Finite Group Databasebeta

← #13453617

Subgroups — #13453617

Order 1792.

978 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740698C211yesmaximal, characteristic, complemented
2#11733738C211yesmaximal, characteristic
3C7×#2023C211yesmaximal, characteristic, complemented
4#11729546C211yesmaximal, characteristic, complemented
5#11727911C212yesmaximal
6#11727907C212yesmaximal, complemented
7#11727392C211yesmaximal, characteristic, complemented
8#11726646C212yesmaximal, complemented
9#11726601C212yesmaximal, complemented
10#11726432C211yesmaximal, characteristic, complemented
11#11726356C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×#6507—21no—
14C2×#6507—21no—
15C2×#6503C2211yescharacteristic
16C2×D4×Dic7C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×(C22⋊D7)⋊C4—21no—
19C2×(C22⋊D7)⋊C4—21no—
20C2×#6432C2211yescharacteristic
21C2×C7×#102C2212yes—
22C7×#390C2211yescharacteristic
23C2×#6406C2212yes—
24C2×#6395C2212yes—
25C2×Dic7⋊C8C2212yes—
26C2×C8×Dic7C2212yes—
27#89748—21no—
28#89748—21no—
29#89748C2211yescharacteristic
30#89748C2212yes—
31#89748C2212yes—
32C2×#6383C2212yes—
33#89729—21no—
34#89729—21no—
35#89711C2211yescharacteristic
36C42×Dic7C2211yescharacteristic
37C7×#340C2212yes—
38C7×#336C2211yescharacteristic
39#89346C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2211yescharacteristic
43#29139—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×D4C2311yescharacteristic
46C22×C22⋊D7—21no—
47C22×C22⋊D7—21no—
48C23×Dic7—21no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×Dic14—21no—
52C22×C7×C8C2312yes—
53C7×C4⋊Q8—42no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C7×C4⋊C4C2312yes—
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C2×C22⋊Dic7—21no—
61#6507—44no—
62#6507—44no—
63#6503—22no—
64#6503Q814yescomplemented
65D4×Dic7—24no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2312yes—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4—22no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85#6456—24no—
86(C22⋊D7)⋊C4—44no—
87(C22⋊D7)⋊C4—44no—
88#6432—42no—
89C7×#102—42no—
90#6406—42no—
91#6395—42no—
92Dic7⋊C8—24no—
93C8×Dic7—24no—
94#6383—42no—
95#2868—71no—
96C4×#127—71no—
97#2023—71no—
98#2023—71no—
99#2023—71no—
100#2023D711yescharacteristic, complemented
101#1957—72no—
102#1834—72no—
103#1802—72no—
104#1802—72no—
105C4×C4⋊C8—71no—
106C24×C7—21no—
107C23×D7—21no—
108C2×C7×Q8—42no—
109C2×C7×Q8—42no—
110C2×C7×Q8—42no—
111C2×C7×D4—22no—
112C2×C7×D4—21no—
113C2×C7×D4—22no—
114C2×C7×D4C2×Q812yes—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2411yescharacteristic
120C2×C22⋊D7—24no—
121C2×C22⋊D7—24no—
122C2×C22⋊D7—42no—
123C2×C22⋊D7—42no—
124C2×C22⋊D7—42no—
125C2×C22⋊D7—42no—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—42no—
129C22×Dic7—42no—
130C22×Dic7—42no—
131C22×Dic7C2×D411yescharacteristic
132C22×Dic7C2×D411yescharacteristic
133C22×Dic7C2×D411yescharacteristic
134C22×Dic7C2×D411yescharacteristic
135C2×D28—22no—
136C2×D28—22no—
137C2×D28—21no—
138C2×D28C2×Q812yes—
139C2×C4×D7—42no—
140C2×C4×D7—42no—
141C2×C4×D7—42no—
142C2×Dic14—42no—
143C2×Dic14—42no—
144C2×Dic14—42no—
145C2×C7×C8—22no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—42no—
152C7×C22⋊C4—42no—
153C7×C22⋊C4—42no—
154C42×C7—22no—
155C22⋊Dic7—42no—
156(C2×D7)⋊C4—42no—
157(C2×D7)⋊C4—42no—
158(C2×D7)⋊C4—42no—
159C4⋊Dic7—22no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7—42no—
163Dic7⋊C4—44no—
164Dic7⋊C4—42no—
165Dic7⋊C4—44no—
166Dic7⋊C4—44no—
167Dic7⋊C4—42no—
168Dic7⋊C4—44no—
169C4×Dic7—22no—
170C4×Dic7—22no—
171C4×Dic7D814yes—
172C4×Dic7SD1614yes—
