Finite Group Databasebeta

← #13276975

Subgroups — #13276975

Order 1792.

2038 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#390C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3C2×#89993C211yesmaximal, characteristic
4C2×#89748C211yesmaximal, characteristic, complemented
5C2×#89716C214yesmaximal, complemented
6C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
7#11729317C211yesmaximal, characteristic, complemented
8#11729301C214yesmaximal, complemented
9C4×C7⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2212yes—
13C22×C7×C22⋊C4C2212yes—
14C22×C42×C7C2211yescharacteristic
15C22×C22⋊Dic7C2214yes—
16C22×C22⋊Dic7C2214yes—
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2214yes—
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C7×#390—24no—
24#89993—24no—
25#89993C418yescomplemented
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2214yes—
29C2×C7⋊#95C2212yes—
30#89748—24no—
31#89716—216no—
32C4×C4⋊Dic7—24no—
33#14001—71nomaximal, Sylow
34C25×C7C2312yes—
35C22×C7×D4—22no—
36C22×C7×D4—22no—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4C2×C414yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2312yes—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C22⋊C4—28no—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—44no—
57C2×C42×C7—22no—
58C2×C42×C7—22no—
59C2×C42×C7D414yes—
60C2×C22⋊Dic7—28no—
61C2×C22⋊Dic7—28no—
62C2×C22⋊Dic7—28no—
63C2×C22⋊Dic7—28no—
64C2×C22⋊Dic7—44no—
65C2×C22⋊Dic7C2×C4116yes—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7C2×C414yes—
71C2×C4⋊Dic7D414yes—
72C2×Dic7⋊C4—28no—
73C2×Dic7⋊C4—28no—
74C2×Dic7⋊C4—44no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—28no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7C2×C414yes—
83C7⋊#95—24no—
84C7⋊#95—28no—
85C7⋊#95—24no—
86C7⋊#95D418yescomplemented
87C2×#390—71no—
88C2×#390—71no—
89C2×#390D711yescharacteristic
90C2×#388—74no—
91C2×#384—71no—
92C2×C4×C4⋊C4—71no—
93#1439—74no—
94C4⋊#95—71no—
95C4×#95—71no—
96C24×C7—24no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—24no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—24no—
105C24×C7—24no—
106C24×C7C22×C414yes—
107C24×C7C2411yescharacteristic
108C2×C7×D4—28no—
109C2×C7×D4—28no—
110C2×C7×D4—28no—
111C2×C7×D4—44no—
112C2×C7×D4—44no—
113C2×C7×D4—44no—
114C2×C7×D4—44no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—24no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—24no—
127C22×C4×C7C22×C412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D414yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D412yes—
132C22×C4×C7C2×D412yes—
133C22×C4×C7#4112yes—
134C22×C4×C7#4112yes—
135C22×C4×C7#4112yes—
136C22×C4×C7#4112yes—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—24no—
144C22×Dic7—24no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—24no—
152C22×Dic7—24no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7C22×C414yes—
156C22×Dic7C22×C414yes—
157C22×Dic7C2×D414yes—
158C22×Dic7C2×D414yes—
159C22×Dic7C2×D414yes—
160C22×Dic7#4114yes—
161C7×C4⋊C4—44no—
162C7×C22⋊C4—48no—
163C7×C22⋊C4—48no—
164C42×C7—28no—
165C22⋊Dic7—232no—
166C22⋊Dic7—416no—
167C4⋊Dic7—28no—
168C4⋊Dic7—28no—
169Dic7⋊C4—416no—
170C4×Dic7—28no—
171C4×Dic7—44no—
172C4×Dic7—48no—
173C23×D4C2×D711yescharacteristic
174C22×C4⋊C4—71no—
175C22×C4⋊C4—71no—
176C22×C4⋊C4—74no—
177C22×C4⋊C4C2×D711yescharacteristic
178C22×C22⋊C4—74no—
179C22×C22⋊C4—74no—
180C22×C22⋊C4C2×D712yes—
181C22×C22⋊C4C2×D712yes—
182C22×C42—71no—
183C22×C42—71no—
184C22×C42—72no—
185C22×C42C2×D711yescharacteristic
186#390—24no—
187#390—78no—
188#390—144no—
189#388—1416no—
