Finite Group Databasebeta

← #13412961

Subgroups — #13412961

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×#2552C211yesmaximal, characteristic, complemented
2#11731090C211yesmaximal, characteristic
3#11727384C211yesmaximal, characteristic
4#11727383C211yesmaximal, characteristic
5#11727357C211yesmaximal, characteristic, complemented
6#11727355C211yesmaximal, characteristic, complemented
7#11727341C211yesmaximal, characteristic
8#11727340C211yesmaximal, characteristic
9#11727338C211yesmaximal, characteristic
10#11727332C211yesmaximal, characteristic, complemented
11#11727322C211yesmaximal, characteristic, complemented
12#11727321C211yesmaximal, characteristic, complemented
13#11727239C211yesmaximal, characteristic, complemented
14#11727228C211yesmaximal, characteristic
15#11727222C211yesmaximal, characteristic, complemented
16C2×C7×#125—21no—
17C2×C4×D28C2211yescharacteristic
18C7×#398—21no—
19C7×#398—21no—
20C7×#389—21no—
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#384C2211yescharacteristic
24C7×#384C2211yescharacteristic
25C7×#384C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28#89762—21no—
29#89762—21no—
30C2×#6383C2211yescharacteristic
31C2×#6383C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89711C2211yescharacteristic
41#89711C2211yescharacteristic
42#89708C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49#89438—21no—
50#89436—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26102—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C7×#125—44no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C4×D28—22no—
96C4×D28Q814yes—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106#6383—22no—
107#6383—22no—
108#6383—41no—
109#6383—41no—
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#2552D711yescharacteristic, complemented
121#2357—71no—
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—
135C23×D7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28C2×Q812yes—
156C2×C4×D7—24no—
157C2×C4×D7—81no—
158C2×C4×D7—81no—
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⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4—81no—
176C7×C4⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—42no—
182C42×C7C2×Q812yes—
183C42×C7#4112yes—
184(C2×D7)⋊C4—24no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—81no—
188(C2×D7)⋊C4—81no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7C2×Q812yes—
195C4×Dic7—81no—
196C4×Dic7—81no—
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—21no—
210C2×C4×D4—71no—
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—21no—
226#398—21no—
227#390—71no—
228#390—71no—
229#389—21no—
230#389C2×D711yescharacteristic
231#389C2×D711yescharacteristic
232#388—141no—
233#388—141no—
234#384C2×D711yescharacteristic
235#384C2×D711yescharacteristic
236#384C2×D711yescharacteristic
237#382—71no—
238#381—141no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4C2×D711yescharacteristic
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C22⋊C4—141no—
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×C7C22×D411yescharacteristic
253C22×D7—22no—
254C22×D7—41no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
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—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
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×C7C22⋊Q812yes—
287C2×C4×C7C22⋊Q812yes—
288C2×C4×C7AGL(1,ℤ/8)12yes—
289C2×C4×C7#13712yes—
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—22no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305D28—22no—
306D28—22no—
307D28—41no—
308D28—41no—
309C4×D7—84no—
310C7⋊C8—22no—
311C7⋊C8—22no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C22×D4—71no—
315C23×C4—141no—
316C22×C8—71no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8—71no—
320#125—44no—
321C4×D4—74no—
322C4×D4—142no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4C22⋊D711yescharacteristic
335C2×C4⋊C4C22⋊D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C42—21no—
346C2×C42—141no—
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—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95C22⋊D711yescharacteristic
373#95C22⋊D711yescharacteristic
374C22×C7C2×C22⋊Q811yescharacteristic
375C22×C7#54711yescharacteristic
376C22×C7#55011yescharacteristic
377C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C7C2×#13711yescharacteristic
379C22×C7#57611yescharacteristic
380C22×C7#57711yescharacteristic
381C2×D7—42no—
382C2×D7—42no—
383C2×D7—42no—
384C2×D7—81no—
385C2×D7—81no—
386C2×D7—81no—
387C2×D7—81no—
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7#53311yescharacteristic
399C4×C7#54411yescharacteristic
400C4×C7#55211yescharacteristic
401C4×C7#55311yescharacteristic
402Dic7—42no—
403Dic7—81no—
404Dic7—81no—
405C24—141no—
406C2×D4—72no—
407C2×D4—141no—
408C2×D4—141no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—141no—
417C22×C4—144no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—561no—
421C22×C4—561no—
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—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—72no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C22⋊C4—144no—
466C22⋊C4—282no—
467C22⋊C4—282no—
468C22⋊C4—561no—
469C22⋊C4—561no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—561no—
476C42—561no—
477C42Q8×D712yes—
478C42#116112yes—
479C2×C7#244111yescharacteristic
480C2×C7#307911yescharacteristic
481C2×C7#308111yescharacteristic
482C2×C7#334511yescharacteristic
483C2×C7#334611yescharacteristic
484C2×C7#339911yescharacteristic
485C2×C7#340011yescharacteristic
486D7—82no—
487C23—142no—
488C23—281no—
489C23—281no—
490C23—281no—
491C23—281no—
492C23—281no—
493C23—281no—
494C23C22×C22⋊D711yescharacteristic
495D4—142no—
496D4—142no—
497D4—281no—
498D4—281no—
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—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
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—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—142no—
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—564no—
535C2×C4#649512yes—
536C2×C4#650512yes—
537C2×C4#650912yes—
538C2×C4#650912yes—
539C2×C4C2×Q8×D711yescharacteristic
540C2×C4C2×#116111yescharacteristic
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—282no—
551C22—282no—
552C22—282no—
553C22—561no—
554C22—561no—
555C22—561no—
556C22—561no—
557C22C2×#649511yescharacteristic
558C22C2×#650511yescharacteristic
559C22C2×#650911yescharacteristic
560C22#9051511yescharacteristic
561C22#9052511yescharacteristic
562C22#9052611yescharacteristic
563C22#9052811yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—282no—
575C4—561no—
576C4—561no—
577C4#9037911yescharacteristic
578C4#9038011yescharacteristic
579C4#9038511yescharacteristic
580C4#9038611yescharacteristic
581C2—562no—
582C2#1174050311yescharacteristic
583C2#1174065711yescharacteristic
584C2#1174091911yescharacteristic
585C2#1174096411yescharacteristic
586C2#1174103411yescharacteristic
587C2#1174107011yescharacteristic
588C2#1174113211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2552C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2552D7
hypocenterC7#26102
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2552

Complements

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

NH ≅ G/Ntype
1. C7×#2552C2semidirect
5. #11727357C2semidirect
6. #11727355C2semidirect
10. #11727332C2semidirect
11. #11727322C2semidirect
12. #11727321C2semidirect
13. #11727239C2semidirect
15. #11727222C2semidirect
120. #2552D7semidirect
549. C7#26102semidirect

Maximal subgroups

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

subgroupindex
C7×#25522
#117310902
#117273842
#117273832
#117273572
#117273552
#117273412
#117273402
#117273382
#117273322
#117273222
#117273212
#117272392
#117272282
#117272222
#261027