Finite Group Databasebeta

← #13456920

Subgroups — #13456920

Order 1792.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2295C211yesmaximal, characteristic, complemented
2#11730759C211yesmaximal, characteristic
3#11729387C211yesmaximal, characteristic
4#11729382C211yesmaximal, characteristic, complemented
5#11727437C211yesmaximal, characteristic, complemented
6#11727434C211yesmaximal, characteristic
7#11727430C211yesmaximal, characteristic
8#11727428C211yesmaximal, characteristic, complemented
9#11727384C211yesmaximal, characteristic
10#11727378C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727369C211yesmaximal, characteristic
13#11727216C211yesmaximal, characteristic
14#11727214C211yesmaximal, characteristic, complemented
15C4×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C22×#1143C2211yescharacteristic
18C2×#6439—21no—
19C7×#391C2211yescharacteristic
20C7×#391C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#387—21no—
24C7×#387—21no—
25C4×C7×C4⋊C4—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#376—21no—
28C43×C7C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38#89712—21no—
39#89712—21no—
40#89712—21no—
41#89711C2211yescharacteristic
42#89711C2211yescharacteristic
43#89711C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4×C7⋊C8C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#29435—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C22×D28C2311yescharacteristic
62C4×C7×Q8—22no—
63C4×C7×Q8—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7—21no—
72C2×C42×C7D411yescharacteristic
73C2×C42×C7D411yescharacteristic
74C2×C42×C7D411yescharacteristic
75C2×C42×C7D411yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×#1143—22no—
80C2×#1143—22no—
81C2×#1143—41no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98#6439—44no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106#6409—22no—
107#6409—22no—
108#6409—41no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112#6383—22no—
113#6383—22no—
114#6383—41no—
115#6382—22no—
116#6381—22no—
117#6381—41no—
118C4⋊(C7⋊C8)—22no—
119C4⋊(C7⋊C8)—22no—
120C4×C7⋊C8—41no—
121C4×C7⋊C8—41no—
122#2860—71no—
123#2295D711yescharacteristic, complemented
124#2088—71no—
125#2068—71no—
126#2067—71no—
127#2055—71no—
128#1968—71no—
129#1942—71no—
130#1934—71no—
131#1933—71no—
132#1925—71no—
133#1920—71no—
134#1880—71no—
135#1863—71no—
136C4×C4⋊C8—71no—
137C23×D7—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8#4112yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
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—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28—21no—
163C2×D28—21no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4#4112yes—
173C42×C7—24no—
174C42×C7—22no—
175C42×C7—22no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—42no—
185#1143—44no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—42no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—42no—
190(C2×D7)⋊C4—42no—
191(C2×D7)⋊C4—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
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—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8#4112yes—
212C2×C7⋊C8#4112yes—
213C22×SD16—71no—
214C2×#126—141no—
215C2×C4×Q8C2×D711yescharacteristic
216C2×#107—71no—
217C2×#107—71no—
218C2×#106—71no—
219C2×C4⋊C8—71no—
220C2×C4⋊C8—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#102—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×C4×C8—71no—
228C2×C4×C8—71no—
229#391C2×D711yescharacteristic
230#391C2×D711yescharacteristic
231#390—71no—
232#390—71no—
233#390—71no—
234#389—71no—
235#389—71no—
236#389—71no—
237#389C2×D711yescharacteristic
238#389C2×D711yescharacteristic
239#388—141no—
240#388—141no—
241#388—141no—
242#387—21no—
243#387—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2×D711yescharacteristic
246#376—21no—
247C43C2×D711yescharacteristic
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#336—71no—
255#336—71no—
256C23×C7C22×D411yescharacteristic
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C22×D7—41no—
263C22×D7—41no—
264C7×Q8—22no—
265C7×Q8—22no—
266C7×Q8—41no—
267C7×Q8—41no—
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—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7#13512yes—
295C2×C4×C7#13512yes—
296C2×C4×C7AGL(1,ℤ/8)12yes—
297C2×C4×C7#13712yes—
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7H(5,𝔽2)11yescharacteristic
303C2×C4×C72−1+411yescharacteristic
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313D28—41no—
314D28—41no—
