Finite Group Databasebeta

← #13364972

Subgroups — #13364972

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×#90002C211yesmaximal, characteristic
2C2×#90000C211yesmaximal, characteristic
3#11731311C211yesmaximal, characteristic
4C7×#1787C211yesmaximal, characteristic
5#11728035C212yesmaximal, complemented
6#11728012C212yesmaximal, complemented
7#11728008C212yesmaximal, complemented
8#11727996C212yesmaximal, complemented
9#11727684C211yesmaximal, characteristic
10Dic7×C22⋊C8C211yesmaximal, characteristic
11#11727258C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×#1144C2212yes—
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6487C2211yescharacteristic
16C22×C4×Dic7C2211yescharacteristic
17C2×#6450C2211yescharacteristic
18C2×C7×#103C2212yes—
19C2×C7×#103C2212yes—
20C2×C7×C22⋊C8C2211yescharacteristic
21C7×#380C2211yescharacteristic
22#90002—21no—
23#90002—21no—
24#90002C2212yes—
25#90002C2212yes—
26#90000C418yescomplemented
27C2×#6409C2212yes—
28C2×C7⋊#95—21no—
29C2×C7⋊#95—21no—
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×#6390C2212yes—
32C2×C8×Dic7C2212yes—
33C4⋊C4×Dic7C2211yescharacteristic
34#89746C2211yescharacteristic
35#89745C2211yescharacteristic
36C2×#6381C2212yes—
37#89718—21no—
38C22⋊C4×Dic7—21no—
39#89424—21no—
40#89421—21no—
41#89346C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22044—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yes—
50C22×C7×Q8C2312yes—
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C23×Dic7—21no—
54C23×Dic7D411yescharacteristic
55C23×Dic7D411yescharacteristic
56C22×Dic14C2311yescharacteristic
57C22×C7×C8C2312yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C22⋊Dic7—21no—
62C2×#1144—42no—
63C2×#1144—42no—
64C2×#1144C2×C418yescomplemented
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—42no—
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69#6487—22no—
70#6487C2×C414yes—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—42no—
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C22×C7⋊C8C2312yes—
82#6450—42no—
83C7×#103—42no—
84C7×#103—42no—
85C7×C22⋊C8—22no—
86C7×#95—21no—
87#6409C2×C418yescomplemented
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C22⋊(C7⋊C8)C2×C414yes—
97#6390—42no—
98C8×Dic7—42no—
99#6381—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#1852—71no—
109#1787D711yescharacteristic
110C4×C22⋊C8—71no—
111C24×C7C2×D411yescharacteristic
112C2×C7×Q8—24no—
113C2×C7×Q8—22no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—48no—
119C2×C7×Q8—42no—
120C2×C7×Q8—42no—
121C2×C7×Q8—48no—
122C2×C7×Q8C22×C414yes—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—24no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7C22×C412yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7#4112yes—
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—22no—
136C22×Dic7—21no—
137C22×Dic7—21no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—41no—
141C22×Dic7—41no—
142C22×Dic7—41no—
143C22×Dic7—42no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C22×Dic7Q1614yescomplemented
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C2×Dic14—22no—
152C2×Dic14—22no—
153C2×Dic14—21no—
154C2×Dic14C22×C412yes—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C22⋊C4—42no—
161C22⋊Dic7—24no—
162#1144—48no—
163C4⋊Dic7—21no—
164C4⋊Dic7—22no—
165C4⋊Dic7—22no—
166C4⋊Dic7—42no—
167C4⋊Dic7C22×C412yes—
168Dic7⋊C4—24no—
169Dic7⋊C4—42no—
170Dic7⋊C4—42no—
171C4×Dic7—22no—
172C4×Dic7—22no—
173C4×Dic7—41no—
174C4×Dic7—41no—
175C4×Dic7—42no—
176C4×Dic7—42no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8C22×C414yes—
180C23×Q8C2×D711yescharacteristic
181C22×Q16—72no—
182C2×C22⋊Q8—71no—
183C2×C4×Q8—71no—
184C22×C4⋊C4—71no—
185C22×C42—71no—
186#451—78no—
187C2×#107—72no—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103C2×D712yes—
191C2×#103C2×D712yes—
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#340—72no—
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
220C7×Q8—24no—
221C7×Q8—24no—
222C7×Q8—48no—
223C7×Q8—42no—
224C7×Q8—42no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—48no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7C4×D412yes—
