Finite Group Databasebeta

← #13370901

Subgroups — #13370901

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#11731324C211yesmaximal, characteristic
3#11731321C211yesmaximal, characteristic
4C7×#1807C211yesmaximal, characteristic
5#11729721C211yesmaximal, characteristic
6#11728342C212yesmaximal, complemented
7#11728324C212yesmaximal, complemented
8#11727279C211yesmaximal, characteristic
9#11727256C211yesmaximal, characteristic
10#11726608C212yesmaximal, complemented
11#11726537C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6450C2212yes—
16C2×C7×#106C2212yes—
17C7×#380C2211yescharacteristic
18C7×#380C2211yescharacteristic
19C4×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2211yescharacteristic
21C2×#6390C418yescomplemented
22C2×#6390C2211yescharacteristic
23C2×#6390C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×#6390C2211yescharacteristic
26C2×#6389C2212yes—
27C4⋊C4×Dic7C2212yes—
28#89746C2212yes—
29#89745C2212yes—
30C2×#6384C2212yes—
31C2×#6384C2212yes—
32#89714—21no—
33#89714—21no—
34C22⋊C4×Dic7—21no—
35C22⋊C4×Dic7—21no—
36#89425—22no—
37#89418—22no—
38C7×#340C2212yes—
39#89345C2212yes—
40#89324C2212yes—
41#89324C2212yes—
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×C8C2311yescharacteristic
56C22×C7×C8C2311yescharacteristic
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C22⋊Dic7D411yescharacteristic
62C2×C22⋊Dic7D411yescharacteristic
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7C2×C414yes—
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—22no—
70C2×C4×Dic7—22no—
71C2×C4×Dic7—22no—
72C2×C4×Dic7C2312yes—
73C22×C7⋊C8C2312yes—
74#6450—24no—
75#6450—42no—
76C7×#106—42no—
77C7×#95—21no—
78C7×#95—21no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95D412yes—
84C22⋊(C7⋊C8)—22no—
85#6390C2×C414yes—
86#6390C2×C414yes—
87#6390C2×C414yes—
88#6390C2×C414yes—
89#6389—42no—
90#6384—24no—
91#6384—42no—
92C22×#103—71no—
93#2345—71no—
94Q8×C22⋊C4—71no—
95#1935—72no—
96#1927—72no—
97#1865—72no—
98#1824—71no—
99#1807D711yescharacteristic
100#1787—71no—
101#1763—72no—
102#1752—71no—
103C24×C7C2×D411yescharacteristic
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7C22×C412yes—
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—44no—
119C22×Dic7—82no—
120C22×Dic7C2×D412yes—
121C22×Dic7C2×D412yes—
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—44no—
138C2×C7×C8C22×C412yes—
139C2×C7×C8C22×C412yes—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C22⋊C4—42no—
145C7×C22⋊C4—42no—
146C22⋊Dic7—22no—
147C22⋊Dic7—22no—
148C4⋊Dic7—24no—
149C4⋊Dic7—21no—
150C4⋊Dic7—21no—
151C4⋊Dic7C22×C412yes—
152C4⋊Dic7C22×C412yes—
153Dic7⋊C4—44no—
154Dic7⋊C4—82no—
155C4×Dic7—22no—
156C4×Dic7—22no—
157C4×Dic7—44no—
158C4×Dic7—82no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—22no—
162C23×Q8—71no—
163C23×C8C2×D711yescharacteristic
164C2×C4×Q8—72no—
165C22×C4⋊C4—71no—
166C2×#106C2×D712yes—
167C2×#103—78no—
168C2×#103—71no—
169C2×#103—71no—
170C2×#103—71no—
171C2×#103—71no—
172C2×#103—72no—
173C2×#103—72no—
174C2×C22⋊C8—71no—
175C2×#97—72no—
176#391—72no—
177#391—142no—
178#389—72no—
179#387—142no—
180#380—141no—
181#380—141no—
182#380C2×D711yescharacteristic
183#380C2×D711yescharacteristic
184C4×C4⋊C4—72no—
185C4×C22⋊C4—71no—
186C4×C22⋊C4—141no—
187C4×C22⋊C4—141no—
188#340—72no—
189#340—72no—
190#340C2×D712yes—
191#336—72no—
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—42no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
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—22no—
219C2×Dic7—22no—
220C2×Dic7—22no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—48no—
