141no—
498C42—141no—
499C42—141no—
500C42—141no—
501C42—282no—
502C42—282no—
503C42—561no—
504C42—561no—
505C42#115912yes—
506C42C22×Dic712yes—
507C2×C7—82no—
508C2×C7#235211yescharacteristic
509C2×C7C4×#13711yescharacteristic
510C2×C7#294611yescharacteristic
511C2×C7#330911yescharacteristic
512C2×C7#332011yescharacteristic
513C2×C7#337111yescharacteristic
514C2×C7#337511yescharacteristic
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D711yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8#648914yes—
527D4—22no—
528D4—22no—
529D4—41no—
530D4—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—72no—
545C2×C4—84no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4D4×Dic712yes—
570C2×C4D4×Dic712yes—
571C2×C4C2×#115711yescharacteristic
572C2×C4C2×#115711yescharacteristic
573C2×C4C2×#115911yescharacteristic
574C2×C4#655311yescharacteristic
575C2×C4#655411yescharacteristic
576C2×C4C23×Dic711yescharacteristic
577C8—22no—
578C8—22no—
579C8—281no—
580C8—281no—
581C7#2277511yescharacteristic, complemented, Sylow
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22#9024111yescharacteristic
590C22#9024311yescharacteristic
591C22#9046511yescharacteristic
592C22#9046611yescharacteristic
593C22D7×#13711yescharacteristic
594C22#9046811yescharacteristic
595C22C2×D4×Dic711yescharacteristic
596C4—24no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—42no—
601C4—142no—
602C4—281no—
603C4—281no—
604C4—281no—
605C4—561no—
606C4—561no—
607C4—561no—
608C4—561no—
609C4#9023211yescharacteristic
610C4#9025611yescharacteristic
611C4C2×#648911yescharacteristic
612C4#9051811yescharacteristic
613C2—82no—
614C2#1173719911yescharacteristic
615C2#1173720111yescharacteristic
616C2#1173724411yescharacteristic
617C2#1173724911yescharacteristic
618C2#1173978011yescharacteristic
619C2#1174137511yescharacteristic
620C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#440D7
hypocenterC7#22775
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic7C2semidirect
6. #11729381C2semidirect
8. #11728364C2semidirect
9. #11728363C2semidirect
10. #11727974C2semidirect
11. Dic7×#103C2semidirect
14. #11727569C2semidirect
15. #11727564C2semidirect
21. C7×#440C4semidirect
226. #440Dic7semidirect
581. C7#22775semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
Q8×C4⋊Dic72
#117397832
#117298022
#117293812
#117293582
#117283642
#117283632
#117279742
Dic7×#1032
#117279412
#117279392
#117275692
#117275642
#227757