Finite Group Databasebeta

← #13422392

Subgroups — #13422392

Order 1792.

672 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740674C211yesmaximal, characteristic
2#11733480C211yesmaximal, characteristic
3C7×#2043C211yesmaximal, characteristic
4#11729535C211yesmaximal, characteristic
5#11728016C211yesmaximal, characteristic
6#11728007C211yesmaximal, characteristic
7#11727996C211yesmaximal, characteristic
8#11727990C211yesmaximal, characteristic
9#11727990C211yesmaximal, characteristic
10#11727987C211yesmaximal, characteristic
11#11727975C211yesmaximal, characteristic
12Dic7×#103C211yesmaximal, characteristic
13#11727357C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic
15#11726363C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×C7×#103C2211yescharacteristic
20C2×C7×#103C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C7×#389C2211yescharacteristic
24#90002C2211yescharacteristic
25#90002C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89757—21no—
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89756—21no—
37#89753—21no—
38#89753—21no—
39#89748C2211yescharacteristic
40C4⋊C4×Dic7—21no—
41C4⋊C4×Dic7C2211yescharacteristic
42#89745—21no—
43C2×#6381C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45C4×C4⋊Dic7C2211yescharacteristic
46#89700C2211yescharacteristic
47#89433—21no—
48#89426—21no—
49C7×#336C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26758—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C22×C7×Q8C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72Q8×Dic7—22no—
73Q8×Dic7—22no—
74Q8×Dic7—41no—
75Q8×Dic7—41no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7D411yescharacteristic
81C2×C4⋊Dic7D411yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C7×#103—22no—
98C7×#103—22no—
99C7×#103—22no—
100C7×#103—22no—
101C7×#95—21no—
102#6409—22no—
103#6409—22no—
104#6409—41no—
105#6409—41no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95D411yescharacteristic
116C7⋊#95D411yescharacteristic
117#6392—22no—
118Dic7⋊C8—41no—
119C8×Dic7—41no—
120#6381—22no—
121#2507—71no—
122#2365—71no—
123#2072—71no—
124#2068—71no—
125#2043—71no—
126#2043D711yescharacteristic
127#1933—71no—
128#1928—71no—
129#1921—71no—
130#1921—71no—
131#1916—71no—
132#1863—71no—
133#1863—71no—
134#1803—71no—
135C4×#103—71no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8#4112yes—
147C2×C7×Q8#4112yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4#4112yes—
176C42×C7—21no—
177C42×C7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—41no—
186C4⋊Dic7—42no—
187C4⋊Dic7—42no—
188C4⋊Dic7—42no—
189C4⋊Dic7—81no—
190C4⋊Dic7Q1614yes—
191C4⋊Dic7#4112yes—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—42no—
200C4×Dic7—42no—
201C4×Dic7—42no—
202C4×Dic7—81no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C4⋊Q8C2×D711yescharacteristic
210C2×C4×Q8—71no—
211C2×C4×Q8—71no—
212C2×#107—71no—
213C2×#107—71no—
214C2×C4⋊C8—71no—
215C2×#103—71no—
216C2×#103—71no—
217C2×#103—71no—
218C2×#103—71no—
219C2×#103C2×D711yescharacteristic
220C2×#103C2×D711yescharacteristic
221C2×#103C2×D711yescharacteristic
222C2×#103C2×D711yescharacteristic
223C2×C4×C8—71no—
224#398—141no—
225#398—141no—
226#391—71no—
227#391—71no—
228#389—71no—
229#389—71no—
230#389—71no—
231#389—141no—
232#389—141no—
233#389C2×D711yescharacteristic
234#387—141no—
235#387—141no—
236#387—141no—
237#384—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—141no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#336—71no—
249#336—71no—
250#336—71no—
251#336C2×D711yescharacteristic
252C23×C7C22×D411yescharacteristic
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8—22no—
257C7×Q8—41no—
258C7×Q8—41no—
259C7×Q8—41no—
260C7×Q8—41no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7C2×Q1612yes—
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7H(5,𝔽2)11yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—22no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—41no—
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×Dic7C2×Q1612yes—
308C2×Dic7C2×Q1612yes—
309C7×C8—41no—
