Finite Group Databasebeta

← #13395178

Subgroups — #13395178

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2323C211yesmaximal, characteristic
2C2×#89794C211yesmaximal, characteristic, complemented
3#11739710C211yesmaximal, characteristic
4C2×#89772C211yesmaximal, characteristic
5#11729449C211yesmaximal, characteristic
6#11729449C211yesmaximal, characteristic
7#11729380C211yesmaximal, characteristic, complemented
8#11727429C211yesmaximal, characteristic, complemented
9#11727428C211yesmaximal, characteristic, complemented
10#11727374C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727272C211yesmaximal, characteristic
13#11727261C211yesmaximal, characteristic
14#11727245C211yesmaximal, characteristic, complemented
15#11727220C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×#6433C2211yescharacteristic
20C7×#391C2211yescharacteristic
21C7×#389—21no—
22C7×#389C2211yescharacteristic
23C7×#388—21no—
24C7×#388—21no—
25C7×#387—21no—
26C7×#380C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32#89984C2211yescharacteristic
33#89984C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36#89794—24no—
37#89772—24no—
38#89756—21no—
39#89756—21no—
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89720—21no—
45#89720—21no—
46#89707C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#24715—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C7×#117—22no—
69C7×#117—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6433—22no—
103#6433—22no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109#6409—22no—
110#6409—22no—
111#6409—22no—
112#6409—22no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C22⋊(C7⋊C8)—22no—
118C22⋊(C7⋊C8)—22no—
119#6382—22no—
120#6381—22no—
121#6381—22no—
122#6381—22no—
123C4⋊(C7⋊C8)—22no—
124C4⋊(C7⋊C8)D414yes—
125#6377—22no—
126C4×C7⋊C8—22no—
127C2×#479—71no—
128C2×#433—71no—
129#2629—71no—
130#2323D711yescharacteristic
131#2077—71no—
132#2075—71no—
133#2036—71no—
134#2036—71no—
135#1989—71no—
136#1947—71no—
137#1943—71no—
138#1942—71no—
139#1935—71no—
140#1933—71no—
141#1803—71no—
142C24×C7C2×D411yescharacteristic
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8#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×C7—21no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
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—81no—
179C7×C4⋊C4#4112yes—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—81no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—42no—
187C42×C7—81no—
188C42×C7C2×D412yes—
189C42×C7#4112yes—
190C22⋊Dic7—81no—
191C22⋊Dic7—81no—
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—81no—
201C4⋊Dic7—81no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—81no—
205Dic7⋊C4—81no—
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⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8C2×D412yes—
217C2×C7⋊C8C2×D412yes—
218C2×#125—71no—
219C2×C22⋊Q8C2×D711yescharacteristic
220C2×C4×Q8C2×D711yescharacteristic
221C2×#117C2×D711yescharacteristic
222#479—144no—
223#433—144no—
224C2×#107—71no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#106—71no—
228C2×C4⋊C8—71no—
229C2×C4⋊C8—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×C22⋊C8—71no—
235C2×C22⋊C8—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#398—141no—
239#398—141no—
240#396—141no—
241#396—141no—
242#391C2×D711yescharacteristic
243#389—21no—
244#389—71no—
245#389—71no—
246#389C2×D711yescharacteristic
247#388—21no—
248#388—21no—
249#387—21no—
250#384—71no—
251#380—71no—
252#380—71no—
253#380C2×D711yescharacteristic
254C4×C4⋊C4C2×D711yescharacteristic
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7C22×D411yescharacteristic
271C7×Q8—22no—
272C7×Q8—22no—
273C7×Q8—41no—
274C7×Q8—41no—
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—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7#12112yes—
305C2×C4×C7#12112yes—
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C72−1+411yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C2×Dic7—41no—
313C2×Dic7—41no—
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—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C7⋊C8—41no—
327C7⋊C8—41no—
