Finite Group Databasebeta

← #13370945

Subgroups — #13370945

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6390C211yesmaximal, characteristic
2#11731323C211yesmaximal, characteristic
3#11731321C211yesmaximal, characteristic
4C7×#1807C211yesmaximal, characteristic
5C8×C22⋊Dic7C211yesmaximal, characteristic
6#11728342C212yesmaximal, complemented
7#11728324C212yesmaximal, complemented
8#11727279C211yesmaximal, characteristic
9#11727255C211yesmaximal, characteristic
10#11726604C212yesmaximal, complemented
11#11726535C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6450C2212yes—
16C2×#6440—21no—
17C2×C7×#106C2212yes—
18C7×#380C2211yescharacteristic
19C7×#380C2211yescharacteristic
20C4×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊(C7⋊C8)C2211yescharacteristic
22C2×#6390C418yescomplemented
23C2×#6390C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×#6390C2211yescharacteristic
26C2×#6390C2211yescharacteristic
27C2×C8×Dic7C2212yes—
28C4⋊C4×Dic7C2212yes—
29#89746C2212yes—
30#89745C2212yes—
31C2×#6384C2212yes—
32C2×#6384C2212yes—
33#89715—21no—
34#89714—21no—
35C22⋊C4×Dic7—21no—
36#89426—22no—
37#89418—22no—
38C7×#340C2212yes—
39#89345C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#22571—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—
76#6440—44no—
77C7×#106—42no—
78C7×#95—21no—
79C7×#95—21no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C7⋊#95D412yes—
85C22⋊(C7⋊C8)—22no—
86#6390C2×C414yes—
87#6390C2×C414yes—
88#6390C2×C414yes—
89#6390C2×C414yes—
90C8×Dic7—42no—
91#6384—24no—
92#6384—42no—
93C22×#103—71no—
94#2344—71no—
95Q8×C22⋊C4—71no—
96#1933—72no—
97#1927—72no—
98#1865—72no—
99#1823—71no—
100#1807D711yescharacteristic
101#1787—71no—
102C4×#103—72no—
103C8×C22⋊C4—71no—
104C24×C7C2×D411yescharacteristic
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7C22×C412yes—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7#4112yes—
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—21no—
114C22×Dic7—24no—
115C22×Dic7—24no—
116C22×Dic7—21no—
117C22×Dic7—21no—
118C22×Dic7—21no—
119C22×Dic7—44no—
120C22×Dic7—42no—
121C22×Dic7—42no—
122C22×Dic7C2×D412yes—
123C22×Dic7C2×D412yes—
124C2×Dic14—21no—
125C2×Dic14—22no—
126C2×Dic14—22no—
127C2×Dic14—22no—
128C2×Dic14—28no—
129C2×Dic14—24no—
130C2×Dic14—21no—
131C2×Dic14—44no—
132C2×Dic14—48no—
133C2×Dic14—44no—
134C2×Dic14C22×C412yes—
135C2×Dic14C22×C412yes—
136C2×Dic14#4114yes—
137C2×C7×C8—22no—
138C2×C7×C8—22no—
139C2×C7×C8—44no—
140C2×C7×C8C22×C412yes—
141C2×C7×C8C22×C412yes—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C22⋊C4—42no—
147C7×C22⋊C4—42no—
148C22⋊Dic7—22no—
149C22⋊Dic7SD1614yes—
150C4⋊Dic7—24no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C4⋊Dic7C22×C412yes—
154C4⋊Dic7C22×C412yes—
155Dic7⋊C4—44no—
156Dic7⋊C4—44no—
157C4×Dic7—22no—
158C4×Dic7—22no—
159C4×Dic7—44no—
160C4×Dic7—44no—
