Finite Group Databasebeta

← #13370907

Subgroups — #13370907

Order 1792.

962 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6390C211yesmaximal, characteristic
2#11730902C211yesmaximal, characteristic
3C22⋊C4×Dic14C211yesmaximal, characteristic
4C7×#1752C211yesmaximal, characteristic
5#11729724C211yesmaximal, characteristic
6#11727717C211yesmaximal, characteristic
7#11727679C211yesmaximal, characteristic
8#11727571C212yesmaximal, complemented
9#11727567C212yesmaximal, complemented
10#11726438C212yesmaximal, complemented
11#11726381C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×C4×Dic14C2212yes—
16C2×C7×C22⋊C8C2211yescharacteristic
17C2×C7×#97C2212yes—
18C4×C7×C22⋊C4C2211yescharacteristic
19#89984C2211yescharacteristic
20#89984C2211yescharacteristic
21C2×#6391C2212yes—
22C2×#6390C418yescomplemented
23C2×#6390C2212yes—
24C2×#6390C2212yes—
25C2×#6390C2211yescharacteristic
26C2×#6390C2211yescharacteristic
27C2×#6390C2211yescharacteristic
28C2×#6390C2211yescharacteristic
29#89714—21no—
30#89714—21no—
31C22⋊C4×Dic7—21no—
32C22⋊C4×Dic7—21no—
33C4×C4⋊Dic7C2212yes—
34#89701C2212yes—
35#89700C2212yes—
36#89426—22no—
37#89418—22no—
38C7×#336C2212yes—
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#22567—71nomaximal, Sylow
43C23×C4×C7C2311yescharacteristic
44C23×Dic7—21no—
45C23×Dic7—21no—
46C22×Dic14—24no—
47C22×Dic14—21no—
48C22×Dic14—22no—
49C22×Dic14—42no—
50C22×Dic14C2×C414yes—
51C22×Dic14C2311yescharacteristic
52C22×Dic14C2311yescharacteristic
53C22×C7×C8—42no—
54C22×C7×C8C2×C414yes—
55C22×C7×C8C2312yes—
56C22×C7×C8C2311yescharacteristic
57C22×C7×C8C2311yescharacteristic
58C2×C7×C22⋊C4D411yescharacteristic
59C2×C7×C22⋊C4D411yescharacteristic
60C2×C42×C7C2312yes—
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7C2×C414yes—
66C2×C4⋊Dic7C2312yes—
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2311yescharacteristic
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—22no—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C4×Dic14—24no—
75C4×Dic14—42no—
76C7×C22⋊C8—22no—
77C7×#97—42no—
78C7×#95D412yes—
79C7⋊#95—21no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85#6391—42no—
86#6390—24no—
87#6390—42no—
88#6390C2×C414yes—
89#6390C2×C414yes—
90#6390C2×C414yes—
91#6390C2×C414yes—
92C22×#103—71no—
93#2345—71no—
94Q8×C22⋊C4—71no—
95#1935—72no—
96#1927—72no—
97#1865—72no—
98#1824—71no—
99#1807—71no—
100#1787—71no—
101#1763—72no—
102#1752D711yescharacteristic
103C24×C7C2×D411yescharacteristic
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7C22×C412yes—
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4112yes—
111C22×C4×C7C2411yescharacteristic
112C22×Dic7—21no—
113C22×Dic7—24no—
114C22×Dic7—21no—
115C22×Dic7—21no—
116C22×Dic7—21no—
117C22×Dic7—24no—
118C22×Dic7—22no—
119C22×Dic7—22no—
120C22×Dic7—44no—
121C22×Dic7—82no—
122C2×Dic14—21no—
123C2×Dic14—22no—
124C2×Dic14—22no—
125C2×Dic14—22no—
126C2×Dic14—28no—
127C2×Dic14—24no—
128C2×Dic14—21no—
129C2×Dic14—44no—
130C2×Dic14—48no—
131C2×Dic14—44no—
132C2×Dic14C22×C412yes—
133C2×Dic14C22×C412yes—
134C2×Dic14#4114yes—
135C2×C7×C8—22no—
136C2×C7×C8—22no—