315D28—41no—
316D28—41no—
317C7⋊C8—22no—
318C7⋊C8—22no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C22×Q8C22×D711yescharacteristic
322C22×D4—71no—
323C2×SD16—142no—
324C2×SD16—142no—
325C2×SD16—281no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330#126—284no—
331C4×Q8—22no—
332C4×Q8—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C22⋊D711yescharacteristic
354C2×C42C22⋊D711yescharacteristic
355C2×C42C22⋊D711yescharacteristic
356C2×C42C22⋊D711yescharacteristic
357C2×C42C22×D711yescharacteristic
358C2×C42C22×D711yescharacteristic
359C2×C42C22×D711yescharacteristic
360#107—142no—
361#107—281no—
362#106—142no—
363C4⋊C8—142no—
364C4⋊C8—142no—
365#103—142no—
366#103—142no—
367#103—281no—
368#102—142no—
369#102—142no—
370#102—281no—
371C4×C8—281no—
372C4×C8—281no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—141no—
381#95—141no—
382#95—141no—
383C22×C7#52511yescharacteristic
384C22×C7#53211yescharacteristic
385C22×C7#53711yescharacteristic
386C22×C7C2×#13511yescharacteristic
387C22×C7C2×#13511yescharacteristic
388C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
389C22×C7C2×#13711yescharacteristic
390C2×D7—81no—
391C2×D7—81no—
392C2×D7—81no—
393C2×D7—81no—
394C2×D7—81no—
395C2×D7—81no—
396C2×D7—81no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—41no—
402C4×C7—42no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—81no—
406C4×C7—81no—
407C4×C7#55111yescharacteristic
408C4×C7#55211yescharacteristic
409C4×C7#55311yescharacteristic
410C4×C7#55611yescharacteristic
411Dic7—81no—
412Dic7—81no—
413Dic7—81no—
414C24—141no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8#115712yes—
421C2×D4—141no—
422C2×D4—141no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C23×D711yescharacteristic
441SD16—284no—
442C2×C8—72no—
443C2×C8—72no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—282no—
470C4⋊C4—282no—
471C4⋊C4#115712yes—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C22⋊C4—282no—
475C22⋊C4—282no—
476C22⋊C4—282no—
477C22⋊C4—282no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C2×C7#260111yescharacteristic
491C2×C7#310611yescharacteristic
492C2×C7#310811yescharacteristic
493C2×C7#316811yescharacteristic
494C2×C7#318211yescharacteristic
495C2×C7#324711yescharacteristic
496C2×C7#324811yescharacteristic
497D7—161no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23—281no—
502C23—281no—
503C23—281no—
504C23—281no—
505C23C22×C22⋊D711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510D4—281no—
511D4—281no—
512D4—281no—
513D4—281no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4#650512yes—
550C2×C4#651212yes—
551C2×C4#651312yes—
552C2×C4#651312yes—
553C2×C4C2×#115711yescharacteristic
554C2×C4C2×#115711yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4D7×#4111yescharacteristic
557C2×C4#655311yescharacteristic
558C2×C4#655411yescharacteristic
559C8—142no—
560C8—142no—
561C8—281no—
562C8—281no—
563C7#2943511yescharacteristic, complemented, Sylow
564C22—561no—
565C22—561no—
566C22—561no—
567C22—561no—
568C22—561no—
569C22—561no—
570C22—561no—
571C22#9048211yescharacteristic
572C22C2×#650511yescharacteristic
573C22C2×#651211yescharacteristic
574C22C2×#651311yescharacteristic
575C22C2×#651311yescharacteristic
576C22#9052311yescharacteristic
577C22#9052411yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—561no—
589C4—561no—
590C4—561no—
591C4#9021811yescharacteristic
592C4#9022111yescharacteristic
593C4#9022211yescharacteristic
594C4#9022411yescharacteristic
595C2—1121no—
596C2#1173970411yescharacteristic
597C2#1173988811yescharacteristic
598C2#1173997411yescharacteristic
599C2#1174057911yescharacteristic
600C2#1174059311yescharacteristic
601C2#1174079211yescharacteristic
602C2#1174080911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2295C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2295D7
hypocenterC7#29435
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

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

NH ≅ G/Ntype
1. C7×#2295C2semidirect
4. #11729382C2semidirect
5. #11727437C2semidirect
8. #11727428C2semidirect
10. #11727378C2semidirect
11. #11727373C2semidirect
14. #11727214C2semidirect
15. C4×C4⋊(C7⋊C8)C2semidirect
123. #2295D7semidirect
563. C7#29435semidirect

Maximal subgroups

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

subgroupindex
C7×#22952
#117307592
#117293872
#117293822
#117274372
#117274342
#117274302
#117274282
#117273842
#117273782
#117273732
#117273692
#117272162
#117272142
C4×C4⋊(C7⋊C8)2
#294357