Finite Group Databasebeta

← #24913380

Subgroups — #24913380

Order 10624.

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

normal subgroups only

This group has 520 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×#14581771C211yesmaximal, characteristic, complemented
2#19127631C211yesmaximal, characteristic, complemented
3#19127629C212yesmaximal, complemented
4#19127626C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127615C212yesmaximal
7#19127613C211yesmaximal, characteristic, complemented
8#19127585C212yesmaximal, complemented
9#19127574C212yesmaximal, complemented
10#19127564C211yesmaximal, characteristic, complemented
11C2×C4×D4×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13#14581771—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2211yescharacteristic
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C22×D4×C83C2211yescharacteristic
30C23×C4×C83C2211yescharacteristic
31C23×C4×C83C2211yescharacteristic
32C4×D4×C83—24no—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C42×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—42no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C4⋊Dic83—22no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—42no—
47C22⋊Dic83—24no—
48C22⋊Dic83—24no—
49C22⋊Dic83—42no—
50C22⋊Dic83—42no—
51C24×C83C2311yescharacteristic
52C24×C83C2311yescharacteristic
53C2×D4×C83—22no—
54C2×D4×C83—22no—
55C2×D4×C83—22no—
56C22×C4×C83—22no—
57C22×C4×C83—42no—
58C22×C4×C83—42no—
59C22×C4×C83D414yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—22no—
68C42×C83—22no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83—21no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83C2×D412yes—
78C23×C83#4112yes—
79C23×C83C2411yescharacteristic
80D4×C83—28no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—44no—
90C2×C4×C83—44no—
91C2×C4×C83C2×D412yes—
92C2×C4×C83C2×D412yes—
93C2×C4×C83#4112yes—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106Dic166—44no—
107C22×C83—22no—
108C22×C83—22no—
109C22×C83—22no—
110C22×C83—22no—
111C22×C83—24no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83C22×D411yescharacteristic
118C22×C83C2×#4111yescharacteristic
119C22×C83C2×#4111yescharacteristic
120C22×C83H(5,𝔽2)12yes—
121C22×C832−1+412yes—
122C4×C83—22no—
123C4×C83—22no—
124C4×C83—22no—
125C4×C83—41no—
126C4×C83—41no—
127Dic83—42no—
128Dic83—42no—
129Dic83—42no—
130Dic83—42no—
131C2×C83—42no—
132C2×C83—42no—
133C2×C83#52511yescharacteristic
134C2×C83#53411yescharacteristic
135C2×C83#53611yescharacteristic
136C2×C83#53912yes—
137C2×C83#54212yes—
138#2638—831nomaximal, Sylow
139C83#263811yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—831no—
141C2×C4×D4D8311yescharacteristic
142#398—831no—
143#397—831no—
144#397—832no—
145#395—831no—
146#394—832no—
147#393—831no—
148#387—832no—
149#385—832no—
150#381—831no—
151C22×Q8—831no—
152C22×D4C2×D8311yescharacteristic
153C23×C4C2×D8311yescharacteristic
154C23×C4C2×D8311yescharacteristic
155C22⋊Q8—1664no—
156C4×D4—24no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4—832no—
159C2×C4⋊C4—831no—
160C2×C4⋊C4—832no—
161C2×C4⋊C4C2×D8311yescharacteristic
162C2×C22⋊C4—832no—
163C2×C22⋊C4—832no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4C2×D8311yescharacteristic
167C2×C22⋊C4C2×D8311yescharacteristic
168C2×C42C2×D8311yescharacteristic
169#95—832no—
170#95—831no—
171#95—831no—
172#95—832no—
173#95—832no—
174#95—832no—
175#95—832no—
176C24C22×D8311yescharacteristic
177C24C22×D8311yescharacteristic
178C2×Q8—3322no—
179C2×Q8—1662no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—832no—
187C22×C4—832no—
188C22×C4—832no—
189C22×C4—832no—
190C22×C4C22⋊D8314yes—
191C22×C4C22×D8311yescharacteristic
192C22×C4C22×D8311yescharacteristic
193C22×C4C22×D8311yescharacteristic
194C22×C4C22×D8311yescharacteristic
195C22×C4C22×D8311yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—3322no—
198C4⋊C4—1664no—
199C4⋊C4—1662no—
200C4⋊C4—1664no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—3322no—
204C22⋊C4—3322no—
205C22⋊C4—1664no—
206C22⋊C4—1664no—
207C42—22no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23C2×C22⋊D8312yes—
217C23#1305059012yes—
218C23C23×D8311yescharacteristic
219Q8—3324no—
220D4—28no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4C2×C22⋊D8312yes—
244C2×C4C2×C22⋊D8312yes—
245C2×C4#1305059012yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—24no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22C22×C22⋊D8311yescharacteristic
257C22C2×#1305059011yescharacteristic
258C22C2×#1305059011yescharacteristic
259C22#1458183712yes—
260C22#1458183912yes—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—3322no—
267C4—3322no—
268C4—3322no—
269C4—3322no—
270C2—42no—
271C2—42no—
272C2#1912821211yescharacteristic
273C2#1912826011yescharacteristic
274C2#1912827211yescharacteristic
275C2#1912829312yes—
276C2#1912831612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4×D4×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×C4×D4D83
hypocenterC83#2638
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. #19127631C2semidirect
3. #19127629C2semidirect
4. #19127626C2semidirect
5. #19127621C2semidirect
7. #19127613C2semidirect
8. #19127585C2semidirect
9. #19127574C2semidirect
10. #19127564C2semidirect
139. C83#2638semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
#191276312
#191276292
#191276262
#191276212
#191276152
#191276132
#191275852
#191275742
#191275642
C2×C4×D4×C832
#263883