137C2×C7×C8—22no—
138C2×C7×C8—22no—
139C2×C7×C8—22no—
140C2×C7×C8—44no—
141C2×C7×C8C22×C412yes—
142C2×C7×C8C22×C412yes—
143C7×C22⋊C4—22no—
144C7×C22⋊C4—22no—
145C42×C7—22no—
146C42×C7—22no—
147C22⋊Dic7—42no—
148C22⋊Dic7—42no—
149C4⋊Dic7—24no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7C22×C412yes—
157C4⋊Dic7C22×C412yes—
158Dic7⋊C4—44no—
159Dic7⋊C4—82no—
160C4×Dic7—44no—
161C4×Dic7—82no—
162C23×Q8—71no—
163C23×C8C2×D711yescharacteristic
164C2×C4×Q8—72no—
165C22×C4⋊C4—71no—
166C2×#106—72no—
167C2×#103—78no—
168C2×#103—71no—
169C2×#103—71no—
170C2×#103—71no—
171C2×#103—71no—
172C2×#103—72no—
173C2×#103—72no—
174C2×C22⋊C8C2×D711yescharacteristic
175C2×#97C2×D712yes—
176#391—72no—
177#391—142no—
178#389—72no—
179#387—142no—
180#380—71no—
181#380—71no—
182#380—141no—
183#380—141no—
184C4×C4⋊C4—72no—
185C4×C22⋊C4—141no—
186C4×C22⋊C4—141no—
187C4×C22⋊C4C2×D711yescharacteristic
188#340—72no—
189#340—72no—
190#340—72no—
191#336C2×D712yes—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7C4×D412yes—
195C23×C7#13712yes—
196C23×C7#13712yes—
197C23×C7C22×D411yescharacteristic
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—22no—
201C2×C4×C7—24no—
202C2×C4×C7—22no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7C4×D412yes—
209C2×C4×C7C23×C411yescharacteristic
210C2×C4×C7C2×#4111yescharacteristic
211C2×C4×C7C2×#4112yes—
212C2×C4×C7H(5,𝔽2)12yes—
213C2×Dic7—22no—
214C2×Dic7—24no—
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—22no—
218C2×Dic7—48no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7—42no—
223C2×Dic7—42no—
224C2×Dic7—42no—
225C2×Dic7—42no—
226C2×Dic7—42no—
227C2×Dic7—42no—
228C2×Dic7—84no—
229Dic14—22no—
230Dic14—24no—
231Dic14—22no—
232Dic14—22no—
233Dic14—48no—
234Dic14—42no—
235Dic14#11714yes—
236C7×C8—22no—
237C7×C8—22no—
238C7×C8—42no—
239C22×Q8—74no—
240C22×Q8—71no—
241C22×Q8—71no—
242C22×Q8—144no—
243C22×Q8—141no—
244C22×Q8—142no—
245C22×Q8—282no—
246C23×C4—141no—
247C23×C4—141no—
248C23×C4C22×D711yescharacteristic
249C22×C8—42no—
250C22×C8C4×D714yes—
251C22×C8C22×D712yes—
252C22×C8C22×D711yescharacteristic
253C22×C8C22×D711yescharacteristic
254C4×Q8—144no—
255C4×Q8—282no—
256C2×C4⋊C4—74no—
257C2×C4⋊C4—71no—
258C2×C4⋊C4—71no—
259C2×C4⋊C4—72no—
260C2×C4⋊C4—72no—
261C2×C4⋊C4—141no—
262C2×C4⋊C4—142no—
263C2×C4⋊C4—142no—
264C2×C4⋊C4—142no—
265C2×C22⋊C4—141no—
266C2×C22⋊C4—141no—
267C2×C22⋊C4D2811yescharacteristic
268C2×C22⋊C4D2811yescharacteristic
269C2×C42—142no—
270C2×C42—142no—
271C2×C42C22×D712yes—
272#106—282no—
273#103—74no—
274#103—74no—
275#103—74no—
276#103—74no—
277#103—144no—
278#103—282no—
279C22⋊C8—22no—
280#97—42no—
281#95—142no—
282#95—142no—
283#95—142no—
284#95—141no—
285#95—141no—
286#95—142no—
287#95D2812yes—
288C22×C7—22no—
289C22×C7—22no—
290C22×C7—24no—
291C22×C7—24no—
292C22×C7#44114yes—
293C22×C7C2×C4×D411yescharacteristic
294C22×C7#54612yes—
295C22×C7C2×#13711yescharacteristic
296C22×C7C2×#13711yescharacteristic
297C22×C7#57712yes—
298C4×C7—22no—
299C4×C7—42no—
300C4×C7—42no—
301C4×C7C2×#11711yescharacteristic
302C4×C7#51811yescharacteristic
303C4×C7#55112yes—
304Dic7—24no—
