Finite Group Databasebeta

← #24917861

Subgroups — #24917861

Order 10624.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128247C211yesmaximal, characteristic
2C2×#14581783C211yesmaximal, characteristic, complemented
3#19128068C212yesmaximal, complemented
4#19128064C212yesmaximal, complemented
5#19128057C212yesmaximal, complemented
6#19128054C212yesmaximal, complemented
7#19127703C211yesmaximal, characteristic, complemented
8#19127702C211yesmaximal, characteristic, complemented
9#19127700C211yesmaximal, characteristic, complemented
10M16×Dic83C211yesmaximal, characteristic, complemented
11#417×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14#14581790C2211yescharacteristic
15#14581788C2211yescharacteristic
16#14581783—21no—
17#14581783—21no—
18#14581783C2212yes—
19#14581783C2212yes—
20#14581781C2211yescharacteristic
21#14581779C2211yescharacteristic
22#14581764—21no—
23#14581764—21no—
24C4×C22⋊D83—21no—
25C4×C22⋊D83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×C4×Dic83C2211yescharacteristic
30#14581717C2212yes—
31#14581716C2212yes—
32#14581715C2212yes—
33#14581714C2212yes—
34#14581713C2212yes—
35#14581712C2212yes—
36#14581702C2212yes—
37C8×Dic83C2212yes—
38C2×#41×C83C2211yescharacteristic
39C2×M16×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#103×C83C2212yes—
42#102×C83C2212yes—
43#13050590—22no—
44#13050590—22no—
45C2×Dic166C2311yescharacteristic
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C22⋊D83—21no—
49C2×C22⋊D83—21no—
50C22×Dic83—21no—
51C22×Dic83D411yescharacteristic
52C22×Dic83D411yescharacteristic
53C2×C4×D83—21no—
54#13050574—22no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83—22no—
58C4⋊Dic83—22no—
59C4⋊Dic83C2312yes—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—22no—
63Dic83⋊C4—22no—
64C22⋊Dic83—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C4×Dic83C2311yescharacteristic
72#41×C83—22no—
73#41×C83—22no—
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83C2311yescharacteristic
80M16×C83—22no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2312yes—
84C23×C83—21no—
85C23×C83C2×D411yescharacteristic
86C22×D83—21no—
87Q8×C83—21no—
88Q8×C83—21no—
89D4×C83—21no—
90D4×C83—21no—
91D4×C83—42no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—42no—
95C2×C4×C83—42no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83#4112yes—
98C2×C4×C83#4112yes—
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—42no—
101C22⋊D83—42no—
102C22⋊D83—42no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—41no—
106C2×Dic83—41no—
107C2×Dic83—42no—
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110C2×Dic83C2×D411yescharacteristic
111C2×Dic83C2×D411yescharacteristic
112D332—21no—
113D332—21no—
114C4×D83—42no—
115Dic166—21no—
116Dic166—21no—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83#12412yes—
125C22×C83C22×D411yescharacteristic
126C2×D83—41no—
127C2×D83—41no—
128C2×D83—41no—
129C4×C83—41no—
130C4×C83—42no—
131C4×C83#12412yes—
132C4×C83C2×#4111yescharacteristic
133C4×C83C2×#4111yescharacteristic
134Dic83—21no—
135Dic83—21no—
136Dic83—21no—
137Dic83—21no—
138Dic83—41no—
139Dic83—42no—
140C2×C83—22no—
141C2×C83—81no—
142C2×C83C2×#12411yescharacteristic
143C2×C83#57511yescharacteristic
144C2×C83#57511yescharacteristic
145D83—81no—
146#3142—831nomaximal, Sylow