190#384—144no—
191C4×C4⋊C4—144no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—74no—
195C2×#95—72no—
196C23×C7—24no—
197C23×C7—28no—
198C23×C7—24no—
199C23×C7—24no—
200C23×C7—28no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—44no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—44no—
211C23×C7—44no—
212C23×C7—44no—
213C23×C7—44no—
214C23×C7—44no—
215C23×C7—44no—
216C23×C7—44no—
217C23×C7—44no—
218C23×C7—44no—
219C23×C7C23×C411yescharacteristic
220C23×C7C22×D411yescharacteristic
221C23×C7C22×D412yes—
222C23×C7C22×D411yescharacteristic
223C23×C7C2×#4111yescharacteristic
224C23×C7C2×#4111yescharacteristic
225C23×C7C2×#4111yescharacteristic
226C23×C7C2×#4111yescharacteristic
227C23×C7C2×#4112yes—
228C23×C7H(5,𝔽2)12yes—
229C23×C72−1+412yes—
230C7×D4—416no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—44no—
244C2×C4×C7—44no—
245C2×C4×C7#11714yes—
246C2×C4×C7C4×D414yes—
247C2×C4×C7C4×D414yes—
248C2×C4×C7#12114yes—
249C2×C4×C7#12314yes—
250C2×C4×C7#12414yes—
251C2×C4×C7#12614yes—
252C2×C4×C7#12714yes—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—28no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—28no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—44no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—44no—
275C2×Dic7—42no—
276C2×Dic7C4×D418yes—
277C25C22×D712yes—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—22no—
281C22×D4—22no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4C4×D714yes—
285C23×C4—72no—
286C23×C4—72no—
287C23×C4—72no—
288C23×C4—72no—
289C23×C4C22×D712yes—
290C23×C4C22×D711yescharacteristic
291C23×C4C22×D711yescharacteristic
292C23×C4C22×D711yescharacteristic
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—74no—
297C2×C4⋊C4—74no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4—142no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—148no—
303C2×C4⋊C4—148no—
304C2×C4⋊C4—284no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—716no—
310C2×C22⋊C4—148no—
311C2×C22⋊C4—148no—
312C2×C22⋊C4—148no—
313C2×C22⋊C4—148no—
314C2×C22⋊C4—284no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—74no—
318C2×C42—142no—
319C2×C42—142no—
320C2×C42—142no—
321C2×C42—148no—
322C2×C42—142no—
323C2×C42—142no—
324C2×C42—144no—
325C2×C42C22⋊D714yes—
326#95—78no—
327#95—144no—
328#95—148no—
329#95—144no—
330C22×C7—44no—
331C22×C7—44no—
332C22×C7—48no—
333C22×C7—48no—
334C22×C7—44no—
335C22×C7—84no—
336C22×C7—82no—
337C22×C7—82no—
338C22×C7—84no—
339C22×C7—82no—
340C22×C7—82no—
341C22×C7C2×#11711yescharacteristic
342C22×C7C2×C4×D411yescharacteristic
343C22×C7C2×C4×D411yescharacteristic
344C22×C7C2×C4×D412yes—
345C22×C7#52012yes—
346C22×C7C2×#12111yescharacteristic
347C22×C7C2×#12311yescharacteristic
348C22×C7C2×#12411yescharacteristic
349C22×C7C2×#12611yescharacteristic
350C22×C7C2×#12711yescharacteristic
351C22×C7#53212yes—
352C22×C7#53211yescharacteristic
353C22×C7#53511yescharacteristic
354C22×C7#53611yescharacteristic
355C22×C7#54212yes—
356C22×C7D4212yes—
357C22×C7#54712yes—
358C22×C7#54712yes—
359C22×C7#54812yes—
360C22×C7#55012yes—
361C22×C7#55312yes—
362C22×C7#55512yes—
363C22×C7#55612yes—
364C4×C7—24no—
365C4×C7—42no—
366C4×C7—82no—
367C4×C7#38818yescomplemented
368Dic7—28no—
369Dic7—44no—
370Dic7—44no—
371Dic7—44no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24C2×C4×D714yes—
383C24C23×D711yescharacteristic
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—74no—
404C22×C4—74no—
405C22×C4—74no—
406C22×C4—74no—
407C22×C4—74no—
408C22×C4—74no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—144no—
416C22×C4—142no—
