Finite Group Databasebeta

← #13410758

Subgroups — #13410758

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×#89861C211yesmaximal, characteristic
3#11739929C211yesmaximal, characteristic
4C2×#89800C211yesmaximal, characteristic
5#11729533C211yesmaximal, characteristic, complemented
6#11729498C212yesmaximal, complemented
7#11729497C212yesmaximal
8#11729389C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal
10#11729386C212yesmaximal, complemented
11#11729385C211yesmaximal, characteristic, complemented
12C7×#557—24no—
13C7×#557—24no—
14C2×C7×C4⋊Q8—21no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—22no—
17C2×C7×#125C2212yes—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C22×#1144C2211yescharacteristic
23C22×#1143C2211yescharacteristic
24C2×#6438C2211yescharacteristic
25#90003—21no—
26C2×#6409C2212yes—
27#89861—24no—
28#89800—24no—
29#89765—21no—
30#89762—22no—
31C2×#6384C2211yescharacteristic
32C2×#6384C2212yes—
33C2×#6383C2211yescharacteristic
34C2×#6383C2212yes—
35#89711C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37#89701C2212yes—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#25941—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×Q8C2311yescharacteristic
49C22×D28C2311yescharacteristic
50C22×Dic14C2311yescharacteristic
51C7×C4⋊Q8—28no—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8D414yes—
54C7×#125—216no—
55C7×#125—24no—
56C7×#125—24no—
57C4×C7×Q8—28no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8—22no—
62C4×C7×Q8—22no—
63C4×C7×Q8D414yes—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4D412yes—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2312yes—
73C2×C42×C7—22no—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77C2×#1144—22no—
78C2×#1144—24no—
79C2×#1143—22no—
80C2×#1143—24no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—22no—
84C2×(C2×D7)⋊C4—22no—
85C2×C4⋊Dic7C2312yes—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90#6438—22no—
91#6438—22no—
92#6409—24no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—21no—
96#6384—24no—
97#6384—22no—
98#6383—24no—
99#6383—22no—
100C4⋊(C7⋊C8)—22no—
101#6377—22no—
102C2×#557D711yescharacteristic, complemented
103C2×#489—71no—
104C2×#464—71no—
105#2592—71no—
106#2026—72no—
107#2025—72no—
108#1987—71no—
109#1985—71no—
110#1880—72no—
111#1878—72no—
112#1763—71no—
113C23×D7—21no—
114C2×C7×Q8—22no—
115C2×C7×Q8—24no—
116C2×C7×Q8—21no—
117C2×C7×Q8—22no—
118C2×C7×Q8—42no—
119C2×C7×Q8—44no—
120C2×C7×Q8C2×D412yes—
121C2×C7×Q8#4114yes—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—22no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2411yescharacteristic
130C22×Dic7—21no—
131C22×Dic7—22no—
132C2×D28—21no—
133C2×D28—21no—
134C2×D28—22no—
135C2×D28—22no—
136C2×Dic14—21no—
137C2×Dic14—21no—
138C2×Dic14—22no—
139C2×Dic14—22no—
140C7×C4⋊C4—28no—
141C7×C4⋊C4—28no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—24no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4C2×D414yes—
150C7×C4⋊C4C2×D412yes—
151C42×C7—28no—
152C42×C7—24no—
153C42×C7—22no—
154C42×C7—22no—
155C42×C7C2×D412yes—
156C42×C7C2×D412yes—
157#1144—44no—
158#1143—44no—
159(C2×D7)⋊C4—44no—
160(C2×D7)⋊C4—42no—
161(C2×D7)⋊C4—42no—
162(C2×D7)⋊C4—82no—
163C4⋊Dic7—22no—
164C4⋊Dic7—22no—
165C4×Dic7—81no—
166C2×C7⋊C8—21no—
167C2×C7⋊C8—21no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C22×Q16—71no—
174C22×SD16—71no—
175#557—24no—
176#557—24no—
177C2×C4⋊Q8—21no—
178C2×C4⋊Q8C2×D711yescharacteristic
179C2×#125—22no—
180C2×#125C2×D712yes—
181C2×#124—71no—
182C2×C4×Q8—21no—
183C2×C4×Q8C2×D712yes—
184C2×C4×Q8C2×D711yescharacteristic
185C2×C4×Q8C2×D711yescharacteristic
186#489—144no—
187#464—144no—
188C2×C4⋊C8—71no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#103—71no—
192C2×#102—72no—
193C2×#102—71no—
194C2×#97—71no—
195#397—141no—
196#393—141no—
197#391—72no—
198#390—72no—
199#388—142no—
200C4×C4⋊C4—71no—
201#340—72no—
202#340—72no—
203#340—72no—
204#336—72no—
205#336—72no—
206C23×C7C22×D411yescharacteristic
207C22×D7—41no—
208C22×D7—42no—
209C22×D7—42no—
210C22×D7—41no—
211C22×D7—41no—
212C7×Q8—22no—
213C7×Q8—24no—
214C7×Q8—44no—
215C7×Q8—48no—
216C7×Q8#12414yes—
217C2×C4×C7—22no—
218C2×C4×C7—24no—
219C2×C4×C7—22no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—44no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7#12012yes—
234C2×C4×C7#12012yes—
235C2×C4×C7C22×D411yescharacteristic