305Dic7—42no—
306Dic7—42no—
307Dic7—42no—
308Dic7—82no—
309Dic7—82no—
310C24C2×D2811yescharacteristic
311C2×Q8—72no—
312C2×Q8—72no—
313C2×Q8—74no—
314C2×Q8—148no—
315C2×Q8—144no—
316C2×Q8—141no—
317C2×Q8—141no—
318C2×Q8—142no—
319C2×Q8—142no—
320C2×Q8—142no—
321C2×Q8—284no—
322C2×Q8—284no—
323C2×Q8—288no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—141no—
327C22×C4—144no—
328C22×C4—141no—
329C22×C4—141no—
330C22×C4—141no—
331C22×C4—144no—
332C22×C4—142no—
333C22×C4—142no—
334C22×C4—284no—
335C22×C4—562no—
336C22×C4C2×C4×D712yes—
337C22×C4C2×D2812yes—
338C22×C4C2×D2812yes—
339C22×C4C2×D2811yescharacteristic
340C22×C4#115712yes—
341C22×C4C23×D711yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—44no—
348C2×C8C2×C4×D712yes—
349C2×C8C2×C4×D712yes—
350C4⋊C4—72no—
351C4⋊C4—72no—
352C4⋊C4—144no—
353C4⋊C4—141no—
354C4⋊C4—141no—
355C4⋊C4—142no—
356C4⋊C4—142no—
357C4⋊C4—142no—
358C4⋊C4—142no—
359C4⋊C4—284no—
360C4⋊C4—562no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—282no—
364C22⋊C4—282no—
365C42—22no—
366C42—22no—
367C42—284no—
368C42—562no—
369C2×C7—24no—
370C2×C7#234211yescharacteristic
371C2×C7C2×#44111yescharacteristic
372C2×C7#294911yescharacteristic
373C2×C7#331612yes—
374C2×C7#332012yes—
375C23—22no—
376C23—22no—
377C23C4×D2812yes—
378C23#647212yes—
379C23#647212yes—
380C23C22×D2811yescharacteristic
381Q8—74no—
382Q8—142no—
383Q8—144no—
384Q8—142no—
385Q8—142no—
386Q8—288no—
387Q8—282no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—142no—
399C2×C4—144no—
400C2×C4—142no—
401C2×C4—142no—
402C2×C4—142no—
403C2×C4—288no—
404C2×C4—282no—
405C2×C4—282no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4—282no—
409C2×C4—282no—
410C2×C4—282no—
411C2×C4—282no—
412C2×C4—282no—
413C2×C4—564no—
414C2×C4C4×D2812yes—
415C2×C4C22×C4×D711yescharacteristic
416C2×C4C2×#115711yescharacteristic
417C2×C4#654812yes—
418C2×C4D7×#4112yes—
419C8—22no—
420C8—22no—
421C8—42no—
422C7#2256711yescharacteristic, complemented, Sylow
423C22—22no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22#8948814yes—
428C22C2×C4×D2811yescharacteristic
429C22#9024512yes—
430C22C2×#647211yescharacteristic
431C22C2×#647211yescharacteristic
432C22#9044212yes—
433C4—22no—
434C4—42no—
435C4—42no—
436C4—144no—
437C4—282no—
438C4—282no—
439C4—282no—
440C4—562no—
441C4—562no—
442C4#9017711yescharacteristic
443C4D7×#11711yescharacteristic
444C4#9024712yes—
445C2—24no—
446C2#1173102011yescharacteristic
447C2C2×#8948811yescharacteristic
448C2#1173601711yescharacteristic
449C2#1173695812yes—
450C2#1173696012yes—
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1752C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1752D7
hypocenterC7#22567
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
8. #11727571C2semidirect
9. #11727567C2semidirect
10. #11726438C2semidirect
11. #11726381C2semidirect
22. C2×#6390C4semidirect
422. C7#22567semidirect

Maximal subgroups

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

subgroupindex
C22×#63902
#117309022
C22⋊C4×Dic142
C7×#17522
#117297242
#117277172
#117276792
#117275712
#117275672
#117264382
#117263812
#225677