Finite Group Databasebeta

← #13383787

Subgroups — #13383787

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6493C211yesmaximal, characteristic
2#11739507C211yesmaximal, characteristic, complemented
3C4×C7×C22⋊C8C211yesmaximal, characteristic, complemented
4#11727692C212yesmaximal, complemented
5#11727688C212yesmaximal
6C4×#6392C212yesmaximal, complemented
7C4×#6395C212yesmaximal, complemented
8#11726435C211yesmaximal, characteristic, complemented
9#11726396C211yesmaximal, characteristic, complemented
10#11726393C211yesmaximal, characteristic, complemented
11#11726389C211yesmaximal, characteristic, complemented
12C22×C42×C7C2211yescharacteristic
13C2×#6493C2211yescharacteristic
14C2×C4×C22⋊D7—21no—
15C2×#6489—21no—
16C2×C4×D28C2211yescharacteristic
17C2×C7×C22⋊C8C2212yes—
18C2×C4×C7×C8C2212yes—
19#89985—21no—
20#89984—21no—
21#89984C2212yes—
22#89984C2211yescharacteristic
23C2×#6395C2212yes—
24C2×#6395C2212yes—
25C2×#6392C2212yes—
26C2×#6392C2212yes—
27#89712—21no—
28#89711C2211yescharacteristic
29C4×(C2×D7)⋊C4—21no—
30#89708—21no—
31#89707C2211yescharacteristic
32#89707C2211yescharacteristic
33C4×C4⋊Dic7—21no—
34C4×C4⋊Dic7C2212yes—
35C4×C4⋊Dic7C2211yescharacteristic
36#89700C2211yescharacteristic
37C7×#336C2211yescharacteristic
38C7×#336C2211yescharacteristic
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2212yes—
43#23754—71nomaximal, Sylow
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C22×C22⋊D7—21no—
48C22×D28C2311yescharacteristic
49C22×C4×D7—21no—
50C22×C7×C8C2312yes—
51C22×C7×C8C2312yes—
52C2×C42×C7—24no—
53C2×C42×C7D411yescharacteristic
54C2×C42×C7D411yescharacteristic
55C2×C42×C7C2311yescharacteristic
56C2×C42×C7C2311yescharacteristic
57C2×C22⋊Dic7—22no—
58C2×C22⋊Dic7—21no—
59#6493—22no—
60#6493—22no—
61C4×C22⋊D7—44no—
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—21no—
64C2×(C2×D7)⋊C4—21no—
65C2×(C2×D7)⋊C4—21no—
66#6489—44no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2312yes—
74C2×C4⋊Dic7C2312yes—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×Dic7⋊C4—21no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—21no—
79C4×D28—24no—
80C7×C22⋊C8—24no—
81C4×C7×C8—24no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87#6395—24no—
88#6395—24no—
89#6392—24no—
90#6392—24no—
91C4×#121—71no—
92#2285—71no—
93#2099—71no—
94#2093—71no—
95#2083—71no—
96#2075—72no—
97#2071—71no—
98#2033—72no—
99C4×#107—72no—
100C4×#102—72no—
101C4×C22⋊C8D711yescharacteristic, complemented
102C24×C7C2×D411yescharacteristic
103C23×D7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—22no—
112C22×C4×C7—41no—
113C22×C4×C7D814yes—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2411yescharacteristic
120C2×C22⋊D7—24no—
121C2×C22⋊D7—42no—
122C2×C22⋊D7—42no—
123C22×Dic7—21no—
124C22×Dic7—22no—
125C22×Dic7—22no—
126C22×Dic7—21no—
127C22×Dic7—21no—
128C2×D28—22no—
129C2×D28—22no—
130C2×D28—21no—
131C2×D28#4112yes—
132C2×C4×D7—42no—
133C2×C4×D7—42no—
134C2×C4×D7—42no—
135C2×C7×C8—22no—
136C2×C7×C8—22no—
137C2×C7×C8—22no—
138C2×C7×C8—22no—
139C2×C7×C8—22no—
140C2×C7×C8#4114yes—
141C42×C7—22no—
142C42×C7—22no—
143C42×C7—22no—
144C42×C7#4114yes—
145C22⋊Dic7—44no—
146C22⋊Dic7—42no—
147(C2×D7)⋊C4—24no—
148(C2×D7)⋊C4—42no—
149(C2×D7)⋊C4—42no—
150(C2×D7)⋊C4—42no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—22no—
158C4⋊Dic7—42no—
159C4⋊Dic7—42no—
160C4⋊Dic7—42no—
161C4⋊Dic7—42no—
162C4⋊Dic7#4112yes—
