Finite Group Databasebeta

← #24914228

Subgroups — #24914228

Order 10624.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581779C212yesmaximal, complemented
2#19127626C211yesmaximal, characteristic, complemented
3#19127615C212yesmaximal, complemented
4#19127613C212yesmaximal, complemented
5#19127605C212yesmaximal, complemented
6#19127594C212yesmaximal, complemented
7#19127581C212yesmaximal, complemented
8#19127573C211yesmaximal, characteristic, complemented
9#392×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2211yescharacteristic
12#14581779—28no—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C4×Dic83C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2211yescharacteristic
27C22×D4×C83C2212yes—
28C22×D4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2211yescharacteristic
31#95×C83C2211yescharacteristic
32C2×Dic166—24no—
33C2×Dic166—24no—
34C2×Dic166—42no—
35C22×Dic83—41no—
36C22×Dic83—42no—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C4⋊Dic83—41no—
42Dic83⋊C4—42no—
43C22⋊Dic83—24no—
44C22⋊Dic83—24no—
45C22⋊Dic83—24no—
46C22⋊Dic83—42no—
47C22⋊Dic83—42no—
48C22⋊Dic83D414yescomplemented
49C4×Dic83—24no—
50C24×C83C2312yes—
51C24×C83C2311yescharacteristic
52C2×D4×C83—24no—
53C2×D4×C83—24no—
54C2×D4×C83—42no—
55C2×D4×C83—41no—
56C2×D4×C83—42no—
57C22×C4×C83C2311yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22⋊C4×C83—24no—
61C22⋊C4×C83—41no—
62C23×C83—22no—
63C23×C83—21no—
64C23×C83—22no—
65C23×C83—21no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2×D412yes—
74C23×C83C2411yescharacteristic
75D4×C83—48no—
76D4×C83—44no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83#4114yes—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—44no—
96C2×Dic83C2×D414yes—
97Dic166—48no—
98C22×C83—22no—
99C22×C83—24no—
100C22×C83—42no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—42no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)11yescharacteristic
113C22×C83H(5,𝔽2)12yes—
114C22×C832−1+411yescharacteristic
115C4×C83—24no—
116C4×C83—41no—
117C4×C83—41no—
118Dic83—24no—
119Dic83—42no—
120Dic83—42no—
121Dic83—42no—
122C2×C83—42no—
123C2×C83—82no—
124C2×C83#53711yescharacteristic
125C2×C83#54212yes—
126C2×C83#54612yes—
127C2×C83#55911yescharacteristic
128C2×C83H(3,𝔽4)11yescharacteristic
129#2712—831nomaximal, Sylow
130C83#271211yescharacteristic, complemented, Sylow
131C2×#124—832no—
132#397—831no—
133#394—832no—
134#393—832no—
135#392D8311yescharacteristic
136#391—832no—
137#388—832no—
138#386—832no—
139#385—831no—
140C22×Q8—832no—
141C22×D4C2×D8312yes—
142C22×D4C2×D8311yescharacteristic
143C23×C4—831no—
144#124—1668no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4—831no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—831no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4C2×D8312yes—
154C2×C22⋊C4C2×D8311yescharacteristic
155C2×C42—832no—
156#95—832no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95—831no—
161#95C2×D8311yescharacteristic
162C24C22×D8312yes—
163C24C22×D8311yescharacteristic
164C2×Q8—3322no—
165C2×Q8—1664no—
166C2×Q8—1664no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—3321no—
173C22×C4—3322no—
174C22×C4—832no—
175C22×C4—832no—
176C22×C4—832no—
177C22×C4—832no—
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8311yescharacteristic
180C22×C4C22×D8312yes—
181C4⋊C4—3322no—
182C4⋊C4—3321no—
183C22⋊C4—24no—
184C22⋊C4—41no—
185C22⋊C4—3322no—
186C22⋊C4—3322no—
187C22⋊C4—834no—
188C22⋊C4—1664no—
189C22⋊C4—1664no—
190C22⋊C4—1664no—
191C42—1664no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23D4×D8312yes—
204C23C23×D8311yescharacteristic
205Q8—3328no—
206D4—48no—
207D4—44no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—3324no—
215C2×C4—834no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4#1305058914yes—
228C22—24no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C2×D4×D8311yescharacteristic
241C22C2×#1305058912yes—
242C22#1458183712yes—
243C22#1458183711yescharacteristic
244C22#1458183911yescharacteristic
245C4—24no—
246C4—41no—
247C4—41no—
248C4—3322no—
249C4—3322no—
250C4—3322no—
251C4—1664no—
252C2—42no—
253C2—82no—
254C2#1912827811yescharacteristic
255C2#1912831812yes—
256C2#1912833812yes—
257C2#1912843711yescharacteristic
258C2#1912844411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radical#392×C83C2
hypercenter#392D83
hypocenterC83#2712
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581779C2semidirect
2. #19127626C2semidirect
3. #19127615C2semidirect
4. #19127613C2semidirect
5. #19127605C2semidirect
6. #19127594C2semidirect
7. #19127581C2semidirect
8. #19127573C2semidirect
48. C22⋊Dic83D4semidirect
130. C83#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
#191276262
#191276152
#191276132
#191276052
#191275942
#191275812
#191275732
#392×C832
#271283