Finite Group Databasebeta

← #13405869

Subgroups — #13405869

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2584C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic
3#11739945C211yesmaximal, characteristic
4C2×#89796C211yesmaximal, characteristic, complemented
5#11729495C211yesmaximal, characteristic, complemented
6#11727383C211yesmaximal, characteristic
7#11727383C211yesmaximal, characteristic
8#11727378C211yesmaximal, characteristic, complemented
9#11727376C211yesmaximal, characteristic, complemented
10#11727375C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11727371C211yesmaximal, characteristic
13#11727369C211yesmaximal, characteristic
14#11727228C211yesmaximal, characteristic
15#11727221C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×#6509—21no—
19C2×C4×D28C2211yescharacteristic
20C7×#398—21no—
21C7×#395—21no—
22C7×#389—21no—
23C7×#389—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29#89838—24no—
30#89796—24no—
31#89766—21no—
32#89766—21no—
33#89759—21no—
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89711C2211yescharacteristic
43#89711C2211yescharacteristic
44#89707C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25486—71nomaximal, Sylow
62C22×C7×Q8D411yescharacteristic
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C7×#125—22no—
66C7×#125—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86#6509—44no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C4×D28—22no—
104C4×D28—22no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112#6383—22no—
113#6383—22no—
114#6383—22no—
115#6383Q814yescomplemented
116#6382—22no—
117#6382—22no—
118#6382—22no—
119#6381—22no—
120C4⋊(C7⋊C8)—22no—
121C4⋊(C7⋊C8)—22no—
122C4⋊(C7⋊C8)D414yes—
123#6377—22no—
124#6377—22no—
125C4×C7⋊C8—22no—
126C2×#478—71no—
127C2×#446—71no—
128#2584D711yescharacteristic, complemented
129#2538—71no—
130#2088—71no—
131#2086—71no—
132#2065—71no—
133#2065—71no—
134#1992—71no—
135#1948—71no—
136#1944—71no—
137#1941—71no—
138#1936—71no—
139#1934—71no—
140#1762—71no—
141C23×D7—21no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—41no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28—21no—
163C2×D28—21no—
164C2×D28C2×Q812yes—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—81no—
182C7×C4⋊C4—81no—
183C7×C4⋊C4C2×Q812yes—
184C7×C4⋊C4#4112yes—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—81no—
190C42×C7C2×D412yes—
191C42×C7#4112yes—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—81no—
196(C2×D7)⋊C4—81no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—42no—
204C4⋊Dic7—81no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8C2×D412yes—
217C2×C7⋊C8C2×D412yes—
218C2×C7⋊C8C2×Q812yes—
219C2×#125C2×D711yescharacteristic
220C2×C22⋊Q8—141no—
221C2×C4×Q8C2×D711yescharacteristic
222C2×C4×D4—71no—
223#478—144no—
224#446—144no—
225C2×#107—71no—
226C2×#106—71no—
227C2×#106—71no—
228C2×#106—71no—
229C2×C4⋊C8—71no—
230C2×C4⋊C8—71no—
231C2×C4⋊C8—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#97—71no—
237C2×#97—71no—
238C2×C4×C8—71no—
239#398—21no—
240#396—141no—
241#396—141no—
242#395—21no—
243#390—71no—
244#390—71no—
245#389—21no—
246#389—21no—
247#389—71no—
248#389—71no—
249#389C2×D711yescharacteristic
250#389C2×D711yescharacteristic
251#389C2×D711yescharacteristic
252#389C2×D711yescharacteristic
253#384—71no—
254#380—141no—
255C4×C4⋊C4C2×D711yescharacteristic
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264C23×C7C22×D411yescharacteristic
265C22×D7—22no—
266C22×D7—41no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C7×Q8—44no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
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—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
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#12112yes—
301C2×C4×C7#12112yes—
302C2×C4×C7AGL(1,ℤ/8)12yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×Q811yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×C4×C72−1+411yescharacteristic
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318D28—22no—
319D28—22no—
320D28—41no—
321D28—41no—
322C4×D7—84no—
323C7⋊C8—22no—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C7⋊C8—41no—
327C22×Q8C22⋊D711yescharacteristic
328C22×D4—71no—
329C23×C4—141no—
330C22×C8—71no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8—71no—
334#125—22no—
335#125—22no—
