Finite Group Databasebeta

← #13372178

Subgroups — #13372178

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×C4×C7×Q16C211yesmaximal, characteristic
2C2×#89956C211yesmaximal, characteristic
3C2×#89956C211yesmaximal, characteristic
4#11739906C211yesmaximal, characteristic
5Q8×Dic7⋊C4C211yesmaximal, characteristic
6#11729383C211yesmaximal, characteristic
7#11729382C211yesmaximal, characteristic
8#11728368C211yesmaximal, characteristic
9#11728362C211yesmaximal, characteristic
10#11727991C211yesmaximal, characteristic
11#11727989C211yesmaximal, characteristic
12#11727976C211yesmaximal, characteristic
13#11727975C211yesmaximal, characteristic
14#11727537C211yesmaximal, characteristic
15#11727523C211yesmaximal, characteristic
16C22×C7×Q16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23C4×C7×Q16—24no—
24C2×C7×#107C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C4×C7×C8C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32#89956—24no—
33#89956C418yescomplemented
34C2×#6391C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6389C2211yescharacteristic
37C2×#6389C2211yescharacteristic
38C2×Dic7⋊C8C2211yescharacteristic
39#89749—21no—
40#89749C2211yescharacteristic
41#89748—21no—
42#89748C2211yescharacteristic
43#89745—21no—
44#89745—21no—
45#89745C2211yescharacteristic
46#89745C2211yescharacteristic
47C2×#6382C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49C4×Dic7⋊C4—21no—
50C4×Dic7⋊C4—21no—
51C4×Dic7⋊C4C2211yescharacteristic
52#89700—21no—
53#89700—21no—
54#89700C2211yescharacteristic
55#89700C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89319C2211yescharacteristic
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×Q8—41no—
73C4×C7×Q8—41no—
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×Dic7C2×C414yes—
88Q8×Dic7C2×C414yes—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4D411yescharacteristic
104C2×Dic7⋊C4D411yescharacteristic
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×#107—22no—
114C7×#103—22no—
115C7×#103—22no—
116C4×C7×C8—22no—
117#6409—22no—
118#6409—22no—
119#6409C2×C414yes—
120#6409C2×C414yes—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95D411yescharacteristic
128C7⋊#95D411yescharacteristic
129#6391—22no—
130#6391C2×C414yes—
131#6389—22no—
132#6389C2×C414yes—
133Dic7⋊C8—22no—
134#6382—41no—
135#6381—41no—
136C2×#441—71no—
137C2×#441—71no—
138C2×C4×Q16D711yescharacteristic
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×C8—21no—
182C2×C7×C8C22×C412yes—
183C2×C7×C8C2×D412yes—
184C2×C7×C8C2×Q812yes—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—42no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—42no—
194C42×C7—42no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—24no—
198C4⋊Dic7—24no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—42no—
204C4⋊Dic7—42no—
205C4⋊Dic7C22×C412yes—
206C4⋊Dic7C22×C412yes—
207Dic7⋊C4—22no—
208Dic7⋊C4—41no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211Dic7⋊C4—42no—
212Dic7⋊C4—81no—
213Dic7⋊C4—81no—
214Dic7⋊C4—81no—
215C4×Dic7—21no—
216C4×Dic7—21no—
217C4×Dic7—24no—
218C4×Dic7—24no—
219C4×Dic7—42no—
220C4×Dic7—42no—
221C4×Dic7C22×C412yes—
222C4×Dic7#4112yes—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8C22×C412yes—
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—78no—
237#441—144no—
238C4×Q16—24no—
239C2×#107—71no—
240C2×#107C2×D711yescharacteristic
241C2×#106—71no—
242C2×#106—71no—
243C2×#106—71no—
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×#97—71no—
253C2×C4×C8C2×D711yescharacteristic
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—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
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#13712yes—
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—22no—
313C2×Dic7—22no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