163Dic7⋊C4—42no—
164C4×Dic7—44no—
165C4×Dic7—42no—
166C2×#121—71no—
167C2×C4×D4—71no—
168C2×C4×D4—141no—
169C2×#117—141no—
170C22×C42C2×D711yescharacteristic
171C2×#107—72no—
172C2×#107—72no—
173C2×#102—72no—
174C2×#102—72no—
175C2×C22⋊C8C2×D712yes—
176C2×C4×C8C2×D712yes—
177#390—71no—
178#389—71no—
179#388—141no—
180#386—141no—
181#384—71no—
182#384—71no—
183#382—141no—
184#380—71no—
185#380—72no—
186#380—141no—
187C4×C4⋊C4—72no—
188C4×C4⋊C4—71no—
189C4×C4⋊C4—141no—
190C4×C22⋊C4—141no—
191#340—72no—
192#340—72no—
193#340—72no—
194#340—72no—
195#336C2×D711yescharacteristic
196#336C2×D711yescharacteristic
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7C2×D812yes—
204C23×C7C22×D411yescharacteristic
205C22×D7—22no—
206C22×D7—41no—
207C22×D7—41no—
208C22×D7—41no—
209C22×D7—41no—
210C22×D7—41no—
211C22×D7—41no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—28no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—44no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7#12312yes—
226C2×C4×C7#12312yes—
227C2×C4×C7C2×D812yes—
228C2×C4×C7C2×D812yes—
229C2×C4×C7#13512yes—
230C2×C4×C7#13712yes—
231C2×C4×C7C2×#4112yes—
232C2×C4×C7C2×#4112yes—
233C2×C4×C7C2×#4111yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C22⋊D7—48no—
236C2×Dic7—22no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—42no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—42no—
251D28—24no—
252D28—42no—
253C4×D7—84no—
254C7×C8—24no—
255C7×C8—42no—
256C22×D4—71no—
257C22×D4—141no—
258C23×C4—141no—
259C23×C4D2811yescharacteristic
260C23×C4D2811yescharacteristic
261C23×C4C22×D711yescharacteristic
262C22×C8C22×D712yes—
263C22×C8C22×D712yes—
264#121—142no—
265#121—142no—
266C4×D4—144no—
267C4×D4—284no—
268#117—284no—
269C2×C4⋊C4—71no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—72no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—71no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—141no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4—141no—
279C2×C22⋊C4—141no—
280C2×C22⋊C4—142no—
281C2×C22⋊C4—141no—
282C2×C22⋊C4—141no—
283C2×C22⋊C4—141no—
284C2×C22⋊C4—141no—
285C2×C42—24no—
286C2×C42—142no—
287C2×C42—141no—
288C2×C42D2811yescharacteristic
289C2×C42D2811yescharacteristic
290C2×C42C22×D711yescharacteristic
291C2×C42C22×D711yescharacteristic
292#107—144no—
293#107—144no—
294#102—144no—
295#102—144no—
296C22⋊C8—24no—
297C4×C8—24no—
298#95—141no—
299#95—141no—
300#95—142no—
301#95—141no—
302#95—142no—
303C22×C7—22no—
304C22×C7—22no—
305C22×C7—22no—
306C22×C7—41no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7C2×#12311yescharacteristic
311C22×C7#52511yescharacteristic
312C22×C7#53211yescharacteristic
313C22×C7C22×D811yescharacteristic
314C22×C7C2×#13511yescharacteristic
315C22×C7C2×#13711yescharacteristic
316C22×C7#57511yescharacteristic
317C2×D7—42no—
318C2×D7—42no—
319C2×D7—42no—
320C2×D7—81no—
321C2×D7—81no—
322C2×D7—81no—
323C2×D7—81no—
324C4×C7—22no—
325C4×C7—22no—
326C4×C7—22no—
327C4×C7—41no—
328C4×C7#48014yes—
329C4×C7#52912yes—
330C4×C7#52912yes—
331Dic7—42no—
332Dic7—81no—
333Dic7—82no—
334Dic7—82no—
335Dic7—81no—
336C24—141no—
337C24C2×D2811yescharacteristic
338C2×D4—72no—
339C2×D4—144no—
340C2×D4—142no—
341C2×D4—142no—
342C2×D4—141no—
343C2×D4—282no—
344C2×D4—282no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—141no—
355C22×C4—141no—
356C22×C4—142no—
357C22×C4—142no—
358C22×C4—141no—
