Finite Group Databasebeta

← #24918070

Subgroups — #24918070

Order 10624.

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

normal subgroups only

This group has 364 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#19128258C211yesmaximal, characteristic, complemented
2#19128111C211yesmaximal, characteristic, complemented
3#19128096C211yesmaximal, characteristic, complemented
4#19128071C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19127957C212yesmaximal, complemented
7#19127945C212yesmaximal
8C4×#13050580C212yesmaximal, complemented
9C4×#13050578C212yesmaximal, complemented
10C4×#13050574C211yesmaximal, characteristic, complemented
11#529×C83C211yesmaximal, characteristic
12C2×#13050580C2212yes—
13C2×#13050578C2212yes—
14C2×#13050574C2212yes—
15#14581792C2211yescharacteristic
16#14581792C2211yescharacteristic
17#14581784—21no—
18#14581771—21no—
19#14581771C2212yes—
20C4×Dic166—21no—
21C4×Dic166C2212yes—
22#14581751—21no—
23#14581725C2212yes—
24#14581723C2212yes—
25#14581716C2212yes—
26#14581715C2212yes—
27#14581715C2212yes—
28#14581713C2212yes—
29#14581703C2211yescharacteristic
30#14581703C2211yescharacteristic
31C4×C83⋊C8C2212yes—
32#126×C83—21no—
33#125×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#123×C83—21no—
36C22⋊Q8×C83C2211yescharacteristic
37#121×C83C2211yescharacteristic
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83—21no—
40C4×D4×C83C2211yescharacteristic
41#117×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166C2312yes—
45#13050580—24no—
46#13050578—24no—
47#13050574—24no—
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83C2312yes—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56C22⋊Dic83—22no—
57C4×Dic83—22no—
58C2×Q8×C83C2311yescharacteristic
59C2×D4×C83—21no—
60C2×D4×C83C2311yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83D411yescharacteristic
63C22×C4×C83D411yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—21no—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C22⋊C4×C83—21no—
74C22⋊C4×C83—21no—
75C22⋊C4×C83—21no—
76C42×C83—21no—
77C42×C83D411yescharacteristic
78C42×C83D411yescharacteristic
79C42×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C23×C83C2×D411yescharacteristic
83Q8×C83—22no—
84D4×C83—22no—
85D4×C83—42no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—41no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C2×Dic83—22no—
100C2×Dic83—22no—
101Dic166—22no—
102Dic166—22no—
103Dic166—42no—
104C83⋊C8—22no—
105C83⋊C8#4114yes—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83—21no—
109C22×C83—41no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83C22×D411yescharacteristic
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—21no—
116C4×C83—21no—
117C4×C83—41no—
118C4×C83—41no—
119C4×C83—41no—
120C4×C83#13712yes—
121C4×C83C2×#4112yes—
122Dic83—42no—
123Dic83—42no—
124C2×C83—41no—
125C2×C83—81no—
126C2×C83#53211yescharacteristic
127C2×C83C2×#13711yescharacteristic
128C2×C83#57511yescharacteristic
129#3172—831nomaximal, Sylow
130C83#317211yescharacteristic, complemented, Sylow
131#533—831no—
132#529D8311yescharacteristic
133#501—831no—
134#497—831no—
135#491—831no—
136#489—831no—
137#470—832no—
138#468—832no—
139C4×Q16—832no—
140C4×SD16—832no—
141C4×M16—831no—
142C2×Q16—832no—
143C2×SD16—832no—
144C2×M16—832no—
145C4⋊Q8—831no—
146C4⋊Q8—831no—
147#126—21no—
148#125—1661no—
149#125C2×D8311yescharacteristic
150#124C2×D8311yescharacteristic
151#123—21no—
152C22⋊Q8—832no—
153C22⋊Q8—1661no—
154C22⋊Q8C2×D8311yescharacteristic
155#121C2×D8311yescharacteristic
156C4×Q8—832no—
157C4×Q8—1661no—
158C4×Q8C2×D8311yescharacteristic
159C4×D4—21no—
160C4×D4C2×D8311yescharacteristic
161#117—21no—
162#117—1661no—
163C2×C42C2×D8311yescharacteristic
164#107—832no—
165#106—832no—
166#103—832no—
167#103—832no—
168#103—832no—
169#102—832no—
170#97—831no—
171#97—831no—
172C4×C8—832no—
173C2×Q8—832no—
174C2×Q8—1661no—
175C2×Q8C22×D8311yescharacteristic
176C2×D4—21no—
177C2×D4C22×D8311yescharacteristic
178C22×C4—21no—
179C22×C4C22⋊D8311yescharacteristic
180C22×C4C22⋊D8311yescharacteristic
181C22×C4C22×D8311yescharacteristic
182Q16—1664no—
183SD16—1664no—
184M16—1664no—
185C2×C8—832no—
186C2×C8—832no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—832no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C4⋊C4—1661no—
197C4⋊C4C22×D8311yescharacteristic
198C4⋊C4C22×D8311yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—1662no—
205C42—21no—
206C42—1662no—
207C42C22⋊D8311yescharacteristic
208C42C22⋊D8311yescharacteristic
209C42C22×D8311yescharacteristic
210C42C22×D8311yescharacteristic
211C23—21no—
212C23C2×C22⋊D8311yescharacteristic
213Q8—22no—
214Q8—3322no—
215Q8—1662no—
216Q8—1662no—
217D4—22no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4C2×C22⋊D8311yescharacteristic
229C2×C4C2×C22⋊D8311yescharacteristic
230C2×C4C2×C22⋊D8311yescharacteristic
231C2×C4C2×C22⋊D8311yescharacteristic
232C2×C4C2×C22⋊D8311yescharacteristic
233C2×C4C23×D8311yescharacteristic
234C8—834no—
235C8—1662no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22C22×C22⋊D8311yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—3322no—
251C4—3322no—
252C4#1458182212yes—
253C4#41×D8312yes—
254C2—41no—
255C2—81no—
256C2#1912824711yescharacteristic
257C2C2×#1458182211yescharacteristic
258C2#1912849911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#529×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#529D83
hypocenterC83#3172
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
2. #19128111C2semidirect
3. #19128096C2semidirect
4. #19128071C2semidirect
5. #19128060C2semidirect
6. #19127957C2semidirect
8. C4×#13050580C2semidirect
9. C4×#13050578C2semidirect
10. C4×#13050574C2semidirect
130. C83#3172semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191281112
#191280962
#191280712
#191280602
#191279572
#191279452
C4×#130505802
C4×#130505782
C4×#130505742
#529×C832
#317283