Finite Group Databasebeta

← #13426567

Subgroups — #13426567

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2363C211yesmaximal, characteristic
2#11730956C211yesmaximal, characteristic, complemented
3#11729434C211yesmaximal, characteristic
4#11729433C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic, complemented
6#11729361C211yesmaximal, characteristic, complemented
7#11729330C211yesmaximal, characteristic, complemented
8#11729329C211yesmaximal, characteristic, complemented
9#11727407C211yesmaximal, characteristic, complemented
10#11727395C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727356C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C7×#390—21no—
20C7×#390C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#386—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C22⋊C4—21no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31#89984C2211yescharacteristic
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89724—21no—
40#89716—21no—
41#89708C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89430—21no—
48#89423—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27038—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C7×#121—22no—
65C7×#121—22no—
66C4×C7×D4—22no—
67C4×C7×D4—41no—
68C7×#117—22no—
69C7×#117—41no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C7×#95—21no—
99C7×#95—21no—
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102#6406—22no—
103#6406—22no—
104#6406—22no—
105#6406—41no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)—22no—
114C22⋊(C7⋊C8)—41no—
115#6382—22no—
116#6382—22no—
117#6381—22no—
118#6381—41no—
119C4⋊(C7⋊C8)—22no—
120C4×C7⋊C8—41no—
121#2495—71no—
122#2363D711yescharacteristic
123#2073—71no—
124#2067—71no—
125#2034—71no—
126#2033—71no—
127#1949—71no—
128#1942—71no—
129#1923—71no—
130#1920—71no—
131#1920—71no—
132#1864—71no—
133#1854—71no—
134#1825—71no—
135C4×#102—71no—
136C24×C7—21no—
137C24×C7C2×D411yescharacteristic
138C2×C7×D4—24no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—81no—
145C2×C7×D4—81no—
146C2×C7×D4#4112yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—81no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4#4112yes—
175C7×C4⋊C4#4112yes—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—41no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—81no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—42no—
188C42×C7—81no—
189C22⋊Dic7—42no—
190C22⋊Dic7—81no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199Dic7⋊C4—81no—
200Dic7⋊C4—81no—
201C4×Dic7—42no—
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—21no—
213C2×C7⋊C8#4112yes—
214C2×#121C2×D711yescharacteristic
215C2×C4×D4C2×D711yescharacteristic
216C2×#117C2×D711yescharacteristic
217C2×#107—71no—
218C2×#107—71no—
219C2×#106—71no—
220C2×#106—71no—
221C2×C4⋊C8—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#102—71no—
225C2×#102—71no—
226C2×C22⋊C8—71no—
227C2×C22⋊C8—71no—
228C2×C4×C8—71no—
229#400—141no—
230#397—141no—
231#390—21no—
232#390C2×D711yescharacteristic
233#389—71no—
234#389—71no—
235#389C2×D711yescharacteristic
236#388—21no—
237#388—141no—
238#386—21no—
239#383—141no—
240#382—71no—
241#380—71no—
242#380—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2×D711yescharacteristic
247#340—71no—
248#340—71no—
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—22no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7C22×D411yescharacteristic
272C7×D4—22no—
273C7×D4—22no—
274C7×D4—41no—
275C7×D4—41no—
276C7×D4—84no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
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—41no—
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×C7—41no—
304C2×C4×C7—84no—
305C2×C4×C7#13512yes—
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C7⋊C8—22no—
325C7⋊C8—41no—
326C7⋊C8—41no—
327C7⋊C8—41no—
328C22×D4—21no—
329C22×D4C22×D711yescharacteristic
330C23×C4—21no—
331C23×C4C22×D711yescharacteristic
332C22×C8—71no—
333C22×C8—71no—
334C22×C8—71no—
335C22×C8—71no—
336#121—22no—
337#121—22no—
338C4×D4—22no—
339C4×D4—41no—
340#117—22no—
341#117—41no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4C22⋊D711yescharacteristic
360C2×C22⋊C4C22⋊D711yescharacteristic
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—141no—
