Finite Group Databasebeta

← #13364974

Subgroups — #13364974

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89746C211yesmaximal, characteristic
2C2×#89511C211yesmaximal, characteristic
3#11731311C211yesmaximal, characteristic
4C7×#1852C211yesmaximal, characteristic
5#11729313C211yesmaximal, characteristic
6#11728381C212yesmaximal, complemented
7#11728378C212yesmaximal, complemented
8Dic7×C22⋊C8C211yesmaximal, characteristic
9#11727279C211yesmaximal, characteristic
10#11726662C212yesmaximal, complemented
11#11726604C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C2×#6487C2211yescharacteristic
15C22×C4×Dic7C2211yescharacteristic
16C22×Dic28C2212yes—
17C2×#6450C2211yescharacteristic
18C2×C7×#107C2212yes—
19C2×C7×C22⋊C8C2211yescharacteristic
20C7×#380C2211yescharacteristic
21C2×C7⋊#95—21no—
22C2×C7⋊#95—21no—
23C2×C22⋊(C7⋊C8)C2211yescharacteristic
24C2×#6390C2212yes—
25C2×C8×Dic7C2212yes—
26C4⋊C4×Dic7C2211yescharacteristic
27#89746—21no—
28#89746—21no—
29#89746C2212yes—
30#89746C2212yes—
31#89746C2211yescharacteristic
32#89745C2211yescharacteristic
33C2×#6384C2212yes—
34C2×#6384C2212yes—
35C2×#6384C2212yes—
36#89718—21no—
37C22⋊C4×Dic7—21no—
38#89511C418yescomplemented
39#89424—21no—
40#89421—21no—
41C7×#340C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22044—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C22×Dic14—21no—
51C22×Dic14—22no—
52C22×Dic14—42no—
53C22×Dic14—42no—
54C22×Dic14C2×C414yes—
55C22×Dic14C2312yes—
56C22×Dic14C2311yescharacteristic
57C22×C7×C8C2312yes—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—42no—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C22⋊C4—21no—
64C2×C22⋊Dic7—21no—
65C2×C4⋊Dic7C2311yescharacteristic
66#6487—22no—
67#6487C2×C414yes—
68C2×Dic7⋊C4—21no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—21no—
71C2×C4×Dic7—21no—
72C2×C4×Dic7—21no—
73C2×C4×Dic7—42no—
74C2×C4×Dic7D411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7C2311yescharacteristic
77C2×C4×Dic7C2311yescharacteristic
78C22×C7⋊C8C2312yes—
79C2×Dic28—42no—
80C2×Dic28—42no—
81C2×Dic28C2×C418yescomplemented
82#6450—42no—
83C7×#107—42no—
84C7×C22⋊C8C2×C414yes—
85C7×#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—22no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94C22⋊(C7⋊C8)—22no—
95#6390C2×C418yescomplemented
96C8×Dic7—42no—
97#6384—42no—
98#6384—42no—
99#6384—42no—
100C2×#451—71no—
101C4×C22⋊Q8—71no—
102C2×#391—71no—
103#1969—72no—
104#1933—72no—
105#1928—72no—
106#1878—72no—
107#1872—71no—
108#1852D711yescharacteristic
109#1787—71no—
110C4×C22⋊C8—71no—
111C24×C7C2×D411yescharacteristic
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—42no—
118C22×C4×C7—42no—
119C22×C4×C7—42no—
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7#4112yes—
123C22×C4×C7C2411yescharacteristic
124C22×Dic7—24no—
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—22no—
130C22×Dic7—41no—
131C22×Dic7—41no—
132C22×Dic7—41no—
133C22×Dic7—42no—
134C22×Dic7—42no—
135C22×Dic7Q1614yescomplemented
136C22×Dic7C2×D411yescharacteristic
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C2×Dic14—22no—
141C2×Dic14—22no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C2×Dic14—22no—
145C2×Dic14—22no—
146C2×Dic14—24no—
147C2×Dic14—21no—
148C2×Dic14—22no—
149C2×Dic14—48no—
150C2×Dic14—42no—
151C2×Dic14—42no—
152C2×Dic14—48no—
153C2×Dic14C22×C412yes—
