Finite Group Databasebeta

← #13366002

Subgroups — #13366002

Order 1792.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#476C211yesmaximal, characteristic
2#11740673C211yesmaximal, characteristic, complemented
3#11739770C211yesmaximal, characteristic
4#11729534C211yesmaximal, characteristic, complemented
5#11729359C211yesmaximal, characteristic
6#11728316C211yesmaximal, characteristic, complemented
7#11728314C211yesmaximal, characteristic, complemented
8#11728312C211yesmaximal, characteristic, complemented
9#11728311C211yesmaximal, characteristic, complemented
10#11728209C211yesmaximal, characteristic, complemented
11#11728196C211yesmaximal, characteristic, complemented
12#11727941C211yesmaximal, characteristic
13#11727939C211yesmaximal, characteristic
14#11727910C211yesmaximal, characteristic
15#11727909C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C7×#476C418yescomplemented
19C2×C7×#106C2211yescharacteristic
20C2×C7×#106C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24#90002—21no—
25#90002—21no—
26#89993C2211yescharacteristic
27#89993C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30#89984—21no—
31#89983—21no—
32C2×#6392C2211yescharacteristic
33C2×#6391C2211yescharacteristic
34C2×#6389C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89757C2211yescharacteristic
39#89757C2211yescharacteristic
40#89753—21no—
41#89753—21no—
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44#89723—21no—
45#89723—21no—
46#89708C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48#89704C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22123—71nomaximal, Sylow
61C22×C7×Q8—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8C2×C414yes—
68C4×C7×D4—22no—
69C4×C7×D4C2×C414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C7×#106C2×C414yes—
96C7×#106C2×C414yes—
97C7×C4⋊C8C2×C414yes—
98C7×#102C2×C414yes—
99C7×#102C2×C414yes—
100#6406—41no—
101#6406—41no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115#6392—41no—
116#6391—41no—
117#6389—41no—
118C8×Dic7—41no—
119C4⋊(C7⋊C8)—22no—
120C2×#476D711yescharacteristic
121#2331—71no—
122#2319—71no—
123#1988—71no—
124#1949—71no—
125#1944—71no—
126#1941—71no—
127#1936—71no—
128#1934—71no—
129#1926—71no—
130#1925—71no—
131#1920—71no—
132#1919—71no—
133#1804—71no—
134#1768—71no—
135C24×C7—21no—
136C2×C7×Q8—24no—
137C2×C7×Q8—81no—
138C2×C7×Q8—81no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4C22×C412yes—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—24no—
149C22×C4×C7—81no—
150C22×C4×C7—81no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8C22×C412yes—
167C2×C7×C8C22×C412yes—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C22×C412yes—
174C7×C4⋊C4C22×C412yes—
175C7×C22⋊C4—24no—
176C42×C7C22×C412yes—
177C42×C7#4112yes—
178C22⋊Dic7—81no—
179C22⋊Dic7—81no—
180C22⋊Dic7—81no—
181C22⋊Dic7—81no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—81no—
191C4⋊Dic7—81no—
192C4⋊Dic7—81no—
193C4⋊Dic7—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C4×Dic7—81no—
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×C4⋊Q8C2×D711yescharacteristic
208C2×C4×D4C2×D711yescharacteristic
209#476Dic718yescomplemented
210C2×#107—71no—
211C2×#106—71no—
212C2×#106C2×D711yescharacteristic
213C2×#106C2×D711yescharacteristic
214C2×C4⋊C8—71no—
215C2×C4⋊C8C2×D711yescharacteristic
216C2×#102—71no—
217C2×#102—71no—
218C2×#102C2×D711yescharacteristic
219C2×#102C2×D711yescharacteristic
220C2×#97—71no—
221C2×C4×C8—71no—
222#391—141no—
223#391—141no—
224#390—71no—
225#390—71no—
226#389—71no—
227#389—71no—
228#389—71no—
229#389—71no—
230#388—141no—
231#388—141no—
232#387—141no—
233#387—141no—
234#382—71no—
235#381—141no—
236#380—141no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240#376—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—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×Q8—84no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—41no—
263C7×D4—41no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—84no—
278C2×C4×C7C4×D412yes—
279C2×C4×C7C4×D412yes—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—41no—
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—
