Finite Group Databasebeta

← #13412607

Subgroups — #13412607

Order 1792.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739617C211yesmaximal, characteristic
2C2×#89736C211yesmaximal, characteristic
3C2×#89630C211yesmaximal, characteristic, complemented
4#11731345C211yesmaximal, characteristic
5C7×#1853C211yesmaximal, characteristic
6#11728321C211yesmaximal, characteristic, complemented
7#11728320C211yesmaximal, characteristic, complemented
8#11728310C211yesmaximal, characteristic, complemented
9#11727703C211yesmaximal, characteristic
10#11727683C211yesmaximal, characteristic
11#11727267C211yesmaximal, characteristic
12#11726614C211yesmaximal, characteristic, complemented
13#11726612C211yesmaximal, characteristic, complemented
14#11726590C211yesmaximal, characteristic, complemented
15#11726583C211yesmaximal, characteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6453C2211yescharacteristic
20C2×#6453C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×#106C2211yescharacteristic
23C2×C7×C22⋊C8C2211yescharacteristic
24C7×#380C2211yescharacteristic
25#89984—21no—
26#89984C2211yescharacteristic
27#89982—21no—
28C4×C22⋊Dic7C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89758—21no—
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89756—21no—
37#89749C2211yescharacteristic
38#89748C2211yescharacteristic
39#89745C2211yescharacteristic
40#89745C2211yescharacteristic
41#89736—24no—
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48#89721—21no—
49#89720—21no—
50#89630—24no—
51#89427—21no—
52#89419—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26075—71nomaximal, Sylow
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—41no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7D411yescharacteristic
77C2×C22⋊Dic7D411yescharacteristic
78#6489—22no—
79#6489—41no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136Q814yes—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6453—22no—
103#6453—22no—
104#6453—41no—
105#6453—41no—
106C7×#106—22no—
107C7×#106—41no—
108C7×C22⋊C8Q814yes—
109C7×#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6392—41no—
120#6391—22no—
121#6389—22no—
122C8×Dic7—22no—
123#6382—22no—
124#6382—22no—
125#6382—22no—
126#6381—22no—
127#6381—22no—
128#6381D414yes—
129C2×#499—71no—
130C2×#428—71no—
131#2442—71no—
132#2441—71no—
133#2085—71no—
134#2084—71no—
135#2059—71no—
136#2059—71no—
137#1943—71no—
138#1941—71no—
139#1854—71no—
140#1853D711yescharacteristic
141#1847—71no—
142#1809—71no—
143#1753—71no—
144C24×C7C2×D411yescharacteristic
145C22×C4×C7—21no—
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—42no—
154C22×C4×C7—42no—
155C22×C4×C7—81no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×Q812yes—
158C22×C4×C7C2×Q812yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8C2×Q812yes—
173C2×C7×C8C2×Q812yes—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4C2×D412yes—
183C7×C22⋊C4—81no—
184C22⋊Dic7—41no—
185C22⋊Dic7—41no—
186C22⋊Dic7—42no—
187C22⋊Dic7—42no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—42no—
196C4⋊Dic7—42no—
197C4⋊Dic7—81no—
198C4⋊Dic7—81no—
199C4⋊Dic7C2×D412yes—
200Dic7⋊C4—42no—
201Dic7⋊C4—42no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—81no—
205Dic7⋊C4—81no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—42no—
211#1136—28no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C2×D412yes—
215C2×C7⋊C8C2×Q812yes—
216C2×C7⋊C8C2×Q812yes—
217C2×C7⋊C8#4112yes—
218C22×M16—71no—
219C2×#125—71no—
220C2×#125—71no—
221C2×#117—71no—
222C22×C4⋊C4C2×D711yescharacteristic
223#499—144no—
224#428—144no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#106—71no—
230C2×#106—71no—
231C2×#106—71no—
232C2×#106—71no—
233C2×#106C2×D711yescharacteristic
234C2×#106C2×D711yescharacteristic
235C2×C22⋊C8C2×D711yescharacteristic
236C2×#97—71no—
237C2×C4×C8—71no—
238#400—141no—
239#399—141no—
240#398—141no—
241#396—141no—
242#389—71no—
243#389—71no—
244#389—71no—
245#389—71no—
246#389—141no—
247#387—141no—
248#385—141no—
249#384—71no—
250#382—71no—
251#380—71no—
252#380—141no—
253#380C2×D711yescharacteristic
254C4×C22⋊C4—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261#336—71no—
262#336—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7C4⋊Q812yes—
270C23×C7C22×D411yescharacteristic
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
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—84no—
290C2×C4×C7C4⋊Q812yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C22×Q811yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—21no—
