Finite Group Databasebeta

← #13363345

Subgroups — #13363345

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90210C211yesmaximal, characteristic
2C2×#89492C211yesmaximal, characteristic
3#11731313C211yesmaximal, characteristic
4C7×#1825C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic
6#11727679C211yesmaximal, characteristic
7#11727255C211yesmaximal, characteristic
8#11726560C211yesmaximal, characteristic, complemented
9#11726559C211yesmaximal, characteristic, complemented
10#11726558C211yesmaximal, characteristic, complemented
11#11726557C211yesmaximal, characteristic, complemented
12#11726556C211yesmaximal, characteristic, complemented
13#11726555C211yesmaximal, characteristic, complemented
14#11726554C211yesmaximal, characteristic, complemented
15#11726553C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19#90210—24no—
20#90210C418yescomplemented
21C2×#6453C2211yescharacteristic
22C2×#6453C2211yescharacteristic
23C2×#6451C2211yescharacteristic
24C2×#6451C2211yescharacteristic
25C2×#6441—21no—
26C2×#6441—21no—
27C2×#6433—21no—
28C2×#6432—21no—
29C2×C7×C22⋊C8C2211yescharacteristic
30C7×#380C2211yescharacteristic
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36C2×C8×Dic7C2211yescharacteristic
37#89749C2211yescharacteristic
38#89748C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41#89717—21no—
42#89716—21no—
43#89492—24no—
44#89426—21no—
45#89425—21no—
46C7×#340C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21843—71nomaximal, Sylow
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C22×Dic14—21no—
66C22×Dic14—21no—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×#117—22no—
70C7×#117C2×C414yes—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7D411yescharacteristic
80C2×C22⋊Dic7D411yescharacteristic
81#6489C2×C414yes—
82#6489D414yes—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2×C414yes—
88C2×C4⋊Dic7D411yescharacteristic
89C2×C4⋊Dic7D411yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7C2311yescharacteristic
103C2×C4×Dic7C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106#6453—22no—
107#6453—22no—
108#6453C2×C414yes—
109#6453C2×C414yes—
110#6451—22no—
111#6451—22no—
112#6451C2×C414yes—
113#6451C2×C414yes—
114#6441—28no—
115#6441—28no—
116#6433—28no—
117#6432—28no—
118C7×C22⋊C8—22no—
119C7×#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C22⋊(C7⋊C8)—22no—
126#6389—22no—
127Dic7⋊C8—22no—
128Dic7⋊C8—22no—
129C8×Dic7—22no—
130C2×#544—71no—
131C2×#433—71no—
132#2314—71no—
133#1993—71no—
134#1992—71no—
135#1991—71no—
136#1990—71no—
137#1989—71no—
138#1989—71no—
139#1988—71no—
140#1987—71no—
141#1825—71no—
142#1825D711yescharacteristic
143#1824—71no—
144#1823—71no—
145C24×C7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7C22×C412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—24no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—81no—
164C22×Dic7—81no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C2×Dic14—24no—
170C2×Dic14—24no—
171C2×Dic14—81no—
172C2×Dic14—81no—
173C2×Dic14—81no—
174C2×Dic14—81no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4C22×C412yes—
188C7×C4⋊C4C22×C412yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—81no—
191C42×C7—24no—
192C22⋊Dic7—81no—
193C22⋊Dic7C22⋊C414yes—
194C22⋊Dic7C22⋊C414yes—
195C4⋊Dic7—24no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—24no—
199C4⋊Dic7—24no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7C22⋊C414yes—
205C4⋊Dic7C22⋊C414yes—
206C4⋊Dic7C22×C412yes—
207C4⋊Dic7C22×C412yes—
208Dic7⋊C4—24no—
209Dic7⋊C4—24no—
210Dic7⋊C4—24no—
211Dic7⋊C4—24no—
212Dic7⋊C4—81no—
213Dic7⋊C4—81no—
214C4×Dic7—81no—
215C4×Dic7—81no—
216C4×Dic7C22×C412yes—
217C4×Dic7C22×C412yes—
218C4×Dic7C2×D412yes—
219C4×Dic7C2×D412yes—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226#544—78no—
227#544—144no—
228C2×C4⋊Q8—71no—
229C2×C4⋊Q8—71no—
230C2×C4⋊Q8—141no—
231C2×#125—71no—
232C2×#125—71no—
233C2×#125—141no—
234C2×C22⋊Q8—141no—
235C2×C22⋊Q8—141no—
236C2×#117—71no—
237C2×#117C2×D711yescharacteristic
238C22×C4⋊C4—71no—
239#433—144no—
240C2×C4⋊C8—71no—
241C2×C4⋊C8—71no—
242C2×C22⋊C8—71no—
243C2×C22⋊C8C2×D711yescharacteristic
244C2×#97—71no—
245C2×C4×C8—71no—
246#388—141no—
247#388—141no—
248#387—141no—
249#387—141no—
250#384—71no—
251#382—71no—
252#380—71no—
253#380C2×D711yescharacteristic
254C4×C4⋊C4—71no—
255C4×C4⋊C4—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340C2×D711yescharacteristic
269#340C2×D711yescharacteristic
270#340C2×D711yescharacteristic
271#340C2×D711yescharacteristic
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7C2×C22⋊C412yes—
279C23×C7C22×D411yescharacteristic
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7C2×C22⋊C412yes—
299C2×C4×C7C23×C411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C72−1+411yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×C4×C72−1+411yescharacteristic
304C2×C4×C72−1+411yescharacteristic
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