161C2×C7⋊C8—22no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—22no—
164C23×Q8—71no—
165C23×C8C2×D711yescharacteristic
166C2×C4×Q8—72no—
167C2×#117—141no—
168C22×C4⋊C4—71no—
169C2×#106C2×D712yes—
170C2×#103—78no—
171C2×#103—71no—
172C2×#103—71no—
173C2×#103—71no—
174C2×#103—71no—
175C2×#103—72no—
176C2×#103—72no—
177C2×C22⋊C8—71no—
178C2×C4×C8—72no—
179#391—72no—
180#391—142no—
181#389—72no—
182#387—142no—
183#381—141no—
184#380—141no—
185#380C2×D711yescharacteristic
186#380C2×D711yescharacteristic
187C4×C4⋊C4—72no—
188C4×C22⋊C4—71no—
189C4×C22⋊C4—141no—
190#340—72no—
191#340—72no—
192#340C2×D712yes—
193#336—72no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7C4×D412yes—
197C23×C7C2×SD1612yes—
198C23×C7#13512yes—
199C23×C7C22×D411yescharacteristic
200C2×C4×C7—22no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—24no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7C4×D412yes—
211C2×C4×C7C23×C411yescharacteristic
212C2×C4×C7C2×#4111yescharacteristic
213C2×C4×C7C2×#4112yes—
214C2×C4×C7H(5,𝔽2)12yes—
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—24no—
218C2×Dic7—22no—
219C2×Dic7—22no—
220C2×Dic7—22no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—48no—
226C2×Dic7—42no—
227C2×Dic7—41no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—48no—
232C2×Dic7C2×SD1614yes—
233Dic14—22no—
234Dic14—24no—
235Dic14—22no—
236Dic14—22no—
237Dic14—48no—
238Dic14—42no—
239Dic14#11714yescomplemented
240C7×C8—22no—
241C7×C8—22no—
242C7⋊C8—42no—
243C22×Q8—74no—
244C22×Q8—71no—
245C22×Q8—71no—
246C22×Q8—144no—
247C22×Q8—141no—
248C22×Q8—142no—
249C22×Q8—282no—
250C23×C4—141no—
251C23×C4—141no—
252C23×C4C22×D711yescharacteristic
253C22×C8—42no—
254C22×C8—72no—
255C22×C8C4×D714yes—
256C22×C8C22×D711yescharacteristic
257C22×C8C22×D711yescharacteristic
258C4×Q8—144no—
259C4×Q8—282no—
260#117—284no—
261C2×C4⋊C4—74no—
262C2×C4⋊C4—71no—
263C2×C4⋊C4—71no—
264C2×C4⋊C4—141no—
265C2×C4⋊C4—142no—
266C2×C4⋊C4—142no—
267C2×C4⋊C4—142no—
268C2×C4⋊C4C22×D712yes—
269C2×C4⋊C4C22×D712yes—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—71no—
273C2×C22⋊C4—71no—
274C2×C42—72no—
275C2×C42—142no—
276C2×C42—142no—
277#106—42no—
278#103—74no—
279#103—74no—
280#103—74no—
281#103—74no—
282#103—144no—
283#103—282no—
284C22⋊C8—142no—
285C4×C8—282no—
286#95—21no—
287#95—21no—
288#95—72no—
289#95—142no—
290#95—142no—
291#95—142no—
292#95—142no—
293C22×C7—22no—
294C22×C7—22no—
295C22×C7—24no—
296C22×C7—24no—
297C22×C7C4×SD1614yes—
298C22×C7C2×C4×D411yescharacteristic
299C22×C7#54612yes—
300C22×C7C22×SD1611yescharacteristic
301C22×C7C2×#13511yescharacteristic
302C22×C7#57812yes—
303C4×C7—22no—
304C4×C7—82no—
305C4×C7—82no—
306C4×C7C2×#11711yescharacteristic
307C4×C7#51811yescharacteristic
308C4×C7#55112yes—
309Dic7—24no—
310Dic7—24no—
311Dic7—42no—
312Dic7—42no—
313Dic7—42no—
314Dic7—42no—
315C24D4×D711yescharacteristic
316C2×Q8—72no—
317C2×Q8—72no—
