Finite Group Databasebeta

← #13456995

Subgroups — #13456995

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2319C211yesmaximal, characteristic
2#11730853C211yesmaximal, characteristic, complemented
3#11729366C211yesmaximal, characteristic
4#11729361C211yesmaximal, characteristic
5#11727406C211yesmaximal, characteristic
6#11727405C211yesmaximal, characteristic
7#11727400C211yesmaximal, characteristic
8#11727399C211yesmaximal, characteristic
9#11727388C211yesmaximal, characteristic, complemented
10#11727372C211yesmaximal, characteristic, complemented
11#11727370C211yesmaximal, characteristic, complemented
12#11727364C211yesmaximal, characteristic, complemented
13#11727218C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727190C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C22×#1142C2211yescharacteristic
18C7×#390C2211yescharacteristic
19C7×#390C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#388—21no—
24C7×#381—21no—
25C7×#380—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#376C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6384C2211yescharacteristic
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37#89701C2211yescharacteristic
38#89701C2211yescharacteristic
39#89701C2211yescharacteristic
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47C7⋊#401—21no—
48#89430—21no—
49#89430—21no—
50#89430—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#29440—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C22×Dic14C2311yescharacteristic
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×#1142—22no—
80C2×#1142—22no—
81C2×#1142—41no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C7×#95—21no—
93C7×#95—21no—
94C7×#95—21no—
95C7×#95D411yescharacteristic
96C7×#95D411yescharacteristic
97C7×#95D411yescharacteristic
98C7×#95D411yescharacteristic
99#6406—22no—
100#6406—22no—
101#6406—41no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111#6384—22no—
112#6384—22no—
113#6384—41no—
114#6382—22no—
115#6382—22no—
116#6382—41no—
117C4⋊(C7⋊C8)—22no—
118C4⋊(C7⋊C8)—22no—
119#6377—41no—
120C4×C7⋊C8—41no—
121#2645—71no—
122#2319D711yescharacteristic
123#2088—71no—
124#2066—71no—
125#2065—71no—
126#2055—71no—
127#1968—71no—
128#1941—71no—
129#1936—71no—
130#1935—71no—
131#1927—71no—
132#1919—71no—
133#1877—71no—
134#1864—71no—
135#1768—71no—
136C24×C7—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4#4112yes—
143C22×C4×C7—21no—
144C22×C4×C7—24no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—81no—
148C22×C4×C7—81no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4#4112yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—81no—
176C7×C22⋊C4—81no—
177C7×C22⋊C4—81no—
178C7×C22⋊C4—81no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—81no—
186#1142—44no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195Dic7⋊C4—81no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8#4112yes—
207C2×C7⋊C8#4112yes—
208C22×SD16—71no—
209C2×C4×D4C2×D711yescharacteristic
210C2×#106—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×C4⋊C8—71no—
214C2×C4⋊C8—71no—
215C2×#103—71no—
216C2×#103—71no—
217C2×#103—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×#102—71no—
221C2×#97—71no—
222C2×C4×C8—71no—
223#401—141no—
224#400—141no—
225#400—141no—
226#400—141no—
227#391—71no—
228#391—71no—
229#391—71no—
230#390C2×D711yescharacteristic
231#390C2×D711yescharacteristic
232#389—71no—
233#389—71no—
234#389—71no—
235#389C2×D711yescharacteristic
236#389C2×D711yescharacteristic
237#388—21no—
238#388—21no—
239#381—21no—
240#380—21no—
241C4×C4⋊C4C2×D711yescharacteristic
242#376C2×D711yescharacteristic
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336—71no—
250#336—71no—
251C23×C7—22no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
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—84no—
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309Dic14—41no—
310Dic14—41no—
311Dic14—41no—
312Dic14—41no—
