Finite Group Databasebeta

← #24913140

Subgroups — #24913140

Order 10624.

540 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581749C212yesmaximal, complemented
2#19127631C211yesmaximal, characteristic, complemented
3#19127623C212yesmaximal, complemented
4#19127618C211yesmaximal, characteristic
5#19127593C212yesmaximal, complemented
6#19127586C212yesmaximal, complemented
7#19127579C212yesmaximal
8#19127575C212yesmaximal, complemented
9#382×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C4×D83C2212yes—
12#14581749—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×C4×Dic83C2212yes—
23C83⋊#95C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2212yes—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2212yes—
31#95×C83C2212yes—
32C23×D83C2312yes—
33C2×D332—24no—
34C2×D332—41no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C2×C4×D83—24no—
39C2×C4×D83—24no—
40C2×C4×D83—24no—
41C4⋊Dic83—24no—
42C4⋊Dic83—41no—
43Dic83⋊C4—24no—
44Dic83⋊C4—24no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—24no—
47(C2×D83)⋊C4—24no—
48(C2×D83)⋊C4—24no—
49(C2×D83)⋊C4—42no—
50C4×Dic83—24no—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2312yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—41no—
57C4⋊C4×C83—41no—
58C42×C83D414yescomplemented
59C23×C83C2411yescharacteristic
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—22no—
65C22×D83—22no—
66C22×D83#4114yes—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83C2×D414yes—
79C2×C4×C83C2×D412yes—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83#4114yes—
89D332—44no—
90C4×D83—216no—
91C22×C83C22×D411yescharacteristic
92C22×C83C2×#4112yes—
93C22×C83C2×#4112yes—
94C22×C83H(5,𝔽2)11yescharacteristic
95C22×C832−1+411yescharacteristic
96C2×D83—24no—
97C2×D83—24no—
98C2×D83—24no—
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—42no—
102C2×D83—42no—
103C4×C83—24no—
104C4×C83—22no—
105C4×C83—41no—
106C4×C83—41no—
107C4×C83—42no—
108Dic83—24no—
109Dic83—42no—
110Dic83—42no—
111C2×C83#52512yes—
112C2×C83#53711yescharacteristic
113C2×C83#55111yescharacteristic
114C2×C83#55212yes—
115C2×C83#55511yescharacteristic
116D83—44no—
117#2609—831nomaximal, Sylow
118C83#260911yescharacteristic, complemented, Sylow
119C2×#117—832no—
120#398—831no—
121#396—832no—
122#394—831no—
123#388—832no—
124#387—832no—
125#386—832no—
126#385—832no—
127#382D8311yescharacteristic, complemented
128C22×D4—831no—
129C23×C4—832no—
130#117—1668no—
131C2×C4⋊C4—831no—
132C2×C4⋊C4—832no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4C2×D8311yescharacteristic
136C2×C4⋊C4C2×D8311yescharacteristic
137C2×C22⋊C4—832no—
138C2×C22⋊C4—832no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C42—832no—
143C2×C42C2×D8311yescharacteristic
144#95—831no—
145#95—832no—
146#95—831no—
147#95—832no—
148#95C2×D8312yes—
149#95C2×D8312yes—
150C24—832no—
151C2×D4—3321no—
152C2×D4—1664no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4—1664no—
157C22×C4—1664no—
158C22×C4—1664no—
159C22×C4C22×D8311yescharacteristic
160C22×C4C22×D8311yescharacteristic
161C22×C4C22×D8312yes—
162C22×C4C22×D8312yes—
163C22×C4C22×D8311yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—3321no—
167C4⋊C4—1664no—
168C4⋊C4—1664no—
169C4⋊C4—1664no—
170C22⋊C4—3322no—
171C22⋊C4—1664no—
172C22⋊C4—1664no—
173C22⋊C4—1664no—
174C22⋊C4—1664no—
175C42—1664no—
176C42D33214yescomplemented
177C23—834no—
178C23—1662no—
179C23—1662no—
180C23—1662no—
181C23—1662no—
182C23—1662no—
183C23—1662no—
184C23C23×D8311yescharacteristic
185D4—3324no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—834no—
198C2×C4—16616no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4C2×D33214yes—
208C2×C4C2×D33212yes—
209C22—3322no—
210C22—3322no—
211C22—3322no—
212C22—3322no—
213C22—1664no—
214C22—1664no—
215C22—1664no—
216C22C22×D33211yescharacteristic
217C22#41×D8312yes—
218C22#41×D8312yes—
219C22#1458183711yescharacteristic
220C22#1458184011yescharacteristic
221C4—24no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—42no—
226C4—3322no—
227C4—3322no—
228C4—1664no—
229C2—3324no—
230C2#1912821312yes—
231C2#1912828211yescharacteristic
232C2#1912838111yescharacteristic
233C2#1912838312yes—
234C2#1912841311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#382×C83C2
prime radicalC22×D332C22
hypercenter#382D83
hypocenterC83#2609
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#14581749C2semidirect
2. #19127631C2semidirect
3. #19127623C2semidirect
5. #19127593C2semidirect
6. #19127586C2semidirect
8. #19127575C2semidirect
9. #382×C83C2semidirect
58. C42×C83D4semidirect
118. C83#2609semidirect
127. #382D83semidirect
176. C42D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145817492
#191276312
#191276232
#191276182
#191275932
#191275862
#191275792
#191275752
#382×C832
#260983