Finite Group Databasebeta

← #13412778

Subgroups — #13412778

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11733511C211yesmaximal, characteristic
2#11731086C211yesmaximal, characteristic
3C7×#1919C211yesmaximal, characteristic
4#11728344C211yesmaximal, characteristic
5#11728308C211yesmaximal, characteristic
6#11727388C211yesmaximal, characteristic
7#11727221C211yesmaximal, characteristic
8#11726666C211yesmaximal, characteristic
9#11726658C211yesmaximal, characteristic
10#11726570C211yesmaximal, characteristic
11#11726550C211yesmaximal, characteristic
12#11726550C211yesmaximal, characteristic
13#11726535C211yesmaximal, characteristic
14#11726446C211yesmaximal, characteristic
15#11726371C211yesmaximal, characteristic
16C2×#6452—21no—
17C2×#6450C2211yescharacteristic
18C2×C7×#106C2211yescharacteristic
19C7×#389C2211yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C2×#6391C2211yescharacteristic
22C2×#6391C2211yescharacteristic
23C2×#6390C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×Dic7⋊C8C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27C2×C8×Dic7C2211yescharacteristic
28#89755—21no—
29#89748C2211yescharacteristic
30#89748C2211yescharacteristic
31#89748C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34#89746C2211yescharacteristic
35#89745C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41#89707C2211yescharacteristic
42#89705—21no—
43C4×Dic7⋊C4—21no—
44C4×Dic7⋊C4C2211yescharacteristic
45#89701C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47#89429—21no—
48#89424—21no—
49#89422—21no—
50#89419—21no—
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54C7×#336C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26088—71nomaximal, Sylow
60C22×Dic14C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C42×C7—21no—
69C2×C42×C7C2311yescharacteristic
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2311yescharacteristic
74C2×C4⋊Dic7C2311yescharacteristic
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4D411yescharacteristic
84C2×Dic7⋊C4D411yescharacteristic
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92#6452—44no—
93#6450—22no—
94#6450Q814yes—
95C7×#106—22no—
96C7×#95—21no—
97C7×#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95D411yescharacteristic
106C7⋊#95D411yescharacteristic
107#6391—22no—
108#6391—41no—
109#6390—22no—
110#6390—41no—
111Dic7⋊C8—22no—
112Dic7⋊C8—22no—
113C8×Dic7—22no—
114#6384—22no—
115#6384—41no—
116#6382—41no—
117#6382—41no—
118#6382—41no—
119C4⋊(C7⋊C8)—22no—
120#2552—71no—
121#2355—71no—
122#2096—71no—
123#2096—71no—
124#2066—71no—
125#2066—71no—
126#1948—71no—
127#1948—71no—
128#1948—71no—
129#1939—71no—
130#1919—71no—
131#1919D711yescharacteristic
132#1844—71no—
133#1803—71no—
134C4×#103—71no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14C2×Q812yes—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8#4112yes—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4C2×Q812yes—
171C42×C7—21no—
172C42×C7—21no—
173C42×C7—81no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180C4⋊Dic7—42no—
181C4⋊Dic7—81no—
182Dic7⋊C4—24no—
183Dic7⋊C4—22no—
184Dic7⋊C4—42no—
185Dic7⋊C4—42no—
186Dic7⋊C4—42no—
187Dic7⋊C4—42no—
188Dic7⋊C4—81no—
189Dic7⋊C4—81no—
190Dic7⋊C4—81no—
191Dic7⋊C4SD1614yes—
192C4×Dic7—21no—
193C4×Dic7—21no—
194C4×Dic7—24no—
195C4×Dic7—42no—
196C4×Dic7—81no—
197C4×Dic7C2×Q812yes—
198C4×Dic7#4112yes—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8#4112yes—
205C2×#125—141no—
206C2×C4×Q8—71no—
207C2×#106—71no—
208C2×#106—71no—
209C2×#106—71no—
210C2×#106—71no—
211C2×#106—71no—
212C2×#106C2×D711yescharacteristic
213C2×C4⋊C8—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#103—71no—
217C2×#103—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×C4×C8—71no—
221#398—141no—
222#398—141no—
223#391—71no—
224#391—71no—
225#389—71no—
226#389—141no—
227#389C2×D711yescharacteristic
228#387—141no—
229#387—141no—
230#384—71no—
231#384—71no—
232#384—71no—
233#384—71no—
234#382—141no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—141no—
239C4×C4⋊C4C2×D711yescharacteristic
240#340—71no—
241#340—71no—
242#340—71no—
243#340C2×D711yescharacteristic
244#340C2×D711yescharacteristic
245#340C2×D711yescharacteristic
246#336—71no—
247#336C2×D711yescharacteristic
248C23×C7C22×D411yescharacteristic
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7C2×SD1612yes—
267C2×C4×C7C22×Q811yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7H(5,𝔽2)11yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C2×Dic7—21no—
274C2×Dic7—21no—
275C2×Dic7—21no—
276C2×Dic7—21no—
277C2×Dic7—22no—
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7C22⋊Q812yes—
293C2×Dic7C22⋊Q812yes—
294C2×Dic7C2×SD1612yes—
