Finite Group Databasebeta

← #24917656

Subgroups — #24917656

Order 10624.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128181C211yesmaximal, characteristic, complemented
2#19128017C211yesmaximal, characteristic, complemented
3#19128012C211yesmaximal, characteristic, complemented
4#19128005C211yesmaximal, characteristic, complemented
5#19127998C211yesmaximal, characteristic, complemented
6#19127898C211yesmaximal, characteristic, complemented
7#19127893C211yesmaximal, characteristic, complemented
8#19127844C211yesmaximal, characteristic, complemented
9#19127844C211yesmaximal, characteristic, complemented
10#19127811C211yesmaximal, characteristic, complemented
11#19127801C211yesmaximal, characteristic, complemented
12C4×#13050578C211yesmaximal, characteristic, complemented
13C4×#13050575C211yesmaximal, characteristic, complemented
14#19127762C211yesmaximal, characteristic, complemented
15#540×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C2×#13050578C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22C4×Dic166C2211yescharacteristic
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25(C22⋊D83)⋊C4—21no—
26(C22⋊D83)⋊C4—21no—
27#14581750—21no—
28C2×C4⋊Dic83C2211yescharacteristic
29C4⋊C4×D83—21no—
30#14581725C2211yescharacteristic
31#14581725C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35C4⋊(C83⋊C8)C2211yescharacteristic
36#14581715C2211yescharacteristic
37#14581715C2211yescharacteristic
38#14581713C2211yescharacteristic
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46#14581703C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48#127×C83C2211yescharacteristic
49#125×C83C2211yescharacteristic
50#123×C83—21no—
51#123×C83—21no—
52#121×C83—21no—
53#121×C83—21no—
54#121×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56C4×D4×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58#117×C83C2211yescharacteristic
59#13050590—22no—
60#13050590—22no—
61C2×Dic166C2311yescharacteristic
62C2×D332C2311yescharacteristic
63C2×C22⋊D83—21no—
64C22×Dic83—21no—
65C2×C4×D83—21no—
66C2×C4×D83—21no—
67C2×C4×D83—21no—
68#13050578—22no—
69#13050578—22no—
70#13050575—22no—
71#13050575—22no—
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83—21no—
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83C4×Dic83—21no—
84C4×Dic83—21no—
85C2×D4×C83—21no—
86C2×D4×C83—21no—
87C2×D4×C83—21no—
88C2×D4×C83—21no—
89C2×D4×C83C2311yescharacteristic
90C2×D4×C83C2311yescharacteristic
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83C2311yescharacteristic
94C4⋊C4×C83—21no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83D411yescharacteristic
97C4⋊C4×C83D411yescharacteristic
98C4⋊C4×C83C2311yescharacteristic
99C4⋊C4×C83C2311yescharacteristic
100C22⋊C4×C83—21no—
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83—21no—
104C22⋊C4×C83D411yescharacteristic
105C22⋊C4×C83D411yescharacteristic
106C42×C83C2311yescharacteristic
107C42×C83C2311yescharacteristic
108C23×C83—21no—
109C23×C83—21no—
110C23×C83C2×D411yescharacteristic
111C22×D83—21no—
112D4×C83—21no—
113D4×C83—21no—
114D4×C83—21no—
115D4×C83—21no—
116D4×C83—81no—
117D4×C83—81no—
118D4×C83—81no—
119D4×C83—81no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—81no—
125C2×C4×C83—81no—
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C22⋊D83—24no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—81no—
137D332—21no—
138D332#4112yes—
139C4×D83—24no—
140C4×D83—81no—
141C4×D83—81no—
142Dic166—21no—
143Dic166#4112yes—
144C83⋊C8—21no—
145C83⋊C8—21no—
146C83⋊C8—21no—
147C83⋊C8—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83C22×D411yescharacteristic
158C2×D83—22no—
159C2×D83—41no—
160C2×D83—41no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83C2×#4111yescharacteristic
169C4×C83C2×#4111yescharacteristic
170Dic83—22no—
171Dic83—41no—
172Dic83—41no—
173C2×C83—41no—
174C2×C83—81no—
175C2×C83—81no—
176C2×C83#52511yescharacteristic
177C2×C83#57611yescharacteristic
