Finite Group Databasebeta

← #13373232

Subgroups — #13373232

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic
2#11739905C211yesmaximal, characteristic, complemented
3#11739782C211yesmaximal, characteristic
4#11729802C211yesmaximal, characteristic
5#11729802C211yesmaximal, characteristic
6#11729380C211yesmaximal, characteristic, complemented
7#11729359C211yesmaximal, 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
21C2×#6489—21no—
22C7×#440C418yescomplemented
23C2×C7×#107C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29#89993C2211yescharacteristic
30#89993C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89984—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#89708—21no—
45#89708C2211yescharacteristic
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#22781—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—
87#6489—44no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7D411yescharacteristic
95C2×C4⋊Dic7D411yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7C2311yescharacteristic
105C2×C4×Dic7C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C7×#107C2×C414yes—
109C7×#103C2×C414yes—
110C7×#102C2×C414yes—
111C7×#97C2×C414yes—
112#6409—22no—
113#6409—22no—
114#6406—41no—
115#6406—41no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95D411yescharacteristic
123C7⋊#95D411yescharacteristic
124#6389—41no—
125C8×Dic7—41no—
126C4⋊(C7⋊C8)—22no—
127C2×#440D711yescharacteristic
128#2354—71no—
129#2351—71no—
130#1936—71no—
131#1934—71no—
132#1921—71no—
133#1918—71no—
134#1880—71no—
135#1880—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771—71no—
140#1763—71no—
141C4×#103—71no—
142C24×C7—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C22×C412yes—
147C2×C7×Q8#4112yes—
148C2×C7×Q8#4112yes—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C22×C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C7×SD16—28no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8C22×C412yes—
178C2×C7×C8C22×C412yes—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4C22×C412yes—
183C7×C4⋊C4C22×C412yes—
184C7×C22⋊C4—24no—
185C42×C7—24no—
186C42×C7C22×C412yes—
187C42×C7#4112yes—
188C22⋊Dic7—42no—
189C22⋊Dic7—42no—
190C22⋊Dic7—81no—
191C22⋊Dic7—81no—
192C4⋊Dic7—22no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—41no—
202C4⋊Dic7—42no—
203C4⋊Dic7—42no—
204C4⋊Dic7—42no—
205C4⋊Dic7—42no—
206C4⋊Dic7—42no—
207C4⋊Dic7—81no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C22×SD16C2×D711yescharacteristic
223C2×C4×Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8C2×D711yescharacteristic
226C2×C4×D4C2×D711yescharacteristic
227C2×#117—141no—
228#440Dic718yescomplemented
229C2×#107C2×D711yescharacteristic
230C2×C4⋊C8—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103C2×D711yescharacteristic
234C2×#102—71no—
235C2×#102—71no—
236C2×#102C2×D711yescharacteristic
237C2×#97—71no—
238C2×#97C2×D711yescharacteristic
239C2×C4×C8—71no—
240#390—71no—
241#390—71no—
242#389—71no—
243#389—71no—
244#389—141no—
245#389—141no—
246#388—141no—
247#388—141no—
248#384—71no—
249#382—71no—
250#382—141no—
251#380—141no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—141no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336—71no—
265#336—71no—
266C23×C7—22no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7C22×D411yescharacteristic
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8—22no—
277C7×Q8#11714yescomplemented
278C7×D4—22no—
279C7×D4—22no—
280C7×D4—41no—
281C7×D4—41no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—48no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7C4×D412yes—
297C2×C4×C7C4×D412yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7H(5,𝔽2)11yescharacteristic
303C2×C4×C72−1+411yescharacteristic
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—21no—
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—41no—
327C2×Dic7AGL(1,ℤ/8)12yes—
328C7×C8—22no—
329C7×C8—22no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×Q8C22×D711yescharacteristic
333C22×D4C22×D711yescharacteristic
334C23×C4—21no—
335C2×SD16—22no—
336C2×SD16—22no—
337C2×SD16C2×Dic714yes—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C4×Q8—22no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×Q8—281no—
346C4×Q8—281no—
347C4×Q8C2×Dic714yes—
348C4×D4—22no—
349C4×D4C2×Dic714yes—
350#117—284no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—21no—