154C2×Dic14C22×C414yes—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8C22×C414yes—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—42no—
163C7×C22⋊C4—42no—
164C22⋊Dic7—24no—
165C4⋊Dic7—21no—
166C4⋊Dic7C22×C412yes—
167Dic7⋊C4—24no—
168Dic7⋊C4—42no—
169Dic7⋊C4—42no—
170C4×Dic7—22no—
171C4×Dic7—22no—
172C4×Dic7—41no—
173C4×Dic7—41no—
174C4×Dic7—42no—
175C4×Dic7—42no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179Dic28—48no—
180C23×Q8—71no—
181C22×Q16—72no—
182C2×C22⋊Q8—71no—
183C2×C4×Q8—71no—
184C22×C4⋊C4C2×D711yescharacteristic
185C22×C42—71no—
186#451—78no—
187C2×#107C2×D712yes—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#103—72no—
192C2×C22⋊C8—71no—
193C2×C22⋊C8C2×D711yescharacteristic
194C2×C4×C8—72no—
195#391—72no—
196#391—72no—
197#391—71no—
198#391—141no—
199#391—141no—
200#389—71no—
201#387—141no—
202#385—141no—
203#380C2×D711yescharacteristic
204C4×C4⋊C4—71no—
205C4×C4⋊C4—141no—
206C4×C22⋊C4—141no—
207C2×#95—141no—
208C2×#95—141no—
209#340—72no—
210#340C2×D712yes—
211#336—72no—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7C4×D412yes—
217C23×C7C2×Q1612yes—
218C23×C7#13512yes—
219C23×C7C22×D411yescharacteristic
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—42no—
233C2×C4×C7—84no—
234C2×C4×C7C4×D412yes—
235C2×C4×C7C23×C411yescharacteristic
236C2×C4×C7C2×#4111yescharacteristic
237C2×C4×C7H(5,𝔽2)12yes—
238C2×C4×C7H(5,𝔽2)12yes—
239C2×Dic7—21no—
240C2×Dic7—21no—
241C2×Dic7—21no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—22no—
246C2×Dic7—22no—
247C2×Dic7—24no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—44no—
252C2×Dic7—44no—
253C2×Dic7—48no—
254C2×Dic7—41no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7C2×Q1612yes—
261C2×Dic7C2×Q1612yes—
262C2×Dic7#13712yes—
263Dic14—24no—
264Dic14—24no—
265Dic14—24no—
266Dic14—48no—
267Dic14—42no—
268Dic14—42no—
269Dic14—42no—
270C7×C8—24no—
271C7⋊C8—42no—
272C22×Q8—74no—
273C22×Q8—71no—
274C22×Q8—72no—
275C22×Q8—141no—
276C22×Q8—142no—
277C22×Q8—282no—
278C22×Q8—282no—
279C23×C4—21no—
280C23×C4—71no—
281C23×C4—71no—
282C23×C4—141no—
283C23×C4C22×D711yescharacteristic
284C2×Q16—78no—
285C2×Q16—282no—
286C2×Q16—282no—
287C22×C8—72no—
288C22×C8C22×D712yes—
289C22⋊Q8—74no—
290C22⋊Q8—142no—
291C4×Q8—282no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4C22×D712yes—
299C2×C4⋊C4C22×D711yescharacteristic
300C2×C4⋊C4C22×D711yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—141no—
303C2×C42—71no—
304C2×C42—71no—
305C2×C42—71no—
306C2×C42—71no—
307C2×C42—141no—
308C2×C42—141no—
309C2×C42—282no—
310#107—42no—
311#103—78no—
312#103—282no—
313#103—282no—
314#103—282no—
315C22⋊C8—142no—
316C22⋊C8C4×D714yes—
317C4×C8—282no—
318#95—21no—
319#95—142no—
320#95—142no—
321#95—142no—
322#95—142no—
323#95—141no—
324#95—141no—
325#95—141no—
326#95—141no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7C4×Q1614yes—
334C22×C7C2×C4×D411yescharacteristic
335C22×C7#53411yescharacteristic
336C22×C7#53511yescharacteristic
337C22×C7C22×Q1611yescharacteristic
338C22×C7C2×#13511yescharacteristic
339C22×C7C2×#13711yescharacteristic
340C22×C7#57511yescharacteristic
341C4×C7—22no—
342C4×C7—81no—
343C4×C7—81no—
344C4×C7—82no—
345C4×C7#51812yes—
346C4×C7#53812yes—
347Dic7—22no—
348Dic7—22no—
349Dic7—22no—
350Dic7—44no—
351Dic7—42no—
352Dic7—41no—
353Dic7—42no—
354C24D4×D711yescharacteristic
355C2×Q8—74no—
356C2×Q8—72no—
357C2×Q8—144no—
358C2×Q8—141no—
359C2×Q8—142no—
360C2×Q8—142no—