178C2×C83#57711yescharacteristic
179D83—42no—
180#3119—831nomaximal, Sylow
181C83#311911yescharacteristic, complemented, Sylow
182#540D8311yescharacteristic, complemented
183#520—831no—
184#481—831no—
185#480—831no—
186#479—831no—
187#478—831no—
188#460—831no—
189#459—831no—
190#449—831no—
191#449—831no—
192#442—831no—
193#440—831no—
194C4×SD16—831no—
195C4×D8—831no—
196#433—831no—
197C2×#41—831no—
198C2×SD16—831no—
199C2×SD16—831no—
200C2×D8—831no—
201C2×D8—831no—
202#127C2×D8311yescharacteristic
203#125C2×D8311yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D8311yescharacteristic
209#121C2×D8311yescharacteristic
210C4×Q8—831no—
211C4×Q8—831no—
212C4×D4—831no—
213C4×D4—831no—
214C4×D4—1661no—
215C4×D4—1661no—
216C4×D4C2×D8311yescharacteristic
217C4×D4C2×D8311yescharacteristic
218#117—1661no—
219#117C2×D8311yescharacteristic
220C2×C4⋊C4—831no—
221C2×C4⋊C4—1661no—
222#107—831no—
223#107—831no—
224#106—831no—
225#106—831no—
226C4⋊C8—831no—
227C4⋊C8—831no—
228#103—831no—
229#103—831no—
230#102—831no—
231#102—831no—
232#102—831no—
233#102—831no—
234#102—831no—
235#102—831no—
236C22⋊C8—831no—
237C22⋊C8—831no—
238#97—831no—
239C4×C8—831no—
240#41—1662no—
241#41—1662no—
242C2×Q8—831no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—831no—
248C2×D4—1661no—
249C2×D4C22×D8311yescharacteristic
250C2×D4C22×D8311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1661no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4C22×D8311yescharacteristic
258SD16—1662no—
259SD16—1662no—
260D8—1662no—
261D8—1662no—
262C2×C8—831no—
263C2×C8—831no—
264C2×C8—831no—
265C2×C8—831no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—831no—
269C4⋊C4—831no—
270C4⋊C4—1661no—
271C4⋊C4—1661no—
272C4⋊C4—1661no—
273C4⋊C4C22⋊D8311yescharacteristic
274C4⋊C4C22⋊D8311yescharacteristic
275C4⋊C4C22×D8311yescharacteristic
276C4⋊C4C22×D8311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C22⋊C4C22⋊D8311yescharacteristic
284C22⋊C4C22⋊D8311yescharacteristic
285C42—1661no—
286C42—1661no—
287C42C22×D8311yescharacteristic
288C42C22×D8311yescharacteristic
289C23—21no—
290C23—21no—
291C23—1661no—
292C23C2×C22⋊D8311yescharacteristic
293Q8—832no—
294Q8—1661no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—832no—
304D4—1664no—
305D4—1661no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—6641no—
313C2×C4—6641no—
314C2×C4—6641no—
315C2×C4—1664no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4C2×C22⋊D8311yescharacteristic
320C2×C4C2×C22⋊D8311yescharacteristic
321C2×C4C2×C22⋊D8311yescharacteristic
322C2×C4C2×C22⋊D8311yescharacteristic
323C2×C4C2×C22⋊D8311yescharacteristic
324C2×C4C23×D8311yescharacteristic
325C8—1661no—
326C8—1661no—
327C8—1661no—
328C8—1661no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3321no—
339C22—3321no—
340C22—1662no—
341C22C22×C22⋊D8311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—3321no—
350C4—3321no—
351C4—1662no—
352C4#41×D8311yescharacteristic
353C4#41×D8311yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—3322no—
358C2#1912820811yescharacteristic
359C2#1912850411yescharacteristic
360C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#540×C83C2
socleC22×C83C22×D4
hypercenter#540D83
hypocenterC83#3119
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #19128181C2semidirect
2. #19128017C2semidirect
3. #19128012C2semidirect
4. #19128005C2semidirect
5. #19127998C2semidirect
6. #19127898C2semidirect
7. #19127893C2semidirect
8. #19127844C2semidirect
9. #19127844C2semidirect
10. #19127811C2semidirect
11. #19127801C2semidirect
12. C4×#13050578C2semidirect
13. C4×#13050575C2semidirect
14. #19127762C2semidirect
15. #540×C83C2semidirect
181. C83#3119semidirect
182. #540D83semidirect

Maximal subgroups

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

subgroupindex
#191281812
#191280172
#191280122
#191280052
#191279982
#191278982
#191278932
#191278442
#191278112
#191278012
C4×#130505782
C4×#130505752
#191277622
#540×C832
#311983