Finite Group Databasebeta

← #24910046

Subgroups — #24910046

Order 10624.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×Q8×D83C211yesmaximal, characteristic
2C2×#14581750C211yesmaximal, characteristic
3C2×#14581749C211yesmaximal, characteristic
4#19127608C211yesmaximal, characteristic, complemented
5#19127607C211yesmaximal, characteristic, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127605C214yesmaximal, complemented
8#19127563C214yesmaximal
9#391×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2212yes—
11C22×Dic166C2211yescharacteristic
12C2×Q8×D83—24no—
13C2×Q8×D83—24no—
14C2×Q8×D83C418yescomplemented
15C22×C4×D83C2212yes—
16C22×C4×D83C2211yescharacteristic
17#14581750—24no—
18#14581749—24no—
19C2×C4⋊Dic83C2211yescharacteristic
20C2×Dic83⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2214yes—
26C83⋊#95C2214yes—
27C83⋊#95C2214yes—
28C22×Q8×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2211yescharacteristic
30C2×C42×C83C2211yescharacteristic
31#95×C83C2214yes—
32C23×D83C2311yescharacteristic
33C2×Dic166—22no—
34C2×Dic166—22no—
35C2×Dic166—24no—
36C2×Dic166—24no—
37C2×Dic166C2×C414yes—
38C2×Dic166C2×C418yes—
39Q8×D83—232no—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C2×C4×D83—24no—
45C2×C4×D83—24no—
46C2×C4×D83C2×C418yes—
47C2×C4×D83C2×C414yes—
48C2×C4×D83D414yes—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51Dic83⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—24no—
54C4×Dic83—22no—
55C4×Dic83—24no—
56C2×Q8×C83—22no—
57C2×Q8×C83—22no—
58C2×Q8×C83C2×C414yes—
59C22×C4×C83C2312yes—
60C22×C4×C83C2312yes—
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—22no—
64C42×C83—22no—
65C23×C83C2411yescharacteristic
66C22×D83—22no—
67C22×D83—22no—
68C22×D83C22×C412yes—
69C22×D83C2×D412yes—
70C22×D83C2×D412yes—
71Q8×C83—28no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—24no—
78C2×C4×C83—22no—
79C2×C4×C83C22×C414yes—
80C2×C4×C83C22×C412yes—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D412yes—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—24no—
90C2×Dic83C22×C414yes—
91C2×Dic83C22×C412yes—
92C2×Dic83C2×D412yes—
93C2×Dic83C2×D412yes—
94C4×D83—216no—
95C4×D83C22⋊C4116yes—
96Dic166—28no—
97Dic166—216no—
98C22×C83C23×C411yescharacteristic
99C22×C83C22×D411yescharacteristic
100C22×C83C22×D411yescharacteristic
101C22×C832−1+412yes—
102C22×C832−1+412yes—
103C2×D83—24no—
104C2×D83—22no—
105C2×D83—24no—
106C2×D83—22no—
107C2×D83C2×C22⋊C414yes—
108C4×C83—24no—
109C4×C83—42no—
110C4×C83—42no—
111C4×C83C2×C22⋊C414yes—
112Dic83—24no—
113Dic83—42no—
114Dic83—42no—
115Dic83C2×C22⋊C414yes—
116C2×C83C22×C22⋊C411yescharacteristic
117C2×C83#51912yes—
118C2×C83#53611yescharacteristic
119C2×C83#53612yes—
120C2×C83#53611yescharacteristic
121D83—24no—
122#2310—831nomaximal, Sylow
123C83#231011yescharacteristic, complemented, Sylow
124C23×Q8—831no—
125C2×#117—831no—
126C2×#117—831no—
127#391—831no—
128#391—831no—
129#391—831no—
130#391—834no—
131#391D8311yescharacteristic, complemented
132#381—834no—
133C22×Q8—838no—
134C22×Q8—832no—
135C22×Q8—831no—
136C22×Q8—1664no—
137C22×Q8—1664no—
138C22×Q8C2×D8311yescharacteristic
139C23×C4—831no—
140C23×C4—832no—
141#117—1664no—
142#117—1664no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—831no—
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C42—832no—
149C2×C42—831no—
150C2×C42C2×D8311yescharacteristic
151#95—834no—
152#95—834no—
153#95—834no—
154#95C2×D8314yes—
155C24—831no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—838no—
159C2×Q8—834no—
160C2×Q8—1664no—
161C2×Q8—1664no—
162C2×Q8—16632no—
163C2×Q8—1662no—
164C2×Q8—1662no—
165C2×Q8C4×D8314yes—
166C22×C4—831no—
167C22×C4—834no—
168C22×C4—834no—
169C22×C4—834no—
170C22×C4—832no—
171C22×C4—838no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4—1664no—
175C22×C4—1664no—
176C22×C4C22×D8312yes—
177C22×C4C22×D8312yes—
178C22×C4C22×D8312yes—
179C22×C4C22×D8311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—1664no—
182C4⋊C4—1662no—
183C22⋊C4—1664no—
184C22⋊C4—1664no—
185C42—22no—
186C42—1664no—
187C42—1662no—
188C23—832no—
189C23—832no—
190C23—832no—
191C23—1662no—
192C23—1662no—
193C23C23×D8311yescharacteristic
194Q8—28no—
195Q8—16616no—
196Q8—1668no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—832no—
205C2×C4—832no—
206C2×C4—832no—
207C2×C4—8316no—
208C2×C4—834no—
209C2×C4—16616no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1664no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4C2×C4×D8314yes—
218C2×C4C2×C4×D8312yes—
219C2×C4D4×D8312yes—
220C2×C4D4×D8312yes—
221C22—834no—
222C22—1664no—
223C22—1662no—
224C22—1664no—
225C22—1662no—
226C22C22×C4×D8311yescharacteristic
227C22C2×D4×D8311yescharacteristic
228C22C2×D4×D8311yescharacteristic
229C22#1458183912yes—
230C22#1458184012yes—
231C4—24no—
232C4—42no—
233C4—42no—
234C4—3322no—
235C4—3322no—
236C4—834no—
237C4—1664no—
238C4C22⋊C4×D8314yes—
239C2—1664no—
240C2C2×C22⋊C4×D8311yescharacteristic
241C2#1912817712yes—
242C2#1912827111yescharacteristic
243C2#1912827412yes—
244C2#1912827611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutatorC22×C83C23×C4
Fitting#391×C83C2
socle, cosocleC23×C83C24
prime radicalC23×D83C23
hypercenter#391D83
hypocenterC83#2310
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #19127608C2semidirect
5. #19127607C2semidirect
6. #19127606C2semidirect
7. #19127605C2semidirect
9. #391×C83C2semidirect
14. C2×Q8×D83C4semidirect
123. C83#2310semidirect
131. #391D83semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D832
C2×#145817502
C2×#145817492
#191276082
#191276072
#191276062
#191276052
#191275632
#391×C832
#231083