Finite Group Databasebeta

← #13411942

Subgroups — #13411942

Order 1792.

710 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740687C211yesmaximal, characteristic
2#11731086C211yesmaximal, characteristic
3C7×#1987C211yesmaximal, characteristic
4#11729548C211yesmaximal, characteristic
5#11728243C211yesmaximal, characteristic
6#11728205C211yesmaximal, characteristic
7#11727215C211yesmaximal, characteristic
8#11726667C211yesmaximal, characteristic
9#11726664C211yesmaximal, characteristic
10#11726658C211yesmaximal, characteristic
11#11726601C211yesmaximal, characteristic
12#11726574C211yesmaximal, characteristic
13#11726553C211yesmaximal, characteristic
14#11726549C211yesmaximal, characteristic
15#11726535C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×#6507—21no—
18C2×#6451C2211yescharacteristic
19C2×#6450C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22#90001—21no—
23C2×#6392C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×#6390C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29#89751—21no—
30#89748C2211yescharacteristic
31#89748C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34#89746C2211yescharacteristic
35#89745C2211yescharacteristic
36#89744—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89705—21no—
41C4×Dic7⋊C4—21no—
42C4×Dic7⋊C4C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45#89424—21no—
46#89422—21no—
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26026—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×Dic14—21no—
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—41no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74#6507—44no—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4D411yescharacteristic
87C2×Dic7⋊C4D411yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6451—22no—
96#6451—41no—
97#6450—22no—
98#6450Q814yescomplemented
99C7×C4⋊C8—41no—
100C7×#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95D411yescharacteristic
110C7⋊#95D411yescharacteristic
111#6392—22no—
112#6390—22no—
113#6390—41no—
114Dic7⋊C8—22no—
115Dic7⋊C8—41no—
116C8×Dic7—22no—
117#6384—22no—
118#6384—41no—
119#6381—22no—
120#2551—71no—
121#2355—71no—
122#2068—71no—
123#2068—71no—
124#2024—71no—
125#2024—71no—
126#1987—71no—
127#1987D711yescharacteristic
128#1947—71no—
129#1939—71no—
130#1921—71no—
131#1921—71no—
132#1834—71no—
133#1803—71no—
134C4×#103—71no—
135C2×C7×Q8—42no—
136C2×C7×Q8—42no—
137C2×C7×Q8—81no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2411yescharacteristic
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14—21no—
158C2×Dic14—42no—
159C2×Dic14—42no—
160C2×Dic14—81no—
161C2×Dic14C2×Q812yes—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4C2×Q812yes—
176C7×C4⋊C4#4112yes—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—81no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7#4112yes—
186Dic7⋊C4—24no—
187Dic7⋊C4—22no—
188Dic7⋊C4—42no—
189Dic7⋊C4—42no—
190Dic7⋊C4—42no—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195Dic7⋊C4Q1614yes—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—24no—
199C4×Dic7—42no—
200C4×Dic7—81no—
201C4×Dic7C2×Q812yes—
202C4×Dic7#4112yes—
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⋊Q8—71no—
210C2×C4⋊Q8—141no—
211C2×C4⋊Q8C2×D711yescharacteristic
212C2×C4×Q8—71no—
213C2×#107—71no—
214C2×#107—71no—
215C2×C4⋊C8—71no—
216C2×C4⋊C8—71no—
217C2×C4⋊C8C2×D711yescharacteristic
218C2×#103—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×C4×C8—71no—
223#395—141no—
224#395—141no—
225#391—71no—
226#391—71no—
227#389—71no—
228#389—71no—
229#389—141no—
230#387—141no—
231#387—141no—
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#340—71no—
244#340—71no—
245#340—71no—
246#340C2×D711yescharacteristic
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#340C2×D711yescharacteristic
250#336—71no—
251#336—71no—
252C23×C7C22×D411yescharacteristic
253C7×Q8—84no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7C2×Q1612yes—
272C2×C4×C7C22×Q811yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C72−1+411yescharacteristic
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—22no—
283C2×Dic7—21no—
284C2×Dic7—22no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7C22⋊Q812yes—
298C2×Dic7C22⋊Q812yes—
299C2×Dic7C2×Q1612yes—
300C2×Dic7C2×Q1612yes—
301C2×Dic7AGL(1,ℤ/8)12yes—
302Dic14—22no—
303Dic14—22no—