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—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C7×C8—22no—
322C7×C8—22no—
323C7⋊C8—22no—
324C7⋊C8C22⋊Q814yes—
325C23×C4—21no—
326C23×C4C22×D711yescharacteristic
327C2×M16—74no—
328C2×M16—142no—
329C2×M16—142no—
330C22×C8—71no—
331C22×C8—71no—
332C22×C8C22×D711yescharacteristic
333C22×C8C22×D711yescharacteristic
334#125—142no—
335#125—142no—
336#125—281no—
337#125—281no—
338#117—142no—
339#117—281no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—41no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—71no—
363C2×C22⋊C4—71no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4—141no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—141no—
369#107—74no—
370#107—142no—
371#107—142no—
372#107—281no—
373#106—22no—
374#106—41no—
375#106—142no—
376#106—142no—
377#106—142no—
378#106—142no—
379C22⋊C8Dic1414yes—
380#97—142no—
381C4×C8—142no—
382#95—21no—
383#95—71no—
384#95—71no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392C22×C7—22no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7C2×C4⋊Q811yescharacteristic
400C22×C7D4211yescharacteristic
401C22×C7#54711yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57511yescharacteristic
404C22×C7#57711yescharacteristic
405C22×C7#57711yescharacteristic
406C4×C7—22no—
407C4×C7—42no—
408C4×C7—81no—
409C4×C7—81no—
410C4×C7—81no—
411C4×C7C2×C22⋊Q811yescharacteristic
412C4×C7#54411yescharacteristic
413C4×C7D4×Q811yescharacteristic
414C4×C7#55411yescharacteristic
415Dic7—41no—
416Dic7—41no—
417Dic7—41no—
418Dic7—42no—
419Dic7—41no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423C24D4×D711yescharacteristic
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—81no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4C2×Dic1412yes—
444C22×C4D4×D711yescharacteristic
445C22×C4Q8×D712yes—
446C22×C4C23×D711yescharacteristic
447M16—148no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—72no—
453C2×C8—72no—
454C2×C8—72no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8C2×Dic1412yes—
459C2×C8C2×Dic1412yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—72no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—282no—
479C4⋊C4—282no—
480C4⋊C4—282no—
481C4⋊C4—282no—
482C4⋊C4—561no—
483C4⋊C4—561no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4D4×D712yes—
487C22⋊C4—81no—
488C22⋊C4—281no—
489C22⋊C4—281no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C42—141no—
493C42—141no—
494C42—141no—
495C42—141no—
496C42—282no—
497C2×C7—42no—
498C2×C7#244011yescharacteristic
499C2×C7#316311yescharacteristic
500C2×C7#316611yescharacteristic
501C2×C7#326711yescharacteristic
502C2×C7#327111yescharacteristic
503C2×C7#335111yescharacteristic
504C2×C7#335111yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23#645112yes—
512C23C2×D4×D711yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—142no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4#645112yes—
557C2×C4C22×Dic1411yescharacteristic
558C2×C4C2×D4×D711yescharacteristic
559C2×C4#654811yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655411yescharacteristic
563C8—22no—
564C8—22no—
565C8—74no—
566C8—142no—
567C7#2607511yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#645111yescharacteristic
576C22#9028111yescharacteristic
577C22#9034611yescharacteristic
578C22#9045311yescharacteristic
579C22#9045311yescharacteristic
580C22#9046411yescharacteristic
581C22#9046811yescharacteristic
582C4—22no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—281no—
588C4—282no—
589C4—281no—
590C4—281no—
591C4—281no—
592C4—561no—
593C4—561no—
594C4—561no—
595C4D4×Dic1411yescharacteristic
596C4#9023011yescharacteristic
597C4#9031411yescharacteristic
598C4D7×C22⋊Q811yescharacteristic
599C2—42no—
600C2#1173186111yescharacteristic
601C2#1173655511yescharacteristic
602C2#1173664911yescharacteristic
603C2#1173751011yescharacteristic
604C2#1173760211yescharacteristic
605C2#1173878211yescharacteristic
606C2#1173888511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1853D7
hypocenterC7#26075
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. C2×#89630C2semidirect
6. #11728321C2semidirect
7. #11728320C2semidirect
8. #11728310C2semidirect
12. #11726614C2semidirect
13. #11726612C2semidirect
14. #11726590C2semidirect
15. #11726583C2semidirect
567. C7#26075semidirect

Maximal subgroups

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

subgroupindex
#117396172
C2×#897362
C2×#896302
#117313452
C7×#18532
#117283212
#117283202
#117283102
#117277032
#117276832
#117272672
#117266142
#117266122
#117265902
#117265832
#260757