Finite Group Databasebeta

← #24915767

Subgroups — #24915767

Order 10624.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581781C211yesmaximal, characteristic, complemented
2C2×#14581768C211yesmaximal, characteristic, complemented
3C2×#14581767C211yesmaximal, characteristic
4C2×C42×D83C211yesmaximal, characteristic, complemented
5#19127603C211yesmaximal, characteristic, complemented
6#19127590C214yesmaximal, complemented
7#19127579C214yesmaximal, complemented
8#19127566C211yesmaximal, characteristic, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×D332C2212yes—
11C22×D332C2211yescharacteristic
12C22×C4×D83C2212yes—
13C22×C4×D83C2211yescharacteristic
14#14581781—24no—
15#14581768—24no—
16#14581767—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2214yes—
21C2×(C2×D83)⋊C4C2214yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C42×D83—24no—
26C83⋊#95C2214yes—
27C22×Q8×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2214yes—
31C23×D83C2312yes—
32C23×D83C2311yescharacteristic
33C2×D332—24no—
34C2×D332—28no—
35C2×D332—22no—
36C2×D332—22no—
37C2×D332—22no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2311yescharacteristic
41C2×C4×D83—24no—
42C2×C4×D83—22no—
43C2×C4×D83—22no—
44C2×C4×D83—24no—
45C2×C4×D83—24no—
46C2×C4×D83D414yes—
47C4⋊Dic83—22no—
48Dic83⋊C4—24no—
49(C2×D83)⋊C4—24no—
50(C2×D83)⋊C4—28no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—28no—
53C4×Dic83—24no—
54C4×Dic83D414yescomplemented
55C2×Q8×C83—22no—
56C2×Q8×C83—22no—
57C2×Q8×C83—22no—
58C22×C4×C83C2312yes—
59C22×C4×C83C2312yes—
60C22×C4×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—22no—
63C42×C83—22no—
64C23×C83C2411yescharacteristic
65C22×D83—21no—
66C22×D83—21no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—24no—
71C22×D83—24no—
72C22×D83—22no—
73C22×D83C2×D412yes—
74C22×D83#4114yes—
75Q8×C83—44no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—24no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D412yes—
85C2×C4×C83#4112yes—
86C2×C4×C83#4114yes—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—24no—
93C2×Dic83C2×D414yes—
94C2×Dic83C2×D412yes—
95D332—48no—
96D332—44no—
97C4×D83—216no—
98C4×D83—28no—
99C22×C83C22×D411yescharacteristic
100C22×C83C22×D411yescharacteristic
101C22×C83C2×#4112yes—
102C22×C83C2×#4111yescharacteristic
103C22×C83C2×#4112yes—
104C2×D83—22no—
105C2×D83—24no—
106C2×D83—24no—
107C2×D83—22no—
108C2×D83—42no—
109C2×D83—44no—
110C2×D83—42no—
111C2×D83—44no—
112C2×D83—42no—
113C2×D83#12414yescomplemented
114C4×C83—24no—
115C4×C83—42no—
116C4×C83—42no—
117C4×C83#12114yes—
118Dic83—22no—
119Dic83—24no—
120Dic83—42no—
121C2×C83C2×#12111yescharacteristic
122C2×C83#52512yes—
123C2×C83C2×#12411yescharacteristic
124C2×C83#52912yes—
125C2×C83#53211yescharacteristic
126D83—24no—
127D83—82no—
128#2850—831nomaximal, Sylow
129C83#285011yescharacteristic, complemented, Sylow
130C2×#124—831no—
131C2×C22⋊Q8—831no—
132C2×#121—831no—
133C22×C42—831no—
134#391D8311yescharacteristic, complemented
135#390—831no—
136#388—834no—
137#386—834no—
138#382—831no—
139C22×Q8C2×D8311yescharacteristic
140C22×D4—831no—
141C22×D4—832no—
142C23×C4—831no—
143C23×C4—832no—
144#124—1664no—
145C22⋊Q8—1664no—
146#121—1664no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4C2×D8311yescharacteristic
150C2×C22⋊C4—832no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—834no—
153C2×C22⋊C4—834no—
154C2×C42—831no—
155C2×C42—832no—
156C2×C42—1664no—
157C2×C42C2×D8311yescharacteristic
158#95—834no—
159#95C2×D8314yes—
160C24—831no—
161C24—832no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—1664no—
166C2×D4—1668no—
167C2×D4—1662no—
168C2×D4—1662no—
169C2×D4—1662no—
170C22×C4—832no—
171C22×C4—831no—
172C22×C4—834no—
173C22×C4—832no—
174C22×C4—1664no—
175C22×C4—1664no—
176C22×C4—1662no—
177C22×C4—1662no—
178C22×C4—1664no—
179C22×C4C22×D8312yes—
180C22×C4C22×D8312yes—
181C22×C4C22×D8312yes—
182C22×C4C22×D8311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—1664no—
185C4⋊C4—1662no—
186C22⋊C4—1664no—
187C22⋊C4—1668no—
188C22⋊C4—1664no—
189C22⋊C4—1668no—
190C42—22no—
191C42—834no—
192C42—1664no—
193C23—832no—
194C23—834no—
195C23—1662no—
196C23—1662no—
197C23—1664no—
198C23—1662no—
199C23—1664no—
200C23—1661no—
201C23—1661no—
202C23—1662no—
203C23C23×D8311yescharacteristic
204Q8—44no—
205D4—3328no—
206D4—3324no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—834no—
215C2×C4—832no—
216C2×C4—16616no—
217C2×C4—1662no—
218C2×C4—1664no—
219C2×C4—1661no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1668no—
224C2×C4D4×D8312yes—
225C2×C4D4×D8312yes—
226C2×C4#1305059012yes—
227C2×C4#1305059114yes—
228C22—3322no—
229C22—3324no—
230C22—3322no—
231C22—3324no—
232C22—3322no—
233C22—834no—
234C22—1662no—
235C22—1664no—
236C22—1664no—
237C22—1662no—
238C22C2×D4×D8311yescharacteristic
239C22C2×D4×D8311yescharacteristic
240C22#41×D8312yes—
241C22C2×#1305059011yescharacteristic
242C22C2×#1305059112yes—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—3322no—
247C4—1662no—
248C4—1664no—
249C4#1458176514yes—
250C2—6642no—
251C2—1664no—
252C2C2×#1458176511yescharacteristic
253C2#1912821412yes—
254C2#124×D8311yescharacteristic
255C2#1912823812yes—
256C2#1912825211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC2×#14581767C2
hypercenter#391D83
hypocenterC83#2850
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14581781C2semidirect
2. C2×#14581768C2semidirect
4. C2×C42×D83C2semidirect
5. #19127603C2semidirect
6. #19127590C2semidirect
7. #19127579C2semidirect
8. #19127566C2semidirect
9. #391×C83C2semidirect
54. C4×Dic83D4semidirect
113. C2×D83#124semidirect
129. C83#2850semidirect
134. #391D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×#145817682
C2×#145817672
C2×C42×D832
#191276032
#191275902
#191275792
#191275662
#391×C832
#285083