Finite Group Databasebeta

← #13412914

Subgroups — #13412914

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2354C211yesmaximal, characteristic
2#11731168C211yesmaximal, characteristic, complemented
3#11729363C211yesmaximal, characteristic
4#11729359C211yesmaximal, characteristic
5#11727410C211yesmaximal, characteristic
6#11727408C211yesmaximal, characteristic
7#11727407C211yesmaximal, characteristic
8#11727406C211yesmaximal, characteristic
9#11727332C211yesmaximal, characteristic, complemented
10#11727322C211yesmaximal, characteristic, complemented
11#11727239C211yesmaximal, characteristic, complemented
12#11727222C211yesmaximal, characteristic, complemented
13#11727221C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727212C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C7×#117—21no—
18C7×#390C2211yescharacteristic
19C7×#390C2211yescharacteristic
20C7×#388—21no—
21C7×#388—21no—
22C7×#384C2211yescharacteristic
23C7×#380—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31#89758—21no—
32#89756—21no—
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89708C2211yescharacteristic
40#89708C2211yescharacteristic
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#89427—21no—
50#89426—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26099—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C4×C7×D4—22no—
63C4×C7×D4Q814yes—
64C7×#117—44no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C7×#95—21no—
96C7×#95—21no—
97C7×#95D411yescharacteristic
98C7×#95D411yescharacteristic
99#6406—22no—
100#6406—22no—
101#6406—41no—
102#6406—41no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110#6382—22no—
111#6382—41no—
112#6382—41no—
113#6382—41no—
114#6381—22no—
115#6381—41no—
116C4⋊(C7⋊C8)—22no—
117C4⋊(C7⋊C8)—22no—
118C4⋊(C7⋊C8)—22no—
119#6377—22no—
120#2570—71no—
121#2354D711yescharacteristic
122#2094—71no—
123#2092—71no—
124#2067—71no—
125#2065—71no—
126#1949—71no—
127#1948—71no—
128#1948—71no—
129#1940—71no—
130#1919—71no—
131#1918—71no—
132#1847—71no—
133#1804—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—42no—
147C22×C4×C7—42no—
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—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4—81no—
168C7×C4⋊C4C2×Q812yes—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—42no—
171C7×C22⋊C4—42no—
172C7×C22⋊C4—81no—
173C7×C22⋊C4—81no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—42no—
179C42×C7—42no—
180C42×C7C2×Q812yes—
181C42×C7#4112yes—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—42no—
191C4⋊Dic7—81no—
192Dic7⋊C4—42no—
193Dic7⋊C4—81no—
194C4×Dic7—81no—
195C2×C7⋊C8—21no—
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#4112yes—
206C2×C7⋊C8#4112yes—
207C2×C4×D4C2×D711yescharacteristic
208C2×#117—21no—
209C2×#107—71no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×#106—71no—
215C2×C4⋊C8—71no—
216C2×C4⋊C8—71no—
217C2×C4⋊C8—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×#102—71no—
221C2×#102—71no—
222C2×#97—71no—
223#400—141no—
224#398—141no—
225#390C2×D711yescharacteristic
226#390C2×D711yescharacteristic
227#389—71no—
228#389—71no—
229#388—21no—
230#388—21no—
231#387—141no—
232#387—141no—
233#384—71no—
234#384C2×D711yescharacteristic
235#382—71no—
236#382—71no—
237#380—21no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4C2×D711yescharacteristic
240C4×C4⋊C4C2×D711yescharacteristic
241C4×C4⋊C4C2×D711yescharacteristic
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#336—71no—
249#336—71no—
250C23×C7—22no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7C22×D411yescharacteristic
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—41no—
261C7×D4—41no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
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—48no—
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×C7C22⋊Q812yes—
288C2×C4×C7C22⋊Q812yes—
289C2×C4×C7AGL(1,ℤ/8)12yes—
290C2×C4×C7C22×Q811yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
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×Dic7—41no—
308C7⋊C8—22no—
309C7⋊C8—22no—
310C7⋊C8—41no—
311C7⋊C8—41no—
312C22×D4C22×D711yescharacteristic
