Finite Group Databasebeta

← #13363507

Subgroups — #13363507

Order 1792.

1106 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90313C211yesmaximal, characteristic
2#11739669C211yesmaximal, characteristic
3C2×#89490C211yesmaximal, characteristic
4C7×#1873C211yesmaximal, characteristic
5#11729447C211yesmaximal, characteristic
6#11729339C211yesmaximal, characteristic
7#11727979C212yesmaximal, complemented
8#11727976C212yesmaximal, complemented
9#11727685C211yesmaximal, characteristic
10#11726659C212yesmaximal, complemented
11#11726557C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C2×#6507C2211yescharacteristic
16C2×#6489C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19#90313—24no—
20#90313C418yescomplemented
21C2×#6452—21no—
22C2×#6451C2211yescharacteristic
23C2×#6450—21no—
24C2×#6450C2211yescharacteristic
25C2×#6441—21no—
26C2×#6440—21no—
27C2×C7×#103C2212yes—
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×#6409C2212yes—
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×#6390C2212yes—
32C2×Dic7⋊C8C2212yes—
33C2×C8×Dic7C2212yes—
34#89749C2212yes—
35#89748C2212yes—
36C2×#6384C2212yes—
37#89714—21no—
38#89714—21no—
39C4×Dic7⋊C4—21no—
40C4×Dic7⋊C4—21no—
41#89490—24no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21864—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7D411yescharacteristic
50C23×Dic7D411yescharacteristic
51C22×Dic14—21no—
52C22×Dic14C2311yescharacteristic
53C22×C7×C8C2312yes—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8C2×C414yes—
56C7×#117—42no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7—21no—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—21no—
65Q8×Dic7—22no—
66Q8×Dic7C2×C414yes—
67#6507—22no—
68#6507C2×C414yes—
69#6489—42no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7C2312yes—
72C2×C4⋊Dic7C2311yescharacteristic
73#6487—22no—
74#6487C2×C414yes—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7C2×C414yes—
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D414yes—
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2312yes—
92#6452—28no—
93#6451—22no—
94#6451C2×C414yes—
95#6450—28no—
96#6450—22no—
97#6450C2×C414yes—
98#6441—28no—
99#6440—28no—
100C7×#103—42no—
101C7×C22⋊C8—22no—
102#6409—42no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105C22⋊(C7⋊C8)—22no—
106#6390—42no—
107Dic7⋊C8—24no—
108C8×Dic7—24no—
109#6384—42no—
110C2×#533—71no—
111C2×#431—71no—
112#2285—71no—
113#1990—72no—
114#1956—72no—
115#1873—71no—
116#1873—71no—
117#1873—71no—
118#1873D711yescharacteristic
119#1805—72no—
120#1805—72no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7C22×C412yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—24no—
136C22×Dic7—24no—
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—22no—
140C22×Dic7C22⋊C414yes—
141C22×Dic7C22⋊C414yes—
142C22×Dic7C2×D411yescharacteristic
143C22×Dic7C2×D411yescharacteristic
144C22×Dic7C2×D411yescharacteristic
145C22×Dic7C2×D411yescharacteristic
146C2×Dic14—22no—
147C2×Dic14—22no—
148C2×Dic14—21no—
149C2×Dic14—24no—
150C2×Dic14—44no—
151C2×Dic14C22×C412yes—
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4C22×C412yes—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—42no—