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C2×#4112yes—
238C2×C4×C72−1+412yes—
239C2×Dic7—41no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244D28—41no—
245D28—41no—
246D28—42no—
247Dic14—41no—
248Dic14—41no—
249Dic14—42no—
250C7⋊C8—41no—
251C7⋊C8—41no—
252C7⋊C8—42no—
253C22×Q8—21no—
254C22×Q8—71no—
255C22×Q8C22⋊D711yescharacteristic
256C22×Q8C22×D711yescharacteristic
257C22×D4—71no—
258C2×Q16—144no—
259C2×Q16—142no—
260C2×SD16—144no—
261C2×SD16—142no—
262C22×C8—71no—
263C22×C8—71no—
264C22×C8—72no—
265C4⋊Q8—28no—
266C4⋊Q8—22no—
267C4⋊Q8C22⋊D714yes—
268#125—216no—
269#125—24no—
270#125—24no—
271#124—142no—
272#124—142no—
273C4×Q8—28no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8C22⋊D714yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4C22⋊D711yescharacteristic
287C2×C4⋊C4C22⋊D712yes—
288C2×C4⋊C4C22×D711yescharacteristic
289C2×C4⋊C4C22×D712yes—
290C2×C22⋊C4—141no—
291C2×C22⋊C4—141no—
292C2×C22⋊C4—142no—
293C2×C22⋊C4—142no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—141no—
297C2×C42C22×D712yes—
298C2×C42C22×D711yescharacteristic
299C4⋊C8—142no—
300#103—144no—
301#103—142no—
302#103—144no—
303#102—142no—
304#102—144no—
305#97—142no—
306#95—141no—
307#95—142no—
308#95—142no—
309C22×C7C2×#12011yescharacteristic
310C22×C7#54812yes—
311C22×C7#57512yes—
312C22×C7#57711yescharacteristic
313C22×C7#57811yescharacteristic
314C2×D7—82no—
315C2×D7—82no—
316C2×D7—81no—
317C2×D7—81no—
318C2×D7—81no—
319C4×C7—24no—
320C4×C7—22no—
321C4×C7—22no—
322C4×C7—42no—
323C4×C7—44no—
324C4×C7—42no—
325C4×C7—42no—
326C4×C7C2×#12411yescharacteristic
327C4×C7#53611yescharacteristic
328C4×C7#54812yes—
329Dic7—82no—
330Dic7—81no—
331C24—141no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—142no—
339C2×Q8—142no—
340C2×Q8—141no—
341C2×Q8—141no—
342C2×Q8#116114yes—
343C2×Q8C2×C22⋊D712yes—
344C2×D4—142no—
345C2×D4—142no—
346C2×D4—141no—
347C2×D4—141no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—141no—
352C22×C4—142no—
353C22×C4C2×C22⋊D712yes—
354C22×C4C2×C22⋊D712yes—
355C22×C4C2×C22⋊D711yescharacteristic
356C22×C4C2×C22⋊D711yescharacteristic
357C22×C4C23×D711yescharacteristic
358Q16—284no—
359SD16—284no—
360C2×C8—142no—
361C2×C8—142no—
362C2×C8—142no—
363C2×C8—142no—
364C2×C8—142no—
365C2×C8—141no—
366C2×C8—141no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—142no—
377C4⋊C4—142no—
378C4⋊C4C2×C22⋊D714yes—
379C4⋊C4C2×C22⋊D712yes—
380C22⋊C4—284no—
381C22⋊C4—282no—
382C22⋊C4—282no—
383C22⋊C4—562no—
384C42—28no—
385C42—24no—
386C42—22no—
387C42—22no—
388C42—561no—
389C42C2×C22⋊D712yes—
390C42C2×C22⋊D712yes—
391C2×C7#243611yescharacteristic
392C2×C7#300511yescharacteristic
393C2×C7#301311yescharacteristic
394C2×C7#337112yes—
395C2×C7#337712yes—
396D7—161no—
397C23—282no—
398C23—282no—
399C23—281no—
400C23—281no—
401C23—281no—
402C23C22×C22⋊D711yescharacteristic
403Q8—22no—
404Q8—24no—
405Q8—44no—
406Q8—48no—
407Q8—281no—
408Q8—281no—
409Q8—282no—
410Q8#651014yes—
411D4—281no—
412D4—281no—
413D4—282no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—281no—
435C2×C4#651612yes—
436C2×C4#651612yes—
437C2×C4C2×#116112yes—
438C2×C4#655112yes—
439C2×C4C22×C22⋊D711yescharacteristic
440C2×C4C22×C22⋊D711yescharacteristic
441C8—281no—
442C8—281no—
443C8—282no—
444C7#2594111yescharacteristic, complemented, Sylow
445C22—561no—
446C22—561no—
447C22—561no—
448C22—562no—
449C22—562no—
450C22#9050812yes—
451C22#9051312yes—
452C22#9052811yescharacteristic
453C22#9052911yescharacteristic
454C22C2×#651611yescharacteristic
455C4—24no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—44no—
460C4—42no—
461C4—42no—
462C4—561no—
463C4—562no—
464C4C2×#651011yescharacteristic
465C4#9050711yescharacteristic
466C4#9050812yes—
467C2—1121no—
468C2#1174113612yes—
469C2#1174114512yes—
470C2#1174151111yescharacteristic
471C2#1174152611yescharacteristic
472C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#557C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterC2×#557D7
hypocenterC7#25941
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
5. #11729533C2semidirect
6. #11729498C2semidirect
8. #11729389C2semidirect
10. #11729386C2semidirect
11. #11729385C2semidirect
102. C2×#557D7semidirect
444. C7#25941semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×#898612
#117399292
C2×#898002
#117295332
#117294982
#117294972
#117293892
#117293872
#117293862
#117293852
#259417