Finite Group Databasebeta

← #13370481

Subgroups — #13370481

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90412C211yesmaximal, characteristic
2C2×#89473C211yesmaximal, characteristic
3#11731603C211yesmaximal, characteristic
4C7×#1993C211yesmaximal, characteristic
5#11729533C211yesmaximal, characteristic
6#11727567C211yesmaximal, characteristic
7#11727313C211yesmaximal, characteristic
8#11726659C211yesmaximal, characteristic
9#11726658C211yesmaximal, characteristic
10#11726559C211yesmaximal, characteristic
11#11726558C211yesmaximal, characteristic
12#11726556C211yesmaximal, characteristic
13#11726555C211yesmaximal, characteristic
14#11726537C211yesmaximal, characteristic
15#11726535C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×Q8×Dic7—21no—
18C2×#6507—21no—
19#90412—24no—
20#90412C418yescomplemented
21C2×#6453C2211yescharacteristic
22C2×#6453C2211yescharacteristic
23C2×#6452—21no—
24C2×#6452—21no—
25C2×#6450C2211yescharacteristic
26C2×#6450C2211yescharacteristic
27C2×C4×Dic14C2211yescharacteristic
28C2×C4×Dic14C2211yescharacteristic
29C2×C7×#97C2211yescharacteristic
30C7×#382C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36C2×C8×Dic7C2211yescharacteristic
37#89749C2211yescharacteristic
38#89748C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45#89473—24no—
46#89425—21no—
47#89424—21no—
48#89421—21no—
49#89418—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#22527—71nomaximal, Sylow
63C22×C7×Q8—21no—
64C22×Dic14D411yescharacteristic
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2×C414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76Q8×Dic7—28no—
77#6507—28no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7D411yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4D411yescharacteristic
92C2×Dic7⋊C4D411yescharacteristic
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×Dic7C2311yescharacteristic
100C2×C4×Dic7C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103#6453—22no—
104#6453—22no—
105#6453C2×C414yes—
106#6453C2×C414yes—
107#6452—28no—
108#6452—28no—
109#6450—22no—
110#6450—22no—
111#6450C2×C414yes—
112#6450C2×C414yes—
113C4×Dic14—22no—
114C4×Dic14C2×C414yes—
115C4×Dic14C2×C414yes—
116C4×Dic14D414yes—
117C7×#97—22no—
118C7×#95—21no—
119C7×#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125#6390—22no—
126#6390Q814yescomplemented
127#6389—22no—
128Dic7⋊C8—22no—
129Dic7⋊C8—22no—
130C8×Dic7—22no—
131#6384—22no—
132#6384—22no—
133C4⋊(C7⋊C8)—22no—
134C2×#557—71no—
135C2×#446—71no—
136#2355—71no—
137#1993D711yescharacteristic
138#1992—71no—
139#1991—71no—
140#1989—71no—
141#1989—71no—
142#1987—71no—
143#1805—71no—
144#1805—71no—
145#1803—71no—
146#1763—71no—
147#1763—71no—
148C4×#103—71no—
149C2×C7×Q8—24no—
150C2×C7×Q8—42no—
151C2×C7×Q8—42no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—41no—
171C2×Dic14—41no—
172C2×Dic14C22⋊C414yes—
173C2×Dic14C22×C412yes—
174C2×Dic14C2×Q812yes—
175C2×Dic14#4112yes—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8C2×Q812yes—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4C22×C412yes—
190C7×C4⋊C4C22×C412yes—
191C42×C7C22×C412yes—
192C42×C7C2×D412yes—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—24no—
196C4⋊Dic7—24no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7C22⋊C414yes—
200C4⋊Dic7C22×C412yes—
201C4⋊Dic7C2×Q812yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—24no—
204Dic7⋊C4—24no—
205Dic7⋊C4—24no—
206Dic7⋊C4—24no—
207Dic7⋊C4—42no—
208Dic7⋊C4—81no—
209Dic7⋊C4C22⋊C414yes—
210Dic7⋊C4C22⋊C414yes—
211C4×Dic7—24no—
212C4×Dic7—24no—
213C4×Dic7—24no—
214C4×Dic7—42no—
215C4×Dic7—81no—
216C4×Dic7—81no—
217C4×Dic7C22×C412yes—
218C4×Dic7C22×C412yes—
219C4×Dic7C2×D412yes—
220C4×Dic7C2×D412yes—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227#557—78no—
228#557—144no—
229C2×C4⋊Q8—141no—
230C2×C4⋊Q8C2×D711yescharacteristic
231C2×#125—71no—
232C2×#125—71no—
233C2×#125—141no—
234C2×#125—141no—
235C2×C4×Q8—71no—
236C2×C4×Q8—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8—141no—
240#446—144no—
241C2×C4⋊C8—71no—
242C2×C4⋊C8—71no—
243C2×C4⋊C8—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#97—71no—
249C2×#97C2×D711yescharacteristic
250C2×C4×C8—71no—
251#391—141no—
252#387—141no—
253#387—141no—
254#384—71no—
255#382—71no—
256#382C2×D711yescharacteristic
257C4×C4⋊C4—71no—
258C4×C4⋊C4—71no—
259C4×C4⋊C4—71no—
260C4×C4⋊C4—141no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340C2×D711yescharacteristic
270#340C2×D711yescharacteristic
271#340C2×D711yescharacteristic
272#340C2×D711yescharacteristic
273C23×C7C22×D411yescharacteristic
274C7×Q8—48no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7C2×C22⋊C412yes—
292C2×C4×C7C2×C22⋊C412yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×Q811yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7C2×C22⋊C412yes—
