Finite Group Databasebeta

← #24913152

Subgroups — #24913152

Order 10624.

576 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14581749C211yesmaximal, characteristic, complemented
2#19127620C211yesmaximal, characteristic, complemented
3#19127615C211yesmaximal, characteristic, complemented
4#19127612C211yesmaximal, characteristic, complemented
5#19127609C211yesmaximal, characteristic
6#19127601C212yesmaximal, complemented
7#19127595C212yesmaximal, complemented
8#19127589C212yesmaximal, complemented
9#19127577C212yesmaximal, complemented
10#19127547C211yesmaximal, characteristic, complemented
11C2×#124×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2212yes—
13C22×C4×D83C2211yescharacteristic
14#14581749—24no—
15C2×Dic83⋊C4C2211yescharacteristic
16C2×Dic83⋊C4C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C2×C4×Dic83C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2211yescharacteristic
31C22×Q8×C83C2211yescharacteristic
32C22×D4×C83C2211yescharacteristic
33#124×C83—24no—
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36C2×C42×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—24no—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—42no—
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—42no—
52C22⋊Dic83—24no—
53C22⋊Dic83—24no—
54C22⋊Dic83—42no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—42no—
58C4×Dic83—22no—
59C4×Dic83—24no—
60C24×C83C2312yes—
61C2×Q8×C83—22no—
62C2×Q8×C83—42no—
63C2×D4×C83—22no—
64C2×D4×C83—42no—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C22⋊C4×C83—24no—
70C22⋊C4×C83—24no—
71C42×C83D414yescomplemented
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83C2411yescharacteristic
80C22×D83—21no—
81C22×D83—21no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83#4112yes—
85Q8×C83—44no—
86D4×C83—44no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2×D412yes—
97C22⋊D83—48no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83#4114yes—
110C2×Dic83#4112yes—
111C4×D83—28no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83C22×D411yescharacteristic
120C22×C83C2×#4111yescharacteristic
121C22×C83C2×#4111yescharacteristic
122C22×C83C2×#4112yes—
123C22×C83H(5,𝔽2)12yes—
124C2×D83—22no—
125C2×D83—24no—
126C2×D83—41no—
127C2×D83—41no—
128C2×D83—42no—
129C4×C83—22no—
130C4×C83—22no—
131C4×C83—22no—
132C4×C83—42no—
133Dic83—22no—
134Dic83—24no—
135Dic83—41no—
136Dic83—41no—
137Dic83—42no—
138C2×C83—82no—
139C2×C83#52511yescharacteristic
140C2×C83#53211yescharacteristic
141C2×C83#53511yescharacteristic
142C2×C83#55112yes—
143C2×C83#55512yes—
144D83—42no—
145#2610—831nomaximal, Sylow
146C83#261011yescharacteristic, complemented, Sylow
147C2×#124D8311yescharacteristic, complemented
148C2×#117—831no—
149#395—831no—
150#394—831no—
151#393—831no—
152#392—831no—
153#390—832no—
154#388—832no—
155#388—832no—
156#386—832no—
157#376—831no—
158C22×Q8C2×D8311yescharacteristic
159C22×D4—832no—
160C22×D4C2×D8311yescharacteristic
161C23×C4—831no—
162#124—24no—
163#117—1664no—
164C2×C4⋊C4—832no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—831no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—831no—
172C2×C22⋊C4—831no—
173C2×C22⋊C4C2×D8312yes—
174C2×C22⋊C4C2×D8312yes—
175C2×C42—831no—
176C2×C42—832no—
177C2×C42C2×D8311yescharacteristic
178#95—832no—
179#95—831no—
180#95—831no—
181#95—832no—
182#95—832no—
183C24—831no—
184C24C22×D8312yes—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—3322no—
190C2×D4—1664no—
191C2×D4—1664no—
192C22×C4—832no—
193C22×C4—831no—
194C22×C4—832no—
195C22×C4—831no—
196C22×C4—831no—
197C22×C4—1662no—
198C22×C4—1662no—
199C22×C4—1662no—
200C22×C4C22×D8312yes—
201C22×C4C22×D8311yescharacteristic
202C22×C4C22×D8311yescharacteristic
203C22×C4C22×D8311yescharacteristic
204C4⋊C4—3322no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—3322no—
210C22⋊C4—3322no—
211C22⋊C4—1664no—
212C22⋊C4—1664no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C42—1664no—
216C42—1662no—
217C42C22⋊D8314yescomplemented
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—832no—
226C23—1661no—
227C23—1661no—
228C23—1662no—
229C23—1662no—
230C23C23×D8311yescharacteristic
231Q8—44no—
232D4—44no—
233D4—3328no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—834no—
242C2×C4—832no—
243C2×C4—1668no—
244C2×C4—1661no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1661no—
252C2×C4—1661no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4C2×C22⋊D8312yes—
256C2×C4C2×C22⋊D8312yes—
257C2×C4C2×C22⋊D8312yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—3322no—
266C22—3321no—
267C22—3321no—
268C22—1664no—
269C22—1662no—
270C22C22×C22⋊D8311yescharacteristic
271C22#41×D8311yescharacteristic
272C22#41×D8312yes—
273C22#41×D8311yescharacteristic
274C22#1458183712yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—3322no—
280C4—3321no—
281C4—3321no—
282C4—1664no—
283C4—1662no—
284C2—82no—
285C2—3322no—
286C2#1912820811yescharacteristic
287C2#1912824711yescharacteristic
288C2#1912826811yescharacteristic
289C2#1912838112yes—
290C2#1912841312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
prime radical#19127609C2
hypercenterC2×#124D83
hypocenterC83#2610
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581749C2semidirect
2. #19127620C2semidirect
3. #19127615C2semidirect
4. #19127612C2semidirect
6. #19127601C2semidirect
7. #19127595C2semidirect
8. #19127589C2semidirect
9. #19127577C2semidirect
10. #19127547C2semidirect
11. C2×#124×C83C2semidirect
71. C42×C83D4semidirect
146. C83#2610semidirect
147. C2×#124D83semidirect
217. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817492
#191276202
#191276152
#191276122
#191276092
#191276012
#191275952
#191275892
#191275772
#191275472
C2×#124×C832
#261083