361C2×Q8—142no—
362C2×Q8—142no—
363C2×Q8—142no—
364C2×Q8—142no—
365C2×Q8—142no—
366C2×Q8—288no—
367C2×Q8—282no—
368C2×Q8—282no—
369C2×Q8—288no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—71no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—74no—
382C22×C4—71no—
383C22×C4—142no—
384C22×C4—144no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4—141no—
388C22×C4—141no—
389C22×C4—281no—
390C22×C4—281no—
391C22×C4—281no—
392C22×C4—282no—
393C22×C4—282no—
394C22×C4C2×C4×D712yes—
395C22×C4D4×D711yescharacteristic
396C22×C4#116112yes—
397C22×C4C23×D711yescharacteristic
398Q16—288no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—142no—
402C2×C8—142no—
403C2×C8—142no—
404C2×C8C2×C4×D714yes—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—72no—
411C4⋊C4—144no—
412C4⋊C4—141no—
413C4⋊C4—282no—
414C4⋊C4—282no—
415C22⋊C4—42no—
416C22⋊C4—144no—
417C42—142no—
418C42—142no—
419C42—281no—
420C42—281no—
421C42—282no—
422C42—282no—
423C2×C7—24no—
424C2×C7#232111yescharacteristic
425C2×C7C2×C4×Q1611yescharacteristic
426C2×C7C4×#13711yescharacteristic
427C2×C7#318911yescharacteristic
428C2×C7#319011yescharacteristic
429C2×C7#320911yescharacteristic
430C2×C7#321111yescharacteristic
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23#645612yes—
436C23#647512yes—
437C23D7×Q1612yes—
438C23C2×D4×D711yescharacteristic
439Q8—144no—
440Q8—144no—
441Q8—144no—
442Q8—288no—
443Q8—282no—
444Q8—282no—
445Q8—282no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—72no—
460C2×C4—72no—
461C2×C4—72no—
462C2×C4—84no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—141no—
468C2×C4—141no—
469C2×C4—141no—
470C2×C4—142no—
471C2×C4—142no—
472C2×C4—144no—
473C2×C4—141no—
474C2×C4—141no—
475C2×C4—284no—
476C2×C4—284no—
477C2×C4—288no—
478C2×C4—281no—
479C2×C4—281no—
480C2×C4—282no—
481C2×C4—281no—
482C2×C4—281no—
483C2×C4—282no—
484C2×C4#645612yes—
485C2×C4C22×C4×D711yescharacteristic
486C2×C4#654812yes—
487C2×C4C2×#116111yescharacteristic
488C2×C4#655312yes—
489C8—24no—
490C8—282no—
491C7#2204411yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22#8964514yes—
499C22C2×#645611yescharacteristic
500C22#9028311yescharacteristic
501C22#9034411yescharacteristic
502C22C2×#647511yescharacteristic
503C22C2×D7×Q1611yescharacteristic
504C22#9046411yescharacteristic
505C22D7×#13711yescharacteristic
506C4—22no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4—142no—
511C4—142no—
512C4—142no—
513C4—284no—
514C4—282no—
515C4—282no—
516C4—281no—
517C4#9023612yes—
518C4#9033512yes—
519C2—24no—
520C2#1173203711yescharacteristic
521C2C2×#8964511yescharacteristic
522C2#1173664511yescharacteristic
523C2#1173744011yescharacteristic
524C2#1173762311yescharacteristic
525C2#1173874911yescharacteristic
526C2#1173893311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1852C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1852D7
hypocenterC7#22044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
6. #11728381C2semidirect
7. #11728378C2semidirect
10. #11726662C2semidirect
11. #11726604C2semidirect
38. #89511C4semidirect
81. C2×Dic28C2×C4semidirect
95. #6390C2×C4semidirect
135. C22×Dic7Q16semidirect
491. C7#22044semidirect

Maximal subgroups

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

subgroupindex
C2×#897462
C2×#895112
#117313112
C7×#18522
#117293132
#117283812
#117283782
Dic7×C22⋊C82
#117272792
#117266622
#117266042
#220447