310C7×C8—41no—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C22×Q8C22×D711yescharacteristic
314C22×Q8C22×D711yescharacteristic
315C22×C8—71no—
316C22×C8—71no—
317C22×C8C22×D711yescharacteristic
318C22×C8C22×D711yescharacteristic
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4×Q8—142no—
322C4×Q8—142no—
323C4×Q8—281no—
324C4×Q8—281no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
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⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C42—71no—
344C2×C42—71no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42C22×D711yescharacteristic
350#107—142no—
351#107—142no—
352C4⋊C8—281no—
353#103—22no—
354#103—22no—
355#103—22no—
356#103—22no—
357#103—142no—
358#103—142no—
359#103—281no—
360#103—281no—
361C4×C8—281no—
362#95—21no—
363#95—71no—
364#95—71no—
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—
374C22×C7#52511yescharacteristic
375C22×C7#54611yescharacteristic
376C22×C7#54811yescharacteristic
377C22×C7C22×Q1611yescharacteristic
378C22×C7#57511yescharacteristic
379C22×C7#57711yescharacteristic
380C22×C7#57711yescharacteristic
381C4×C7—22no—
382C4×C7—42no—
383C4×C7—41no—
384C4×C7—42no—
385C4×C7—42no—
386C4×C7—81no—
387C4×C7#52911yescharacteristic
388C4×C7#53911yescharacteristic
389C4×C7#55211yescharacteristic
390C4×C7#55311yescharacteristic
391Dic7—22no—
392Dic7—22no—
393Dic7—41no—
394Dic7—41no—
395Dic7—42no—
396Dic7—81no—
397Dic7—81no—
398Dic7—81no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8#115912yes—
410C2×Q8#115912yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4D4×D711yescharacteristic
424C22×C4D4×D711yescharacteristic
425C22×C4C23×D711yescharacteristic
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—74no—
444C4⋊C4—72no—
445C4⋊C4—81no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—281no—
454C4⋊C4—282no—
455C4⋊C4—282no—
456C4⋊C4—282no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4—561no—
460C4⋊C4—561no—
461C4⋊C4#115912yes—
462C42—21no—
463C42—21no—
464C42—141no—
465C42—141no—
466C42—141no—
467C42—141no—
468C42—282no—
469C42—282no—
470C42—282no—
471C42—561no—
472C2×C7#246011yescharacteristic
473C2×C7#310411yescharacteristic
474C2×C7#312611yescharacteristic
475C2×C7#329911yescharacteristic
476C2×C7#330711yescharacteristic
477C2×C7#336311yescharacteristic
478C2×C7#337611yescharacteristic
479C23C2×D4×D711yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—72no—
506C2×C4—72no—
507C2×C4—141no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—142no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4D7×Q1612yes—
530C2×C4C2×#115911yescharacteristic
531C2×C4C2×#115911yescharacteristic
532C2×C4C2×#115911yescharacteristic
533C2×C4D7×#4111yescharacteristic
534C2×C4#655311yescharacteristic
535C2×C4#655411yescharacteristic
536C8—41no—
537C8—41no—
538C8—281no—
539C8—281no—
540C7#2675811yescharacteristic, complemented, Sylow
541C22#9027711yescharacteristic
542C22#9028911yescharacteristic
543C22#9036711yescharacteristic
544C22#9045311yescharacteristic
545C22C2×D7×Q1611yescharacteristic
546C22#9046811yescharacteristic
547C22#9046911yescharacteristic
548C4—22no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—81no—
554C4—142no—
555C4—142no—
556C4—281no—
557C4—281no—
558C4—282no—
559C4—561no—
560C4—561no—
561C4—561no—
562C4#9029111yescharacteristic
563C4#9030911yescharacteristic
564C4#9034111yescharacteristic
565C4#9035611yescharacteristic
566C2#1173435911yescharacteristic
567C2#1173747811yescharacteristic
568C2#1173761711yescharacteristic
569C2#1173786411yescharacteristic
570C2#1173790711yescharacteristic
571C2#1173905811yescharacteristic
572C2#1173919311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2043C2
socle, prime radicalC23×C7C22×D4
hypercenter#2043D7
hypocenterC7#26758
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
540. C7#26758semidirect

Maximal subgroups

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

subgroupindex
#117406742
#117334802
C7×#20432
#117295352
#117280162
#117280072
#117279962
#117279902
#117279872
#117279752
Dic7×#1032
#117273572
#117263782
#117263632
#267587