Finite Group Databasebeta

← #24917499

Subgroups — #24917499

Order 10624.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128253C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic
3#19128036C211yesmaximal, characteristic
4#19127995C212yesmaximal, complemented
5#19127910C212yesmaximal, complemented
6#19127859C211yesmaximal, characteristic
7#19127856C211yesmaximal, characteristic
8#19127795C212yesmaximal, complemented
9#19127793C212yesmaximal, complemented
10#19127751C211yesmaximal, characteristic
11#484×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C2×Dic332C2212yes—
14C2×#13050580C2212yes—
15#14581791C2211yescharacteristic
16#14581790C2211yescharacteristic
17#14581782—21no—
18#14581771C2211yescharacteristic
19#14581769—21no—
20Q8×Dic83C2211yescharacteristic
21#14581757—21no—
22#14581757—21no—
23#14581757—21no—
24#14581757C2212yes—
25#14581757C2212yes—
26#14581757C2211yescharacteristic
27#14581748—21no—
28C2×Dic83⋊C4—21no—
29C2×C4×Dic83—21no—
30C2×C4×Dic83C2211yescharacteristic
31#14581725C2212yes—
32Dic83⋊C8C2212yes—
33#14581717C2212yes—
34#14581715C2212yes—
35#14581715C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#107×C83C2212yes—
41#103×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—24no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47C2×Dic166C2312yes—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51Dic332—24no—
52#13050580—24no—
53C2×C83⋊C8C2312yes—
54C4⋊Dic83—21no—
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—22no—
62Dic83⋊C4—22no—
63C22⋊Dic83—21no—
64C4×Dic83—22no—
65C4×Dic83—22no—
66C4×Dic83—21no—
67C4×Dic83—21no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×Q8×C83C2311yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83C2311yescharacteristic
75C2×C8×C83C2312yes—
76C4⋊C4×C83—21no—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—21no—
81C23×C83C2×D411yescharacteristic
82Q8×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—42no—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C2×Dic83—24no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—41no—
95C2×Dic83Q1614yescomplemented
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100Dic166—24no—
101Dic166—22no—
102Dic166—22no—
103Dic166—42no—
104Dic166#4114yes—
105C8×C83—22no—
106C83⋊C8—22no—
107C22×C83—21no—
108C22×C83—21no—
109C22×C83C2×Q1612yes—
110C22×C83C22×D411yescharacteristic
111C4×C83—21no—
112C4×C83—41no—
113C4×C83—41no—
114C4×C83—42no—
115C4×C83C2×#4112yes—
116Dic83—24no—
117Dic83—21no—
118Dic83—21no—
119Dic83—41no—
120Dic83—41no—
121Dic83C2×Q1612yes—
122Dic83C2×Q1612yes—
123C2×C83—22no—
124C2×C83#52511yescharacteristic
125C2×C83C22×Q1611yescharacteristic
126C2×C83#57511yescharacteristic
127#3104—831nomaximal, Sylow
128C83#310411yescharacteristic, complemented, Sylow
129#533—831no—
130C2×C4×Q8—831no—
131#484—831no—
132#484D8311yescharacteristic
133#477—832no—
134#462—832no—
135#451—831no—
136#451—831no—
137C4×Q16—832no—
138C4×Q16—832no—
139#431—831no—
140C22×Q8—831no—
141C2×Q16—832no—
142C2×Q16—832no—
143C4⋊Q8—831no—
144C4⋊Q8—831no—
145#125—1661no—
146C22⋊Q8—831no—
147C22⋊Q8—1661no—
148C22⋊Q8C2×D8311yescharacteristic
149C4×Q8—831no—
150C4×Q8—832no—
151C4×Q8—832no—
152C4×Q8—831no—
153C4×Q8—1661no—
154C4×Q8—1661no—
155C4×Q8—1661no—
156#117—1661no—
157C2×C4⋊C4—1661no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C42—831no—
160C2×C42—1661no—
161#107—832no—
162#107C2×D8312yes—
163C4⋊C8—832no—
164#103—832no—
165#103—832no—
166#103—832no—
167#103C2×D8312yes—
168C22⋊C8—831no—
169C22⋊C8C2×D8311yescharacteristic
170C4×C8—832no—
171C2×Q8—832no—
172C2×Q8—831no—
173C2×Q8—1661no—
174C2×Q8—1664no—
175C2×Q8—1661no—
176C2×Q8C22×D8311yescharacteristic
177C22×C4—21no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—1661no—
181C22×C4C22×D8311yescharacteristic
182Q16—1664no—
183Q16—1664no—
184C2×C8—832no—
185C2×C8C22×D8312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—831no—
189C4⋊C4—1662no—
190C4⋊C4—1662no—
191C4⋊C4—1661no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4C22×D8311yescharacteristic
198C4⋊C4C22×D8312yes—
199C22⋊C4—21no—
200C22⋊C4—1661no—
201C42—831no—
202C42—831no—
203C42—831no—
204C42—831no—
205C42—1662no—
206C42—1662no—
207C42—1661no—
208C42—1661no—
209C23D4×D8311yescharacteristic
210Q8—22no—
211Q8—3322no—
212Q8—834no—
213Q8—1664no—
214Q8—1662no—
215Q8—1662no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—3321no—
222C2×C4—834no—
223C2×C4—831no—
224C2×C4—831no—
225C2×C4—831no—
226C2×C4—831no—
227C2×C4—1661no—
228C2×C4—1661no—
229C2×C4—1664no—
230C2×C4—1662no—
231C2×C4D4×D8311yescharacteristic
232C2×C4C23×D8311yescharacteristic
233C8—22no—
234C8—1662no—
235C22—21no—
236C22—21no—
237C22Q16×D8312yes—
238C22C2×D4×D8311yescharacteristic
239C4—21no—
240C4—41no—
241C4—42no—
242C4—41no—
243C4—3321no—
244C4—3321no—
245C4—832no—
246C4—832no—
247C4—1664no—
248C4—1661no—
249C4—1661no—
250C4#41×D8312yes—
251C2—22no—
252C2#1912820911yescharacteristic
253C2C2×Q16×D8311yescharacteristic
254C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#484×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenter#484D83
hypocenterC83#3104
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
4. #19127995C2semidirect
5. #19127910C2semidirect
8. #19127795C2semidirect
9. #19127793C2semidirect
95. C2×Dic83Q16semidirect
128. C83#3104semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×#145817572
#191280362
#191279952
#191279102
#191278592
#191278562
#191277952
#191277932
#191277512
#484×C832
#310483