173C4×Dic7C2×Q812yes—
174C4×Dic7#4112yes—
175C4×Dic7#4114yes—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C4⋊Q8—71no—
180C2×C4⋊Q8—141no—
181C2×C4⋊Q8—141no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#127—71no—
184C2×C4×D4—71no—
185C2×C4×D4—71no—
186C2×C4×D4—141no—
187C2×C4×D4—141no—
188C2×C4⋊C8—72no—
189C2×#102—72no—
190C2×#102—72no—
191C2×#102—72no—
192C2×#102C2×D712yes—
193C2×C4×C8—72no—
194#390—71no—
195#390—141no—
196#390—141no—
197#390C2×D711yescharacteristic
198#384—71no—
199#384—72no—
200#384—72no—
201#384—141no—
202#384—141no—
203C43—71no—
204#340—72no—
205#340—72no—
206#340—72no—
207#340C2×D712yes—
208#336—71no—
209#336C2×D711yescharacteristic
210C23×C7—22no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7C22×D411yescharacteristic
218C22×D7—22no—
219C22×D7—41no—
220C22×D7—41no—
221C22×D7—41no—
222C22×D7—41no—
223C22×D7—41no—
224C22×D7—41no—
225C7×Q8—84no—
226C7×D4—24no—
227C7×D4—42no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—42no—
238C2×C4×C7C2×D812yes—
239C2×C4×C7C2×SD1612yes—
240C2×C4×C7C22×Q811yescharacteristic
241C2×C4×C7C2×#4112yes—
242C2×C4×C7C2×#4111yescharacteristic
243C2×C4×C7H(5,𝔽2)12yes—
244C22⋊D7—48no—
245C22⋊D7—48no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—41no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—41no—
252C2×Dic7—41no—
253C2×Dic7—42no—
254C2×Dic7—84no—
255C2×Dic7C22⋊Q812yes—
256C2×Dic7C22⋊Q812yes—
257C2×Dic7#12412yes—
258C2×Dic7#12412yes—
259C2×Dic7C2×D812yes—
260C2×Dic7C2×D812yes—
261C2×Dic7C2×SD1612yes—
262C2×Dic7C2×SD1612yes—
263C2×Dic7AGL(1,ℤ/8)12yes—
264C2×Dic7#13712yes—
265D28—24no—
266D28—42no—
267C4×D7—84no—
268Dic14—84no—
269C7×C8—42no—
270C7⋊C8—42no—
271C22×Q8—21no—
272C22×Q8—141no—
273C22×D4—71no—
274C22×D4—141no—
275C22×D4—141no—
276C22×D4C22×D711yescharacteristic
277C23×C4—141no—
278C23×C4—141no—
279C22×C8—72no—
280C22×C8C22×D712yes—
281C4⋊Q8—42no—
282C4⋊Q8—282no—
283C4⋊Q8—284no—
284C4⋊Q8—284no—
285#127—74no—
286#127—142no—
287C4×D4—144no—
288C4×D4—144no—
289C4×D4—284no—
290C4×D4—284no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—71no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—141no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—142no—
297C2×C4⋊C4—142no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4C22×D711yescharacteristic
302C2×C4⋊C4C22×D712yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—141no—
306C2×C22⋊C4—141no—
307C2×C22⋊C4—141no—
308C2×C22⋊C4—141no—
309C2×C42—71no—
310C2×C42—71no—
311C2×C42—71no—
312C2×C42—71no—
313C2×C42—71no—
314C2×C42—71no—
315C2×C42C22×D711yescharacteristic
316C4⋊C8—144no—
317#102—42no—
318#102—282no—
319#102—282no—
320#102—282no—
321C4×C8—144no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7—42no—
325C22×C7—81no—
326C22×C7—81no—
327C22×C7—81no—
328C22×C7—81no—
329C22×C7C2×C22⋊Q811yescharacteristic
330C22×C7C2×#12411yescharacteristic
331C22×C7#53511yescharacteristic
332C22×C7C22×D811yescharacteristic
333C22×C7C22×SD1611yescharacteristic
334C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
335C22×C7C2×#13711yescharacteristic
336C2×D7—42no—
337C2×D7—42no—
338C2×D7—42no—
339C2×D7—81no—
340C2×D7—81no—
341C2×D7—81no—
342C2×D7—81no—
343C4×C7—22no—
344C4×C7—22no—
345C4×C7—81no—
346C4×C7—82no—
347C4×C7#55412yes—
348C4×C7#55512yes—
349Dic7—22no—
350Dic7—81no—
351Dic7—82no—
352Dic7#47414yes—
353Dic7#47614yes—
354Dic7#48614yes—
355C24—21no—
356C24—141no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—282no—
361C2×Q8—282no—
362C2×Q8—282no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—72no—
367C2×D4—144no—
368C2×D4—144no—
369C2×D4—142no—
