Finite Group Databasebeta

← #13372176

Subgroups — #13372176

Order 1792.

906 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
1C2×C7×#441C211yesmaximal, characteristic
2C2×#89956C211yesmaximal, characteristic
3C2×Q16×Dic7C211yesmaximal, characteristic
4#11739905C211yesmaximal, characteristic
5Q8×C4⋊Dic7C211yesmaximal, characteristic
6#11729381C211yesmaximal, characteristic
7#11729380C211yesmaximal, characteristic
8#11728316C211yesmaximal, characteristic
9#11728314C211yesmaximal, characteristic
10#11727992C211yesmaximal, characteristic
11#11727992C211yesmaximal, characteristic
12#11727990C211yesmaximal, characteristic
13#11727990C211yesmaximal, characteristic
14#11727570C211yesmaximal, characteristic
15#11727564C211yesmaximal, characteristic
16C22×C7×Q16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23C7×#441C418yescomplemented
24C2×C7×#106C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32#89956—24no—
33Q16×Dic7—24no—
34C2×#6392C2211yescharacteristic
35C2×#6392C2211yescharacteristic
36C2×#6391C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6389C2211yescharacteristic
39C2×C8×Dic7C2211yescharacteristic
40#89757—21no—
41#89757—21no—
42#89757—21no—
43#89757—21no—
44#89757C2211yescharacteristic
45#89757C2211yescharacteristic
46#89757C2211yescharacteristic
47#89757C2211yescharacteristic
48#89708—21no—
49#89708C2211yescharacteristic
50#89707—21no—
51#89707C2211yescharacteristic
52C4×C4⋊Dic7—21no—
53C4×C4⋊Dic7—21no—
54C4×C4⋊Dic7C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22679—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2311yescharacteristic
65C2×C7×Q16C2×C414yes—
66C2×C7×Q16D414yes—
67C2×C7×Q16Q814yes—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×Q8C2×C414yes—
73C4×C7×Q8C2×C414yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7—21no—
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81Q8×Dic7—22no—
82Q8×Dic7—22no—
83Q8×Dic7—22no—
84Q8×Dic7—22no—
85Q8×Dic7—41no—
86Q8×Dic7—41no—
87Q8×Dic7—41no—
88Q8×Dic7—41no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7D411yescharacteristic
100C2×C4⋊Dic7D411yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4⋊Dic7C2311yescharacteristic
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2311yescharacteristic
113C7×#106C2×C414yes—
114C7×#103C2×C414yes—
115C7×#103C2×C414yes—
116C7×#97C2×C414yes—
117#6409—22no—
118#6409—22no—
119#6409—22no—
120#6409—22no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95D411yescharacteristic
128C7⋊#95D411yescharacteristic
129#6392—22no—
130#6392—41no—
131#6391—22no—
132#6391—41no—
133#6389—22no—
134C8×Dic7—22no—
135C4⋊(C7⋊C8)—22no—
136C2×#441—71no—
137C2×#441D711yescharacteristic
138C2×C4×Q16—71no—
139#2351—71no—
140Q8×C4⋊C4—71no—
141#1944—71no—
142#1941—71no—
143#1865—71no—
144#1865—71no—
145#1863—71no—
146#1863—71no—
147#1847—71no—
148#1805—71no—
149#1803—71no—
150#1771—71no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×Q8C22×C412yes—
159C2×C7×Q8C2×D412yes—
160C2×C7×Q8C2×D412yes—
161C2×C7×Q8C2×Q812yes—
162C2×C7×Q8C2×Q812yes—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C7×Q16C4⋊C4116yescomplemented
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8C22×C412yes—
182C2×C7×C8C22×C412yes—
183C2×C7×C8C2×D412yes—
184C2×C7×C8C2×Q812yes—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4C22×C412yes—
190C7×C4⋊C4C22×C412yes—
191C42×C7—24no—
192C42×C7—24no—
193C42×C7C22×C412yes—
194C42×C7#4112yes—
195C4⋊Dic7—22no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—41no—
205C4⋊Dic7—42no—
206C4⋊Dic7—42no—
207C4⋊Dic7—42no—
208C4⋊Dic7—42no—
209C4⋊Dic7—42no—
210C4⋊Dic7—42no—
211C4⋊Dic7—42no—
212C4⋊Dic7—81no—
213C4⋊Dic7—81no—
214C4⋊Dic7—81no—
215C4×Dic7—21no—
216C4×Dic7—21no—
217C4×Dic7—21no—
218C4×Dic7—21no—
219C4×Dic7—42no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C22×Q16C2×D711yescharacteristic
230C2×C4×Q8—71no—
231C2×C4×Q8—71no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×Q8C2×D711yescharacteristic
235C2×C4×Q8C2×D711yescharacteristic
236#441—144no—
237#441Dic718yescomplemented
238C4×Q16—144no—
239C2×#107—71no—
240C2×#107—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×#106C2×D711yescharacteristic
244C2×C4⋊C8—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103C2×D711yescharacteristic
250C2×#103C2×D711yescharacteristic
251C2×#97—71no—
252C2×#97C2×D711yescharacteristic
253C2×C4×C8—71no—
254#389—71no—
255#389—71no—
256#389—71no—
257#389—71no—
258#389—141no—
259#389—141no—
260#389—141no—
261#389—141no—
262#384—71no—
263#384—141no—
264#382—71no—
265#382—141no—
266C4×C4⋊C4—71no—
267C4×C4⋊C4—141no—
268C4×C4⋊C4—141no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#336—71no—
274#336—71no—
275C23×C7C22×D411yescharacteristic
276C7×Q8—22no—
277C7×Q8—22no—
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8—22no—
282C7×Q8C2×C4⋊C414yes—
283C7×Q8C2×C4⋊C414yes—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7C4×D412yes—