240C2×C4×C7C23×C411yescharacteristic
241C2×C4×C7C2×#4111yescharacteristic
242C2×C4×C7H(5,𝔽2)12yes—
243C2×C4×C7H(5,𝔽2)12yes—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—21no—
253C2×Dic7—21no—
254C2×Dic7—21no—
255C2×Dic7—44no—
256C2×Dic7—44no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—84no—
265C2×Dic7C2×Q1612yes—
266C2×Dic7C2×Q1612yes—
267C2×Dic7#13712yes—
268Dic14—24no—
269Dic14—42no—
270C7×C8—42no—
271C7⋊C8—24no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—42no—
275C22×Q8—42no—
276C22×Q8—71no—
277C22×Q8C4×D714yes—
278C22×Q8C22×D712yes—
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—71no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—141no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—282no—
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—282no—
311#103—42no—
312#103—42no—
313#103—78no—
314#103—282no—
315C22⋊C8—22no—
316C22⋊C8—74no—
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—44no—
343C4×C7—81no—
344C4×C7—81no—
345C4×C7#51812yes—
346C4×C7#53812yes—
347Dic7—22no—
348Dic7—22no—
349Dic7—22no—
350Dic7—42no—
351Dic7—41no—
352Dic7—42no—
353Dic7—82no—
354C24D4×D711yescharacteristic
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—48no—
365C2×Q8—72no—
366C2×Q8—141no—
367C2×Q8—142no—
368C2×Q8—142no—
369C2×Q8C2×C4×D714yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
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—142no—
385C22×C4—141no—
386C22×C4—141no—
387C22×C4—142no—
388C22×C4—281no—
389C22×C4—281no—
390C22×C4—281no—
391C22×C4—282no—
392C22×C4—282no—
393C22×C4—282no—
394C22×C4C2×C4×D712yes—
395C22×C4D4×D711yescharacteristic
396C22×C4#115912yes—
397C22×C4C23×D711yescharacteristic
398Q16—288no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—74no—
403C2×C8—142no—
404C2×C8—142no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—72no—
408C4⋊C4—144no—
409C4⋊C4—141no—
410C4⋊C4—142no—
411C4⋊C4—142no—
412C4⋊C4—282no—
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(C22⋊D7)⋊C412yes—
436C23#647912yes—
437C23D7×Q1612yes—
438C23C2×D4×D711yescharacteristic
439Q8—24no—
440Q8—24no—
441Q8—48no—
442Q8—42no—
443Q8—42no—
444Q8—144no—
445Q8—282no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—72no—
461C2×C4—72no—
462C2×C4—72no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—141no—
468C2×C4—141no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—141no—
472C2×C4—141no—
473C2×C4—141no—
474C2×C4—284no—
475C2×C4—284no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4—281no—
480C2×C4—282no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4—564no—
484C2×C4(C22⋊D7)⋊C412yes—
485C2×C4C22×C4×D711yescharacteristic
486C2×C4C2×#115911yescharacteristic
487C2×C4#654812yes—
488C2×C4#654812yes—
489C8—42no—
490C8—144no—
491C7#2204411yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22#8956214yes—
499C22C2×(C22⋊D7)⋊C411yescharacteristic
500C22#9028011yescharacteristic
501C22#9028311yescharacteristic
502C22C2×#647911yescharacteristic
503C22C2×D7×Q1611yescharacteristic
504C22D7×#13711yescharacteristic
505C22#9046811yescharacteristic
506C4—22no—
507C4—44no—
508C4—81no—
509C4—81no—
510C4—142no—
511C4—142no—
512C4—142no—
513C4—282no—
514C4—282no—
515C4—281no—
516C4—562no—
517C4#9017712yes—
518C4#9028412yes—
519C2—24no—
520C2#1173126611yescharacteristic
521C2C2×#8956211yescharacteristic
522C2#1173639111yescharacteristic
523C2#1173748911yescharacteristic
524C2#1173750211yescharacteristic
525C2#1173762311yescharacteristic
526C2#1173763211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1787C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1787D7
hypocenterC7#22044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
5. #11728035C2semidirect
6. #11728012C2semidirect
7. #11728008C2semidirect
8. #11727996C2semidirect
26. #90000C4semidirect
64. C2×#1144C2×C4semidirect
87. #6409C2×C4semidirect
146. 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×#900022
C2×#900002
#117313112
C7×#17872
#117280352
#117280122
#117280082
#117279962
#117276842
Dic7×C22⋊C82
#117272582
#220447