Finite Group Databasebeta

← #13456952

Subgroups — #13456952

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2644C211yesmaximal, characteristic, complemented
2#11730860C211yesmaximal, characteristic, complemented
3#11727411C211yesmaximal, characteristic
4#11727406C211yesmaximal, characteristic, complemented
5#11727404C211yesmaximal, characteristic
6#11727401C211yesmaximal, characteristic
7#11727399C211yesmaximal, characteristic, complemented
8#11727391C211yesmaximal, characteristic, complemented
9#11727383C211yesmaximal, characteristic, complemented
10#11727380C211yesmaximal, characteristic, complemented
11#11727376C211yesmaximal, characteristic, complemented
12#11727372C211yesmaximal, characteristic, complemented
13#11727371C211yesmaximal, characteristic, complemented
14#11727354C211yesmaximal, characteristic, complemented
15#11727190C211yesmaximal, characteristic, complemented
16C22×#1141C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C7×#397—21no—
19C7×#397—21no—
20C7×#397—21no—
21C7×#396—21no—
22C7×#390C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#390C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#376C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38#89711C2211yescharacteristic
39#89711C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89441—21no—
48#89441—21no—
49#89439—21no—
50#89436—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#29437—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
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×C7×C22⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×#1141—22no—
79C2×#1141—22no—
80C2×#1141—41no—
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×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C4×D28—22no—
97C4×D28—22no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105#6406—22no—
106#6406—22no—
107#6406—41no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111#6383—22no—
112#6383—22no—
113#6383—41no—
114#6382—22no—
115#6382—41no—
116#6381—22no—
117C4⋊(C7⋊C8)—22no—
118C4⋊(C7⋊C8)—22no—
119#6377—41no—
120C4×C7⋊C8—41no—
121#2644D711yescharacteristic, complemented
122#2319—71no—
123#2086—71no—
124#2065—71no—
125#2065—71no—
126#2054—71no—
127#1970—71no—
128#1941—71no—
129#1936—71no—
130#1936—71no—
131#1926—71no—
132#1918—71no—
133#1879—71no—
134#1864—71no—
135#1768—71no—
136C24×C7—21no—
137C23×D7—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28#4112yes—
163C2×C4×D7—24no—
164C2×C4×D7—81no—
165C2×C4×D7—81no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4—81no—
175C7×C22⋊C4—81no—
176C7×C22⋊C4—81no—
177C7×C22⋊C4—81no—
178C7×C22⋊C4—81no—
179C7×C22⋊C4—81no—
180C7×C22⋊C4—81no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187#1141—44no—
188(C2×D7)⋊C4—24no—
189(C2×D7)⋊C4—81no—
190(C2×D7)⋊C4—81no—
191(C2×D7)⋊C4—81no—
192(C2×D7)⋊C4—81no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7#4112yes—
199Dic7⋊C4—81no—
200Dic7⋊C4—81no—
201C4×Dic7—81no—
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—21no—
212C2×C7⋊C8#4112yes—
213C2×C7⋊C8#4112yes—
214C22×D8—71no—
215C2×C4×D4—71no—
216C2×#107—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×C4⋊C8—71no—
220C2×C4⋊C8—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#102—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#97—71no—
228C2×C4×C8—71no—
229#397—21no—
230#397—21no—
231#397—21no—
232#396—21no—
233#390—71no—
234#390—71no—
235#390C2×D711yescharacteristic
236#390C2×D711yescharacteristic
237#390C2×D711yescharacteristic
238#389—71no—
239#389—71no—
240#389C2×D711yescharacteristic
241#389C2×D711yescharacteristic
242#389C2×D711yescharacteristic
243#388—141no—
244#388—141no—
245#381—141no—
246#380—141no—
247C4×C4⋊C4—71no—
248#376C2×D711yescharacteristic
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#336—71no—
256#336—71no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7C22×D411yescharacteristic
265C22×D7—22no—
266C22×D7—41no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C7×D4—41no—
273C7×D4—41no—
274C7×D4—41no—
275C7×D4—41no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7H(5,𝔽2)11yescharacteristic
308C2×C4×C72−1+411yescharacteristic
309C2×Dic7—22no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318D28—22no—
319D28—22no—
320D28—41no—
321D28—41no—
322C4×D7—84no—
323C7⋊C8—22no—
324C7⋊C8—22no—
325C7⋊C8—41no—
326C7⋊C8—41no—
327C22×D4—71no—
328C22×D4C22×D711yescharacteristic
329C23×C4—141no—
330C2×D8—142no—
331C2×D8—142no—
332C2×D8—281no—
333C22×C8—71no—
334C22×C8—71no—
335C22×C8—71no—
336C22×C8—71no—
337C4×D4—142no—
338C4×D4—142no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C42—141no—