320C2×Dic7C2×C22⋊C412yes—
321C2×Dic7C2×C22⋊C412yes—
322C2×Dic7C2×C22⋊C412yes—
323Dic14—22no—
324Dic14—22no—
325Dic14—22no—
326Dic14—44no—
327Dic14C4×Q814yes—
328C7×C8—22no—
329C7×C8—41no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×Q8—21no—
333C22×Q8—71no—
334C22×Q8—71no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C4⋊Q8—22no—
340C4⋊Q8—148no—
341C4⋊Q8C4×D714yes—
342#125—74no—
343#125—74no—
344#125—142no—
345#125—142no—
346#125—148no—
347#125—148no—
348C4×Q8—74no—
349C4×Q8—74no—
350C4×Q8—74no—
351C4×Q8—74no—
352C4×Q8—74no—
353C4×Q8—142no—
354C4×Q8—142no—
355C4×Q8—148no—
356C4×Q8—142no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—141no—
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⋊C4C22×D711yescharacteristic
374C2×C4⋊C4C22×D711yescharacteristic
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42C22×D711yescharacteristic
386C4⋊C8—142no—
387C4⋊C8—142no—
388C4⋊C8—142no—
389#103—74no—
390#103—142no—
391#103—142no—
392#103—142no—
393#97—22no—
394#97—142no—
395C4×C8—142no—
396#95—21no—
397#95—21no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403C22×C7C22×C22⋊C411yescharacteristic
404C22×C7#54811yescharacteristic
405C22×C7D4×Q811yescharacteristic
406C22×C7#57511yescharacteristic
407C22×C7#57511yescharacteristic
408C22×C7#57711yescharacteristic
409C22×C7#57811yescharacteristic
410C4×C7—22no—
411C4×C7—22no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7—81no—
415C4×C7—81no—
416C4×C7C2×C4×Q811yescharacteristic
417C4×C7#51911yescharacteristic
418C4×C7#54811yescharacteristic
419C4×C7D4×Q811yescharacteristic
420Dic7—24no—
421Dic7—22no—
422Dic7—22no—
423Dic7—22no—
424Dic7—22no—
425Dic7—42no—
426Dic7—42no—
427Dic7—81no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—74no—
432C2×Q8—72no—
433C2×Q8—72no—
434C2×Q8—72no—
435C2×Q8—141no—
436C2×Q8—141no—
437C2×Q8—141no—
438C2×Q8—281no—
439C2×Q8—281no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4D4×D711yescharacteristic
453C22×C4D4×D711yescharacteristic
454C22×C4C23×D711yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8Q8×D712yes—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—74no—
475C4⋊C4—74no—
476C4⋊C4—74no—
477C4⋊C4—72no—
478C4⋊C4—72no—
479C4⋊C4—144no—
480C4⋊C4—144no—
481C4⋊C4—144no—
482C4⋊C4—144no—
483C4⋊C4—144no—
484C4⋊C4—144no—
485C4⋊C4—144no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—282no—
491C4⋊C4—561no—
492C4⋊C4C2×C4×D712yes—
493C4⋊C4C2×C4×D712yes—
494C42—72no—
495C42—72no—
496C42—72no—
497C42—72no—
498C42—144no—
499C42—144no—
500C42—144no—
501C42—282no—
502C42—561no—
503C42—561no—
504C42C2×C4×D712yes—
505C42D4×D712yes—
506C2×C7Q8×C22⋊C411yescharacteristic
507C2×C7#289611yescharacteristic
508C2×C7#289811yescharacteristic
509C2×C7#337111yescharacteristic
510C2×C7#337711yescharacteristic
511C2×C7#338811yescharacteristic
512C2×C7#339011yescharacteristic
513C23C2×D4×D711yescharacteristic
514Q8—48no—
515Q8—74no—
516Q8—142no—
517Q8—142no—
518Q8—142no—
519Q8—284no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—72no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—72no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4D7×C22⋊C412yes—
561C2×C4D7×C22⋊C412yes—
562C2×C4C22×C4×D711yescharacteristic
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×Q8×D711yescharacteristic
565C2×C4#655111yescharacteristic
566C2×C4D7×#4111yescharacteristic
567C2×C4#655411yescharacteristic
568C8—22no—
569C8—41no—
570C8—281no—
571C8—281no—
572C7#2252711yescharacteristic, complemented, Sylow
573C22C2×D7×C22⋊C411yescharacteristic
574C22#9036811yescharacteristic
575C22#9041811yescharacteristic
576C22#9046411yescharacteristic
577C22#9046411yescharacteristic
578C22#9046911yescharacteristic
579C22#9047011yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—144no—
587C4—142no—
588C4—142no—
589C4—142no—
590C4—142no—
591C4—282no—
592C4—282no—
593C4—561no—
594C4C4×Q8×D711yescharacteristic
595C4#9026311yescharacteristic
596C4#9032311yescharacteristic
597C4#9032411yescharacteristic
598C2#1173405911yescharacteristic
599C2#1173636911yescharacteristic
600C2#1173640211yescharacteristic
601C2#1173911811yescharacteristic
602C2#1173912211yescharacteristic
603C2#1173941311yescharacteristic
604C2#1173941611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1993C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1993D7
hypocenterC7#22527
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
20. #90412C4semidirect
126. #6390Q8semidirect
572. C7#22527semidirect

Maximal subgroups

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

subgroupindex
C2×#904122
C2×#894732
#117316032
C7×#19932
#117295332
#117275672
#117273132
#117266592
#117266582
#117265592
#117265582
#117265562
#117265552
#117265372
#117265352
#225277