Finite Group Databasebeta

← #13355980

Subgroups — #13355980

Order 1792.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89633C211yesmaximal, characteristic
2D7×#389C211yesmaximal, characteristic
3C4×#6455C211yesmaximal, characteristic
4C7×#1920C211yesmaximal, characteristic, complemented
5#11728311C211yesmaximal, characteristic, complemented
6#11727357C211yesmaximal, characteristic, complemented
7#11727214C211yesmaximal, characteristic, complemented
8#11726621C211yesmaximal, characteristic
9#11726621C211yesmaximal, characteristic
10#11726621C211yesmaximal, characteristic
11#11726569C211yesmaximal, characteristic, complemented
12#11726568C211yesmaximal, characteristic, complemented
13#11726542C211yesmaximal, characteristic, complemented
14#11726414C211yesmaximal, characteristic
15#11726378C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4—21no—
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×C42×D7C2211yescharacteristic
24C2×C7×#106C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28#89757C2211yescharacteristic
29C4⋊C4×Dic7C2211yescharacteristic
30C4⋊C4×Dic7C2211yescharacteristic
31#89745C2211yescharacteristic
32#89744—21no—
33C2×#6382C2211yescharacteristic
34C2×#6381C2211yescharacteristic
35C4×(C2×D7)⋊C4—21no—
36C4×C4⋊Dic7C2211yescharacteristic
37C42×Dic7—21no—
38#89700C2211yescharacteristic
39#89633C418yescomplemented
40#89438—21no—
41D7×#95—21no—
42#89420—21no—
43#89419—21no—
44C7×#340C2211yescharacteristic
45C7×#340C2211yescharacteristic
46C7×#340C2211yescharacteristic
47C7×#336C2211yescharacteristic
48#89346C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89345C2211yescharacteristic
52#89345C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#21034—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7D411yescharacteristic
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C2×#1131—22no—
97C2×#1131—22no—
98C2×#1131—22no—
99C2×#1131—22no—
100C2×#1131—22no—
101C2×#1131C2×C414yes—
102#6455—22no—
103#6455—22no—
104#6455C2×C414yes—
105#6455Q814yes—
106D7×C4⋊C4—22no—
107D7×C4⋊C4—22no—
108D7×C4⋊C4—22no—
109D7×C4⋊C4—44no—
110D7×C4⋊C4C2×C414yes—
111C42×D7—22no—
112C42×D7—22no—
113C7×#106C2×C414yescomplemented
114C7×#95—21no—
115C7×#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122#6392C2×C414yescomplemented
123#6382C2×C414yescomplemented
124#6381C2×C414yescomplemented
125C2×#428—71no—
126C2×#389—71no—
127C4×#117—71no—
128#1921—71no—
129#1921—71no—
130#1920—71no—
131#1920D711yescharacteristic, complemented
132#1919—71no—
133#1919—71no—
134#1918—71no—
135#1918—71no—
136#1736—71no—
137#1736—71no—
138#1736—71no—
139#1728—71no—
140C23×D7C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C2×C4×D7—24no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—22no—
160C2×C4×D7—41no—
161C2×C4×D7—41no—
162C2×C4×D7—42no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7C22×C412yes—
168C2×C4×D7C2×Q812yes—
169C2×C4×D7#4112yes—
170C2×C4×D7#4112yes—
171C2×C4×D7#4112yes—
172C2×C4×D7#4112yes—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C412yes—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4C22×C412yes—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4C2×Q812yes—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—42no—
191(C2×D7)⋊C4—24no—
192(C2×D7)⋊C4—24no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7C22×C412yes—
200C4⋊Dic7C22×C412yes—
201C4⋊Dic7C2×Q812yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—24no—
209C4×Dic7—24no—
210C4×Dic7—22no—
211C4×Dic7—22no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221#1131—28no—
222#1131—44no—
223C22×M16—71no—
224C22×M16—71no—
225C2×#117—71no—
226C2×#117—71no—
227C22×C4⋊C4—71no—
228C22×C4⋊C4—71no—
229C22×C4⋊C4—141no—
230C22×C42—71no—
231#428—78no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#106—71no—
235C2×#106C2×D711yescharacteristic
236#389—71no—
237#389—71no—
238#389—71no—
239#389—141no—
240#389—141no—
241#389C2×D711yescharacteristic
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4C2×D711yescharacteristic
246C4×C22⋊C4—141no—
247C4×C22⋊C4—141no—
248#376—141no—
249C2×#95—141no—
250C43—141no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340C2×D711yescharacteristic
261#340C2×D711yescharacteristic
262#340C2×D711yescharacteristic
263#336—71no—
264#336—71no—
265#336—71no—
266#336C2×D711yescharacteristic
267C23×C7C22×D411yescharacteristic
268C22×D7—21no—
269C22×D7—21no—
270C22×D7—21no—
271C22×D7—21no—
272C22×D7C4×D412yes—
273C22×D7C22⋊Q812yes—
274C22×D7#12312yes—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
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×C7—41no—
292C2×C4×C7#13712yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7C4×D412yes—
317C2×Dic7C22⋊Q812yes—
318C2×Dic7#12312yes—
319C2×Dic7AGL(1,ℤ/8)12yes—
320C4×D7—44no—
321C4×D7—44no—
322C4×D7—48no—
323C4×D7—84no—
324C4×D7#11714yes—
325C4×D7C4×Q814yes—
326C4×D7#12514yes—
327C4×D7#12614yes—
328C7×C8—22no—
329C7×C8—41no—
330C7⋊C8—22no—
331C7⋊C8—41no—
332C23×C4—71no—
333C23×C4—71no—
334C23×C4—71no—
335C23×C4—141no—
336C23×C4—141no—