362C2×C42C22×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364C2×C42C22×D711yescharacteristic
365#107—142no—
366#106—142no—
367#106—281no—
368C4⋊C8—142no—
369C4⋊C8—142no—
370#102—142no—
371#102—142no—
372#102—142no—
373#102—142no—
374#102—281no—
375#102—281no—
376#97—281no—
377C4×C8—281no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95C22⋊D711yescharacteristic
385#95C22⋊D711yescharacteristic
386#95C22⋊D711yescharacteristic
387#95C22⋊D711yescharacteristic
388C22×C7—81no—
389C22×C7—81no—
390C22×C7—81no—
391C22×C7—81no—
392C22×C7—81no—
393C22×C7—81no—
394C22×C7—81no—
395C22×C7#52511yescharacteristic
396C22×C7#53211yescharacteristic
397C22×C7#53711yescharacteristic
398C22×C7#57611yescharacteristic
399C22×C7#57611yescharacteristic
400C22×C7#57711yescharacteristic
401C22×C7#57711yescharacteristic
402C2×D7—42no—
403C2×D7—42no—
404C2×D7—42no—
405C2×D7—81no—
406C2×D7—81no—
407C2×D7—81no—
408C2×D7—81no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—41no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7#55111yescharacteristic
419C4×C7#55211yescharacteristic
420C4×C7#55311yescharacteristic
421C4×C7#55611yescharacteristic
422Dic7—42no—
423Dic7—81no—
424Dic7—81no—
425C24—21no—
426C24—141no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—72no—
434C2×D4—141no—
435C2×D4—141no—
436C2×D4—141no—
437C2×D4—141no—
438C2×D4—141no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—144no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—561no—
447C22×C4—561no—
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C23×D711yescharacteristic
455D8—284no—
456C2×C8—72no—
457C2×C8—72no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—72no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—561no—
484C4⋊C4—561no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—144no—
492C22⋊C4—561no—
493C22⋊C4—561no—
494C22⋊C4—561no—
495C22⋊C4—561no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—561no—
503C2×C7—161no—
504C2×C7#260111yescharacteristic
505C2×C7#311811yescharacteristic
506C2×C7#311911yescharacteristic
507C2×C7#317311yescharacteristic
508C2×C7#317611yescharacteristic
509C2×C7#325711yescharacteristic
510C2×C7#325711yescharacteristic
511D7—82no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—142no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23—281no—
525C23—281no—
526C23C22×C22⋊D711yescharacteristic
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—142no—
532D4—142no—
533D4—281no—
534D4—281no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—142no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—564no—
572C2×C4D7×#4111yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4#655311yescharacteristic
577C2×C4#655411yescharacteristic
578C8—142no—
579C8—142no—
580C8—281no—
581C8—281no—
582C7#2943711yescharacteristic, complemented, Sylow
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—282no—
591C22—282no—
592C22—282no—
593C22—561no—
594C22—561no—
595C22—561no—
596C22—561no—
597C22#9051911yescharacteristic
598C22#9052311yescharacteristic
599C22#9052411yescharacteristic
600C22#9052611yescharacteristic
601C22#9052611yescharacteristic
602C22#9052711yescharacteristic
603C22#9052811yescharacteristic
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—282no—
614C4—561no—
615C4—561no—
616C4#9039311yescharacteristic
617C4#9039611yescharacteristic
618C4#9039911yescharacteristic
619C4#9040011yescharacteristic
620C2—161no—
621C2—562no—
622C2#1174055611yescharacteristic
623C2#1174087711yescharacteristic
624C2#1174087911yescharacteristic
625C2#1174088211yescharacteristic
626C2#1174090111yescharacteristic
627C2#1174090511yescharacteristic
628C2#1174091011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2644C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenter#2644D7
hypocenterC7#29437
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2644

Complements

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

NH ≅ G/Ntype
1. C7×#2644C2semidirect
2. #11730860C2semidirect
4. #11727406C2semidirect
7. #11727399C2semidirect
8. #11727391C2semidirect
9. #11727383C2semidirect
10. #11727380C2semidirect
11. #11727376C2semidirect
12. #11727372C2semidirect
13. #11727371C2semidirect
14. #11727354C2semidirect
15. #11727190C2semidirect
121. #2644D7semidirect
582. C7#29437semidirect

Maximal subgroups

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

subgroupindex
C7×#26442
#117308602
#117274112
#117274062
#117274042
#117274012
#117273992
#117273912
#117273832
#117273802
#117273762
#117273722
#117273712
#117273542
#117271902
#294377