Finite Group Databasebeta

← #13370424

Subgroups — #13370424

Order 1792.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90381C211yesmaximal, characteristic
2C2×#89600C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
4C7×#1989C211yesmaximal, characteristic, complemented
5#11729495C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic
7#11727228C211yesmaximal, characteristic
8#11726646C211yesmaximal, characteristic
9#11726646C211yesmaximal, characteristic
10#11726559C211yesmaximal, characteristic, complemented
11#11726557C211yesmaximal, characteristic, complemented
12#11726555C211yesmaximal, characteristic, complemented
13#11726553C211yesmaximal, characteristic, complemented
14#11726536C211yesmaximal, characteristic
15#11726534C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17#90381—24no—
18#90381C418yescomplemented
19C2×#6460—21no—
20C2×#6459—21no—
21C2×#6459—21no—
22C2×#6456C2211yescharacteristic
23C2×#6456C2211yescharacteristic
24C2×D7×C4⋊C4—21no—
25C2×#6453C2211yescharacteristic
26C2×#6451C2211yescharacteristic
27C2×C4×D28C2211yescharacteristic
28C2×C4×Dic14—21no—
29C2×C4×Dic14C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C4×C7×C4⋊C4C2211yescharacteristic
32C2×#6395C2211yescharacteristic
33C2×#6395C2211yescharacteristic
34C2×#6389C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36C2×Dic7⋊C8C2211yescharacteristic
37C2×C8×Dic7C2211yescharacteristic
38#89748C2211yescharacteristic
39#89748C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C2×#6383C2211yescharacteristic
43C2×#6383C2211yescharacteristic
44#89707C2211yescharacteristic
45#89700—21no—
46C2×#6377C2211yescharacteristic
47#89600—24no—
48#89419—21no—
49#89419—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22522—71nomaximal, Sylow
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C22×Dic14—21no—
68C22×Dic14D411yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C7×#125—22no—
72C7×#125C2×C414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7D411yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4D411yescharacteristic
98C2×Dic7⋊C4D411yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6460—28no—
106#6459—28no—
107#6459—28no—
108#6456—22no—
109#6456—22no—
110#6456C2×C414yes—
111#6456C2×C414yes—
112D7×C4⋊C4—28no—
113#6453—22no—
114#6453C2×C414yes—
115#6451—22no—
116#6451C2×C414yes—
117C4×D28—22no—
118C4×D28C2×C414yes—
119C4×Dic14—44no—
120C4×Dic14C2×C414yes—
121C4×Dic14D414yes—
122C7×C4⋊C8—22no—
123C7×#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126#6395—22no—
127#6395—22no—
128#6389—22no—
129Dic7⋊C8—22no—
130Dic7⋊C8—22no—
131C8×Dic7—22no—
132#6383—22no—
133#6383Q814yescomplemented
134#6377—22no—
135C2×#554—71no—
136C2×#446—71no—
137Q8×C4⋊C4—71no—
138#1992—71no—
139#1992—71no—
140#1990—71no—
141#1989—71no—
142#1989D711yescharacteristic, complemented
143#1987—71no—
144#1802—71no—
145#1802—71no—
146#1802—71no—
147#1762—71no—
148#1762—71no—
149C4×#102—71no—
150C23×D7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28C22×C412yes—
169C2×D28C2×Q812yes—
170C2×D28#4112yes—
171C2×C4×D7—24no—
172C2×C4×D7—24no—
173C2×C4×D7—24no—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—81no—
177C2×C4×D7—81no—
178C2×C4×D7—81no—
179C2×C4×D7—81no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×Dic14—24no—
183C2×Dic14—81no—
184C2×Dic14—81no—
185C2×Dic14C22⋊C414yes—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C22×C412yes—
200C7×C4⋊C4C22×C412yes—
201C7×C4⋊C4C2×Q812yes—
202C42×C7—42no—
203C42×C7C22×C412yes—
204C42×C7C2×D412yes—
205(C2×D7)⋊C4—24no—
206(C2×D7)⋊C4—24no—
207(C2×D7)⋊C4—24no—
208C4⋊Dic7—21no—
209C4⋊Dic7—24no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7—81no—
216C4⋊Dic7C22⋊C414yes—
217C4⋊Dic7C22×C412yes—
218Dic7⋊C4—24no—
219Dic7⋊C4—24no—
220Dic7⋊C4—42no—
221Dic7⋊C4—42no—
