Finite Group Databasebeta

← #13404542

Subgroups — #13404542

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2542C211yesmaximal, characteristic
2C2×#89862C211yesmaximal, characteristic, complemented
3C2×#89823C212yesmaximal, complemented
4#11739821C211yesmaximal, characteristic
5C2×D4×C7⋊C8C211yesmaximal, characteristic
6#11729539C211yesmaximal, characteristic, complemented
7#11729454C212yesmaximal, complemented
8#11729450C212yesmaximal, complemented
9#11729449C212yesmaximal, complemented
10C4×#6384C211yesmaximal, characteristic, complemented
11#11727220C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×C22⋊Q8C2212yes—
14C2×C7×C22⋊Q8C2212yes—
15C2×C4×C7×D4C2211yescharacteristic
16C22×#1144C2212yes—
17C2×#6487C2212yes—
18C23×C7⋊C8C2212yes—
19C2×C4×Dic14C2211yescharacteristic
20C7×#399—22no—
21C7×#390—21no—
22C7×#384—21no—
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27#89993—21no—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30#89862—24no—
31#89823—28no—
32D4×C7⋊C8—24no—
33#89745—21no—
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6381C2212yes—
37#89720—22no—
38#89707C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#25386—71nomaximal, Sylow
45C22×C7×Q8C2312yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7C2312yes—
48C22×Dic14C2311yescharacteristic
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8—22no—
51C7×C22⋊Q8—24no—
52C7×C22⋊Q8—24no—
53C7×C22⋊Q8—42no—
54C7×C22⋊Q8—42no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4D412yes—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—22no—
71C2×#1144—24no—
72C2×#1144—24no—
73C2×#1144—42no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2312yes—
78#6487—24no—
79#6487—42no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C22×C7⋊C8—24no—
84C22×C7⋊C8—24no—
85C22×C7⋊C8—42no—
86C22×C7⋊C8C2311yescharacteristic
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C4×Dic14—22no—
90C4×Dic14—22no—
91C7×#95—21no—
92#6409—24no—
93#6409—24no—
94#6409—42no—
95C7⋊#95—22no—
96C22⋊(C7⋊C8)—24no—
97#6384—22no—
98#6384D414yescomplemented
99#6381—24no—
100C4⋊(C7⋊C8)D414yes—
101C4×C7⋊C8—22no—
102C2×#467—72no—
103C2×#462—71no—
104C2×C8×D4—71no—
105#2542D711yescharacteristic
106#2538—71no—
107#2036—72no—
108#2001—72no—
109#1956—71no—
110#1947—71no—
111#1892—72no—
112C4×#103—71no—
113C24×C7C2×D412yes—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4Q1614yescomplemented
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2411yescharacteristic
136C22×Dic7—21no—
137C22×Dic7—21no—
138C22×Dic7—22no—
139C2×Dic14—21no—
140C2×Dic14—21no—
141C2×Dic14—21no—
142C2×Dic14—21no—
143C2×Dic14—21no—
144C2×Dic14C2×D412yes—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—44no—
149C7×C4⋊C4—44no—
150C7×C4⋊C4—42no—
151C7×C4⋊C4—42no—
152C7×C4⋊C4C2×D412yes—
153C7×C4⋊C4#4112yes—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—44no—
156C7×C22⋊C4—44no—
157C42×C7—42no—
158C42×C7C2×D412yes—
159C42×C7#4112yes—
160C22⋊Dic7—44no—
161C22⋊Dic7—82no—
162#1144—48no—
163C4⋊Dic7—21no—
164C4⋊Dic7—22no—
165C4⋊Dic7—21no—
166C4⋊Dic7—22no—
167C4⋊Dic7—42no—
168C4⋊Dic7—81no—
169Dic7⋊C4—44no—
170Dic7⋊C4—42no—
171C4×Dic7—42no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—21no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—21no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—48no—
