Finite Group Databasebeta

← #24920538

Subgroups — #24920538

Order 10624.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128424C211yesmaximal, characteristic, complemented
2#19128096C211yesmaximal, characteristic, complemented
3#19128093C211yesmaximal, characteristic, complemented
4#19128083C211yesmaximal, characteristic
5#19127923C211yesmaximal, characteristic, complemented
6#19127920C211yesmaximal, characteristic, complemented
7#19127916C211yesmaximal, characteristic
8#19127913C211yesmaximal, characteristic
9#19127911C211yesmaximal, characteristic, complemented
10#19127908C211yesmaximal, characteristic, complemented
11#19127832C211yesmaximal, characteristic, complemented
12#19127801C211yesmaximal, characteristic, complemented
13#19127799C211yesmaximal, characteristic, complemented
14C4×#13050580C211yesmaximal, characteristic, complemented
15#556×C83C211yesmaximal, characteristic, complemented
16C2×#13050580C2211yescharacteristic
17C2×#13050580C2211yescharacteristic
18C2×#13050580C2211yescharacteristic
19C2×#13050580C2211yescharacteristic
20C2×#13050578C2211yescharacteristic
21C2×#13050578C2211yescharacteristic
22C2×#13050577C2211yescharacteristic
23C2×#13050577C2211yescharacteristic
24C2×#13050575C2211yescharacteristic
25#14581792C2211yescharacteristic
26#14581787—21no—
27#14581787—21no—
28#14581780C2211yescharacteristic
29#14581780C2211yescharacteristic
30#14581768—21no—
31C4×Dic166C2211yescharacteristic
32C4×Dic166C2211yescharacteristic
33C4×D332C2211yescharacteristic
34#14581750—21no—
35#14581725C2211yescharacteristic
36C4⋊(C83⋊C8)C2211yescharacteristic
37C4⋊(C83⋊C8)C2211yescharacteristic
38C4⋊(C83⋊C8)C2211yescharacteristic
39#14581716C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581715C2211yescharacteristic
42#14581715C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581712C2211yescharacteristic
45#14581703C2211yescharacteristic
46#14581703C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48C4⋊Q8×C83C2211yescharacteristic
49#126×C83—21no—
50#126×C83—21no—
51#124×C83C2211yescharacteristic
52#124×C83C2211yescharacteristic
53C22⋊Q8×C83—21no—
54C4×Q8×C83C2211yescharacteristic
55C4×Q8×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58#117×C83—21no—
59C2×Dic166C2311yescharacteristic
60C2×Dic166C2311yescharacteristic
61C2×D332C2311yescharacteristic
62C2×C4×D83—21no—
63#13050580—22no—
64#13050580—41no—
65#13050580—41no—
66#13050580—41no—
67#13050578—22no—
68#13050578—41no—
69#13050577—22no—
70#13050577—41no—
71#13050575—41no—
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86C4×Dic83—21no—
87C4×Dic83—21no—
88C2×Q8×C83D411yescharacteristic
89C2×Q8×C83C2311yescharacteristic
90C2×Q8×C83C2311yescharacteristic
91C2×D4×C83C2311yescharacteristic
92C22×C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83D411yescharacteristic
98C4⋊C4×C83D411yescharacteristic
99C4⋊C4×C83D411yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C42×C83—21no—
107C42×C83—21no—
108C42×C83C2311yescharacteristic
109C42×C83C2311yescharacteristic
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C22×D83—21no—
113Q8×C83—21no—
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—41no—
118D4×C83—21no—
119D4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—42no—
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2411yescharacteristic
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134D332—21no—
135D332—21no—
136C4×D83—42no—
137Dic166—21no—
138Dic166—21no—
139Dic166—21no—
140Dic166—21no—
141C83⋊C8—21no—
142C83⋊C8—21no—
143C83⋊C8—21no—
144C83⋊C8#4112yes—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83C22×D411yescharacteristic
149C2×D83—41no—
150C2×D83—41no—
151C2×D83—41no—
152C4×C83—21no—
153C4×C83—21no—
154C4×C83—21no—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83C2×#4111yescharacteristic
162C4×C83H(5,𝔽2)11yescharacteristic
163Dic83—41no—
164Dic83—41no—
165Dic83—41no—
166C2×C83—81no—
167C2×C83#55011yescharacteristic