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—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—84no—
326C2×Dic7C2×C22⋊C412yes—
327C2×Dic7C2×C22⋊C412yes—
328C2×Dic7C2×C22⋊C412yes—
329C2×Dic7C2×C22⋊C412yes—
330Dic14—84no—
331Dic14—84no—
332C7×C8—41no—
333C7×C8—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8—141no—
337C22×Q8—141no—
338C23×C4—141no—
339C23×C4C22×D711yescharacteristic
340C22×C8—71no—
341C22×C8—71no—
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C4⋊Q8—74no—
345C4⋊Q8—74no—
346C4⋊Q8—148no—
347C4⋊Q8—142no—
348C4⋊Q8—142no—
349#125—74no—
350#125—74no—
351#125—148no—
352#125—142no—
353#125—142no—
354C22⋊Q8—148no—
355C22⋊Q8—148no—
356#117—22no—
357#117—74no—
358#117—74no—
359#117C4×D714yes—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—74no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
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—142no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—71no—
384C2×C22⋊C4—71no—
385C2×C22⋊C4—141no—
386C2×C42—21no—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—141no—
390C2×C42—141no—
391C4⋊C8—142no—
392C4⋊C8—142no—
393C22⋊C8—22no—
394C22⋊C8—142no—
395#97—142no—
396C4×C8—142no—
397#95—21no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—41no—
407C22×C7—41no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7C22×C22⋊C411yescharacteristic
411C22×C7#53611yescharacteristic
412C22×C7#53611yescharacteristic
413C22×C7#57611yescharacteristic
414C22×C7#57711yescharacteristic
415C22×C7#57711yescharacteristic
416C22×C7#57811yescharacteristic
417C4×C7—22no—
418C4×C7—42no—
419C4×C7—42no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7#51911yescharacteristic
423C4×C7#51911yescharacteristic
424C4×C7#53611yescharacteristic
425C4×C7#53611yescharacteristic
426Dic7—22no—
427Dic7—22no—
428Dic7—22no—
429Dic7—22no—
430Dic7—42no—
431Dic7—42no—
432Dic7—81no—
433Dic7—81no—
434C24D4×D711yescharacteristic
435C2×Q8—144no—
436C2×Q8—144no—
437C2×Q8—561no—
438C2×Q8—561no—
439C2×Q8—561no—
440C2×Q8—561no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—144no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—561no—
459C22×C4—561no—
460C22×C4C2×C4×D712yes—
461C22×C4D4×D711yescharacteristic
462C22×C4D4×D712yes—
463C22×C4C23×D711yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—74no—
483C4⋊C4—74no—
484C4⋊C4—72no—
485C4⋊C4—72no—
486C4⋊C4—144no—
487C4⋊C4—144no—
488C4⋊C4—144no—
489C4⋊C4—144no—
490C4⋊C4—144no—
491C4⋊C4—144no—
492C4⋊C4—144no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—561no—
500C4⋊C4—561no—
501C4⋊C4C2×C4×D712yes—
502C4⋊C4C2×C4×D712yes—
503C22⋊C4—24no—
504C22⋊C4—74no—
505C22⋊C4—74no—
506C22⋊C4—81no—
507C22⋊C4—561no—
508C42—24no—
509C42—72no—
510C42—72no—
511C42—72no—
512C42—72no—
513C42—561no—
514C42—561no—
515C2×C7—42no—
516C2×C7#231011yescharacteristic
517C2×C7#289711yescharacteristic
518C2×C7#289811yescharacteristic
519C2×C7#324111yescharacteristic
520C2×C7#324211yescharacteristic
521C2×C7#324311yescharacteristic
522C2×C7#324411yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23D7×C22⋊C412yes—
530C23C2×D4×D711yescharacteristic
531Q8—564no—
532Q8—564no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—564no—
576C2×C4D7×C22⋊C412yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×D4×D711yescharacteristic
579C2×C4#655411yescharacteristic
580C2×C4#655411yescharacteristic
581C2×C4#655411yescharacteristic
582C2×C4#655411yescharacteristic
583C8—41no—
584C8—41no—
585C8—281no—
586C8—281no—
587C7#2184311yescharacteristic, complemented, Sylow
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×D7×C22⋊C411yescharacteristic
596C22#9033911yescharacteristic
597C22#9033911yescharacteristic
598C22#9046111yescharacteristic
599C22#9046211yescharacteristic
600C22#9046511yescharacteristic
601C22#9046911yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—142no—
608C4—142no—
609C4—142no—
610C4—142no—
611C4—282no—
612C4—282no—
613C4—561no—
614C4—561no—
615C4#9020711yescharacteristic
616C4#9020711yescharacteristic
617C4#9033911yescharacteristic
618C4#9033911yescharacteristic
619C2—42no—
620C2#1173149811yescharacteristic
621C2#1173627711yescharacteristic
622C2#1173632711yescharacteristic
623C2#1173880511yescharacteristic
624C2#1173880711yescharacteristic
625C2#1173894911yescharacteristic
626C2#1173895211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1825C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1825D7
hypocenterC7#21843
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
8. #11726560C2semidirect
9. #11726559C2semidirect
10. #11726558C2semidirect
11. #11726557C2semidirect
12. #11726556C2semidirect
13. #11726555C2semidirect
14. #11726554C2semidirect
15. #11726553C2semidirect
20. #90210C4semidirect
587. C7#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#902102
C2×#894922
#117313132
C7×#18252
#117293292
#117276792
#117272552
#117265602
#117265592
#117265582
#117265572
#117265562
#117265552
#117265542
#117265532
#218437