359C22×C4—282no—
360C22×C4—282no—
361C22×C4—282no—
362C22×C4D5614yes—
363C22×C4C2×D2811yescharacteristic
364C22×C4C2×D2811yescharacteristic
365C22×C4C2×D2811yescharacteristic
366C22×C4C2×D2811yescharacteristic
367C22×C4C2×D2811yescharacteristic
368C22×C4C23×D711yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8#115714yes—
375C4⋊C4—72no—
376C4⋊C4—141no—
377C4⋊C4—142no—
378C4⋊C4—142no—
379C4⋊C4—142no—
380C4⋊C4—142no—
381C4⋊C4—142no—
382C4⋊C4—142no—
383C4⋊C4—282no—
384C4⋊C4—282no—
385C4⋊C4—282no—
386C4⋊C4—282no—
387C4⋊C4—282no—
388C22⋊C4—144no—
389C22⋊C4—284no—
390C22⋊C4—282no—
391C22⋊C4—282no—
392C22⋊C4—282no—
393C22⋊C4—282no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—284no—
398C42—282no—
399C42#115914yes—
400C2×C7—42no—
401C2×C7#239511yescharacteristic
402C2×C7C2×#48011yescharacteristic
403C2×C7#309111yescharacteristic
404C2×C7#310011yescharacteristic
405C2×C7#310711yescharacteristic
406C2×C7#316911yescharacteristic
407C2×C7#317211yescharacteristic
408D7—82no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—142no—
416C23—281no—
417C23—281no—
418C23—281no—
419C23—281no—
420C23—281no—
421C23—281no—
422C23C2×D5612yes—
423C23C22×D2811yescharacteristic
424D4—144no—
425D4—288no—
426D4—282no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—28no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—142no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—282no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—282no—
447C2×C4—282no—
448C2×C4—281no—
449C2×C4—281no—
450C2×C4—282no—
451C2×C4—281no—
452C2×C4—281no—
453C2×C4—282no—
454C2×C4—282no—
455C2×C4—564no—
456C2×C4#644912yes—
457C2×C4#644912yes—
458C2×C4C2×D5612yes—
459C2×C4C2×D5612yes—
460C2×C4#646712yes—
461C2×C4#647212yes—
462C2×C4C2×#115712yes—
463C2×C4C2×#115912yes—
464C2×C4D7×#4111yescharacteristic
465C2×C4D7×#4111yescharacteristic
466C8—24no—
467C8—42no—
468C7#2375411yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—282no—
477C22—282no—
478C22—282no—
479C22—561no—
480C22—561no—
481C22—561no—
482C22—561no—
483C22#9016811yescharacteristic
484C22C2×#644911yescharacteristic
485C22#9021511yescharacteristic
486C22C22×D5611yescharacteristic
487C22C2×#646711yescharacteristic
488C22C2×#647211yescharacteristic
489C22#9043911yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—282no—
495C4—561no—
496C4—562no—
497C4—562no—
498C4—561no—
499C4#8950814yes—
500C4#9022912yes—
501C4#9024812yes—
502C2—42no—
503C2—562no—
504C2#1173102811yescharacteristic
505C2#1173591511yescharacteristic
506C2#1173598411yescharacteristic
507C2C2×#8950811yescharacteristic
508C2#1173603611yescharacteristic
509C2#1173674211yescharacteristic
510C2#1173674711yescharacteristic
subgroupG/H
centerC22×C4C2×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×C22⋊C8C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC4×C22⋊C8D7
hypocenterC7#23754
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #11739507C2semidirect
3. C4×C7×C22⋊C8C2semidirect
4. #11727692C2semidirect
6. C4×#6392C2semidirect
7. C4×#6395C2semidirect
8. #11726435C2semidirect
9. #11726396C2semidirect
10. #11726393C2semidirect
11. #11726389C2semidirect
101. C4×C22⋊C8D7semidirect
468. C7#23754semidirect

Maximal subgroups

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

subgroupindex
C4×#64932
#117395072
C4×C7×C22⋊C82
#117276922
#117276882
C4×#63922
C4×#63952
#117264352
#117263962
#117263932
#117263892
#237547