310C7×C8—22no—
311C7×C8—22no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C22×Q8—21no—
315C22×D4C22×D711yescharacteristic
316C23×C4—21no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C4⋊Q8—22no—
322C4⋊Q8C2×Dic714yes—
323C4×D4—22no—
324C4×D4C2×Dic714yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—141no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C42—71no—
343C2×C42—71no—
344C2×C42—141no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42C22×D711yescharacteristic
348#107—281no—
349#106—281no—
350#106C2×Dic714yes—
351#106C2×Dic714yes—
352C4⋊C8—142no—
353C4⋊C8C2×Dic714yes—
354#102—281no—
355#102—281no—
356#102C2×Dic714yes—
357#102C2×Dic714yes—
358#97—281no—
359C4×C8—281no—
360#95—71no—
361#95—71no—
362#95—71no—
363#95—71no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7C2×C4×D411yescharacteristic
381C22×C7#53611yescharacteristic
382C22×C7#53711yescharacteristic
383C22×C7#57611yescharacteristic
384C22×C7#57711yescharacteristic
385C22×C7#57711yescharacteristic
386C22×C7#57711yescharacteristic
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7#51911yescharacteristic
393C4×C7#52011yescharacteristic
394C4×C7#54111yescharacteristic
395C4×C7#54211yescharacteristic
396Dic7—41no—
397Dic7—41no—
398Dic7—41no—
399Dic7—41no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404C24—21no—
405C2×Q8—24no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C22×Dic712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8C22×Dic712yes—
442C2×C8C22×Dic712yes—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—561no—
456C4⋊C4—561no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4C22×Dic712yes—
460C4⋊C4C22×Dic712yes—
461C4⋊C4C22×Dic712yes—
462C22⋊C4—24no—
463C22⋊C4—561no—
464C22⋊C4—561no—
465C22⋊C4—561no—
466C22⋊C4—561no—
467C42—141no—
468C42—141no—
469C42—141no—
470C42—141no—
471C42—561no—
472C42—561no—
473C42—561no—
474C42#115912yes—
475C42C22×Dic712yes—
476C2×C7—82no—
477C2×C7#232311yescharacteristic
478C2×C7#294611yescharacteristic
479C2×C7#294911yescharacteristic
480C2×C7#324111yescharacteristic
481C2×C7#324411yescharacteristic
482C2×C7#325711yescharacteristic
483C2×C7#326311yescharacteristic
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D711yescharacteristic
492Q8—84no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
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×C4D4×Dic712yes—
536C2×C4D4×Dic712yes—
537C2×C4C2×#115911yescharacteristic
538C2×C4#655111yescharacteristic
539C2×C4#655111yescharacteristic
540C2×C4#655311yescharacteristic
541C2×C4#655411yescharacteristic
542C2×C4C23×Dic711yescharacteristic
543C8—22no—
544C8—22no—
545C8—281no—
546C8—281no—
547C7#2212311yescharacteristic, complemented, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#9031911yescharacteristic
556C22#9032011yescharacteristic
557C22#9045311yescharacteristic
558C22#9045311yescharacteristic
559C22#9046511yescharacteristic
560C22#9046611yescharacteristic
561C22C2×D4×Dic711yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—281no—
568C4—281no—
569C4—281no—
570C4—281no—
571C4—561no—
572C4—561no—
573C4—561no—
574C4—561no—
575C4#9031611yescharacteristic
576C4#9033411yescharacteristic
577C4#9050311yescharacteristic
578C4#9051811yescharacteristic
579C2—82no—
580C2#1173849511yescharacteristic
581C2#1173851311yescharacteristic
582C2#1173851411yescharacteristic
583C2#1173851911yescharacteristic
584C2#1174018911yescharacteristic
585C2#1174128811yescharacteristic
586C2#1174137511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#476C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#476D7
hypocenterC7#22123
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. #11740673C2semidirect
4. #11729534C2semidirect
6. #11728316C2semidirect
7. #11728314C2semidirect
8. #11728312C2semidirect
9. #11728311C2semidirect
10. #11728209C2semidirect
11. #11728196C2semidirect
18. C7×#476C4semidirect
209. #476Dic7semidirect
547. C7#22123semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4762
#117406732
#117397702
#117295342
#117293592
#117283162
#117283142
#117283122
#117283112
#117282092
#117281962
#117279412
#117279392
#117279102
#117279092
#221237