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×Dic7C4×D412yes—
331C2×Dic7C4×D412yes—
332C7×C8—41no—
333C7×C8C2×C4⋊C414yes—
334C7⋊C8—22no—
335C7⋊C8—41no—
336C22×Q8C22×D711yescharacteristic
337C22×Q8C22×D711yescharacteristic
338C2×Q16Dic1414yes—
339C2×Q16C4×D714yes—
340C2×Q16C22⋊D714yes—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—41no—
348C4×Q8—41no—
349C4×Q8—74no—
350C4×Q8—74no—
351C4×Q8—142no—
352C4×Q8—142no—
353C4×Q8—142no—
354C4×Q8—142no—
355C4×Q8—281no—
356C4×Q8—281no—
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—22no—
387#107—281no—
388#106—74no—
389#106—142no—
390#106—281no—
391C4⋊C8—142no—
392#103—22no—
393#103—22no—
394#103—74no—
395#103—74no—
396#103—142no—
397#103—142no—
398#97—74no—
399#97—142no—
400C4×C8—22no—
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—41no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7C22×C4⋊C411yescharacteristic
423C4×C7#52011yescharacteristic
424C4×C7#54711yescharacteristic
425C4×C7#55411yescharacteristic
426Dic7—22no—
427Dic7—22no—
428Dic7—42no—
429Dic7—42no—
430Dic7—41no—
431Dic7—41no—
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×C4×D712yes—
443C2×Q8C2×C4×D712yes—
444C2×Q8C2×C22⋊D712yes—
445C2×Q8C2×C22⋊D712yes—
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
461Q16Dic7⋊C4116yescomplemented
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8C2×Dic1412yes—
472C2×C8C2×C4×D712yes—
473C2×C8C2×C22⋊D712yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—72no—
481C4⋊C4—72no—
482C4⋊C4—144no—
483C4⋊C4—144no—
484C4⋊C4—142no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—281no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—561no—
498C4⋊C4—561no—
499C4⋊C4—561no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—42no—
504C42—72no—
505C42—72no—
506C42—144no—
507C42—144no—
508C42—141no—
509C42—141no—
510C42—282no—
511C42—282no—
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×Dic7⋊C414yes—
527Q8C2×Dic7⋊C414yes—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—72no—
546C2×C4—72no—
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—142no—
556C2×C4—142no—
557C2×C4—141no—
558C2×C4—141no—
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×C4—281no—
569C2×C4#648112yes—
570C2×C4C22×Dic1411yescharacteristic
571C2×C4C22×C4×D711yescharacteristic
572C2×C4#654811yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4#655411yescharacteristic
575C2×C4C22×C22⋊D711yescharacteristic
576C8—41no—
577C8—142no—
578C8—281no—
579C8C2×Dic7⋊C414yes—
580C7#2267911yescharacteristic, complemented, Sylow
581C22D4×Dic1411yescharacteristic
582C22C4×D4×D711yescharacteristic
583C22C2×#648111yescharacteristic
584C22#9045711yescharacteristic
585C22#9045911yescharacteristic
586C22#9046111yescharacteristic
587C22D4×C22⋊D711yescharacteristic
588C4—24no—
589C4—24no—
590C4—22no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—142no—
595C4—142no—
596C4—282no—
597C4—282no—
598C4—281no—
599C4—281no—
600C4—561no—
601C4—561no—
602C4#9023011yescharacteristic
603C4#9023711yescharacteristic
604C4C22×Dic7⋊C411yescharacteristic
605C4#9051511yescharacteristic
606C2#1173715311yescharacteristic
607C2#1173715411yescharacteristic
608C2C4×#648111yescharacteristic
609C2#1173716411yescharacteristic
610C2D4×Dic7⋊C411yescharacteristic
611C2#1174133311yescharacteristic
612C2#1174136811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×Q16C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×C4×Q16D7
hypocenterC7#22679
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
33. #89956C4semidirect
178. C7×Q16C4⋊C4semidirect
461. Q16Dic7⋊C4semidirect
580. C7#22679semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×Q162
C2×#899562
#117399062
Q8×Dic7⋊C42
#117293832
#117293822
#117283682
#117283622
#117279912
#117279892
#117279762
#117279752
#117275372
#117275232
#226797