336C22⋊Q8—284no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—142no—
340C4×D4—142no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22⋊D711yescharacteristic
355C2×C4⋊C4C22⋊D711yescharacteristic
356C2×C4⋊C4C22⋊D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4—141no—
366C2×C42—21no—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369C2×C42C22×D711yescharacteristic
370#107—142no—
371#106—142no—
372#106—142no—
373#106—142no—
374C4⋊C8—74no—
375C4⋊C8—142no—
376C4⋊C8—142no—
377#102—74no—
378#102—142no—
379#102—142no—
380#102—142no—
381#97—142no—
382#97—142no—
383C4×C8—142no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—141no—
390#95—141no—
391C22×C7C2×#12111yescharacteristic
392C22×C7#54811yescharacteristic
393C22×C7D4×Q811yescharacteristic
394C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C7#57511yescharacteristic
396C22×C7#57711yescharacteristic
397C22×C7#57711yescharacteristic
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—42no—
401C2×D7—81no—
402C2×D7—81no—
403C2×D7—81no—
404C2×D7—81no—
405C4×C7—22no—
406C4×C7—22no—
407C4×C7—22no—
408C4×C7—42no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—41no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7#53311yescharacteristic
416C4×C7#54111yescharacteristic
417C4×C7#54711yescharacteristic
418C4×C7D4×Q811yescharacteristic
419Dic7—81no—
420Dic7—81no—
421Dic7—81no—
422C24—141no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—41no—
426C2×D4—72no—
427C2×D4—141no—
428C2×D4—141no—
429C2×D4—141no—
430C2×D4—141no—
431C2×D4—141no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—282no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C23×D711yescharacteristic
449C2×C8—72no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—561no—
486C4⋊C4Q8×D712yes—
487C4⋊C4#116112yes—
488C22⋊C4—282no—
489C22⋊C4—282no—
490C22⋊C4—282no—
491C22⋊C4—561no—
492C22⋊C4—561no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—81no—
498C42D4×D712yes—
499C42#115912yes—
500C2×C7#242911yescharacteristic
501C2×C7#305511yescharacteristic
502C2×C7#306011yescharacteristic
503C2×C7#336811yescharacteristic
504C2×C7#337611yescharacteristic
505C2×C7#338411yescharacteristic
506C2×C7#338811yescharacteristic
507D7—82no—
508C23—142no—
509C23—281no—
510C23—281no—
511C23—281no—
512C23—281no—
513C23—281no—
514C23—281no—
515C23C22×C22⋊D711yescharacteristic
516Q8—44no—
517D4—142no—
518D4—142no—
519D4—281no—
520D4—281no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—564no—
558C2×C4#650212yes—
559C2×C4#650212yes—
560C2×C4#651212yes—
561C2×C4C2×D4×D711yescharacteristic
562C2×C4C2×#115911yescharacteristic
563C2×C4C2×Q8×D711yescharacteristic
564C2×C4C2×#116111yescharacteristic
565C2×C4#655111yescharacteristic
566C2×C4#655111yescharacteristic
567C8—142no—
568C8—142no—
569C8—142no—
570C8—281no—
571C7#2548611yescharacteristic, complemented, Sylow
572C22—282no—
573C22—282no—
574C22—282no—
575C22—561no—
576C22—561no—
577C22—561no—
578C22—561no—
579C22C2×#650211yescharacteristic
580C22Q8×C22⋊D711yescharacteristic
581C22#9050811yescharacteristic
582C22C2×#651211yescharacteristic
583C22#9051311yescharacteristic
584C22#9052711yescharacteristic
585C22#9052811yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—561no—
597C4—561no—
598C4—561no—
599C4#9041711yescharacteristic
600C4#9041811yescharacteristic
601C4#9041911yescharacteristic
602C4#9042211yescharacteristic
603C2—562no—
604C2#1174036911yescharacteristic
605C2#1174069911yescharacteristic
606C2#1174086511yescharacteristic
607C2#1174107511yescharacteristic
608C2#1174109011yescharacteristic
609C2#1174113511yescharacteristic
610C2#1174114011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2584C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2584D7
hypocenterC7#25486
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
1. C7×#2584C2semidirect
4. C2×#89796C2semidirect
5. #11729495C2semidirect
8. #11727378C2semidirect
9. #11727376C2semidirect
10. #11727375C2semidirect
11. #11727372C2semidirect
15. #11727221C2semidirect
115. #6383Q8semidirect
128. #2584D7semidirect
571. C7#25486semidirect

Maximal subgroups

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

subgroupindex
C7×#25842
C2×#898382
#117399452
C2×#897962
#117294952
#117273832
#117273782
#117273762
#117273752
#117273722
#117273712
#117273692
#117272282
#117272212
#254867