328C22×Q8C22×D711yescharacteristic
329C23×C4C22×D711yescharacteristic
330C22×C8—71no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8—71no—
334#125—142no—
335#125—142no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340#117—22no—
341#117—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4C22⋊D711yescharacteristic
356C2×C4⋊C4C22⋊D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4C22⋊D711yescharacteristic
365C2×C22⋊C4C22⋊D711yescharacteristic
366C2×C42—21no—
367C2×C42—21no—
368C2×C42C22×D711yescharacteristic
369C2×C42C22×D711yescharacteristic
370#107—142no—
371#107—142no—
372#107—142no—
373#106—142no—
374C4⋊C8—74no—
375C4⋊C8—142no—
376#103—142no—
377#103—142no—
378#103—142no—
379#103—142no—
380C22⋊C8—142no—
381C22⋊C8—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—
391#95—141no—
392#95—141no—
393C22×C7—41no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7C2×#12111yescharacteristic
401C22×C7#52511yescharacteristic
402C22×C7#53611yescharacteristic
403C22×C7#57611yescharacteristic
404C22×C7#57711yescharacteristic
405C22×C7#57811yescharacteristic
406C22×C7#57811yescharacteristic
407C4×C7—22no—
408C4×C7—22no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—81no—
415C4×C7—81no—
416C4×C7#54711yescharacteristic
417C4×C7#54811yescharacteristic
418C4×C7#55211yescharacteristic
419C4×C7#55611yescharacteristic
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423Dic7—81no—
424C24C2×C22⋊D711yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#115712yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451C2×C8—72no—
452C2×C8—72no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—282no—
482C4⋊C4—282no—
483C4⋊C4—561no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4—561no—
487C4⋊C4#115712yes—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—81no—
492C22⋊C4—561no—
493C22⋊C4—561no—
494C42—21no—
495C42—21no—
496C42—42no—
497C42—81no—
498C42D4×D712yes—
499C42#115912yes—
500C2×C7—81no—
501C2×C7#241511yescharacteristic
502C2×C7#306111yescharacteristic
503C2×C7#306311yescharacteristic
504C2×C7#311911yescharacteristic
505C2×C7#312711yescharacteristic
506C2×C7#324211yescharacteristic
507C2×C7#324311yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23C22×C22⋊D711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
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—41no—
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—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4#650212yes—
562C2×C4#650212yes—
563C2×C4C2×#115711yescharacteristic
564C2×C4C2×#115711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4#655411yescharacteristic
568C2×C4#655411yescharacteristic
569C8—142no—
570C8—142no—
571C8—281no—
572C8—281no—
573C7#2471511yescharacteristic, complemented, Sylow
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22#9048211yescharacteristic
582C22C2×#650211yescharacteristic
583C22#9051611yescharacteristic
584C22#9052611yescharacteristic
585C22#9052811yescharacteristic
586C22#9052911yescharacteristic
587C22#9052911yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—561no—
598C4—561no—
599C4—561no—
600C4—561no—
601C4#9023211yescharacteristic
602C4#9023311yescharacteristic
603C4#9024211yescharacteristic
604C4#9024311yescharacteristic
605C2—81no—
606C2#1173979611yescharacteristic
607C2#1174000011yescharacteristic
608C2#1174001311yescharacteristic
609C2#1174084511yescharacteristic
610C2#1174086611yescharacteristic
611C2#1174088711yescharacteristic
612C2#1174089911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2323C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2323D7
hypocenterC7#24715
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#89794C2semidirect
7. #11729380C2semidirect
8. #11727429C2semidirect
9. #11727428C2semidirect
10. #11727374C2semidirect
11. #11727373C2semidirect
14. #11727245C2semidirect
15. #11727220C2semidirect
573. C7#24715semidirect

Maximal subgroups

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

subgroupindex
C7×#23232
C2×#897942
#117397102
C2×#897722
#117294492
#117293802
#117274292
#117274282
#117273742
#117273732
#117272722
#117272612
#117272452
#117272202
#247157