364C2×C42C22×D711yescharacteristic
365C2×C42C22×D711yescharacteristic
366#107—142no—
367#107—281no—
368#106—142no—
369#106—142no—
370C4⋊C8—142no—
371#102—142no—
372#102—142no—
373#102—142no—
374#102—281no—
375C22⋊C8—142no—
376C22⋊C8—281no—
377C4×C8—281no—
378#95—21no—
379#95—21no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95C22⋊D711yescharacteristic
388#95C22⋊D711yescharacteristic
389C22×C7—41no—
390C22×C7—41no—
391C22×C7—41no—
392C22×C7—42no—
393C22×C7—41no—
394C22×C7—42no—
395C22×C7—41no—
396C22×C7—42no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—81no—
400C22×C7—81no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7#52511yescharacteristic
404C22×C7#54611yescharacteristic
405C22×C7#54711yescharacteristic
406C22×C7C2×#13511yescharacteristic
407C22×C7#57511yescharacteristic
408C22×C7#57611yescharacteristic
409C22×C7#57711yescharacteristic
410C4×C7—22no—
411C4×C7—41no—
412C4×C7—41no—
413C4×C7—42no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—41no—
417C4×C7—81no—
418C4×C7#52911yescharacteristic
419C4×C7#53911yescharacteristic
420C4×C7#55311yescharacteristic
421C4×C7#55611yescharacteristic
422Dic7—81no—
423Dic7—81no—
424Dic7—81no—
425Dic7—81no—
426C24—21no—
427C24C2×C22⋊D711yescharacteristic
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—81no—
435C2×D4—81no—
436C2×D4#115712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459C2×C8—72no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4#115712yes—
487C4⋊C4#115912yes—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—282no—
496C22⋊C4—561no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—42no—
502C42—81no—
503C42—282no—
504C2×C7—81no—
505C2×C7—82no—
506C2×C7#246611yescharacteristic
507C2×C7#310211yescharacteristic
508C2×C7#311811yescharacteristic
509C2×C7#329611yescharacteristic
510C2×C7#330911yescharacteristic
511C2×C7#334111yescharacteristic
512C2×C7#335111yescharacteristic
513C23—22no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C22×C22⋊D711yescharacteristic
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532D4—84no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
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—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—281no—
562C2×C4—281no—
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—281no—
572C2×C4—281no—
573C2×C4#651312yes—
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115711yescharacteristic
576C2×C4C2×#115911yescharacteristic
577C2×C4#654811yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C2×C4#655411yescharacteristic
580C8—142no—
581C8—281no—
582C8—281no—
583C8—281no—
584C7#2703811yescharacteristic, complemented, Sylow
585C22—41no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#9048211yescharacteristic
600C22#9049711yescharacteristic
601C22C2×#651311yescharacteristic
602C22#9051311yescharacteristic
603C22#9051511yescharacteristic
604C22#9052611yescharacteristic
605C22#9052711yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—81no—
614C4—561no—
615C4—561no—
616C4—561no—
617C4—561no—
618C4#9022911yescharacteristic
619C4#9023311yescharacteristic
620C4#9024911yescharacteristic
621C4#9025511yescharacteristic
622C2—81no—
623C2—82no—
624C2#1173981411yescharacteristic
625C2#1173997311yescharacteristic
626C2#1173998611yescharacteristic
627C2#1174093911yescharacteristic
628C2#1174101711yescharacteristic
629C2#1174105711yescharacteristic
630C2#1174109211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2363C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2363D7
hypocenterC7#27038
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #11730956C2semidirect
5. C4×#6406C2semidirect
6. #11729361C2semidirect
7. #11729330C2semidirect
8. #11729329C2semidirect
9. #11727407C2semidirect
10. #11727395C2semidirect
11. #11727373C2semidirect
12. #11727356C2semidirect
13. #11727222C2semidirect
14. #11727214C2semidirect
15. #11727205C2semidirect
584. C7#27038semidirect

Maximal subgroups

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

subgroupindex
C7×#23632
#117309562
#117294342
#117294332
C4×#64062
#117293612
#117293302
#117293292
#117274072
#117273952
#117273732
#117273562
#117272222
#117272142
#117272052
#270387