162C42×C7—42no—
163C22⋊Dic7—24no—
164C22⋊Dic7—42no—
165C4⋊Dic7—21no—
166C4⋊Dic7—24no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7C22×C412yes—
170Dic7⋊C4—24no—
171Dic7⋊C4—24no—
172Dic7⋊C4—24no—
173Dic7⋊C4—24no—
174Dic7⋊C4—24no—
175Dic7⋊C4—44no—
176Dic7⋊C4—44no—
177C4×Dic7—24no—
178C4×Dic7—24no—
179C4×Dic7—42no—
180C4×Dic7C22⋊C414yes—
181C4×Dic7C22⋊C414yes—
182C4×Dic7C22×C412yes—
183C4×Dic7C22×C412yes—
184C4×Dic7C2×D412yes—
185C4×Dic7C2×D412yes—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189#533—78no—
190#533—144no—
191C2×C4⋊Q8—71no—
192C2×C4⋊Q8—71no—
193C2×#125—141no—
194C2×C22⋊Q8—71no—
195C2×C22⋊Q8—141no—
196C2×C22⋊Q8C2×D711yescharacteristic
197C2×C4×Q8—71no—
198C2×C4×Q8—71no—
199C2×C4×Q8—141no—
200C2×#117—71no—
201C2×#117—141no—
202C2×#117C2×D711yescharacteristic
203C22×C42—71no—
204#431—144no—
205C2×C4⋊C8—72no—
206C2×#103—72no—
207C2×#103—72no—
208C2×#103—72no—
209C2×#103C2×D712yes—
210C2×C22⋊C8—71no—
211C2×C22⋊C8C2×D711yescharacteristic
212C2×C4×C8—72no—
213#384—72no—
214#382—72no—
215#380—141no—
216#380—141no—
217C4×C4⋊C4—141no—
218C4×C4⋊C4—141no—
219#340—72no—
220#340—72no—
221#340—72no—
222#340C2×D712yes—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7C2×C22⋊C412yes—
228C23×C7C22×D411yescharacteristic
229C7×Q8—24no—
230C7×Q8—42no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7C2×C22⋊C412yes—
244C2×C4×C7C23×C411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7H(5,𝔽2)12yes—
247C2×C4×C72−1+412yes—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—28no—
251C2×Dic7—28no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7C2×C22⋊C412yes—
261C2×Dic7C2×C22⋊C412yes—
262C2×Dic7C2×C22⋊C412yes—
263C2×Dic7C2×C22⋊C412yes—
264C2×Dic7#13514yes—
265C2×Dic7#13714yes—
266Dic14—24no—
267Dic14—42no—
268Dic14—48no—
269C7×C8—42no—
270C7⋊C8—42no—
271C22×Q8—71no—
272C22×Q8—141no—
273C22×Q8C22×D711yescharacteristic
274C23×C4—71no—
275C23×C4—71no—
276C23×C4C22×D711yescharacteristic
277C22×C8—72no—
278C22×C8C22×D712yes—
279C4⋊Q8—74no—
280C4⋊Q8—74no—
281C4⋊Q8—142no—
282C4⋊Q8—142no—
283#125—148no—
284C22⋊Q8—22no—
285C22⋊Q8—74no—
286C22⋊Q8—148no—
287C22⋊Q8—142no—
288C22⋊Q8C4×D714yes—
289C4×Q8—74no—
290C4×Q8—74no—
291C4×Q8—142no—
292C4×Q8—148no—
293C4×Q8—142no—
294#117—42no—
295#117—148no—
296#117—282no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—71no—
299C2×C4⋊C4—72no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4—141no—
302C2×C4⋊C4—141no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—142no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4C22×D712yes—
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—141no—
313C2×C22⋊C4—141no—
314C2×C42—21no—
315C2×C42—74no—
316C2×C42—71no—
317C2×C42—71no—
318C2×C42—74no—
319C2×C42—71no—
320C2×C42—71no—
321C2×C42—141no—
322C2×C42—141no—
323C2×C42—141no—
324C4⋊C8—144no—
325#103—42no—
326#103—282no—
327#103—282no—
328#103—282no—
329C22⋊C8—22no—
330C22⋊C8—142no—
331C4×C8—144no—
332#95—142no—
333#95—142no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7—22no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7C22×C22⋊C411yescharacteristic
340C22×C7#53511yescharacteristic
341C22×C7#53611yescharacteristic
342C22×C7C2×#13512yes—
343C22×C7C2×#13712yes—