180C2×C7⋊C8C2×D414yes—
181C2×C7⋊C8C2×D412yes—
182C22×Q16—72no—
183C23×C8—72no—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×C22⋊Q8—72no—
186C2×C22⋊Q8C2×D712yes—
187C2×C22⋊Q8C2×D712yes—
188C2×C4×Q8—71no—
189C2×C4×D4C2×D711yescharacteristic
190#467—148no—
191#462—144no—
192C8×D4—144no—
193C2×#107—72no—
194C2×C4⋊C8—71no—
195C2×#103—71no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—71no—
199C2×#103—72no—
200C2×C22⋊C8—72no—
201C2×C4×C8—71no—
202#399—22no—
203#396—142no—
204#390—21no—
205#390—141no—
206#389—141no—
207#384—21no—
208#384—71no—
209#380—72no—
210C4×C4⋊C4C2×D711yescharacteristic
211#340—71no—
212#340—72no—
213#340—71no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7C2×Q1614yes—
221C23×C7C22×D411yescharacteristic
222C7×Q8—42no—
223C7×Q8—42no—
224C7×Q8—42no—
225C7×Q8—42no—
226C7×D4—28no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7#12112yes—
246C2×C4×C7#12112yes—
247C2×C4×C7C2×Q1612yes—
248C2×C4×C7#13512yes—
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7H(5,𝔽2)11yescharacteristic
254C2×C4×C7H(5,𝔽2)11yescharacteristic
255C2×Dic7—22no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264Dic14—22no—
265Dic14—22no—
266Dic14—41no—
267Dic14—41no—
268C7⋊C8—22no—
269C7⋊C8—24no—
270C7⋊C8—41no—
271C22×Q8—71no—
272C22×Q8C22×D712yes—
273C22×D4C22⋊D711yescharacteristic
274C23×C4C22×D712yes—
275C2×Q16—144no—
276C2×Q16—144no—
277C2×Q16—282no—
278C22×C8—71no—
279C22×C8—72no—
280C22×C8—71no—
281C22×C8—144no—
282C22×C8—144no—
283C22×C8—282no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—42no—
289C22⋊Q8—42no—
290C22⋊Q8—144no—
291C22⋊Q8—282no—
292C4×Q8—142no—
293C4×Q8—142no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—71no—
301C2×C4⋊C4—72no—
302C2×C4⋊C4—141no—
303C2×C4⋊C4—142no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4C22⋊D711yescharacteristic
307C2×C4⋊C4C22×D711yescharacteristic
308C2×C4⋊C4C22×D711yescharacteristic
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—142no—
312C2×C22⋊C4—142no—
313C2×C22⋊C4C22⋊D712yes—
314C2×C42—21no—
315C2×C42—141no—
316C2×C42C22×D711yescharacteristic
317#107—144no—
318C4⋊C8—74no—
319#103—74no—
320#103—144no—
321#103—144no—
322#103—142no—
323#103—282no—
324C22⋊C8—144no—
325C4×C8—142no—
326#95—21no—
327#95—142no—
328C22×C7—24no—
329C22×C7—24no—
330C22×C7—24no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7C2×#12111yescharacteristic
336C22×C7D4211yescharacteristic
337C22×C7#54611yescharacteristic
338C22×C7C22×Q1611yescharacteristic
339C22×C7C2×#13511yescharacteristic
340C22×C7#57711yescharacteristic
341C22×C7#57811yescharacteristic
342C4×C7—22no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—82no—
347C4×C7#46714yescomplemented
348C4×C7#53211yescharacteristic
349C4×C7#54011yescharacteristic
350C4×C7D4211yescharacteristic
351C4×C7#55011yescharacteristic
352Dic7—42no—
353Dic7—82no—
354Dic7—81no—
355C24C2×C22⋊D712yes—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—72no—
363C2×Q8—141no—
364C2×Q8—141no—
365C2×Q8—141no—
366C2×Q8—141no—
367C2×Q8—141no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4#114414yescomplemented
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—141no—
381C22×C4—141no—
382C22×C4—142no—