301C2×C4×C7C4×D412yes—
302C2×C4×C7C23×C411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×Q811yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7H(5,𝔽2)11yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
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×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7#13712yes—
332C7×C8—22no—
333C7×C8C2×C4⋊C414yes—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8C22×D711yescharacteristic
337C22×Q8C22×D711yescharacteristic
338C2×Q16Dic1414yes—
339C2×Q16D2814yes—
340C2×Q16C2×Dic714yes—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—142no—
348C4×Q8—142no—
349C4×Q8—142no—
350C4×Q8—142no—
351C4×Q8—281no—
352C4×Q8—281no—
353C4×Q8—281no—
354C4×Q8—281no—
355C4×Q8C2×Dic714yes—
356C4×Q8C2×Dic714yes—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42C22×D711yescharacteristic
386#107—142no—
387#107—281no—
388#106—142no—
389#106—281no—
390#106C2×Dic714yes—
391C4⋊C8—142no—
392#103—142no—
393#103—142no—
394#103—142no—
395#103—142no—
396#103C2×Dic714yes—
397#103C2×Dic714yes—
398#97—142no—
399#97C2×Dic714yes—
400C4×C8—142no—
401#95—71no—
402#95—71no—
403#95—141no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409C22×C7C2×C4×D411yescharacteristic
410C22×C7D4211yescharacteristic
411C22×C7D4×Q811yescharacteristic
412C22×C7C2×#13711yescharacteristic
413C22×C7#57511yescharacteristic
414C22×C7#57711yescharacteristic
415C22×C7#57811yescharacteristic
416C4×C7—24no—
417C4×C7—24no—
418C4×C7—22no—
419C4×C7—22no—
420C4×C7—42no—
421C4×C7—42no—
422C4×C7C22×C4⋊C411yescharacteristic
423C4×C7#52011yescharacteristic
424C4×C7#54711yescharacteristic
425C4×C7#55411yescharacteristic
426Dic7—22no—
427Dic7—41no—
428Dic7—41no—
429Dic7—41no—
430Dic7—81no—
431Dic7—81no—
432Dic7—81no—
433Dic7—81no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×Dic1412yes—
441C2×Q8C2×Dic1412yes—
442C2×Q8C2×D2812yes—
443C2×Q8C2×D2812yes—
444C2×Q8C22×Dic712yes—
445C2×Q8C22×Dic712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4C23×D711yescharacteristic
461Q16C4⋊Dic7116yescomplemented
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×C8C2×Dic1412yes—
471C2×C8C2×D2812yes—
472C2×C8C22×Dic712yes—
473C2×C8C22×Dic712yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—142no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—281no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C4⋊C4—561no—
498C4⋊C4C22×Dic712yes—
499C4⋊C4C22×Dic712yes—
500C42—24no—
501C42—24no—
502C42—141no—
503C42—141no—
504C42—141no—
505C42—141no—
506C42—282no—
507C42—282no—
508C42—282no—
509C42—282no—
510C42#115912yes—
511C42C22×Dic712yes—
512C2×C7D4×C4⋊C411yescharacteristic
513C2×C7C4×#13711yescharacteristic
514C2×C7#294711yescharacteristic
515C2×C7#327011yescharacteristic
516C2×C7#327111yescharacteristic
517C2×C7#338711yescharacteristic
518C2×C7#339011yescharacteristic
519C23C2×D4×D711yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8C2×C4⋊Dic714yes—
527Q8C2×C4⋊Dic714yes—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—72no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
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—281no—
557C2×C4—281no—
558C2×C4—281no—
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×C4D4×Dic712yes—
569C2×C4D4×Dic712yes—
570C2×C4C22×Dic1411yescharacteristic
571C2×C4C22×D2811yescharacteristic
572C2×C4C2×#115911yescharacteristic
573C2×C4#655311yescharacteristic
574C2×C4#655411yescharacteristic
575C2×C4C23×Dic711yescharacteristic
576C8—22no—
577C8—281no—
578C8—281no—
579C8C2×C4⋊Dic714yes—
580C7#2267911yescharacteristic, complemented, Sylow
581C22D4×Dic1411yescharacteristic
582C22D4×D2811yescharacteristic
583C22D7×#13711yescharacteristic
584C22#9046811yescharacteristic
585C22#9046911yescharacteristic
586C22#9047011yescharacteristic
587C22C2×D4×Dic711yescharacteristic
588C4—24no—
589C4—24no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—142no—
595C4—281no—
596C4—281no—
597C4—281no—
598C4—561no—
599C4—561no—
600C4—561no—
601C4—561no—
602C4#9023411yescharacteristic
603C4#9024611yescharacteristic
604C4C22×C4⋊Dic711yescharacteristic
605C4#9051811yescharacteristic
606C2#1173726411yescharacteristic
607C2#1173726511yescharacteristic
608C2#1173728411yescharacteristic
609C2#1173728511yescharacteristic
610C2D4×C4⋊Dic711yescharacteristic
611C2Dic7×#13711yescharacteristic
612C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#441C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#441D7
hypocenterC7#22679
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
23. C7×#441C4semidirect
178. C7×Q16C4⋊C4semidirect
237. #441Dic7semidirect
461. Q16C4⋊Dic7semidirect
580. C7#22679semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4412
C2×#899562
C2×Q16×Dic72
#117399052
Q8×C4⋊Dic72
#117293812
#117293802
#117283162
#117283142
#117279922
#117279902
#117275702
#117275642
#226797