344C4×C7—22no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—82no—
348C4×C7#52012yes—
349C4×C7#53712yes—
350Dic7—22no—
351Dic7—22no—
352Dic7—42no—
353Dic7—42no—
354Dic7—82no—
355Dic7#41614yes—
356Dic7#41914yes—
357C24D4×D711yescharacteristic
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—72no—
362C2×Q8—144no—
363C2×Q8—141no—
364C2×Q8—142no—
365C2×Q8—142no—
366C2×Q8—284no—
367C2×Q8C2×C4×D712yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—71no—
374C22×C4—71no—
375C22×C4—74no—
376C22×C4—74no—
377C22×C4—71no—
378C22×C4—71no—
379C22×C4—144no—
380C22×C4—144no—
381C22×C4—141no—
382C22×C4—141no—
383C22×C4—142no—
384C22×C4C2×C4×D712yes—
385C22×C4D4×D711yescharacteristic
386C22×C4D4×D712yes—
387C22×C4C23×D711yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—142no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—72no—
399C4⋊C4—144no—
400C4⋊C4—144no—
401C4⋊C4—144no—
402C4⋊C4—144no—
403C4⋊C4—144no—
404C4⋊C4—144no—
405C4⋊C4—141no—
406C4⋊C4—142no—
407C4⋊C4—142no—
408C4⋊C4—284no—
409C4⋊C4—284no—
410C4⋊C4C2×C4×D712yes—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—144no—
414C22⋊C4—282no—
415C42—42no—
416C42—74no—
417C42—74no—
418C42—72no—
419C42—72no—
420C42—72no—
421C42—72no—
422C42—144no—
423C42—144no—
424C42—282no—
425C2×C7—42no—
426C2×C7#231111yescharacteristic
427C2×C7C2×#41611yescharacteristic
428C2×C7C2×#41911yescharacteristic
429C2×C7#321012yes—
430C2×C7#323112yes—
431C23—22no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23D7×C22⋊C412yes—
436C23C2×D4×D711yescharacteristic
437Q8—24no—
438Q8—42no—
439Q8—144no—
440Q8—288no—
441Q8—282no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—72no—
455C2×C4—72no—
456C2×C4—74no—
457C2×C4—74no—
458C2×C4—72no—
459C2×C4—72no—
460C2×C4—148no—
461C2×C4—148no—
462C2×C4—142no—
463C2×C4—142no—
464C2×C4—142no—
465C2×C4—142no—
466C2×C4—282no—
467C2×C4—281no—
468C2×C4—281no—
469C2×C4—282no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4D7×C22⋊C412yes—
473C2×C4C22×C4×D711yescharacteristic
474C2×C4C2×D4×D711yescharacteristic
475C2×C4#654812yes—
476C2×C4#655412yes—
477C8—42no—
478C8—282no—
479C7#2186411yescharacteristic, complemented, Sylow
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22C2×D7×C22⋊C411yescharacteristic
486C22#9028311yescharacteristic
487C22#9033911yescharacteristic
488C22D7×#13512yes—
489C22D7×#13712yes—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—74no—
494C4—74no—
495C4—82no—
496C4—142no—
497C4—142no—
498C4—282no—
499C4—282no—
500C4—562no—
501C4#9023712yes—
502C4#9028812yes—
503C2—42no—
504C2#1173210711yescharacteristic
505C2D7×#41611yescharacteristic
506C2D7×#41911yescharacteristic
507C2#1173766512yes—
508C2#1173894512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1873C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1873D7
hypocenterC7#21864
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
7. #11727979C2semidirect
8. #11727976C2semidirect
10. #11726659C2semidirect
11. #11726557C2semidirect
20. #90313C4semidirect
479. C7#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#903132
#117396692
C2×#894902
C7×#18732
#117294472
#117293392
#117279792
#117279762
#117276852
#117266592
#117265572
#218647