Finite Group Databasebeta

← #13413152

Subgroups — #13413152

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic
2#11731168C211yesmaximal, characteristic, complemented
3C4×#6406C211yesmaximal, characteristic
4#11729361C211yesmaximal, characteristic
5#11729360C211yesmaximal, characteristic
6#11729359C211yesmaximal, characteristic
7#11729359C211yesmaximal, characteristic
8#11729358C211yesmaximal, characteristic
9#11727376C211yesmaximal, characteristic, complemented
10#11727367C211yesmaximal, characteristic, complemented
11#11727366C211yesmaximal, characteristic, complemented
12#11727364C211yesmaximal, characteristic, complemented
13#11727301C211yesmaximal, characteristic, complemented
14#11727294C211yesmaximal, characteristic, complemented
15#11727238C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C7×#389C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#385—21no—
22C7×#385—21no—
23C7×#382C2211yescharacteristic
24C7×#380—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C2×C7×#95—21no—
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31#89758—21no—
32#89756—21no—
33C2×#6382C2211yescharacteristic
34C2×#6381C2211yescharacteristic
35#89708C2211yescharacteristic
36#89708C2211yescharacteristic
37#89707C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89427—21no—
46#89426—21no—
47#89324C2211yescharacteristic
48#89324C2211yescharacteristic
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26117—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7—21no—
63C23×C4×C7—21no—
64C4×C7×D4—22no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4—41no—
68C4×C7×D4—41no—
69C4×C7×D4Q814yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C4⋊Dic7—21no—
84C2×C4⋊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#6406—22no—
103#6406—22no—
104#6406—22no—
105#6406—22no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113#6382—41no—
114#6381—41no—
115C4⋊(C7⋊C8)—22no—
116C4⋊(C7⋊C8)—41no—
117C4⋊(C7⋊C8)—41no—
118C4×C7⋊C8—22no—
119#2570—71no—
120#2349D711yescharacteristic
121#2101—71no—
122#2100—71no—
123#2088—71no—
124#2086—71no—
125#1949—71no—
126#1939—71no—
127#1864—71no—
128#1862—71no—
129#1835—71no—
130#1804—71no—
131#1804—71no—
132#1802—71no—
133C4×#102—71no—
134C24×C7—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4C2×Q812yes—
139C2×C7×D4#4112yes—
140C2×C7×D4#4112yes—
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—81no—
150C22×C4×C7—81no—
151C22×C4×C7—81no—
152C22×C4×C7—81no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4C2×Q812yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—42no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7C2×Q812yes—
182C42×C7#4112yes—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—42no—
192C4⋊Dic7—81no—
193Dic7⋊C4—42no—
194Dic7⋊C4—81no—
195C4×Dic7—81no—
196C2×C7⋊C8—21no—
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—21no—
208C2×C4×D4C2×D711yescharacteristic
209C2×C4×D4C2×D711yescharacteristic
210C2×C4×D4C2×D711yescharacteristic
211C2×#107—71no—
212C2×#106—71no—
213C2×C4⋊C8—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×C4×C8—71no—
221#400—141no—
222#398—141no—
223#389—71no—
224#389—71no—
225#389C2×D711yescharacteristic
226#389C2×D711yescharacteristic
227#387—141no—
228#387—141no—
229#385—21no—
230#385—21no—
231#384—71no—
232#382—71no—
233#382—71no—
234#382C2×D711yescharacteristic
235#380—21no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4C2×D711yescharacteristic
238C2×#95—21no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336—71no—
250#336—71no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4#12514yes—
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—22no—
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—21no—
278C2×C4×C7—21no—
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×C7—41no—
287C2×C4×C7—84no—
288C2×C4×C7—84no—
289C2×C4×C7—84no—
290C2×C4×C7C22⋊Q812yes—
291C2×C4×C7C22⋊Q812yes—
292C2×C4×C7#13512yes—
293C2×C4×C7C22×Q811yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7H(5,𝔽2)11yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C7⋊C8—41no—
315C22×D4C22×D711yescharacteristic
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—21no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—71no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4Dic1414yes—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4C22⋊D711yescharacteristic
340C2×C4⋊C4C22⋊D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351#107—281no—
352#106—281no—
353C4⋊C8—142no—
354C4⋊C8—281no—
355C4⋊C8—281no—
356#102—142no—
357#102—142no—
358#102—142no—
359#102—142no—
360C4×C8—142no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95C22⋊D711yescharacteristic
373#95C22⋊D711yescharacteristic
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7#54811yescharacteristic
384C22×C7C2×#13511yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57611yescharacteristic
387C22×C7#57711yescharacteristic
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—42no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—41no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7C2×#12511yescharacteristic
398C4×C7#54411yescharacteristic
399C4×C7#55211yescharacteristic
400C4×C7#55311yescharacteristic
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×Dic1412yes—
410C2×D4#115912yes—
411C2×D4#116112yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—561no—
467C4⋊C4—561no—
468C4⋊C4C2×Dic1412yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—561no—
477C42C2×Dic1412yes—
478C42#115712yes—
479C2×C7—44no—
480C2×C7#244211yescharacteristic
481C2×C7#307611yescharacteristic
482C2×C7#308111yescharacteristic
483C2×C7#329711yescharacteristic
484C2×C7#332311yescharacteristic
485C2×C7#336011yescharacteristic
486C2×C7#337011yescharacteristic
487C23—22no—
488C23—22no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D711yescharacteristic
495D4—22no—
496D4—22no—
497D4—22no—
498D4#645314yes—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—84no—
524C2×C4—84no—
525C2×C4—84no—
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—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4#648712yes—
539C2×C4#648712yes—
540C2×C4#651312yes—
541C2×C4C22×Dic1411yescharacteristic
542C2×C4C2×#115711yescharacteristic
543C2×C4C2×#115911yescharacteristic
544C2×C4#654811yescharacteristic
545C2×C4C2×#116111yescharacteristic
546C2×C4#655111yescharacteristic
547C8—281no—
548C8—281no—
549C8—281no—
550C8—281no—
551C7#2611711yescharacteristic, complemented, Sylow
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22C2×#648711yescharacteristic
560C22#9049711yescharacteristic
561C22#9050811yescharacteristic
562C22C2×#651311yescharacteristic
563C22#9051311yescharacteristic
564C22#9052611yescharacteristic
565C22#9052711yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4—561no—
579C4C2×#645311yescharacteristic
580C4#9019111yescharacteristic
581C4#9020411yescharacteristic
582C4#9020511yescharacteristic
583C2—44no—
584C2#1173961711yescharacteristic
585C2#1173995211yescharacteristic
586C2#1173998411yescharacteristic
587C2#1174094011yescharacteristic
588C2#1174094611yescharacteristic
589C2#1174108211yescharacteristic
590C2#1174108411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2349D7
hypocenterC7#26117
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #11731168C2semidirect
9. #11727376C2semidirect
10. #11727367C2semidirect
11. #11727366C2semidirect
12. #11727364C2semidirect
13. #11727301C2semidirect
14. #11727294C2semidirect
15. #11727238C2semidirect
551. C7#26117semidirect

Maximal subgroups

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

subgroupindex
C7×#23492
#117311682
C4×#64062
#117293612
#117293602
#117293592
#117293582
#117273762
#117273672
#117273662
#117273642
#117273012
#117272942
#117272382
#261177