Finite Group Databasebeta

← #24913223

Subgroups — #24913223

Order 10624.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581780C211yesmaximal, characteristic, complemented
2C2×#14581772C211yesmaximal, characteristic, complemented
3C2×#14581767C211yesmaximal, characteristic
4C2×#14581749C211yesmaximal, characteristic, complemented
5#19127623C214yesmaximal, complemented
6#19127618C214yesmaximal, complemented
7#19127606C211yesmaximal, characteristic, complemented
8#19127602C211yesmaximal, characteristic, complemented
9#384×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C22×D332C2212yes—
13C22×C4×D83C2211yescharacteristic
14#14581780—24no—
15#14581772—24no—
16#14581767—24no—
17#14581749—24no—
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2214yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2214yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2214yes—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2214yes—
30C2×C42×C83C2211yescharacteristic
31C23×D83C2312yes—
32C23×D83C2311yescharacteristic
33C2×Dic166—22no—
34C2×Dic166—22no—
35C2×Dic166—22no—
36C2×D332—24no—
37C2×D332—22no—
38C2×D332—22no—
39C2×D332—22no—
40C2×D332—44no—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C2×C4×D83—22no—
45C2×C4×D83—22no—
46C2×C4×D83D414yes—
47C4⋊Dic83—24no—
48Dic83⋊C4—24no—
49(C2×D83)⋊C4—24no—
50(C2×D83)⋊C4—28no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—44no—
54C4×Dic83—22no—
55C22×C4×C83C2312yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—22no—
60C4⋊C4×C83—22no—
61C4⋊C4×C83—44no—
62C42×C83D414yescomplemented
63C23×C83C2411yescharacteristic
64C22×D83—21no—
65C22×D83—21no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—24no—
72C22×D83—24no—
73C22×D83C2×D412yes—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—24no—
79C2×C4×C83—24no—
80C2×C4×C83C2×D412yes—
81C2×C4×C83C2×D414yes—
82C2×C4×C83C2×D412yes—
83C2×C4×C83#4112yes—
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83C2×D412yes—
93D332—48no—
94D332—44no—
95C4×D83—28no—
96Dic166—44no—
97C22×C83C22×D411yescharacteristic
98C22×C83C22×D411yescharacteristic
99C22×C83C2×#4111yescharacteristic
100C22×C83H(5,𝔽2)12yes—
101C22×C832−1+412yes—
102C2×D83—22no—
103C2×D83—22no—
104C2×D83—22no—
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—44no—
111C2×D83—44no—
112C4×C83—24no—
113C4×C83—42no—
114C4×C83—42no—
115C4×C83#12114yes—
116Dic83—22no—
117Dic83—42no—
118Dic83—42no—
119C2×C83C2×#12111yescharacteristic
120C2×C83#53511yescharacteristic
121C2×C83#53611yescharacteristic
122C2×C83#53712yes—
123C2×C83#54212yes—
124D83—42no—
125D83—82no—
126#2620—831nomaximal, Sylow
127C83#262011yescharacteristic, complemented, Sylow
128C2×#124—831no—
129C2×C22⋊Q8—831no—
130C2×#121—831no—
131C2×#117—831no—
132#396—834no—
133#394—834no—
134#391—831no—
135#390—831no—
136#384D8311yescharacteristic, complemented
137C22×Q8—831no—
138C22×D4—832no—
139C22×D4—831no—
140C23×C4—831no—
141#124—1664no—
142C22⋊Q8—1664no—
143#121—1664no—
144#117—1664no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C4⋊C4C2×D8311yescharacteristic
149C2×C4⋊C4C2×D8314yes—
150C2×C22⋊C4—834no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4—834no—
155C2×C42—831no—
156C2×C42C2×D8311yescharacteristic
157#95—834no—
158C24—831no—
159C24—832no—
160C2×Q8—1662no—
161C2×Q8—1662no—
162C2×Q8—1662no—
163C2×D4—3324no—
164C2×D4—1664no—
165C2×D4—1662no—
166C2×D4—1662no—
167C2×D4—1662no—
168C22×C4—831no—
169C22×C4—834no—
170C22×C4—832no—
171C22×C4—832no—
172C22×C4—1662no—
173C22×C4—1662no—
174C22×C4C22×D8312yes—
175C22×C4C22×D8312yes—
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—44no—
181C4⋊C4—1664no—
182C4⋊C4—1664no—
183C22⋊C4—3324no—
184C22⋊C4—1664no—
185C22⋊C4—1664no—
186C22⋊C4—1664no—
187C22⋊C4—1668no—
188C42—1662no—
189C42D33214yescomplemented
190C23—832no—
191C23—1662no—
192C23—1662no—
193C23—1662no—
194C23—1664no—
195C23—1664no—
196C23—1661no—
197C23—1661no—
198C23—1662no—
199C23—1662no—
200C23C23×D8311yescharacteristic
201Q8—3324no—
202D4—3328no—
203D4—3324no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—832no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1661no—
218C2×C4—1661no—
219C2×C4—1668no—
220C2×C4D4×D8312yes—
221C2×C4C2×D33214yes—
222C2×C4C2×D33212yes—
223C2×C4#1305059112yes—
224C22—3322no—
225C22—3322no—
226C22—3324no—
227C22—3322no—
228C22—3322no—
229C22—3322no—
230C22—3324no—
231C22—1662no—
232C22—1662no—
233C22—1662no—
234C22C2×D4×D8311yescharacteristic
235C22C22×D33211yescharacteristic
236C22C2×#1305059111yescharacteristic
237C22#1458183712yes—
238C22#1458184012yes—
239C4—24no—
240C4—42no—
241C4—42no—
242C4—3322no—
243C4—3322no—
244C4—1662no—
245C4#1458176714yes—
246C2—3322no—
247C2—6642no—
248C2C2×#1458176711yescharacteristic
249C2#1912826411yescharacteristic
250C2#1912827611yescharacteristic
251C2#1912828212yes—
252C2#1912832512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#384×C83C2
prime radicalC2×#14581767C2
hypercenter#384D83
hypocenterC83#2620
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#14581780C2semidirect
2. C2×#14581772C2semidirect
4. C2×#14581749C2semidirect
5. #19127623C2semidirect
6. #19127618C2semidirect
7. #19127606C2semidirect
8. #19127602C2semidirect
9. #384×C83C2semidirect
62. C42×C83D4semidirect
127. C83#2620semidirect
136. #384D83semidirect
189. C42D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145817802
C2×#145817722
C2×#145817672
C2×#145817492
#191276232
#191276182
#191276062
#191276022
#384×C832
#262083