222Dic7⋊C4—81no—
223Dic7⋊C4—81no—
224Dic7⋊C4C22⋊C414yes—
225Dic7⋊C4C22⋊C414yes—
226C4×Dic7—42no—
227C4×Dic7—42no—
228C4×Dic7C22×C412yes—
229C4×Dic7C22×C412yes—
230C4×Dic7C2×D412yes—
231C4×Dic7C2×D412yes—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8C2×Q812yes—
238#554—78no—
239#554—144no—
240C2×C4⋊Q8—71no—
241C2×#125—71no—
242C2×#125C2×D711yescharacteristic
243C2×C22⋊Q8—141no—
244C2×C22⋊Q8—141no—
245C2×C22⋊Q8—141no—
246C2×C4×Q8—71no—
247C2×C4×Q8—141no—
248C2×C4×D4—71no—
249C2×C4×D4—71no—
250C2×C4×D4—71no—
251C22×C4⋊C4—141no—
252#446—144no—
253C2×C4⋊C8—71no—
254C2×C4⋊C8—71no—
255C2×C4⋊C8C2×D711yescharacteristic
256C2×#102—71no—
257C2×#102—71no—
258C2×#102—71no—
259C2×#102—71no—
260C2×#97—71no—
261C2×#97—71no—
262C2×C4×C8—71no—
263#389—141no—
264#389—141no—
265#389—141no—
266#384—71no—
267#384—71no—
268#384—71no—
269C4×C4⋊C4—71no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4C2×D711yescharacteristic
272#340—71no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340C2×D711yescharacteristic
281#340C2×D711yescharacteristic
282#340C2×D711yescharacteristic
283#340C2×D711yescharacteristic
284C23×C7C22×D411yescharacteristic
285C22×D7—22no—
286C22×D7—22no—
287C22×D7—22no—
288C22×D7—41no—
289C22×D7—41no—
290C22×D7—41no—
291C22×D7—41no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7C2×C22⋊C412yes—
309C2×C4×C7C2×C22⋊C412yes—
310C2×C4×C7C23×C411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C22×Q811yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C72−1+411yescharacteristic
315C2×C4×C72−1+411yescharacteristic
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—22no—
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×Dic7C2×C22⋊C412yes—
332C2×Dic7C2×C22⋊C412yes—
333C2×Dic7C2×C22⋊C412yes—
334C2×Dic7C2×C22⋊C412yes—
335C2×Dic7AGL(1,ℤ/8)12yes—
336C2×Dic7#13712yes—
337D28—22no—
338D28—22no—
339D28—22no—
340D28C4×Q814yescomplemented
341C4×D7—48no—
342C4×D7—84no—
343C4×D7—84no—
344Dic14—28no—
345Dic14—84no—
346C7×C8—41no—
347C7×C8—41no—
348C7⋊C8—22no—
349C7⋊C8—41no—
350C22×Q8—71no—
351C22×Q8—141no—
352C22×D4—71no—
353C23×C4—141no—
354C23×C4—141no—
355C23×C4—141no—
356C22×C8—71no—
357C22×C8—71no—
358C22×C8C22×D711yescharacteristic
359C22×C8C22×D711yescharacteristic
360C4⋊Q8—74no—
361C4⋊Q8—142no—
362#125—22no—
363#125—74no—
364#125—142no—
365#125C4×D714yes—
366C22⋊Q8—148no—
367C22⋊Q8—148no—
368C22⋊Q8—148no—
369C4×Q8—74no—
370C4×Q8—74no—
371C4×Q8—284no—
372C4×D4—74no—
373C4×D4—74no—
374C4×D4—74no—
375C4×D4—142no—
376C4×D4—142no—
377C4×D4—142no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—71no—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—148no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4—141no—
396C2×C4⋊C4C22×D711yescharacteristic
397C2×C4⋊C4C22×D711yescharacteristic
398C2×C4⋊C4C22×D711yescharacteristic
399C2×C4⋊C4C22×D711yescharacteristic
400C2×C22⋊C4—141no—
401C2×C22⋊C4—141no—
402C2×C22⋊C4—141no—
403C2×C42—21no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—141no—
407C2×C42—141no—
408C2×C42C22×D711yescharacteristic
409C4⋊C8—22no—
410C4⋊C8—142no—
411C4⋊C8—142no—
412#102—74no—
413#102—142no—
414#102—142no—
415#102—142no—
416#97—142no—
417#97—142no—
418C4×C8—142no—
419#95—21no—
420#95—141no—
421#95—141no—
422C22×C7C22×C22⋊C411yescharacteristic
423C22×C7#54811yescharacteristic
424C22×C7D4×Q811yescharacteristic
425C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
426C22×C7C2×#13711yescharacteristic
427C22×C7#57611yescharacteristic
428C22×C7#57711yescharacteristic
429C2×D7—24no—
430C2×D7—42no—
431C2×D7—42no—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C4×C7—22no—
437C4×C7—22no—
438C4×C7—42no—
439C4×C7—42no—
440C4×C7—42no—
441C4×C7—81no—
442C4×C7C2×C4×Q811yescharacteristic
443C4×C7#51911yescharacteristic
444C4×C7#54811yescharacteristic
445C4×C7D4×Q811yescharacteristic