313C7⋊C8—22no—
314C7⋊C8—22no—
315C7⋊C8—41no—
316C7⋊C8—41no—
317C22×Q8—71no—
318C22×D4C22×D711yescharacteristic
319C23×C4—21no—
320C2×SD16—142no—
321C2×SD16—142no—
322C2×SD16—281no—
323C22×C8—71no—
324C22×C8—71no—
325C22×C8—71no—
326C22×C8—71no—
327C4×D4—22no—
328C4×D4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349#106—142no—
350#106—142no—
351#106—281no—
352C4⋊C8—142no—
353C4⋊C8—142no—
354#103—142no—
355#103—142no—
356#103—281no—
357#102—142no—
358#102—142no—
359#102—281no—
360#97—281no—
361C4×C8—281no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95C22⋊D711yescharacteristic
375#95C22⋊D711yescharacteristic
376#95C22⋊D711yescharacteristic
377#95C22⋊D711yescharacteristic
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7#52511yescharacteristic
386C22×C7#53211yescharacteristic
387C22×C7#53711yescharacteristic
388C22×C7#57711yescharacteristic
389C22×C7#57711yescharacteristic
390C22×C7#57711yescharacteristic
391C22×C7#57711yescharacteristic
392C4×C7—41no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7—41no—
397C4×C7—41no—
398C4×C7—41no—
399C4×C7—81no—
400C4×C7—81no—
401C4×C7#55111yescharacteristic
402C4×C7#55211yescharacteristic
403C4×C7#55311yescharacteristic
404C4×C7#55611yescharacteristic
405Dic7—81no—
406Dic7—81no—
407Dic7—81no—
408Dic7—81no—
409C24—21no—
410C2×Q8—141no—
411C2×Q8—141no—
412C2×Q8—141no—
413C2×Q8—141no—
414C2×Q8—141no—
415C2×Q8—141no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4#115712yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C23×D711yescharacteristic
439SD16—284no—
440C2×C8—72no—
441C2×C8—72no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C4⋊C4—561no—
468C4⋊C4#115712yes—
469C22⋊C4—24no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C2×C7—82no—
482C2×C7#260111yescharacteristic
483C2×C7#312311yescharacteristic
484C2×C7#312511yescharacteristic
485C2×C7#317311yescharacteristic
486C2×C7#317411yescharacteristic
487C2×C7#326011yescharacteristic
488C2×C7#326311yescharacteristic
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×C22⋊D711yescharacteristic
497Q8—281no—
498Q8—281no—
499Q8—281no—
500Q8—281no—
501D4—22no—
502D4—22no—
503D4—41no—
504D4—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4C2×#115711yescharacteristic
546C2×C4C2×#115711yescharacteristic
547C2×C4D7×#4111yescharacteristic
548C2×C4D7×#4111yescharacteristic
549C2×C4#655311yescharacteristic
550C2×C4#655411yescharacteristic
551C8—142no—
552C8—142no—
553C8—281no—
554C8—281no—
555C7#2944011yescharacteristic, complemented, Sylow
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#9048211yescharacteristic
564C22#9052311yescharacteristic
565C22#9052411yescharacteristic
566C22#9052711yescharacteristic
567C22#9052711yescharacteristic
568C22#9052711yescharacteristic
569C22#9052811yescharacteristic
570C4—41no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—561no—
580C4—561no—
581C4—561no—
582C4—561no—
583C4#9021811yescharacteristic
584C4#9022111yescharacteristic
585C4#9022211yescharacteristic
586C4#9022411yescharacteristic
587C2—82no—
588C2#1173970411yescharacteristic
589C2#1173999011yescharacteristic
590C2#1174002111yescharacteristic
591C2#1174087611yescharacteristic
592C2#1174088611yescharacteristic
593C2#1174090111yescharacteristic
594C2#1174091411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2319C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2319D7
hypocenterC7#29440
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #11730853C2semidirect
9. #11727388C2semidirect
10. #11727372C2semidirect
11. #11727370C2semidirect
12. #11727364C2semidirect
13. #11727218C2semidirect
14. #11727213C2semidirect
15. #11727190C2semidirect
555. C7#29440semidirect

Maximal subgroups

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

subgroupindex
C7×#23192
#117308532
#117293662
#117293612
#117274062
#117274052
#117274002
#117273992
#117273882
#117273722
#117273702
#117273642
#117272182
#117272132
#117271902
#294407