Finite Group Databasebeta

← #13426522

Subgroups — #13426522

Order 1792.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2360C211yesmaximal, characteristic
2#11730956C211yesmaximal, characteristic
3#11729449C211yesmaximal, characteristic
4#11729448C211yesmaximal, characteristic
5C4×#6409C211yesmaximal, characteristic, complemented
6#11729382C211yesmaximal, characteristic, complemented
7#11729309C211yesmaximal, characteristic
8#11729307C211yesmaximal, characteristic
9#11727437C211yesmaximal, characteristic, complemented
10#11727425C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727356C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C7×#391C2211yescharacteristic
20C7×#389—21no—
21C7×#389C2211yescharacteristic
22C7×#387—21no—
23C7×#385—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C22⋊C4—21no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31#89984C2211yescharacteristic
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
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#27035—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—41no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—41no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C4⋊Dic7C2311yescharacteristic
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
96C7×#95—21no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102#6409—22no—
103#6409—22no—
104#6409—22no—
105#6409—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#6381—22no—
117#6381—22no—
118#6381—41no—
119C4⋊(C7⋊C8)—22no—
120C4×C7⋊C8—41no—
121#2495—71no—
122#2360D711yescharacteristic
123#2074—71no—
124#2068—71no—
125#2036—71no—
126#2035—71no—
127#1949—71no—
128#1942—71no—
129#1923—71no—
130#1920—71no—
131#1918—71no—
132#1863—71no—
133#1852—71no—
134#1823—71no—
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8#4112yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
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—42no—
154C22×C4×C7—42no—
155C22×C4×C7—81no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4#4112yes—
175C7×C4⋊C4#4112yes—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—42no—
184C42×C7—81no—
185C22⋊Dic7—42no—
186C22⋊Dic7—81no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197C4×Dic7—42no—
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—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8#4112yes—
210C2×C22⋊Q8C2×D711yescharacteristic
211C2×C4×Q8C2×D711yescharacteristic
212C22×C4⋊C4C2×D711yescharacteristic
213C2×#107—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×C4⋊C8—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×C22⋊C8—71no—
223C2×C22⋊C8—71no—
224C2×C4×C8—71no—
225#400—141no—
226#397—141no—
227#391C2×D711yescharacteristic
228#389—21no—
229#389—71no—
230#389—71no—
231#389C2×D711yescharacteristic
232#388—141no—
233#387—21no—
234#385—21no—
235#383—141no—
236#382—71no—
237#380—71no—
238#380—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4C2×D711yescharacteristic
241C4×C22⋊C4—21no—
242C4×C22⋊C4C2×D711yescharacteristic
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#336—71no—
252#336—71no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7#13512yes—
260C23×C7C22×D411yescharacteristic
261C7×Q8—22no—
262C7×Q8—22no—
263C7×Q8—41no—
264C7×Q8—41no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
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—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
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—41no—
295C2×C4×C7—84no—
296C2×C4×C7#13712yes—
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×Dic7—41no—
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—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C7⋊C8—22no—
316C7⋊C8—41no—
317C7⋊C8—41no—
318C7⋊C8—41no—
319C22×Q8C22×D711yescharacteristic
320C23×C4—21no—
321C23×C4C22×D711yescharacteristic
322C22×C8—71no—
323C22×C8—71no—
324C22×C8—71no—
325C22×C8—71no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C4×Q8—22no—
329C4×Q8—41no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4C22⋊D711yescharacteristic
349C2×C22⋊C4C22⋊D711yescharacteristic
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—141no—
353C2×C42C22×D711yescharacteristic
354C2×C42C22×D711yescharacteristic
355#107—142no—
356#107—142no—
357#107—281no—
358#106—142no—
359C4⋊C8—142no—
360#103—142no—
361#103—142no—
362#103—142no—
363#103—281no—
364C22⋊C8—142no—
365C22⋊C8—281no—
366C4×C8—281no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95C22⋊D711yescharacteristic
379#95C22⋊D711yescharacteristic
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—41no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7#52511yescharacteristic
388C22×C7#54611yescharacteristic
389C22×C7#54711yescharacteristic
390C22×C7C2×#13511yescharacteristic
391C22×C7C2×#13711yescharacteristic
392C22×C7#57611yescharacteristic
393C22×C7#57811yescharacteristic
394C4×C7—22no—
395C4×C7—41no—
396C4×C7—41no—
397C4×C7—42no—
398C4×C7—41no—
399C4×C7—42no—
400C4×C7—41no—
401C4×C7—42no—
402C4×C7—81no—
403C4×C7#52911yescharacteristic
404C4×C7#53911yescharacteristic
405C4×C7#55311yescharacteristic
406C4×C7#55611yescharacteristic
407Dic7—81no—
408Dic7—81no—
409Dic7—81no—
410Dic7—81no—
411C24C2×C22⋊D711yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8#115712yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C23×D711yescharacteristic
441C2×C8—72no—
442C2×C8—141no—
443C2×C8—141no—
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—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—561no—
470C4⋊C4—561no—
471C4⋊C4#115712yes—
472C4⋊C4#115912yes—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—282no—
477C22⋊C4—561no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—81no—
484C42—282no—
485C2×C7—42no—
486C2×C7#246611yescharacteristic
487C2×C7#310811yescharacteristic
488C2×C7#312411yescharacteristic
489C2×C7#329711yescharacteristic
490C2×C7#331211yescharacteristic
491C2×C7#333611yescharacteristic
492C2×C7#335711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#651312yes—
500C23C22×C22⋊D711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
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—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
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#650512yes—
549C2×C4C2×#115711yescharacteristic
550C2×C4C2×#115711yescharacteristic
551C2×C4C2×#115911yescharacteristic
552C2×C4#654811yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C2×C4#655411yescharacteristic
555C8—142no—
556C8—281no—
557C8—281no—
558C8—281no—
559C7#2703511yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#9048211yescharacteristic
568C22#9049711yescharacteristic
569C22C2×#650511yescharacteristic
570C22C2×#651311yescharacteristic
571C22#9051511yescharacteristic
572C22#9052611yescharacteristic
573C22#9052911yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—561no—
584C4—561no—
585C4—561no—
586C4—561no—
587C4#9022911yescharacteristic
588C4#9023311yescharacteristic
589C4#9024911yescharacteristic
590C4#9025511yescharacteristic
591C2—42no—
592C2#1173981411yescharacteristic
593C2#1173988811yescharacteristic
594C2#1174001411yescharacteristic
595C2#1174094011yescharacteristic
596C2#1174099711yescharacteristic
597C2#1174103511yescharacteristic
598C2#1174107211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2360C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2360D7
hypocenterC7#27035
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
5. C4×#6409C2semidirect
6. #11729382C2semidirect
9. #11727437C2semidirect
10. #11727425C2semidirect
11. #11727373C2semidirect
12. #11727356C2semidirect
13. #11727222C2semidirect
14. #11727212C2semidirect
559. C7#27035semidirect

Maximal subgroups

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

subgroupindex
C7×#23602
#117309562
#117294492
#117294482
C4×#64092
#117293822
#117293092
#117293072
#117274372
#117274252
#117273732
#117273562
#117272222
#117272122
#117272052
#270357