446Dic7—22no—
447Dic7—22no—
448Dic7—22no—
449Dic7—42no—
450Dic7—81no—
451Dic7—81no—
452Dic7#41714yes—
453C24—141no—
454C2×Q8—74no—
455C2×Q8—142no—
456C2×Q8—142no—
457C2×Q8—144no—
458C2×Q8—561no—
459C2×Q8—561no—
460C2×D4—72no—
461C2×D4—72no—
462C2×D4—72no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—141no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—144no—
475C22×C4—144no—
476C22×C4—141no—
477C22×C4—144no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4—561no—
485C22×C4—561no—
486C22×C4D4×D711yescharacteristic
487C22×C4D4×D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—72no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—74no—
509C4⋊C4—74no—
510C4⋊C4—74no—
511C4⋊C4—72no—
512C4⋊C4—144no—
513C4⋊C4—144no—
514C4⋊C4—144no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—282no—
521C4⋊C4—282no—
522C4⋊C4—282no—
523C4⋊C4—561no—
524C4⋊C4—561no—
525C4⋊C4—561no—
526C4⋊C4C2×C4×D712yes—
527C4⋊C4C2×C4×D712yes—
528C4⋊C4Q8×D712yes—
529C22⋊C4—144no—
530C22⋊C4—144no—
531C22⋊C4—144no—
532C42—42no—
533C42—72no—
534C42—72no—
535C42—72no—
536C42—72no—
537C42—282no—
538C42—282no—
539C42C2×C4×D712yes—
540C42D4×D712yes—
541C2×C7Q8×C22⋊C411yescharacteristic
542C2×C7C2×#41711yescharacteristic
543C2×C7#289711yescharacteristic
544C2×C7#336611yescharacteristic
545C2×C7#337211yescharacteristic
546C2×C7#338411yescharacteristic
547C2×C7#338511yescharacteristic
548D7—44no—
549C23—142no—
550C23—142no—
551C23—142no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23C2×D4×D711yescharacteristic
557Q8—148no—
558Q8—564no—
559D4—74no—
560D4—142no—
561D4—142no—
562D4—142no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—72no—
582C2×C4—72no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—142no—
592C2×C4—288no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—564no—
602C2×C4—564no—
603C2×C4D7×C22⋊C412yes—
604C2×C4D7×C22⋊C412yes—
605C2×C4C22×C4×D711yescharacteristic
606C2×C4C2×D4×D711yescharacteristic
607C2×C4C2×Q8×D711yescharacteristic
608C2×C4#655111yescharacteristic
609C2×C4D7×#4111yescharacteristic
610C2×C4#655411yescharacteristic
611C8—41no—
612C8—41no—
613C8—142no—
614C8—281no—
615C7#2252211yescharacteristic, complemented, Sylow
616C22—144no—
617C22—282no—
618C22—282no—
619C22—282no—
620C22—282no—
621C22—282no—
622C22—282no—
623C22C2×D7×C22⋊C411yescharacteristic
624C22#9032311yescharacteristic
625C22#9032411yescharacteristic
626C22#9046011yescharacteristic
627C22#9046111yescharacteristic
628C22D7×AGL(1,ℤ/8)11yescharacteristic
629C22D7×#13711yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—74no—
636C4—81no—
637C4—142no—
638C4—142no—
639C4—142no—
640C4—282no—
641C4—561no—
642C4—561no—
643C4C4×Q8×D711yescharacteristic
644C4#9026311yescharacteristic
645C4#9036811yescharacteristic
646C4#9041811yescharacteristic
647C2—284no—
648C2#1173405911yescharacteristic
649C2#1173627611yescharacteristic
650C2D7×#41711yescharacteristic
651C2#1173859711yescharacteristic
652C2#1173860311yescharacteristic
653C2#1173866411yescharacteristic
654C2#1173866811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1989C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1989D7
hypocenterC7#22522
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#89600C2semidirect
3. C4⋊C4×Dic14C2semidirect
4. C7×#1989C2semidirect
5. #11729495C2semidirect
10. #11726559C2semidirect
11. #11726557C2semidirect
12. #11726555C2semidirect
13. #11726553C2semidirect
18. #90381C4semidirect
133. #6383Q8semidirect
142. #1989D7semidirect
340. D28C4×Q8semidirect
615. C7#22522semidirect

Maximal subgroups

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

subgroupindex
C2×#903812
C2×#896002
C4⋊C4×Dic142
C7×#19892
#117294952
#117282332
#117272282
#117266462
#117265592
#117265572
#117265552
#117265532
#117265362
#117265342
#225227