Finite Group Databasebeta

← #13412959

Subgroups — #13412959

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2357C211yesmaximal, characteristic
2#11731162C211yesmaximal, characteristic, complemented
3#11729363C211yesmaximal, characteristic
4#11729358C211yesmaximal, characteristic
5#11727410C211yesmaximal, characteristic
6#11727409C211yesmaximal, characteristic
7#11727407C211yesmaximal, characteristic
8#11727406C211yesmaximal, characteristic
9#11727304C211yesmaximal, characteristic, complemented
10#11727294C211yesmaximal, characteristic, complemented
11#11727239C211yesmaximal, characteristic, complemented
12#11727221C211yesmaximal, characteristic, complemented
13#11727220C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727213C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6453—21no—
18C7×#390C2211yescharacteristic
19C7×#390C2211yescharacteristic
20C7×#388—21no—
21C7×#388—21no—
22C7×#382C2211yescharacteristic
23C7×#381—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4—21no—
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89757—21no—
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89707C2211yescharacteristic
40#89707C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49#89433—21no—
50#89433—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26102—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C4×C7×D4—22no—
63C4×C7×D4Q814yes—
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6453—44no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103#6406—22no—
104#6406—22no—
105#6406—41no—
106#6406—41no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110#6382—22no—
111#6382—41no—
112#6382—41no—
113#6381—22no—
114#6381—41no—
115#6381—41no—
116C4⋊(C7⋊C8)—22no—
117C4⋊(C7⋊C8)—22no—
118C4⋊(C7⋊C8)—22no—
119#6377—22no—
120#2552—71no—
121#2357D711yescharacteristic
122#2094—71no—
123#2093—71no—
124#2067—71no—
125#2065—71no—
126#1949—71no—
127#1948—71no—
128#1947—71no—
129#1940—71no—
130#1921—71no—
131#1919—71no—
132#1847—71no—
133#1802—71no—
134#1762—71no—
135C24×C7—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C2×Q812yes—
142C22×C4×C7—21no—
143C22×C4×C7—24no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—81no—
147C22×C4×C7—81no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4C2×Q812yes—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—42no—
169C7×C22⋊C4—42no—
170C7×C22⋊C4—81no—
171C7×C22⋊C4—81no—
172C42×C7—21no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—81no—
177C42×C7—81no—
178C42×C7C2×Q812yes—
179C42×C7#4112yes—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—42no—
189C4⋊Dic7—42no—
190C4⋊Dic7—42no—
191C4⋊Dic7—81no—
192C4⋊Dic7—81no—
193C4⋊Dic7—81no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196C4×Dic7—42no—
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—21no—
207C2×C7⋊C8#4112yes—
208C2×C7⋊C8#4112yes—
209C2×#125—141no—
210C2×C4×D4C2×D711yescharacteristic
211C2×#107—71no—
212C2×#107—71no—
213C2×#107—71no—
214C2×#106—71no—
215C2×#106—71no—
216C2×#106—71no—
217C2×C4⋊C8—71no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8—71no—
220C2×#102—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#97—71no—
225#398—141no—
226#398—141no—
227#390C2×D711yescharacteristic
228#390C2×D711yescharacteristic
229#389—71no—
230#389—71no—
231#389—141no—
232#388—21no—
233#388—21no—
234#384—71no—
235#384—71no—
236#384—71no—
237#382C2×D711yescharacteristic
238#381—21no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4C2×D711yescharacteristic
241C4×C4⋊C4C2×D711yescharacteristic
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C22⋊C4—21no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336—71no—
251#336—71no—
252C23×C7—22no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7C22×D411yescharacteristic
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—41no—
263C7×D4—41no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
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—22no—
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—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—84no—
288C2×C4×C7C22⋊Q812yes—
289C2×C4×C7C22⋊Q812yes—
290C2×C4×C7AGL(1,ℤ/8)12yes—
291C2×C4×C7#13712yes—
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
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—
309C2×Dic7—41no—
310C7⋊C8—22no—
311C7⋊C8—22no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C22×D4C22×D711yescharacteristic
315C23×C4—21no—
316C22×C8—71no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8—71no—
320#125—284no—
321C4×D4—22no—
322C4×D4Dic1414yes—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22⋊D711yescharacteristic
336C2×C4⋊C4C22⋊D711yescharacteristic
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×C42—21no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351#107—142no—
352#107—281no—
353#107—281no—
354#106—142no—
355#106—281no—
356#106—281no—
357C4⋊C8—142no—
358C4⋊C8—142no—
359C4⋊C8—142no—
360#102—142no—
361#102—142no—
362#102—281no—
363#102—281no—
364#97—142no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95C22⋊D711yescharacteristic
373#95C22⋊D711yescharacteristic
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54711yescharacteristic
383C22×C7#55011yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7C2×#13711yescharacteristic
386C22×C7#57611yescharacteristic
387C22×C7#57711yescharacteristic
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—42no—
393C4×C7—41no—
394C4×C7—41no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7#53311yescharacteristic
398C4×C7#54411yescharacteristic
399C4×C7#55211yescharacteristic
400C4×C7#55311yescharacteristic
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4C2×Dic1412yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C23×D711yescharacteristic
429C2×C8—72no—
430C2×C8—72no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—282no—
457C4⋊C4—282no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—561no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4C2×Dic1412yes—
465C22⋊C4—24no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—81no—
475C42—81no—
476C42—282no—
477C42C2×Dic1412yes—
478C42#115712yes—
479C2×C7—82no—
480C2×C7#244111yescharacteristic
481C2×C7#307911yescharacteristic
482C2×C7#308111yescharacteristic
483C2×C7#334511yescharacteristic
484C2×C7#334611yescharacteristic
485C2×C7#339911yescharacteristic
486C2×C7#340011yescharacteristic
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D711yescharacteristic
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4#648712yes—
536C2×C4#648712yes—
537C2×C4#651212yes—
538C2×C4#651412yes—
539C2×C4C22×Dic1411yescharacteristic
540C2×C4C2×#115711yescharacteristic
541C2×C4D7×#4111yescharacteristic
542C2×C4D7×#4111yescharacteristic
543C2×C4#655311yescharacteristic
544C2×C4#655411yescharacteristic
545C8—142no—
546C8—142no—
547C8—281no—
548C8—281no—
549C7#2610211yescharacteristic, complemented, Sylow
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#648711yescharacteristic
558C22C2×#651211yescharacteristic
559C22C2×#651411yescharacteristic
560C22#9051511yescharacteristic
561C22#9052511yescharacteristic
562C22#9052611yescharacteristic
563C22#9052711yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9020811yescharacteristic
578C4#9021011yescharacteristic
579C4#9022011yescharacteristic
580C4#9022111yescharacteristic
581C2—82no—
582C2#1173969411yescharacteristic
583C2#1173995311yescharacteristic
584C2#1173998411yescharacteristic
585C2#1174105311yescharacteristic
586C2#1174105411yescharacteristic
587C2#1174115611yescharacteristic
588C2#1174116611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2357C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2357D7
hypocenterC7#26102
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #11731162C2semidirect
9. #11727304C2semidirect
10. #11727294C2semidirect
11. #11727239C2semidirect
12. #11727221C2semidirect
13. #11727220C2semidirect
14. #11727215C2semidirect
15. #11727213C2semidirect
549. C7#26102semidirect

Maximal subgroups

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

subgroupindex
C7×#23572
#117311622
#117293632
#117293582
#117274102
#117274092
#117274072
#117274062
#117273042
#117272942
#117272392
#117272212
#117272202
#117272152
#117272132
#261027