370C2×D4—142no—
371C2×D4—141no—
372C2×D4—282no—
373C2×D4—282no—
374C2×D4—282no—
375C2×D4—282no—
376C2×D4Q8×D712yes—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—71no—
382C22×C4—71no—
383C22×C4—141no—
384C22×C4—142no—
385C22×C4—282no—
386C22×C4—282no—
387C22×C4—282no—
388C22×C4—282no—
389C22×C4—282no—
390C22×C4—282no—
391C22×C4D4×D711yescharacteristic
392C22×C4D4×D711yescharacteristic
393C22×C4C23×D711yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—142no—
398C2×C8—142no—
399C2×C8—142no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—142no—
405C4⋊C4—142no—
406C4⋊C4—142no—
407C4⋊C4—282no—
408C4⋊C4—282no—
409C4⋊C4—282no—
410C4⋊C4—284no—
411C4⋊C4—284no—
412C4⋊C4—284no—
413C4⋊C4—284no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—282no—
417C22⋊C4—282no—
418C22⋊C4—282no—
419C22⋊C4—282no—
420C42—22no—
421C42—74no—
422C42—74no—
423C42—72no—
424C42—72no—
425C42—74no—
426C42—142no—
427C42—142no—
428C2×C7—82no—
429C2×C7#256311yescharacteristic
430C2×C7C2×#47411yescharacteristic
431C2×C7C2×#47611yescharacteristic
432C2×C7C2×#48611yescharacteristic
433C2×C7#314111yescharacteristic
434C2×C7#320511yescharacteristic
435C2×C7#320611yescharacteristic
436D7—82no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—142no—
445C23—281no—
446C23—281no—
447C23—281no—
448C23—281no—
449C23—281no—
450C23—281no—
451C23C2×D4×D711yescharacteristic
452Q8—84no—
453Q8—564no—
454D4—24no—
455D4—42no—
456D4—144no—
457D4—288no—
458D4—288no—
459D4—282no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—72no—
471C2×C4—72no—
472C2×C4—72no—
473C2×C4—72no—
474C2×C4—72no—
475C2×C4—72no—
476C2×C4—72no—
477C2×C4—72no—
478C2×C4—72no—
479C2×C4—72no—
480C2×C4—141no—
481C2×C4—141no—
482C2×C4—281no—
483C2×C4—282no—
484C2×C4—281no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—281no—
488C2×C4—564no—
489C2×C4—564no—
490C2×C4D7×D812yes—
491C2×C4D7×SD1612yes—
492C2×C4#654812yes—
493C2×C4C2×Q8×D711yescharacteristic
494C2×C4D7×#4112yes—
495C2×C4D7×#4111yescharacteristic
496C8—42no—
497C8—282no—
498C7#2913911yescharacteristic, complemented, Sylow
499C22—42no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—282no—
507C22—282no—
508C22—282no—
509C22—561no—
510C22—561no—
511C22—561no—
512C22—561no—
513C22#9028311yescharacteristic
514C22D7×C22⋊Q811yescharacteristic
515C22D7×#12411yescharacteristic
516C22C2×D7×D811yescharacteristic
517C22C2×D7×SD1611yescharacteristic
518C22D7×AGL(1,ℤ/8)11yescharacteristic
519C22D7×#13711yescharacteristic
520C4—22no—
521C4—22no—
522C4—74no—
523C4—74no—
524C4—74no—
525C4—82no—
526C4—81no—
527C4—142no—
528C4—562no—
529C4—561no—
530C4#9032712yes—
531C4#9036112yes—
532C2—82no—
533C2—562no—
534C2#1173424411yescharacteristic
535C2#1173743911yescharacteristic
536C2#1173747911yescharacteristic
537C2D7×#47411yescharacteristic
538C2D7×#47611yescharacteristic
539C2D7×#48611yescharacteristic
540C2#1173912811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2023C2
socleC23×C7C22×D4
prime radicalC2×#6503C22
hypercenter#2023D7
hypocenterC7#29139
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #11740698C2semidirect
3. C7×#2023C2semidirect
4. #11729546C2semidirect
6. #11727907C2semidirect
7. #11727392C2semidirect
8. #11726646C2semidirect
9. #11726601C2semidirect
10. #11726432C2semidirect
11. #11726356C2semidirect
64. #6503Q8semidirect
100. #2023D7semidirect
498. C7#29139semidirect

Maximal subgroups

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

subgroupindex
#117406982
#117337382
C7×#20232
#117295462
#117279112
#117279072
#117273922
#117266462
#117266012
#117264322
#117263562
#291397