372C2×C42—71no—
373C2×C42—71no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42—141no—
377C2×C42—141no—
378C2×C42C22×D711yescharacteristic
379#107C2×Dic714yes—
380C4⋊C8—142no—
381#103—142no—
382#103—142no—
383#103C2×Dic714yes—
384#102—281no—
385#102—281no—
386#102C2×Dic714yes—
387#97—281no—
388#97C2×Dic714yes—
389C4×C8—281no—
390#95—71no—
391#95—71no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7—81no—
404C22×C7—81no—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7#54611yescharacteristic
407C22×C7#54811yescharacteristic
408C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C7#57511yescharacteristic
410C22×C7#57711yescharacteristic
411C22×C7#57811yescharacteristic
412C4×C7—24no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7C2×#11711yescharacteristic
418C4×C7#52011yescharacteristic
419C4×C7#55211yescharacteristic
420C4×C7#55311yescharacteristic
421Dic7—22no—
422Dic7—41no—
423Dic7—41no—
424Dic7—41no—
425Dic7—81no—
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429C24—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8#115712yes—
434C2×Q8#115712yes—
435C2×Q8C22×Dic712yes—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C22×Dic712yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4D4×D711yescharacteristic
457C22×C4D4×D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459SD16—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8C22×Dic712yes—
471C2×C8C22×Dic712yes—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—142no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—281no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—561no—
491C4⋊C4C22×Dic712yes—
492C4⋊C4C22×Dic712yes—
493C22⋊C4—24no—
494C22⋊C4—282no—
495C22⋊C4—282no—
496C22⋊C4—561no—
497C22⋊C4—561no—
498C42—24no—
499C42—141no—
500C42—141no—
501C42—141no—
502C42—141no—
503C42—282no—
504C42—282no—
505C42—282no—
506C42—282no—
507C42#115912yes—
508C42C22×Dic712yes—
509C2×C7—82no—
510C2×C7#235211yescharacteristic
511C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
512C2×C7#294711yescharacteristic
513C2×C7#330811yescharacteristic
514C2×C7#332111yescharacteristic
515C2×C7#336811yescharacteristic
516C2×C7#337711yescharacteristic
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×D4×D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—22no—
528Q8#648914yescomplemented
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—72no—
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—141no—
558C2×C4—141no—
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×C4—281no—
570C2×C4—281no—
571C2×C4D4×Dic712yes—
572C2×C4D4×Dic712yes—
573C2×C4C2×#115711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115911yescharacteristic
576C2×C4#655311yescharacteristic
577C2×C4#655411yescharacteristic
578C2×C4C23×Dic711yescharacteristic
579C8—22no—
580C8—22no—
581C8—281no—
582C8—281no—
583C7#2278111yescharacteristic, complemented, Sylow
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#9024111yescharacteristic
592C22#9024311yescharacteristic
593C22D7×AGL(1,ℤ/8)11yescharacteristic
594C22#9046411yescharacteristic
595C22#9046911yescharacteristic
596C22#9047011yescharacteristic
597C22C2×D4×Dic711yescharacteristic
598C4—24no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—42no—
603C4—142no—
604C4—281no—
605C4—281no—
606C4—281no—
607C4—561no—
608C4—561no—
609C4—561no—
610C4—561no—
611C4#9023211yescharacteristic
612C4#9025611yescharacteristic
613C4C2×#648911yescharacteristic
614C4#9051811yescharacteristic
615C2—82no—
616C2#1173719711yescharacteristic
617C2#1173720311yescharacteristic
618C2#1173724311yescharacteristic
619C2#1173725011yescharacteristic
620C2#1173978011yescharacteristic
621C2Dic7×AGL(1,ℤ/8)11yescharacteristic
622C2#1174141011yescharacteristic
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#22781
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. #11739905C2semidirect
6. #11729380C2semidirect
8. #11728364C2semidirect
9. #11728363C2semidirect
10. #11727974C2semidirect
11. Dic7×#103C2semidirect
14. #11727569C2semidirect
15. #11727564C2semidirect
22. C7×#440C4semidirect
228. #440Dic7semidirect
277. C7×Q8#117semidirect
528. Q8#6489semidirect
583. C7#22781semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
#117399052
#117397822
#117298022
#117293802
#117293592
#117283642
#117283632
#117279742
Dic7×#1032
#117279412
#117279392
#117275692
#117275642
#227817