313C23×C4—21no—
314C22×C8—71no—
315C22×C8—71no—
316C22×C8—71no—
317C22×C8—71no—
318C4×D4—22no—
319C4×D4Dic1414yes—
320#117—44no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
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⋊C4C22⋊D711yescharacteristic
332C2×C4⋊C4C22⋊D711yescharacteristic
333C2×C4⋊C4C22×D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—141no—
344C2×C42C22×D711yescharacteristic
345C2×C42C22×D711yescharacteristic
346C2×C42C22×D711yescharacteristic
347#107—142no—
348#107—281no—
349#106—142no—
350#106—281no—
351#106—281no—
352#106—281no—
353C4⋊C8—142no—
354C4⋊C8—142no—
355C4⋊C8—142no—
356#102—142no—
357#102—142no—
358#102—281no—
359#102—281no—
360#97—142no—
361#95—21no—
362#95—21no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95C22⋊D711yescharacteristic
371#95C22⋊D711yescharacteristic
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7C2×C22⋊Q811yescharacteristic
380C22×C7#54711yescharacteristic
381C22×C7#55011yescharacteristic
382C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
383C22×C7#57511yescharacteristic
384C22×C7#57711yescharacteristic
385C22×C7#57711yescharacteristic
386C4×C7—22no—
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—81no—
394C4×C7—81no—
395C4×C7#53311yescharacteristic
396C4×C7#54411yescharacteristic
397C4×C7#55211yescharacteristic
398C4×C7#55311yescharacteristic
399Dic7—81no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×Dic1412yes—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427C2×C8—72no—
428C2×C8—72no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—81no—
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—561no—
459C4⋊C4—561no—
460C4⋊C4C2×Dic1412yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—42no—
471C42—42no—
472C42—561no—
473C42C2×Dic1412yes—
474C42#115712yes—
475C2×C7—82no—
476C2×C7#244111yescharacteristic
477C2×C7#307711yescharacteristic
478C2×C7#308411yescharacteristic
479C2×C7#334511yescharacteristic
480C2×C7#334811yescharacteristic
481C2×C7#339711yescharacteristic
482C2×C7#340111yescharacteristic
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C22×C22⋊D711yescharacteristic
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—48no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
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×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4#648712yes—
533C2×C4#648712yes—
534C2×C4#649512yes—
535C2×C4C22×Dic1411yescharacteristic
536C2×C4C2×#115711yescharacteristic
537C2×C4D7×#4111yescharacteristic
538C2×C4D7×#4111yescharacteristic
539C2×C4#655311yescharacteristic
540C2×C4#655411yescharacteristic
541C8—142no—
542C8—142no—
543C8—281no—
544C8—281no—
545C7#2609911yescharacteristic, complemented, Sylow
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#648711yescharacteristic
554C22C2×#649511yescharacteristic
555C22#9051311yescharacteristic
556C22#9051511yescharacteristic
557C22#9052511yescharacteristic
558C22#9052711yescharacteristic
559C22#9052811yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—561no—
570C4—561no—
571C4—561no—
572C4—561no—
573C4#9020811yescharacteristic
574C4#9021011yescharacteristic
575C4#9022011yescharacteristic
576C4#9022111yescharacteristic
577C2—82no—
578C2#1173969411yescharacteristic
579C2#1173974111yescharacteristic
580C2#1174000511yescharacteristic
581C2#1174096411yescharacteristic
582C2#1174102111yescharacteristic
583C2#1174109411yescharacteristic
584C2#1174111711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2354C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2354D7
hypocenterC7#26099
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #11731168C2semidirect
9. #11727332C2semidirect
10. #11727322C2semidirect
11. #11727239C2semidirect
12. #11727222C2semidirect
13. #11727221C2semidirect
14. #11727213C2semidirect
15. #11727212C2semidirect
545. C7#26099semidirect

Maximal subgroups

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

subgroupindex
C7×#23542
#117311682
#117293632
#117293592
#117274102
#117274082
#117274072
#117274062
#117273322
#117273222
#117272392
#117272222
#117272212
#117272132
#117272122
#260997