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×C8—72no—
251C22×C8C4×D714yes—
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—141no—
260C2×C4⋊C4—142no—
261C2×C4⋊C4—142no—
262C2×C4⋊C4—142no—
263C2×C4⋊C4C22×D712yes—
264C2×C4⋊C4C22×D712yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—71no—
268C2×C22⋊C4—71no—
269C2×C42—72no—
270C2×C42—142no—
271C2×C42—142no—
272#106—42no—
273#103—74no—
274#103—74no—
275#103—74no—
276#103—74no—
277#103—144no—
278#103—282no—
279C22⋊C8—142no—
280#97—282no—
281#95—21no—
282#95—21no—
283#95—72no—
284#95—142no—
285#95—142no—
286#95—142no—
287#95—142no—
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—82no—
300C4×C7—82no—
301C4×C7C2×#11711yescharacteristic
302C4×C7#51811yescharacteristic
303C4×C7#55112yes—
304Dic7—24no—
305Dic7—42no—
306Dic7—42no—
307Dic7—42no—
308Dic7—42no—
309Dic7—42no—
310C24D4×D711yescharacteristic
311C2×Q8—72no—
312C2×Q8—72no—
313C2×Q8—74no—
314C2×Q8—148no—
315C2×Q8—144no—
316C2×Q8—142no—
317C2×Q8—141no—
318C2×Q8—141no—
319C2×Q8—142no—
320C2×Q8—142no—
321C2×Q8—284no—
322C2×Q8—284no—
323C2×Q8—288no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—72no—
329C22×C4—72no—
330C22×C4—141no—
331C22×C4—144no—
332C22×C4—141no—
333C22×C4—141no—
334C22×C4—141no—
335C22×C4—144no—
336C22×C4—284no—
337C22×C4—562no—
338C22×C4C2×C4×D712yes—
339C22×C4D4×D711yescharacteristic
340C22×C4#116112yes—
341C22×C4C23×D711yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—44no—
345C2×C8—142no—
346C2×C8—142no—
347C2×C8—142no—
348C2×C8C2×C4×D712yes—
349C2×C8C2×C4×D712yes—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—72no—
355C4⋊C4—72no—
356C4⋊C4—144no—
357C4⋊C4—141no—
358C4⋊C4—141no—
359C4⋊C4—284no—
360C4⋊C4—562no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—142no—
364C22⋊C4—142no—
365C42—142no—
366C42—142no—
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—
377C23#645612yes—
378C23#647812yes—
379C23#647812yes—
380C23C2×D4×D711yescharacteristic
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—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—142no—
399C2×C4—144no—
400C2×C4—142no—
401C2×C4—142no—
402C2×C4—142no—
403C2×C4—142no—
404C2×C4—142no—
405C2×C4—142no—
406C2×C4—142no—
407C2×C4—142no—
408C2×C4—142no—
409C2×C4—288no—
410C2×C4—282no—
411C2×C4—282no—
412C2×C4—282no—
413C2×C4—564no—
414C2×C4#645612yes—
415C2×C4C22×C4×D711yescharacteristic
416C2×C4#654812yes—
417C2×C4C2×#116111yescharacteristic
418C2×C4D7×#4112yes—
419C8—22no—
420C8—22no—
421C8—282no—
422C7#2256711yescharacteristic, complemented, Sylow
423C22—24no—
424C22—24no—
425C22—22no—
426C22—22no—
427C22#8962514yes—
428C22C2×#645611yescharacteristic
429C22#9030112yes—
430C22C2×#647811yescharacteristic
431C22C2×#647811yescharacteristic
432C22#9046112yes—
433C4—22no—
434C4—82no—
435C4—82no—
436C4—144no—
437C4—282no—
438C4—282no—
439C4—282no—
440C4—282no—
441C4—282no—
442C4#9017711yescharacteristic
443C4D7×#11711yescharacteristic
444C4#9032812yes—
445C2—24no—
446C2#1173145411yescharacteristic
447C2C2×#8962511yescharacteristic
448C2#1173661311yescharacteristic
449C2#1173813712yes—
450C2#1173813912yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1807C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1807D7
hypocenterC7#22567
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
6. #11728342C2semidirect
7. #11728324C2semidirect
10. #11726608C2semidirect
11. #11726537C2semidirect
21. 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
#117313242
#117313212
C7×#18072
#117297212
#117283422
#117283242
#117272792
#117272562
#117266082
#117265372
#225677