Finite Group Databasebeta

← #13405690

Subgroups — #13405690

Order 1792.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2584C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic
3#11739946C211yesmaximal, characteristic
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729494C211yesmaximal, characteristic, complemented
6#11727384C211yesmaximal, characteristic
7#11727383C211yesmaximal, characteristic
8#11727378C211yesmaximal, characteristic, complemented
9#11727377C211yesmaximal, characteristic, complemented
10#11727373C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11727369C211yesmaximal, characteristic
13#11727369C211yesmaximal, characteristic
14C4×#6383C211yesmaximal, characteristic
15#11727222C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C7×#398—21no—
20C7×#395—21no—
21C7×#389—21no—
22C7×#389—21no—
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28#90003—21no—
29#89838—24no—
30Q8×C7⋊C8—24no—
31#89764—21no—
32#89761—21no—
33#89761—21no—
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89711C2211yescharacteristic
43#89711C2211yescharacteristic
44#89708C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25473—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
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C4×D28—22no—
101C4×D28—22no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111#6383—22no—
112#6383—22no—
113#6383—22no—
114#6383Q814yescomplemented
115#6382—22no—
116#6382—22no—
117#6381—22no—
118#6381—22no—
119C4⋊(C7⋊C8)—22no—
120C4⋊(C7⋊C8)—22no—
121C4⋊(C7⋊C8)D414yes—
122C4×C7⋊C8—22no—
123C4×C7⋊C8—22no—
124C4×C7⋊C8—22no—
125C2×#478—71no—
126C2×C8×Q8—71no—
127#2584D711yescharacteristic, complemented
128#2580—71no—
129#2088—71no—
130#2087—71no—
131#2067—71no—
132#2065—71no—
133#1991—71no—
134#1949—71no—
135#1942—71no—
136#1941—71no—
137#1934—71no—
138#1934—71no—
139C4×#102—71no—
140C23×D7—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28—21no—
163C2×D28C2×Q812yes—
164C2×C4×D7—42no—
165C2×C4×D7—81no—
166C2×C4×D7—81no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4—81no—
182C7×C4⋊C4C2×Q812yes—
183C7×C4⋊C4#4112yes—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—81no—
189C42×C7C2×D412yes—
190C42×C7#4112yes—
191(C2×D7)⋊C4—42no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—81no—
195(C2×D7)⋊C4—81no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202Dic7⋊C4—81no—
203Dic7⋊C4—81no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
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⋊C8C2×D412yes—
214C2×C7⋊C8C2×D412yes—
215C2×C7⋊C8C2×Q812yes—
216C2×#125C2×D711yescharacteristic
217C2×C4×Q8C2×D711yescharacteristic
218C2×C4×D4—71no—
219#478—144no—
220C8×Q8—144no—
221C2×#107—71no—
222C2×#107—71no—
223C2×#106—71no—
224C2×#106—71no—
225C2×C4⋊C8—71no—
226C2×C4⋊C8—71no—
227C2×C4⋊C8—71no—
228C2×#102—71no—
229C2×#102—71no—
230C2×#102—71no—
231C2×#102—71no—
232C2×C4×C8—71no—
233C2×C4×C8—71no—
234C2×C4×C8—71no—
235#399—141no—
236#398—21no—
237#395—21no—
238#393—141no—
239#390—71no—
240#390—71no—
241#389—21no—
242#389—21no—
243#389—71no—
244#389—71no—
245#389C2×D711yescharacteristic
246#389C2×D711yescharacteristic
247#389C2×D711yescharacteristic
248#389C2×D711yescharacteristic
249#385—141no—
250#385—141no—
251#382—71no—
252C4×C4⋊C4C2×D711yescharacteristic
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261C23×C7C22×D411yescharacteristic
262C22×D7—22no—
263C22×D7—41no—
264C22×D7—41no—
265C22×D7—41no—
266C22×D7—41no—
267C22×D7—41no—
268C22×D7—41no—
269C7×Q8—28no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
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—22no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
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#12112yes—
297C2×C4×C7#12112yes—
298C2×C4×C7#13512yes—
299C2×C4×C7#13512yes—
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×Q811yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C72−1+411yescharacteristic
305C2×C4×C72−1+411yescharacteristic
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315D28—22no—
316D28—22no—
317D28—41no—
318D28—41no—
319C4×D7—84no—
320C7⋊C8—22no—
321C7⋊C8—22no—
322C7⋊C8—22no—
323C7⋊C8—41no—
324C22×Q8C22⋊D711yescharacteristic
325C22×D4—71no—
326C23×C4—141no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330C22×C8—71no—