168C2×C83#57511yescharacteristic
169C2×C83#57811yescharacteristic
170D83—81no—
171#3402—831nomaximal, Sylow
172C83#340211yescharacteristic, complemented, Sylow
173#556—831no—
174#556D8311yescharacteristic, complemented
175#497—831no—
176#497—831no—
177#495—831no—
178#464—831no—
179#464—831no—
180#463—831no—
181#463—831no—
182#462—831no—
183#462—831no—
184#446—831no—
185#440—831no—
186#440—831no—
187C4×Q16—831no—
188C2×Q16—831no—
189C2×Q16—831no—
190C2×Q16—831no—
191C2×Q16—831no—
192C2×SD16—831no—
193C2×SD16—831no—
194C2×SD16—831no—
195C2×SD16—831no—
196C2×D8—831no—
197C4⋊Q8—831no—
198C4⋊Q8C2×D8311yescharacteristic
199#126—21no—
200#126—21no—
201#126—1661no—
202#126—1661no—
203#124—831no—
204#124—831no—
205#124C2×D8311yescharacteristic
206#124C2×D8311yescharacteristic
207C22⋊Q8—21no—
208C22⋊Q8—1661no—
209C4×Q8—831no—
210C4×Q8—831no—
211C4×Q8C2×D8311yescharacteristic
212C4×Q8C2×D8311yescharacteristic
213C4×Q8C2×D8311yescharacteristic
214C4×D4—831no—
215C4×D4C2×D8311yescharacteristic
216#117—21no—
217#117—1661no—
218#107—831no—
219C4⋊C8—831no—
220C4⋊C8—831no—
221C4⋊C8—831no—
222#103—831no—
223#103—831no—
224#103—831no—
225#103—831no—
226#102—831no—
227#102—831no—
228#97—831no—
229#97—831no—
230C4×C8—831no—
231C2×Q8—831no—
232C2×Q8—831no—
233C2×Q8C22⋊D8311yescharacteristic
234C2×Q8C22×D8311yescharacteristic
235C2×Q8C22×D8311yescharacteristic
236C2×D4—831no—
237C2×D4C22×D8311yescharacteristic
238C22×C4—21no—
239C22×C4—1661no—
240Q16—3321no—
241Q16—3321no—
242Q16—3321no—
243Q16—1662no—
244SD16—3321no—
245SD16—3321no—
246SD16—1662no—
247SD16—1662no—
248D8—3321no—
249C2×C8—831no—
250C2×C8—831no—
251C2×C8—831no—
252C2×C8—831no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—831no—
258C4⋊C4—1661no—
259C4⋊C4—1661no—
260C4⋊C4—1661no—
261C4⋊C4—1661no—
262C4⋊C4C22⋊D8311yescharacteristic
263C4⋊C4C22⋊D8311yescharacteristic
264C4⋊C4C22⋊D8311yescharacteristic
265C4⋊C4C22×D8311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1661no—
272C22⋊C4—1661no—
273C22⋊C4—1661no—
274C22⋊C4—1661no—
275C22⋊C4—1661no—
276C42—21no—
277C42—21no—
278C42—1661no—
279C42—1661no—
280C42C22×D8311yescharacteristic
281C42C22×D8311yescharacteristic
282C42C22×D8311yescharacteristic
283C23—21no—
284C23—1661no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—1661no—
291Q8—1661no—
292Q8—1661no—
293Q8—1661no—
294D4—21no—
295D4—21no—
296D4—1661no—
297D4—1661no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3322no—
303C2×C4—1661no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4C2×C22⋊D8311yescharacteristic
307C2×C4C2×C22⋊D8311yescharacteristic
308C2×C4C2×C22⋊D8311yescharacteristic
309C2×C4C2×C22⋊D8311yescharacteristic
310C2×C4C2×C22⋊D8311yescharacteristic
311C2×C4C2×C22⋊D8311yescharacteristic
312C2×C4C23×D8311yescharacteristic
313C8—832no—
314C8—1661no—
315C8—1661no—
316C8—1661no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—3321no—
321C22—3321no—
322C22—3321no—
323C22C22×C22⋊D8311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3321no—
334C4—3321no—
335C4—3321no—
336C4#41×D8311yescharacteristic
337C4#1458183811yescharacteristic
338C2—81no—
339C2—6641no—
340C2#1912837011yescharacteristic
341C2#1912849911yescharacteristic
342C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#556D83
hypocenterC83#3402
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #19128424C2semidirect
2. #19128096C2semidirect
3. #19128093C2semidirect
5. #19127923C2semidirect
6. #19127920C2semidirect
9. #19127911C2semidirect
10. #19127908C2semidirect
11. #19127832C2semidirect
12. #19127801C2semidirect
13. #19127799C2semidirect
14. C4×#13050580C2semidirect
15. #556×C83C2semidirect
172. C83#3402semidirect
174. #556D83semidirect

Maximal subgroups

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

subgroupindex
#191284242
#191280962
#191280932
#191280832
#191279232
#191279202
#191279162
#191279132
#191279112
#191279082
#191278322
#191278012
#191277992
C4×#130505802
#556×C832
#340283