295C2×Dic7C2×SD1612yes—
296C2×Dic7#13712yes—
297Dic14—22no—
298Dic14—22no—
299Dic14—41no—
300Dic14—41no—
301C7×C8—22no—
302C7×C8—41no—
303C7⋊C8—22no—
304C7⋊C8—41no—
305C22×Q8—71no—
306C22×C8—71no—
307C22×C8—71no—
308C22×C8C22×D711yescharacteristic
309C22×C8C22×D711yescharacteristic
310#125—284no—
311C4×Q8—74no—
312C4×Q8—142no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C4⋊C4C22×D711yescharacteristic
331C2×C4⋊C4C22×D711yescharacteristic
332C2×C4⋊C4C22×D711yescharacteristic
333C2×C42—21no—
334C2×C42—71no—
335C2×C42—71no—
336C2×C42—141no—
337C2×C42—141no—
338C2×C42—141no—
339C2×C42C22×D711yescharacteristic
340#106—22no—
341#106—142no—
342#106—281no—
343#106—281no—
344#106—281no—
345#106—281no—
346C4⋊C8—142no—
347C4⋊C8—142no—
348C4⋊C8—142no—
349#103—142no—
350#103—142no—
351#103—281no—
352#103—281no—
353C4×C8—142no—
354#95—21no—
355#95—21no—
356#95—71no—
357#95—71no—
358#95—141no—
359#95—141no—
360#95—141no—
361#95—141no—
362#95—141no—
363#95—141no—
364#95—141no—
365C22×C7C2×C22⋊Q811yescharacteristic
366C22×C7#54711yescharacteristic
367C22×C7#55011yescharacteristic
368C22×C7C22×SD1611yescharacteristic
369C22×C7C2×#13711yescharacteristic
370C22×C7#57711yescharacteristic
371C22×C7#57711yescharacteristic
372C4×C7—22no—
373C4×C7—42no—
374C4×C7—41no—
375C4×C7—81no—
376C4×C7—81no—
377C4×C7—81no—
378C4×C7#53311yescharacteristic
379C4×C7#54411yescharacteristic
380C4×C7#55211yescharacteristic
381C4×C7#55311yescharacteristic
382Dic7—22no—
383Dic7—22no—
384Dic7—41no—
385Dic7—42no—
386Dic7—81no—
387Dic7—81no—
388Dic7—81no—
389Dic7#47514yes—
390C2×Q8—72no—
391C2×Q8—141no—
392C2×Q8—141no—
393C2×Q8—141no—
394C2×Q8—141no—
395C2×Q8—141no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—71no—
401C22×C4—71no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—141no—
405C22×C4—141no—
406C22×C4—141no—
407C22×C4—141no—
408C22×C4D4×D711yescharacteristic
409C22×C4D4×D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—72no—
417C2×C8—141no—
418C2×C8—141no—
419C2×C8—141no—
420C2×C8—141no—
421C2×C8—141no—
422C2×C8#115712yes—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—74no—
432C4⋊C4—142no—
433C4⋊C4—144no—
434C4⋊C4—141no—
435C4⋊C4—141no—
436C4⋊C4—141no—
437C4⋊C4—141no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—282no—
441C4⋊C4—282no—
442C4⋊C4—282no—
443C4⋊C4—282no—
444C4⋊C4—282no—
445C4⋊C4—561no—
446C4⋊C4—561no—
447C4⋊C4—561no—
448C4⋊C4—561no—
449C4⋊C4Q8×D712yes—
450C42—21no—
451C42—21no—
452C42—72no—
453C42—72no—
454C42—81no—
455C42—144no—
456C42—141no—
457C42—141no—
458C42—282no—
459C42—561no—
460C2×C7#244111yescharacteristic
461C2×C7C2×#47511yescharacteristic
462C2×C7#308411yescharacteristic
463C2×C7#333511yescharacteristic
464C2×C7#335611yescharacteristic
465C2×C7#339211yescharacteristic
466C2×C7#340011yescharacteristic
467C23C2×D4×D711yescharacteristic
468Q8—142no—
469Q8—142no—
470Q8—281no—
471Q8—281no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—72no—
490C2×C4—72no—
491C2×C4—72no—
492C2×C4—72no—
493C2×C4—72no—
494C2×C4—141no—
495C2×C4—141no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—142no—
501C2×C4—141no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4D7×SD1612yes—
514C2×C4C2×#115711yescharacteristic
515C2×C4C2×Q8×D711yescharacteristic
516C2×C4D7×#4111yescharacteristic
517C2×C4D7×#4111yescharacteristic
518C2×C4#655311yescharacteristic
519C2×C4#655411yescharacteristic
520C8—22no—
521C8—41no—
522C8—142no—
523C8—281no—
524C7#2608811yescharacteristic, complemented, Sylow
525C22#9024411yescharacteristic
526C22D7×C22⋊Q811yescharacteristic
527C22#9034611yescharacteristic
528C22C2×D7×SD1611yescharacteristic
529C22#9046111yescharacteristic
530C22#9046611yescharacteristic
531C22D7×#13711yescharacteristic
532C4—22no—
533C4—41no—
534C4—42no—
535C4—74no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—142no—
540C4—142no—
541C4—281no—
542C4—282no—
543C4—561no—
544C4—561no—
545C4—561no—
546C4#9022211yescharacteristic
547C4#9026911yescharacteristic
548C4#9038011yescharacteristic
549C4#9040011yescharacteristic
550C2#1173242911yescharacteristic
551C2#1173708211yescharacteristic
552C2#1173720911yescharacteristic
553C2D7×#47511yescharacteristic
554C2#1173861611yescharacteristic
555C2#1173876511yescharacteristic
556C2#1173895711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1919C2
socle, prime radicalC23×C7C22×D4
hypercenter#1919D7
hypocenterC7#26088
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
524. C7#26088semidirect

Maximal subgroups

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

subgroupindex
#117335112
#117310862
C7×#19192
#117283442
#117283082
#117273882
#117272212
#117266662
#117266582
#117265702
#117265502
#117265352
#117264462
#117263712
#260887