Finite Group Databasebeta

← #24911954

Subgroups — #24911954

Order 10624.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581763C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19127629C214yesmaximal, complemented
4#19127604C211yesmaximal, characteristic, complemented
5#19127577C214yesmaximal, complemented
6#19127567C211yesmaximal, characteristic, complemented
7#19127566C211yesmaximal, characteristic, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic, complemented
10C22×C22⋊D83C2212yes—
11C22×C4×D83C2211yescharacteristic
12C22×C4×D83C2212yes—
13#14581763—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C4⋊C4×D83—24no—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2214yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2214yes—
23C83⋊#95C2214yes—
24C83⋊#95C2212yes—
25C22×D4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2214yes—
28C2×C42×C83C2211yescharacteristic
29C23×D83C2311yescharacteristic
30C2×C22⋊D83—24no—
31C2×C22⋊D83—28no—
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2311yescharacteristic
36C2×C4×D83—22no—
37C2×C4×D83—22no—
38C2×C4×D83—24no—
39C2×C4×D83—44no—
40C2×C4×D83D414yes—
41C4⋊Dic83—22no—
42C4⋊Dic83—22no—
43Dic83⋊C4—24no—
44C22⋊Dic83—24no—
45C22⋊Dic83—44no—
46(C2×D83)⋊C4—24no—
47C4×Dic83—22no—
48C24×C83C2312yes—
49C2×D4×C83—22no—
50C2×D4×C83—22no—
51C2×D4×C83—22no—
52C22×C4×C83C2312yes—
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—22no—
56C22⋊C4×C83—44no—
57C42×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83—24no—
62C23×C83—24no—
63C23×C83C2411yescharacteristic
64C22×D83—21no—
65C22×D83—21no—
66C22×D83—22no—
67C22×D83C2×D412yes—
68C22×D83#4114yes—
69D4×C83—44no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83C2×D412yes—
76C2×C4×C83#4112yes—
77C2×C4×C83#4112yes—
78C22⋊D83—48no—
79C2×Dic83—21no—
80C2×Dic83—21no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83—24no—
88C2×Dic83C2×D412yes—
89C4×D83—28no—
90C4×D83—48no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—44no—
95C22×C83—44no—
96C22×C83C22×D411yescharacteristic
97C22×C83C2×#4112yes—
98C22×C83C2×#4111yescharacteristic
99C22×C83C2×#4111yescharacteristic
100C22×C83H(5,𝔽2)12yes—
101C2×D83—22no—
102C2×D83—22no—
103C2×D83—24no—
104C2×D83—24no—
105C2×D83#12414yes—
106C4×C83—42no—
107C4×C83—42no—
108C4×C83#12314yes—
109Dic83—22no—
110Dic83—42no—
111Dic83—42no—
112Dic83—42no—
113C2×C83—82no—
114C2×C83C2×#12311yescharacteristic
115C2×C83C2×#12411yescharacteristic
116C2×C83#53511yescharacteristic
117C2×C83#55312yes—
118C2×C83#55312yes—
119D83—24no—
120#2504—831nomaximal, Sylow
121C83#250411yescharacteristic, complemented, Sylow
122C2×#121—831no—
123C22×C4⋊C4—831no—
124#397—834no—
125#390—831no—
126#390D8311yescharacteristic, complemented
127#386—834no—
128#382—831no—
129#382—831no—
130C4×C22⋊C4—831no—
131C22×D4—832no—
132C22×D4C2×D8311yescharacteristic
133C23×C4—832no—
134C23×C4—831no—
135#121—1664no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4—831no—
139C2×C4⋊C4—1664no—
140C2×C4⋊C4C2×D8311yescharacteristic
141C2×C22⋊C4—834no—
142C2×C22⋊C4—832no—
143C2×C22⋊C4—832no—
144C2×C22⋊C4C2×D8314yes—
145C2×C42—831no—
146C2×C42C2×D8311yescharacteristic
147#95—832no—
148#95—834no—
149#95—834no—
150C24—831no—
151C24C22×D8312yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—1664no—
156C2×D4—1668no—
157C22×C4—3324no—
158C22×C4—831no—
159C22×C4—834no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4—832no—
163C22×C4—1664no—
164C22×C4—1662no—
165C22×C4—1662no—
166C22×C4C22×D8312yes—
167C22×C4C22×D8312yes—
168C22×C4C22×D8311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—1664no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C22⋊C4—44no—
174C22⋊C4—3324no—
175C22⋊C4—1664no—
176C22⋊C4—1664no—
177C42—22no—
178C42—1662no—
179C23—22no—
180C23—24no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—832no—
185C23—834no—
186C23—1661no—
187C23—1661no—
188C23—1662no—
189C23C23×D8311yescharacteristic
190D4—44no—
191D4—3328no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—3328no—
198C2×C4—832no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1664no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1664no—
206C2×C4—1661no—
207C2×C4—1661no—
208C2×C4—1668no—
209C2×C4D4×D8312yes—
210C2×C4#1305058912yes—
211C2×C4#1305059012yes—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—834no—
218C22—1664no—
219C22—1664no—
220C22—1662no—
221C22—1662no—
222C22C2×D4×D8311yescharacteristic
223C22#41×D8312yes—
224C22C2×#1305058911yescharacteristic
225C22C2×#1305059011yescharacteristic
226C22#1458183712yes—
227C4—42no—
228C4—42no—
229C4—3322no—
230C4—3322no—
231C4—3322no—
232C4—1662no—
233C4#1458177414yes—
234C2—82no—
235C2—1664no—
236C2C2×#1458177411yescharacteristic
237C2#124×D8311yescharacteristic
238C2#1912826411yescharacteristic
239C2#1912839812yes—
240C2#1912840112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC2×#14581763C2
hypercenter#390D83
hypocenterC83#2504
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19127629C2semidirect
4. #19127604C2semidirect
5. #19127577C2semidirect
6. #19127567C2semidirect
7. #19127566C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #390×C83C2semidirect
121. C83#2504semidirect
126. #390D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817632
C2×C4⋊C4×D832
#191276292
#191276042
#191275772
#191275672
#191275662
C4×(C2×D83)⋊C42
#390×C832
#250483