Finite Group Databasebeta

← #24915784

Subgroups — #24915784

Order 10624.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14581773C211yesmaximal, characteristic
2C22×C4×Dic83C211yesmaximal, characteristic, complemented
3#19127595C212yesmaximal, complemented
4#19127594C212yesmaximal, complemented
5#19127584C212yesmaximal, complemented
6#19127583C212yesmaximal, complemented
7#19127581C211yesmaximal, characteristic
8#19127576C211yesmaximal, characteristic, complemented
9#19127574C211yesmaximal, characteristic, complemented
10#19127562C211yesmaximal, characteristic, complemented
11#386×C83C211yesmaximal, characteristic
12C23×Dic83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14#14581773—24no—
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C4×Dic83—24no—
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C23×C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2212yes—
37C22×Dic83—22no—
38C22×Dic83—22no—
39C22×Dic83—22no—
40C22×Dic83—24no—
41C22×Dic83D414yes—
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2312yes—
48C4⋊Dic83—24no—
49Dic83⋊C4—24no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52C22⋊Dic83—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—22no—
56C22⋊Dic83—24no—
57C4×Dic83—24no—
58C4×Dic83—24no—
59C24×C83C2311yescharacteristic
60C24×C83C2311yescharacteristic
61C2×D4×C83—22no—
62C2×D4×C83—22no—
63C2×D4×C83—22no—
64C22×C4×C83—22no—
65C22×C4×C83—24no—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2312yes—
68C22×C4×C83C2312yes—
69C22⋊C4×C83—22no—
70C22⋊C4×C83—24no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—21no—
74C23×C83—22no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83C2×D412yes—
79C23×C83#4112yes—
80C23×C83#4112yes—
81C23×C83C2411yescharacteristic
82D4×C83—44no—
83C2×C4×C83—28no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83#4114yes—
93C2×Dic83—28no—
94C2×Dic83—28no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83C2×D412yes—
109C2×Dic83C2×D412yes—
110C2×Dic83#4114yes—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—24no—
114C22×C83—24no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83#12314yes—
121C22×C83C22×D411yescharacteristic
122C22×C83C2×#4112yes—
123C22×C83C2×#4111yescharacteristic
124C22×C83C2×#4112yes—
125C22×C83C2×#4111yescharacteristic
126C4×C83—24no—
127C4×C83—41no—
128C4×C83—42no—
129Dic83—22no—
130Dic83—22no—
131Dic83—24no—
132Dic83—41no—
133Dic83—41no—
134Dic83—42no—
135C2×C83—24no—
136C2×C83—81no—
137C2×C83C2×#12311yescharacteristic
138C2×C83#52511yescharacteristic
139C2×C83#52511yescharacteristic
140C2×C83#52912yes—
141C2×C83#52912yes—
142#2853—831nomaximal, Sylow
143C83#285311yescharacteristic, complemented, Sylow
144C2×#123—831no—
145C22×C42—831no—
146#388—832no—
147#388—832no—
148#387—832no—
149#387—832no—
150#386—831no—
151#386D8311yescharacteristic
152#385—831no—
153#385—831no—
154#381—831no—
155C22×D4C2×D8311yescharacteristic
156C23×C4—831no—
157C23×C4—831no—
158C23×C4C2×D8311yescharacteristic
159#123—1664no—
160C2×C4⋊C4—831no—
161C2×C4⋊C4—831no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—832no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4—831no—
166C2×C22⋊C4—831no—
167C2×C22⋊C4—831no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4C2×D8311yescharacteristic
170C2×C22⋊C4C2×D8312yes—
171C2×C42—832no—
172C2×C42—832no—
173C2×C42—1664no—
174#95—832no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95—832no—
179#95C2×D8312yes—
180C24C22×D8311yescharacteristic
181C24C22×D8311yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—834no—
188C22×C4—831no—
189C22×C4—831no—
190C22×C4—832no—
191C22×C4—832no—
192C22×C4—831no—
193C22×C4—831no—
194C22×C4—1664no—
195C22×C4—1662no—
196C22×C4—1662no—
197C22×C4—1662no—
198C22×C4C22×D8312yes—
199C22×C4C22×D8311yescharacteristic
200C22×C4C22×D8312yes—
201C4⋊C4—1662no—
202C4⋊C4—1662no—
203C4⋊C4—1664no—
204C4⋊C4—1664no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C22⋊C4—1664no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C42—1664no—
213C42—1664no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D8312yes—
222C23#1305058912yes—
223C23#1305059012yes—
224C23C23×D8311yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—832no—
236C2×C4—832no—
237C2×C4—834no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1668no—
246C2×C4—1668no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4—1661no—
253C2×C4#1305058914yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1458177414yes—
264C22C2×D4×D8311yescharacteristic
265C22#41×D8312yes—
266C22C2×#1305058911yescharacteristic
267C22C2×#1305058912yes—
268C22C2×#1305059011yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—3321no—
273C4—3321no—
274C4—3322no—
275C4—1662no—
276C4—1662no—
277C4—1664no—
278C2—24no—
279C2—81no—
280C2C2×#1458177411yescharacteristic
281C2#1912820711yescharacteristic
282C2#1912820911yescharacteristic
283C2#1912823112yes—
284C2#1912823212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2853
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic83C2semidirect
3. #19127595C2semidirect
4. #19127594C2semidirect
5. #19127584C2semidirect
6. #19127583C2semidirect
8. #19127576C2semidirect
9. #19127574C2semidirect
10. #19127562C2semidirect
143. C83#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#145817732
C22×C4×Dic832
#191275952
#191275942
#191275842
#191275832
#191275812
#191275762
#191275742
#191275622
#386×C832
#285383