147C83#314211yescharacteristic, complemented, Sylow
148#532—831no—
149C2×#125—831no—
150#491—832no—
151#490—832no—
152#488—832no—
153#488—832no—
154#417—831no—
155#417—831no—
156#417—831no—
157#417D8311yescharacteristic, complemented
158C4×M16—831no—
159C2×#41—831no—
160C2×#41C2×D8311yescharacteristic
161C2×M16—831no—
162C2×M16C2×D8311yescharacteristic
163C4⋊Q8—831no—
164#127—831no—
165#125—832no—
166#125—832no—
167#125—1661no—
168#125—1661no—
169#124—831no—
170#124—831no—
171#121—1661no—
172#121—1661no—
173C4×D4—1661no—
174C4×D4—1661no—
175C2×C4⋊C4—831no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4C2×D8311yescharacteristic
179C2×C42—831no—
180#103—832no—
181#103—832no—
182#103—832no—
183#103C2×D8312yes—
184#102—832no—
185#102—832no—
186#102—832no—
187#102C2×D8312yes—
188#97—832no—
189C4×C8—832no—
190#41—22no—
191#41—22no—
192#41—1662no—
193#41—1662no—
194C2×Q8—831no—
195C2×Q8C22×D8311yescharacteristic
196C2×D4—21no—
197C2×D4—831no—
198C2×D4—1661no—
199C2×D4—1661no—
200C2×D4—1661no—
201C2×D4C22×D8311yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—831no—
205C22×C4—831no—
206C22×C4—1661no—
207C22×C4—1661no—
208C22×C4C22×D8311yescharacteristic
209M16—22no—
210M16—1662no—
211C2×C8—832no—
212C2×C8C22×D8312yes—
213C4⋊C4—21no—
214C4⋊C4—832no—
215C4⋊C4—1662no—
216C4⋊C4—1662no—
217C4⋊C4—1662no—
218C4⋊C4—1662no—
219C4⋊C4—1661no—
220C4⋊C4—1661no—
221C4⋊C4—1661no—
222C4⋊C4C22×D8312yes—
223C22⋊C4—1661no—
224C22⋊C4—1661no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C42—831no—
228C42—831no—
229C42—831no—
230C42—831no—
231C23—21no—
232C23—1661no—
233C23D4×D8311yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—1661no—
237Q8—1661no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—3322no—
242D4—3322no—
243D4—3322no—
244D4—1661no—
245D4—1661no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—3322no—
251C2×C4—3322no—
252C2×C4—3321no—
253C2×C4—3321no—
254C2×C4—831no—
255C2×C4—831no—
256C2×C4—831no—
257C2×C4—831no—
258C2×C4—1661no—
259C2×C4—1662no—
260C2×C4D4×D8311yescharacteristic
261C2×C4#1305058912yes—
262C2×C4#1305059012yes—
263C2×C4C23×D8311yescharacteristic
264C8—22no—
265C8—1662no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—3321no—
272C22—3321no—
273C22—3321no—
274C22#1458177812yes—
275C22C2×D4×D8311yescharacteristic
276C4—41no—
277C4—42no—
278C4—3321no—
279C4—3322no—
280C4—1661no—
281C4—1661no—
282C4—1661no—
283C4—1661no—
284C4#1458177812yes—
285C4C2×#1305058911yescharacteristic
286C4C2×#1305059011yescharacteristic
287C2—22no—
288C2—81no—
289C2—6641no—
290C2C2×#1458177811yescharacteristic
291C2#1912849711yescharacteristic
292C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
prime radical#19128247C2
hypercenter#417D83
hypocenterC83#3142
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#14581783C2semidirect
3. #19128068C2semidirect
4. #19128064C2semidirect
5. #19128057C2semidirect
6. #19128054C2semidirect
7. #19127703C2semidirect
8. #19127702C2semidirect
9. #19127700C2semidirect
10. M16×Dic83C2semidirect
11. #417×C83C2semidirect
147. C83#3142semidirect
157. #417D83semidirect

Maximal subgroups

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

subgroupindex
#191282472
C2×#145817832
#191280682
#191280642
#191280572
#191280542
#191277032
#191277022
#191277002
M16×Dic832
#417×C832
#314283