Finite Group Databasebeta

← #13373143

Subgroups — #13373143

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
1C7×#2357C211yesmaximal, characteristic
2C2×#89790C211yesmaximal, characteristic
3C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
4#11729358C211yesmaximal, characteristic
5#11727414C211yesmaximal, characteristic
6#11727414C211yesmaximal, characteristic
7#11727399C211yesmaximal, characteristic
8#11727398C211yesmaximal, characteristic
9#11727313C211yesmaximal, characteristic, complemented
10#11727302C211yesmaximal, characteristic, complemented
11#11727232C211yesmaximal, characteristic, complemented
12#11727229C211yesmaximal, characteristic, complemented
13C4×#6384C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727212C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C22×#1142C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C7×#390C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#388—21no—
24C7×#388—21no—
25C7×#382C2211yescharacteristic
26C7×#381—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4—21no—
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89790C418yescomplemented
35#89748—21no—
36C4⋊C4×Dic7—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89707C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89419—21no—
49#89419—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22775—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C22×Dic14C2311yescharacteristic
64C4×C7×D4—22no—
65C4×C7×D4C2×C414yes—
66C2×C7×C4⋊C4D411yescharacteristic
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×#1142—22no—
82C2×#1142—22no—
83C2×#1142C2×C414yes—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C4×Dic14—22no—
102C4×Dic14—22no—
103C4×Dic14—22no—
104C4×Dic14—41no—
105C4×Dic14—41no—
106C4×Dic14C2×C414yes—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95D411yescharacteristic
112C7×#95D411yescharacteristic
113#6406—41no—
114#6406—41no—
115#6406C2×C414yes—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6384—22no—
119#6384—22no—
120#6384C2×C414yes—
121#6381C2×C414yes—
122C4⋊(C7⋊C8)—22no—
123#6377—41no—
124#6377C2×C414yes—
125C4×C7⋊C8—41no—
126C2×#440—71no—
127#2357D711yescharacteristic
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×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4C22×C412yes—
148C22×C4×C7—21no—
149C22×C4×C7—24no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—81no—
153C22×C4×C7—81no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14C22×C412yes—
169C2×Dic14#4112yes—
170C2×Dic14#4112yes—
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—41no—
178C7×C4⋊C4C22×C412yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—81no—
183C7×C22⋊C4—81no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—81no—
189C42×C7—81no—
190C42×C7C22×C412yes—
191C42×C7#4112yes—
192#1142—28no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7—81no—
200C4⋊Dic7C22×C412yes—
201Dic7⋊C4—24no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—81no—
205Dic7⋊C4—81no—
206C4×Dic7—24no—
207C4×Dic7—42no—
208C4×Dic7—42no—
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⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C22×C412yes—
220C2×C7⋊C8C22×C412yes—
221C22×SD16—71no—
222C2×C4×Q8—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×D4C2×D711yescharacteristic
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#97—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#390C2×D711yescharacteristic
239#390C2×D711yescharacteristic
240#389—71no—
241#389—71no—
242#389—141no—
243#389—141no—
244#388—21no—
245#388—21no—
246#384—71no—
247#384—141no—
248#382C2×D711yescharacteristic
249#381—21no—
250C4×C4⋊C4—141no—
251C4×C4⋊C4C2×D711yescharacteristic
252C4×C4⋊C4C2×D711yescharacteristic
253C4×C4⋊C4C2×D711yescharacteristic
254C4×C22⋊C4—21no—
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×D4—22no—
274C7×D4—22no—
275C7×D4—41no—
276C7×D4—41no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
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—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
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—84no—
302C2×C4×C7C4×D412yes—
