Finite Group Databasebeta

← #24911872

Subgroups — #24911872

Order 10624.

412 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#19127639C212yesmaximal, complemented
2#19127630C212yesmaximal
3#19127597C211yesmaximal, characteristic, complemented
4#19127591C212yesmaximal
5#19127568C212yesmaximal, complemented
6#19127567C211yesmaximal, characteristic, complemented
7#19127558C211yesmaximal, characteristic
8#19127558C211yesmaximal, characteristic
9C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
10C4×(C2×D83)⋊C4C211yesmaximal, characteristic
11#389×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2211yescharacteristic
18C2×(C2×D83)⋊C4C2211yescharacteristic
19C2×(C2×D83)⋊C4C2212yes—
20C2×C4×Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C4⋊C4×C83C2212yes—
33C2×C42×C83C2211yescharacteristic
34#95×C83C2212yes—
35C23×D83C2311yescharacteristic
36C22×Dic83C2311yescharacteristic
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2311yescharacteristic
41C2×C4×D83—22no—
42C2×C4×D83—22no—
43C2×C4×D83—22no—
44C4⋊Dic83—22no—
45C4⋊Dic83—22no—
46Dic83⋊C4—42no—
47(C2×D83)⋊C4—24no—
48(C2×D83)⋊C4—22no—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—42no—
51C4×Dic83—22no—
52C4×Dic83—24no—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57C22×C4×C83C2311yescharacteristic
58C4⋊C4×C83—22no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83—42no—
61C42×C83—22no—
62C23×C83C2411yescharacteristic
63C22×D83—21no—
64C22×D83—21no—
65C22×D83—21no—
66C22×D83—21no—
67C22×D83—21no—
68C22×D83—21no—
69C22×D83#4112yes—
70C2×C4×C83—21no—
71C2×C4×C83—21no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83#4112yes—
83C2×C4×C83#4112yes—
84C2×C4×C83#4112yes—
85C2×Dic83—21no—
86C2×Dic83—21no—
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—22no—
99C2×Dic83#4112yes—
100C4×D83—44no—
101C22×C83C2×#4111yescharacteristic
102C22×C83C2×#4111yescharacteristic
103C22×C83C2×#4111yescharacteristic
104C22×C83C2×#4111yescharacteristic
105C22×C83C2×#4111yescharacteristic
106C22×C83H(5,𝔽2)11yescharacteristic
107C22×C832−1+411yescharacteristic
108C2×D83—22no—
109C2×D83—22no—
110C2×D83—22no—
111C2×D83—41no—
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C4×C83—41no—
116C4×C83—42no—
117C4×C83—41no—
118C4×C83—42no—
119C4×C83#12614yes—
120Dic83—22no—
121Dic83—41no—
122Dic83—42no—
123Dic83—42no—
124Dic83—41no—
125C2×C83C2×#12611yescharacteristic
126C2×C83#52911yescharacteristic
127C2×C83#54211yescharacteristic
128C2×C83#55211yescharacteristic
129C2×C83#55211yescharacteristic
130C2×C83#55311yescharacteristic
131C2×C83#55511yescharacteristic
132D83—42no—
133#2495—831nomaximal, Sylow
134C83#249511yescharacteristic, complemented, Sylow
135#400—832no—
136#397—832no—
137#389—831no—
138#389D8311yescharacteristic, complemented
139#388—832no—
140#383—832no—
141#382—831no—
142#380—831no—
143#380—831no—
144C4×C4⋊C4—831no—
145C4×C22⋊C4—831no—
146C23×C4—831no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4C2×D8311yescharacteristic
151C2×C4⋊C4C2×D8311yescharacteristic
152C2×C4⋊C4C2×D8312yes—
153C2×C22⋊C4—832no—
154C2×C22⋊C4—831no—
155C2×C22⋊C4—831no—
156C2×C22⋊C4—832no—
157C2×C42—831no—
158C2×C42—832no—
159C2×C42C2×D8311yescharacteristic
160#95—831no—
161#95—831no—
162#95—832no—
163#95—832no—
164#95—832no—
165#95—832no—
166#95—832no—
167#95—832no—
168#95C2×D8312yes—
169C24—831no—
170C22×C4—831no—
171C22×C4—831no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4—831no—
175C22×C4—1662no—
176C22×C4—1662no—
177C22×C4—1662no—
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8311yescharacteristic
180C22×C4C22×D8312yes—
181C22×C4C22×D8312yes—
182C22×C4C22×D8311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—3322no—
187C4⋊C4—1662no—
188C4⋊C4—1662no—
189C22⋊C4—3322no—
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C22⋊C4—1664no—
193C42—22no—
194C42—1664no—
195C42—1662no—
196C23—832no—
197C23—1661no—
198C23—1661no—
199C23—1661no—
200C23—1661no—
201C23—1661no—
202C23—1661no—
203C23C23×D8311yescharacteristic
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—3324no—
217C2×C4—832no—
218C2×C4—1662no—
219C2×C4—1661no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4—1661no—
223C2×C4—1661no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1661no—
229C2×C4—1662no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4#1305058912yes—
233C2×C4#1305059012yes—
234C2×C4#1305059112yes—
235C22—3321no—
236C22—3321no—
237C22—3321no—
238C22—3321no—
239C22—1662no—
240C22—1662no—
241C22—1662no—
242C22#41×D8311yescharacteristic
243C22#41×D8311yescharacteristic
244C22C2×#1305058911yescharacteristic
245C22C2×#1305059011yescharacteristic
246C22C2×#1305059111yescharacteristic
247C22#1458183711yescharacteristic
248C22#1458184011yescharacteristic
249C4—41no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—3322no—
254C4—3321no—
255C4—3321no—
256C4—3322no—
257C4—1662no—
258C4#1458178714yes—
259C2—3322no—
260C2C2×#1458178711yescharacteristic
261C2#1912823611yescharacteristic
262C2#1912832511yescharacteristic
263C2#1912838511yescharacteristic
264C2#1912839211yescharacteristic
265C2#1912840111yescharacteristic
266C2#1912841511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#389×C83C2
prime radicalC23×D83C23
hypercenter#389D83
hypocenterC83#2495
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #19127639C2semidirect
3. #19127597C2semidirect
5. #19127568C2semidirect
6. #19127567C2semidirect
9. C4⋊C4×Dic83C2semidirect
11. #389×C83C2semidirect
134. C83#2495semidirect
138. #389D83semidirect

Maximal subgroups

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

subgroupindex
#191276392
#191276302
#191275972
#191275912
#191275682
#191275672
#191275582
C4⋊C4×Dic832
C4×(C2×D83)⋊C42
#389×C832
#249583