Finite Group Databasebeta

← #13365991

Subgroups — #13365991

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×#89533C211yesmaximal, characteristic
2#11733740C211yesmaximal, characteristic
3#11733419C211yesmaximal, characteristic, complemented
4C7×#1934C211yesmaximal, characteristic
5#11727927C211yesmaximal, characteristic
6#11727910C211yesmaximal, characteristic
7#11727908C211yesmaximal, characteristic
8#11727523C211yesmaximal, characteristic, complemented
9#11727509C211yesmaximal, characteristic, complemented
10#11727399C211yesmaximal, characteristic
11#11727375C211yesmaximal, characteristic, complemented
12#11726570C211yesmaximal, characteristic, complemented
13#11726554C211yesmaximal, characteristic, complemented
14#11726551C211yesmaximal, characteristic, complemented
15#11726543C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×#6451C2211yescharacteristic
18C2×C7×#102C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C4×C7×C8C2211yescharacteristic
21C7×#390C2211yescharacteristic
22C7×#389C2211yescharacteristic
23#89993C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30#89750—21no—
31#89749C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35#89746—21no—
36#89745C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89717—21no—
40#89716—21no—
41#89715—21no—
42#89714—21no—
43#89704C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89533C418yescomplemented
48#89424—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22123—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×Dic7—21no—
63C22×Dic14—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76D4×Dic7—22no—
77D4×Dic7C2×C414yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93#6451—22no—
94#6451C2×C414yes—
95C7×#102—41no—
96C7×#102C2×C414yes—
97C4×C7×C8—41no—
98C7×#95—21no—
99#6406—41no—
100#6406C2×C414yes—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95D411yescharacteristic
110C7⋊#95D411yescharacteristic
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113#6391—41no—
114#6391C2×C414yes—
115Dic7⋊C8—22no—
116Dic7⋊C8C2×C414yes—
117#6382C2×C414yes—
118#6381—41no—
119#6377—41no—
120C2×#476—71no—
121#2331—71no—
122#2319—71no—
123#1988—71no—
124#1949—71no—
125#1944—71no—
126#1941—71no—
127#1936—71no—
128#1934D711yescharacteristic
129#1926—71no—
130#1925—71no—
131#1920—71no—
132#1919—71no—
133#1804—71no—
134#1768—71no—
135C24×C7—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—24no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—81no—
154C22×Dic7—81no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C2×Dic14—24no—
160C2×Dic14—81no—
161C2×Dic14—81no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8C22×C412yes—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4C22×C412yes—
175C7×C22⋊C4—81no—
176C7×C22⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C22⋊Dic7—24no—
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⋊Dic7C22×C412yes—
189C4⋊Dic7C22×C412yes—
190Dic7⋊C4—24no—
191Dic7⋊C4—81no—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—81no—
197C4×Dic7—81no—
198C4×Dic7—81no—
199C4×Dic7C22×C412yes—
200C4×Dic7#4112yes—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8C22×C412yes—
207C2×C4⋊Q8—71no—
208C2×C4×D4—71no—
209#476—78no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102C2×D711yescharacteristic
219C2×#102C2×D711yescharacteristic
220C2×#97—71no—
221C2×C4×C8C2×D711yescharacteristic
222#391—141no—
223#391—141no—
224#390—71no—
225#390C2×D711yescharacteristic
226#389—71no—
227#389—71no—
228#389—71no—
229#389C2×D711yescharacteristic
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#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
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×D4—22no—
260C7×D4—22no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—22no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—41no—
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—41no—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7H(5,𝔽2)11yescharacteristic
281C2×C4×C72−1+411yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—41no—
297C2×Dic7—41no—
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—84no—
307C2×Dic7C4×D412yes—
308C2×Dic7C4×D412yes—
309Dic14—84no—
310C7×C8—22no—
311C7×C8—41no—
312C7⋊C8—22no—
313C7⋊C8—41no—
314C22×Q8—141no—
315C22×D4C22×D711yescharacteristic
316C23×C4—141no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C4⋊Q8—74no—
322C4⋊Q8—142no—
323C4×D4—74no—
324C4×D4—142no—
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—21no—
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—74no—
350#106—74no—
351#106—281no—
352C4⋊C8—74no—
353C4⋊C8—142no—
354#102—41no—
355#102—74no—
356#102—281no—
357#102C4×D714yes—
358#97—281no—
359C4×C8—41no—
360#95—21no—
361#95—71no—
362#95—71no—
363#95—71no—
364#95—71no—
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—42no—
388C4×C7—41no—
389C4×C7—41no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7#51911yescharacteristic
393C4×C7#52011yescharacteristic
394C4×C7#54111yescharacteristic
395C4×C7#54211yescharacteristic
396Dic7—22no—
397Dic7—22no—
398Dic7—42no—
399Dic7—42no—
400Dic7—41no—
401Dic7—41no—
402Dic7—81no—
403Dic7—81no—
404C24—21no—
405C2×Q8—144no—
406C2×Q8—561no—
407C2×Q8—561no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×C4×D712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—144no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—561no—
427C22×C4—561no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—72no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8C2×C4×D712yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—72no—
449C4⋊C4—72no—
450C4⋊C4—81no—
451C4⋊C4—144no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—561no—
459C4⋊C4—561no—
460C4⋊C4—561no—
461C4⋊C4C2×C4×D712yes—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—144no—
465C22⋊C4—561no—
466C22⋊C4—561no—
467C42—21no—
468C42—21no—
469C42—72no—
470C42—72no—
471C42—141no—
472C42—141no—
473C42—561no—
474C42—561no—
475C42—561no—
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—564no—
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—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—72no—
513C2×C4—72no—
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—142no—
523C2×C4—142no—
524C2×C4—141no—
525C2×C4—141no—
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—564no—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4#654811yescharacteristic
539C2×C4D7×#4111yescharacteristic
540C2×C4#655411yescharacteristic
541C2×C4#655411yescharacteristic
542C2×C4#655411yescharacteristic
543C8—22no—
544C8—41no—
545C8—142no—
546C8—281no—
547C7#2212311yescharacteristic, complemented, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C4×D4×D711yescharacteristic
556C22#9028811yescharacteristic
557C22#9036611yescharacteristic
558C22#9044811yescharacteristic
559C22#9045311yescharacteristic
560C22#9046111yescharacteristic
561C22#9046111yescharacteristic
562C4—41no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—142no—
568C4—142no—
569C4—282no—
570C4—281no—
571C4—282no—
572C4—281no—
573C4—561no—
574C4—561no—
575C4#9020711yescharacteristic
576C4#9023711yescharacteristic
577C4#9035311yescharacteristic
578C4#9039211yescharacteristic
579C2—82no—
580C2#1173290411yescharacteristic
581C2#1173703211yescharacteristic
582C2#1173709411yescharacteristic
583C2#1173809211yescharacteristic
584C2#1173815011yescharacteristic
585C2#1173902011yescharacteristic
586C2#1173909611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1934C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1934D7
hypocenterC7#22123
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
3. #11733419C2semidirect
8. #11727523C2semidirect
9. #11727509C2semidirect
11. #11727375C2semidirect
12. #11726570C2semidirect
13. #11726554C2semidirect
14. #11726551C2semidirect
15. #11726543C2semidirect
47. #89533C4semidirect
547. C7#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#895332
#117337402
#117334192
C7×#19342
#117279272
#117279102
#117279082
#117275232
#117275092
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