318C2×Q8—74no—
319C2×Q8—148no—
320C2×Q8—144no—
321C2×Q8—142no—
322C2×Q8—141no—
323C2×Q8—141no—
324C2×Q8—142no—
325C2×Q8—142no—
326C2×Q8—284no—
327C2×Q8—284no—
328C2×Q8—288no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—72no—
334C22×C4—72no—
335C22×C4—141no—
336C22×C4—144no—
337C22×C4—144no—
338C22×C4—141no—
339C22×C4—141no—
340C22×C4—141no—
341C22×C4—284no—
342C22×C4—282no—
343C22×C4—282no—
344C22×C4C2×C4×D712yes—
345C22×C4D4×D711yescharacteristic
346C22×C4#116112yes—
347C22×C4C23×D711yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—44no—
351C2×C8—142no—
352C2×C8—142no—
353C2×C8—142no—
354C2×C8C2×C4×D712yes—
355C2×C8C2×C4×D712yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—72no—
361C4⋊C4—72no—
362C4⋊C4—144no—
363C4⋊C4—141no—
364C4⋊C4—141no—
365C4⋊C4—284no—
366C4⋊C4—284no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—74no—
370C22⋊C4—142no—
371C42—142no—
372C42—142no—
373C42—284no—
374C42—284no—
375C2×C7—24no—
376C2×C7#234211yescharacteristic
377C2×C7C2×C4×SD1611yescharacteristic
378C2×C7#295011yescharacteristic
379C2×C7#329812yes—
380C2×C7#330012yes—
381C23—22no—
382C23—22no—
383C23#645612yes—
384C23D7×SD1612yes—
385C23#647912yes—
386C23C2×D4×D711yescharacteristic
387Q8—74no—
388Q8—142no—
389Q8—144no—
390Q8—142no—
391Q8—142no—
392Q8—288no—
393Q8—282no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—74no—
405C2×C4—142no—
406C2×C4—142no—
407C2×C4—142no—
408C2×C4—144no—
409C2×C4—142no—
410C2×C4—142no—
411C2×C4—142no—
412C2×C4—142no—
413C2×C4—142no—
414C2×C4—142no—
415C2×C4—288no—
416C2×C4—288no—
417C2×C4—281no—
418C2×C4—281no—
419C2×C4—281no—
420C2×C4—281no—
421C2×C4—282no—
422C2×C4#645612yes—
423C2×C4C22×C4×D711yescharacteristic
424C2×C4#654812yes—
425C2×C4C2×#116111yescharacteristic
426C2×C4D7×#4112yes—
427C8—22no—
428C8—22no—
429C8—282no—
430C7#2257111yescharacteristic, complemented, Sylow
431C22—24no—
432C22—24no—
433C22—22no—
434C22—22no—
435C22#8962414yes—
436C22C2×#645611yescharacteristic
437C22#9030112yes—
438C22C2×D7×SD1611yescharacteristic
439C22C2×#647911yescharacteristic
440C22#9046212yes—
441C4—22no—
442C4—82no—
443C4—82no—
444C4—144no—
445C4—144no—
446C4—282no—
447C4—282no—
448C4—282no—
449C4—282no—
450C4#9017711yescharacteristic
451C4D7×#11711yescharacteristic
452C4#9032812yes—
453C2—24no—
454C2#1173145411yescharacteristic
455C2C2×#8962411yescharacteristic
456C2#1173661411yescharacteristic
457C2#1173813412yes—
458C2#1173813512yes—
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#22571
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. #11726604C2semidirect
11. #11726535C2semidirect
22. C2×#6390C4semidirect
239. Dic14#117semidirect
430. C7#22571semidirect

Maximal subgroups

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

subgroupindex
C22×#63902
#117313232
#117313212
C7×#18072
C8×C22⋊Dic72
#117283422
#117283242
#117272792
#117272552
#117266042
#117265352
#225717