303C2×C4×C7C4×D412yes—
304C2×C4×C7#13712yes—
305C2×C4×C7C23×C411yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7H(5,𝔽2)11yescharacteristic
310C2×C4×C72−1+411yescharacteristic
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323Dic14—22no—
324Dic14—22no—
325Dic14—22no—
326Dic14#11714yes—
327C7⋊C8—22no—
328C7⋊C8—22no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×Q8—71no—
332C22×D4C22×D711yescharacteristic
333C23×C4—21no—
334C2×SD16—74no—
335C2×SD16—142no—
336C2×SD16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C4×Q8—74no—
342C4×Q8—142no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×Q8—281no—
346C4×Q8—281no—
347C4×D4—22no—
348C4×D4C4×D714yes—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4C22⋊D711yescharacteristic
360C2×C4⋊C4C22⋊D711yescharacteristic
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377#107—74no—
378C4⋊C8—142no—
379#103—74no—
380#103—142no—
381#103—142no—
382#102—74no—
383#102—281no—
384#102—281no—
385#97—74no—
386#97—281no—
387C4×C8—281no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—141no—
393#95—141no—
394#95C22⋊D711yescharacteristic
395#95C22⋊D711yescharacteristic
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—22no—
411C4×C7—22no—
412C4×C7—22no—
413C4×C7—42no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7—41no—
417C4×C7—81no—
418C4×C7—81no—
419C4×C7C2×#11711yescharacteristic
420C4×C7#52011yescharacteristic
421C4×C7#55211yescharacteristic
422C4×C7#55311yescharacteristic
423Dic7—24no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427C24—21no—
428C2×Q8—72no—
429C2×Q8—72no—
430C2×Q8—72no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×C4×D712yes—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457SD16—148no—
458C2×C8—72no—
459C2×C8—72no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—72no—
478C4⋊C4—144no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4—561no—
488C4⋊C4—561no—
489C4⋊C4—561no—
490C4⋊C4C2×C4×D712yes—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—81no—
502C42—144no—
503C42—282no—
504C42—282no—
505C42C2×C4×D712yes—
506C42#115712yes—
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—
522C23C22×C22⋊D711yescharacteristic
523Q8—74no—
524Q8—142no—
525Q8—142no—
526Q8—142no—
527D4—22no—
528D4—22no—
529D4—41no—
530D4—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
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×C4C4×C22⋊D712yes—
569C2×C4C4×C22⋊D712yes—
570C2×C4#651412yes—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×#115711yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C2×C4#655311yescharacteristic
576C2×C4#655411yescharacteristic
577C8—142no—
578C8—142no—
579C8—281no—
580C8—281no—
581C7#2277511yescharacteristic, complemented, Sylow
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×C4×C22⋊D711yescharacteristic
590C22#9051311yescharacteristic
591C22C2×#651411yescharacteristic
592C22#9052011yescharacteristic
593C22#9052211yescharacteristic
594C22#9052611yescharacteristic
595C22#9052711yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—144no—
606C4—282no—
607C4—561no—
608C4—561no—
609C4D7×#11711yescharacteristic
610C4#9021411yescharacteristic
611C4#9022011yescharacteristic
612C4#9022111yescharacteristic
613C2—82no—
614C2#1173971711yescharacteristic
615C2C4×#651411yescharacteristic
616C2#1173998211yescharacteristic
617C2#1174106411yescharacteristic
618C2#1174106511yescharacteristic
619C2#1174116311yescharacteristic
620C2#1174116411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2357C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2357D7
hypocenterC7#22775
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×Dic14C2semidirect
9. #11727313C2semidirect
10. #11727302C2semidirect
11. #11727232C2semidirect
12. #11727229C2semidirect
13. C4×#6384C2semidirect
14. #11727215C2semidirect
15. #11727212C2semidirect
34. #89790C4semidirect
581. C7#22775semidirect

Maximal subgroups

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

subgroupindex
C7×#23572
C2×#897902
C4⋊C4×Dic142
#117293582
#117274142
#117273992
#117273982
#117273132
#117273022
#117272322
#117272292
C4×#63842
#117272152
#117272122
#227757