Finite Group Databasebeta

← #24912073

Subgroups — #24912073

Order 10624.

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

normal subgroups only

This group has 372 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
1C83⋊#401C212yesmaximal
2#19127642C212yesmaximal
3#19127608C211yesmaximal, characteristic
4#19127608C211yesmaximal, characteristic
5#19127597C211yesmaximal, characteristic
6#19127583C212yesmaximal
7#19127568C212yesmaximal
8#19127567C211yesmaximal, characteristic
9C4⋊C4×Dic83C211yesmaximal, characteristic
10C4×Dic83⋊C4C211yesmaximal, characteristic
11#391×C83C211yesmaximal, characteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2211yescharacteristic
15C2×Dic83⋊C4C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×C4×Dic83C2211yescharacteristic
18C2×C4×Dic83C2211yescharacteristic
19C2×C4×Dic83C2212yes—
20C83⋊#95C2211yescharacteristic
21C83⋊#95C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C22×Q8×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C42×C83C2211yescharacteristic
33#95×C83C2212yes—
34#95×C83C2212yes—
35C22×Dic83C2311yescharacteristic
36C22×Dic83C2311yescharacteristic
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2311yescharacteristic
41C4⋊Dic83—22no—
42C4⋊Dic83—22no—
43Dic83⋊C4—22no—
44Dic83⋊C4—22no—
45Dic83⋊C4—24no—
46C4×Dic83—22no—
47C4×Dic83—22no—
48C4×Dic83—24no—
49C2×Q8×C83—22no—
50C2×Q8×C83—22no—
51C2×Q8×C83—22no—
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57C4⋊C4×C83—22no—
58C42×C83—22no—
59C23×C83C2411yescharacteristic
60Q8×C83—44no—
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83#4112yes—
74C2×C4×C83#4112yes—
75C2×C4×C83#4112yes—
76C2×Dic83—21no—
77C2×Dic83—21no—
78C2×Dic83—21no—
79C2×Dic83—21no—
80C2×Dic83—22no—
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83#4112yes—
93C2×Dic83#4112yes—
94C22×C83C2×#4111yescharacteristic
95C22×C83C2×#4111yescharacteristic
96C22×C83C2×#4111yescharacteristic
97C22×C83C2×#4111yescharacteristic
98C22×C83C2×#4111yescharacteristic
99C22×C832−1+411yescharacteristic
100C22×C832−1+411yescharacteristic
101C4×C83—42no—
102C4×C83—42no—
103C4×C83—41no—
104C4×C83—41no—
105C4×C83#12614yes—
106Dic83—22no—
107Dic83—22no—
108Dic83—41no—
109Dic83—41no—
110Dic83—42no—
111Dic83—42no—
112C2×C83C2×#12611yescharacteristic
113C2×C83#52911yescharacteristic
114C2×C83#54111yescharacteristic
115C2×C83#55211yescharacteristic
116C2×C83#55611yescharacteristic
117C2×C83#55611yescharacteristic
118C2×C83#55611yescharacteristic
119#2514—831nomaximal, Sylow
120C83#251411yescharacteristic, complemented, Sylow
121#401—832no—
122#400—832no—
123#391—831no—
124#391—831no—
125#391D8311yescharacteristic
126#389—831no—
127#387—832no—
128#383—832no—
129#382—831no—
130C4×C4⋊C4—831no—
131C4×C4⋊C4—831no—
132C22×Q8C2×D8311yescharacteristic
133C2×C4⋊C4—831no—
134C2×C4⋊C4—831no—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C42—831no—
140C2×C42—832no—
141C2×C42—831no—
142C2×C42C2×D8311yescharacteristic
143#95—831no—
144#95—832no—
145#95—831no—
146#95—832no—
147#95—832no—
148#95—832no—
149#95—832no—
150#95—832no—
151#95—832no—
152#95—832no—
153#95C2×D8312yes—
154#95C2×D8312yes—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C22×C4—831no—
159C22×C4—831no—
160C22×C4—831no—
161C22×C4—832no—
162C22×C4—831no—
163C22×C4—832no—
164C22×C4C22×D8312yes—
165C22×C4C22×D8311yescharacteristic
166C22×C4C22×D8311yescharacteristic
167C22×C4C22×D8312yes—
168C22×C4C22×D8311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1664no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C42—22no—
176C42—1664no—
177C42—1662no—
178C42—1662no—
179C23C23×D8311yescharacteristic
180Q8—44no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—832no—
194C2×C4—832no—
195C2×C4—1662no—
196C2×C4—1661no—
197C2×C4—1662no—
198C2×C4—1661no—
199C2×C4—1661no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1661no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4—1661no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4—1661no—
210C2×C4—1661no—
211C2×C4#1305058912yes—
212C2×C4#1305058912yes—
213C2×C4#1305059012yes—
214C22#41×D8311yescharacteristic
215C22#41×D8311yescharacteristic
216C22C2×#1305058911yescharacteristic
217C22C2×#1305058911yescharacteristic
218C22C2×#1305059011yescharacteristic
219C22#1458184011yescharacteristic
220C22#1458184011yescharacteristic
221C4—41no—
222C4—41no—
223C4—42no—
224C4—42no—
225C4—3321no—
226C4—3322no—
227C4—3322no—
228C4—3321no—
229C4—1662no—
230C4—1662no—
231C4#1458178514yes—
232C2C2×#1458178511yescharacteristic
233C2#1912823011yescharacteristic
234C2#1912831211yescharacteristic
235C2#1912838511yescharacteristic
236C2#1912842211yescharacteristic
237C2#1912842311yescharacteristic
238C2#1912842311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#391×C83C2
hypercenter#391D83
hypocenterC83#2514
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
120. C83#2514semidirect

Maximal subgroups

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

subgroupindex
C83⋊#4012
#191276422
#191276082
#191275972
#191275832
#191275682
#191275672
C4⋊C4×Dic832
C4×Dic83⋊C42
#391×C832
#251483