417C22×C4—144no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—144no—
422C22×C4—142no—
423C22×C4—144no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4C2×C4×D712yes—
428C22×C4#115712yes—
429C22×C4D4×D714yes—
430C22×C4D4×D712yes—
431C22×C4#115912yes—
432C22×C4#115912yes—
433C22×C4#115912yes—
434C22×C4C2×C22⋊D712yes—
435C22×C4C2×C22⋊D712yes—
436C22×C4C2×C22⋊D712yes—
437C4⋊C4—44no—
438C4⋊C4—148no—
439C4⋊C4—148no—
440C4⋊C4—2816no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—1432no—
444C22⋊C4—2816no—
445C42—28no—
446C42—148no—
447C42—284no—
448C42—288no—
449C2×C7—84no—
450C2×C7C2×#38811yescharacteristic
451C2×C7C4×#12711yescharacteristic
452C2×C7#232611yescharacteristic
453C2×C7#235412yes—
454C2×C7#236112yes—
455C2×C7#249112yes—
456C2×C7#250011yescharacteristic
457C2×C7#250111yescharacteristic
458C2×C7#251212yes—
459C2×C7#254111yescharacteristic
460C2×C7#254511yescharacteristic
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C22×C4×D711yescharacteristic
485C23C2×#115711yescharacteristic
486C23C2×D4×D711yescharacteristic
487C23C2×D4×D712yes—
488C23C2×#115911yescharacteristic
489C23C2×#115911yescharacteristic
490C23C2×#115911yescharacteristic
491C23#654812yes—
492C23D7×#4112yes—
493C23#655412yes—
494C23C22×C22⋊D711yescharacteristic
495D4—416no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—78no—
511C2×C4—144no—
512C2×C4—144no—
513C2×C4—144no—
514C2×C4—144no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—148no—
519C2×C4—144no—
520C2×C4—148no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—282no—
525C2×C4—284no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—284no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4#644014yes—
535C2×C4#644214yes—
536C2×C4(C22⋊D7)⋊C414yes—
537C2×C4#644814yes—
538C2×C4C4×C22⋊D714yes—
539C2×C4#649814yes—
540C2×C4#650114yes—
541C2×C4#650314yes—
542C7#1400111yescharacteristic, complemented, Sylow
543C22—48no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—48no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C2×#644011yescharacteristic
555C22C2×#644211yescharacteristic
556C22C2×(C22⋊D7)⋊C411yescharacteristic
557C22C2×#644811yescharacteristic
558C22C4×D4×D712yes—
559C22#9023712yes—
560C22#9023912yes—
561C22#9025212yes—
562C22#9030212yes—
563C22#9030212yes—
564C22#9030512yes—
565C22#9030712yes—
566C22#9036212yes—
567C22#9036812yes—
568C22#9040812yes—
569C22#9040912yes—
570C22C2×C4×C22⋊D711yescharacteristic
571C22C2×#649811yescharacteristic
572C22C2×#650111yescharacteristic
573C22C2×#650311yescharacteristic
574C22#9049611yescharacteristic
575C22#9051611yescharacteristic
576C22#9052311yescharacteristic
577C4—24no—
578C4—42no—
579C4—82no—
580C4—148no—
581C4—284no—
582C4—284no—
583C4—284no—
584C4#8971618yescomplemented
585C2—84no—
586C2#1173374712yes—
587C2#1173378312yes—
588C2#1173387812yes—
589C2#1173389612yes—
590C2C2×#8971611yescharacteristic
591C2C4×#650311yescharacteristic
592C2#1173977311yescharacteristic
593C2#1174010811yescharacteristic
594C2#1174010911yescharacteristic
595C2#1174041611yescharacteristic
596C2#1174063311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#14001
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#89748C2semidirect
5. C2×#89716C2semidirect
6. C2×C4×C4⋊Dic7C2semidirect
7. #11729317C2semidirect
8. #11729301C2semidirect
9. C4×C7⋊#95C2semidirect
25. #89993C4semidirect
86. C7⋊#95D4semidirect
367. C4×C7#388semidirect
542. C7#14001semidirect
584. C4#89716semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3902
C2×#899932
C2×#897482
C2×#897162
C2×C4×C4⋊Dic72
#117293172
#117293012
C4×C7⋊#952
#140017