304Dic14—41no—
305Dic14—41no—
306Dic14—84no—
307C7×C8—41no—
308C7×C8—41no—
309C7⋊C8—41no—
310C7⋊C8—41no—
311C22×Q8—21no—
312C22×Q8—71no—
313C22×Q8—141no—
314C22×C8—71no—
315C22×C8—71no—
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D711yescharacteristic
318C4⋊Q8—22no—
319C4⋊Q8—41no—
320C4⋊Q8—142no—
321C4⋊Q8—281no—
322C4⋊Q8—284no—
323C4×Q8—74no—
324C4×Q8—142no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C42—21no—
344C2×C42—71no—
345C2×C42—71no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42C22×D711yescharacteristic
350#107—142no—
351#107—142no—
352C4⋊C8—41no—
353C4⋊C8—142no—
354C4⋊C8—281no—
355#103—142no—
356#103—142no—
357#103—281no—
358#103—281no—
359C4×C8—142no—
360#95—21no—
361#95—71no—
362#95—71no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371C22×C7C2×C22⋊Q811yescharacteristic
372C22×C7#54611yescharacteristic
373C22×C7#54811yescharacteristic
374C22×C7C22×Q1611yescharacteristic
375C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
376C22×C7#57711yescharacteristic
377C22×C7#57711yescharacteristic
378C4×C7—22no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—41no—
382C4×C7—81no—
383C4×C7—81no—
384C4×C7#53311yescharacteristic
385C4×C7#54411yescharacteristic
386C4×C7#55511yescharacteristic
387C4×C7#55611yescharacteristic
388Dic7—22no—
389Dic7—22no—
390Dic7—42no—
391Dic7—42no—
392Dic7—41no—
393Dic7—81no—
394Dic7—81no—
395Dic7#47714yes—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—72no—
399C2×Q8—81no—
400C2×Q8—141no—
401C2×Q8—141no—
402C2×Q8—141no—
403C2×Q8—141no—
404C2×Q8—141no—
405C2×Q8—282no—
406C2×Q8—282no—
407C2×Q8—561no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—71no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4D4×D711yescharacteristic
421C22×C4D4×D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—74no—
443C4⋊C4—72no—
444C4⋊C4—142no—
445C4⋊C4—144no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—282no—
452C4⋊C4—282no—
453C4⋊C4—282no—
454C4⋊C4—282no—
455C4⋊C4—282no—
456C4⋊C4—561no—
457C4⋊C4—561no—
458C4⋊C4#115712yes—
459C4⋊C4Q8×D712yes—
460C42—21no—
461C42—21no—
462C42—72no—
463C42—72no—
464C42—81no—
465C42—144no—
466C42—141no—
467C42—141no—
468C42—282no—
469C42—561no—
470C2×C7#243911yescharacteristic
471C2×C7C2×#47711yescharacteristic
472C2×C7#308411yescharacteristic
473C2×C7#329911yescharacteristic
474C2×C7#332411yescharacteristic
475C2×C7#336311yescharacteristic
476C2×C7#337311yescharacteristic
477C23C2×D4×D711yescharacteristic
478Q8—84no—
479Q8—142no—
480Q8—142no—
481Q8—281no—
482Q8—281no—
483Q8—564no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—72no—
502C2×C4—72no—
503C2×C4—72no—
504C2×C4—72no—
505C2×C4—72no—
506C2×C4—141no—
507C2×C4—141no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4D7×Q1612yes—
526C2×C4C2×#115711yescharacteristic
527C2×C4#654811yescharacteristic
528C2×C4C2×Q8×D711yescharacteristic
529C2×C4#655111yescharacteristic
530C2×C4D7×#4111yescharacteristic
531C2×C4D7×#4111yescharacteristic
532C8—41no—
533C8—41no—
534C8—281no—
535C8—281no—
536C7#2602611yescharacteristic, complemented, Sylow
537C22#9018111yescharacteristic
538C22D7×C22⋊Q811yescharacteristic
539C22#9032411yescharacteristic
540C22#9045311yescharacteristic
541C22C2×D7×Q1611yescharacteristic
542C22D7×AGL(1,ℤ/8)11yescharacteristic
543C22#9046611yescharacteristic
544C4—22no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—74no—
549C4—81no—
550C4—81no—
551C4—142no—
552C4—142no—
553C4—281no—
554C4—282no—
555C4—282no—
556C4—561no—
557C4—561no—
558C4#9025911yescharacteristic
559C4#9026911yescharacteristic
560C4#9036111yescharacteristic
561C4#9042011yescharacteristic
562C2#1173394611yescharacteristic
563C2#1173625111yescharacteristic
564C2#1173638511yescharacteristic
565C2#1173849711yescharacteristic
566C2#1173850811yescharacteristic
567C2D7×#47711yescharacteristic
568C2#1173858311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1987C2
socle, prime radicalC23×C7C22×D4
hypercenter#1987D7
hypocenterC7#26026
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
98. #6450Q8semidirect
536. C7#26026semidirect

Maximal subgroups

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

subgroupindex
#117406872
#117310862
C7×#19872
#117295482
#117282432
#117282052
#117272152
#117266672
#117266642
#117266582
#117266012
#117265742
#117265532
#117265492
#117265352
#260267