337C2×M16—74no—
338C2×M16—142no—
339C2×M16—142no—
340C2×M16—142no—
341C2×M16—142no—
342C2×M16—142no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8C22×D711yescharacteristic
346C22×C8C22×D711yescharacteristic
347#117—74no—
348#117—74no—
349#117—142no—
350#117—142no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—74no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—284no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C42—21no—
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—71no—
376C2×C42—142no—
377C2×C42—142no—
378C2×C42—141no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42C22×D711yescharacteristic
384#107—74no—
385#107—74no—
386#106—74no—
387#106C4×D714yescomplemented
388#95—21no—
389#95—21no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7C2×C4×D411yescharacteristic
397C22×C7C2×C22⋊Q811yescharacteristic
398C22×C7C2×#12311yescharacteristic
399C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C7C2×#13711yescharacteristic
401C22×C7#57511yescharacteristic
402C22×C7#57511yescharacteristic
403C2×D7—22no—
404C2×D7—22no—
405C2×D7—22no—
406C2×D7—22no—
407C2×D7—22no—
408C2×D7—22no—
409C2×D7#38714yes—
410C4×C7—22no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—81no—
416C4×C7C2×#11711yescharacteristic
417C4×C7C2×C4×Q811yescharacteristic
418C4×C7C2×#12511yescharacteristic
419C4×C7C2×#12611yescharacteristic
420Dic7—22no—
421Dic7—42no—
422Dic7—41no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7#38714yes—
427C24—71no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—71no—
433C22×C4—72no—
434C22×C4—72no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—144no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—142no—
449C22×C4—141no—
450C22×C4—281no—
451C22×C4—281no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4—282no—
457C22×C4D4×D711yescharacteristic
458C22×C4D4×D711yescharacteristic
459C22×C4C23×D711yescharacteristic
460M16—148no—
461M16—284no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—72no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8C2×C4×D712yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—72no—
481C4⋊C4—72no—
482C4⋊C4—72no—
483C4⋊C4—144no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4C2×C4×D712yes—
494C4⋊C4C2×C4×D712yes—
495C4⋊C4Q8×D712yes—
496C22⋊C4—144no—
497C22⋊C4—144no—
498C42—21no—
499C42—21no—
500C42—42no—
501C42—142no—
502C42—142no—
503C42—144no—
504C42—144no—
505C42—141no—
506C42—141no—
507C42—282no—
508C42—282no—
509C42—282no—
510C2×C7C2×#38711yescharacteristic
511C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
512C2×C7C4×#13711yescharacteristic
513C2×C7#307911yescharacteristic
514C2×C7#308011yescharacteristic
515C2×C7#309411yescharacteristic
516C2×C7#309511yescharacteristic
517D7—24no—
518C23—72no—
519C23—72no—
520C23—72no—
521C23—141no—
522C23—141no—
523C23—141no—
524C23—141no—
525C23C2×D4×D711yescharacteristic
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—74no—
547C2×C4—74no—
548C2×C4—74no—
549C2×C4—74no—
550C2×C4—72no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—141no—
559C2×C4—284no—
560C2×C4—284no—
561C2×C4—288no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—564no—
572C2×C4#647812yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×Q8×D711yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—142no—
582C8—281no—
583C7#2103411yescharacteristic, complemented, Sylow
584C22—74no—
585C22—142no—
586C22—142no—
587C22—142no—
588C22—142no—
589C22—142no—
590C22—142no—
591C22C4×D4×D711yescharacteristic
592C22D7×C22⋊Q811yescharacteristic
593C22D7×#12311yescharacteristic
594C22C2×#647811yescharacteristic
595C22#9045911yescharacteristic
596C22D7×AGL(1,ℤ/8)11yescharacteristic
597C22#9046811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—74no—
604C4—81no—
605C4—142no—
606C4—282no—
607C4—282no—
608C4—282no—
609C4—281no—
610C4—561no—
611C4D7×#11711yescharacteristic
612C4C4×Q8×D711yescharacteristic
613C4D7×#12511yescharacteristic
614C4D7×#12611yescharacteristic
615C2—144no—
616C2D7×#38711yescharacteristic
617C2C4×#647811yescharacteristic
618C2#1173721811yescharacteristic
619C2#1173850611yescharacteristic
620C2#1173867611yescharacteristic
621C2#1173881811yescharacteristic
622C2#1173887311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1920C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1920D7
hypocenterC7#21034
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. C7×#1920C2semidirect
5. #11728311C2semidirect
6. #11727357C2semidirect
7. #11727214C2semidirect
11. #11726569C2semidirect
12. #11726568C2semidirect
13. #11726542C2semidirect
15. #11726378C2semidirect
39. #89633C4semidirect
113. C7×#106C2×C4semidirect
122. #6392C2×C4semidirect
123. #6382C2×C4semidirect
124. #6381C2×C4semidirect
131. #1920D7semidirect
387. #106C4×D7semidirect
583. C7#21034semidirect

Maximal subgroups

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

subgroupindex
C2×#896332
D7×#3892
C4×#64552
C7×#19202
#117283112
#117273572
#117272142
#117266212
#117265692
#117265682
#117265422
#117264142
#117263782
#210347