331#125—22no—
332#125—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×D4—142no—
336C4×D4—142no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22⋊D711yescharacteristic
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22⋊D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C42—21no—
361C2×C42C22×D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364#107—142no—
365#107—142no—
366#106—142no—
367#106—142no—
368C4⋊C8—74no—
369C4⋊C8—142no—
370C4⋊C8—142no—
371#102—74no—
372#102—142no—
373#102—142no—
374#102—142no—
375C4×C8—142no—
376C4×C8—142no—
377C4×C8—142no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387C22×C7C2×#12111yescharacteristic
388C22×C7#54811yescharacteristic
389C22×C7D4×Q811yescharacteristic
390C22×C7C2×#13511yescharacteristic
391C22×C7C2×#13511yescharacteristic
392C22×C7#57611yescharacteristic
393C22×C7#57711yescharacteristic
394C2×D7—42no—
395C2×D7—42no—
396C2×D7—42no—
397C2×D7—81no—
398C2×D7—81no—
399C2×D7—81no—
400C2×D7—81no—
401C4×C7—22no—
402C4×C7—22no—
403C4×C7—22no—
404C4×C7—22no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—81no—
410C4×C7—81no—
411C4×C7#53311yescharacteristic
412C4×C7#54111yescharacteristic
413C4×C7#54711yescharacteristic
414C4×C7D4×Q811yescharacteristic
415Dic7—81no—
416Dic7—81no—
417Dic7—81no—
418C24—141no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×D4—72no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C2×D4—141no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—282no—
436C22×C4—561no—
437C22×C4—561no—
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C23×D711yescharacteristic
445C2×C8—72no—
446C2×C8—72no—
447C2×C8—72no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—561no—
479C4⋊C4—561no—
480C4⋊C4Q8×D712yes—
481C4⋊C4#116112yes—
482C22⋊C4—282no—
483C22⋊C4—282no—
484C22⋊C4—282no—
485C22⋊C4—561no—
486C22⋊C4—561no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—81no—
492C42D4×D712yes—
493C42#115912yes—
494C2×C7#242911yescharacteristic
495C2×C7#305011yescharacteristic
496C2×C7#306211yescharacteristic
497C2×C7#336111yescharacteristic
498C2×C7#336511yescharacteristic
499C2×C7#338011yescharacteristic
500C2×C7#338111yescharacteristic
501D7—82no—
502C23—142no—
503C23—281no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23—281no—
509C23C22×C22⋊D711yescharacteristic
510Q8—28no—
511D4—142no—
512D4—142no—
513D4—281no—
514D4—281no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—564no—
551C2×C4#650212yes—
552C2×C4#650212yes—
553C2×C4#651312yes—
554C2×C4#651312yes—
555C2×C4C2×D4×D711yescharacteristic
556C2×C4C2×#115911yescharacteristic
557C2×C4C2×Q8×D711yescharacteristic
558C2×C4C2×#116111yescharacteristic
559C2×C4#655111yescharacteristic
560C2×C4#655111yescharacteristic
561C8—142no—
562C8—142no—
563C8—142no—
564C8—281no—
565C7#2547311yescharacteristic, complemented, Sylow
566C22—282no—
567C22—282no—
568C22—282no—
569C22—561no—
570C22—561no—
571C22—561no—
572C22—561no—
573C22C2×#650211yescharacteristic
574C22Q8×C22⋊D711yescharacteristic
575C22#9050811yescharacteristic
576C22C2×#651311yescharacteristic
577C22C2×#651311yescharacteristic
578C22#9052611yescharacteristic
579C22#9052811yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—561no—
591C4—561no—
592C4—561no—
593C4#9041711yescharacteristic
594C4#9041811yescharacteristic
595C4#9041911yescharacteristic
596C4#9042211yescharacteristic
597C2—562no—
598C2#1174036711yescharacteristic
599C2#1174069911yescharacteristic
600C2#1174086711yescharacteristic
601C2#1174107311yescharacteristic
602C2#1174108811yescharacteristic
603C2#1174113811yescharacteristic
604C2#1174114211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2584C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2584D7
hypocenterC7#25473
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×Q8×C7⋊C8C2semidirect
5. #11729494C2semidirect
8. #11727378C2semidirect
9. #11727377C2semidirect
10. #11727373C2semidirect
11. #11727372C2semidirect
15. #11727222C2semidirect
114. #6383Q8semidirect
127. #2584D7semidirect
565. C7#25473semidirect

Maximal subgroups

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

subgroupindex
C7×#25842
C2×#898382
#117399462
C2×Q8×C7⋊C82
#117294942
#117273842
#117273832
#117273782
#117273772
#117273732
#117273722
#117273692
C4×#63832
#117272222
#254737