383C22×C4C2×C22⋊D712yes—
384C22×C4C2×C22⋊D711yescharacteristic
385C22×C4C2×C22⋊D711yescharacteristic
386C22×C4C23×D711yescharacteristic
387Q16—288no—
388C2×C8—74no—
389C2×C8—72no—
390C2×C8—141no—
391C2×C8—142no—
392C2×C8—141no—
393C2×C8—142no—
394C2×C8—141no—
395C2×C8—141no—
396C2×C8—141no—
397C2×C8—288no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—141no—
406C4⋊C4—142no—
407C4⋊C4—141no—
408C4⋊C4—142no—
409C4⋊C4—284no—
410C4⋊C4—282no—
411C4⋊C4—282no—
412C4⋊C4—561no—
413C4⋊C4D4×D712yes—
414C4⋊C4#115912yes—
415C22⋊C4—24no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—284no—
419C22⋊C4—562no—
420C42—42no—
421C42—282no—
422C42D4×D712yes—
423C42#115912yes—
424C2×C7—44no—
425C2×C7#242811yescharacteristic
426C2×C7C2×#46711yescharacteristic
427C2×C7#306311yescharacteristic
428C2×C7#328411yescharacteristic
429C2×C7D4×Q1611yescharacteristic
430C2×C7#329911yescharacteristic
431C2×C7#330211yescharacteristic
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23C2×#114414yes—
439C23C22×C22⋊D711yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—142no—
445Q8—142no—
446Q8—281no—
447Q8—281no—
448D4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—142no—
468C2×C4—281no—
469C2×C4—281no—
470C2×C4—282no—
471C2×C4—281no—
472C2×C4—282no—
473C2×C4—281no—
474C2×C4—282no—
475C2×C4—281no—
476C2×C4#650212yes—
477C2×C4#650212yes—
478C2×C4C2×#114412yes—
479C2×C4#651312yes—
480C2×C4C2×D4×D711yescharacteristic
481C2×C4C2×D4×D711yescharacteristic
482C2×C4C2×#115911yescharacteristic
483C2×C4C2×#115911yescharacteristic
484C2×C4#654811yescharacteristic
485C2×C4#654811yescharacteristic
486C8—142no—
487C8—144no—
488C8—281no—
489C7#2538611yescharacteristic, complemented, Sylow
490C22—24no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22D4×C22⋊D711yescharacteristic
498C22C2×#650211yescharacteristic
499C22#9049711yescharacteristic
500C22C22×#114411yescharacteristic
501C22C2×#651311yescharacteristic
502C22#9052811yescharacteristic
503C22#9052911yescharacteristic
504C4—22no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—282no—
510C4—562no—
511C4—561no—
512C4#8982314yescomplemented
513C4#9040711yescharacteristic
514C4#9040811yescharacteristic
515C4#9040911yescharacteristic
516C4#9041011yescharacteristic
517C2—44no—
518C2C2×#8982311yescharacteristic
519C2#1174063111yescharacteristic
520C2#1174084511yescharacteristic
521C2#1174095511yescharacteristic
522C2#1174097511yescharacteristic
523C2#1174099611yescharacteristic
524C2D4×#114411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2542C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenter#2542D7
hypocenterC7#25386
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. C2×#89862C2semidirect
3. C2×#89823C2semidirect
6. #11729539C2semidirect
7. #11729454C2semidirect
8. #11729450C2semidirect
9. #11729449C2semidirect
10. C4×#6384C2semidirect
11. #11727220C2semidirect
98. #6384D4semidirect
122. C2×C7×D4Q16semidirect
347. C4×C7#467semidirect
370. C2×D4#1144semidirect
489. C7#25386semidirect
512. C4#89823semidirect

Maximal subgroups

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

subgroupindex
C7×#25422
C2×#898622
C2×#898232
#117398212
C2×D4×C7⋊C82
#117295392
#117294542
#117294502
#117294492
C4×#63842
#117272202
#253867