Finite Group Databasebeta

← #24913603

Subgroups — #24913603

Order 10624.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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
2#19127623C212yesmaximal, complemented
3#19127621C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127617C212yesmaximal, complemented
6#19127613C211yesmaximal, characteristic, complemented
7#19127610C211yesmaximal, characteristic
8#19127601C212yesmaximal, complemented
9#19127591C212yesmaximal, complemented
10C4×C4⋊Dic83C211yesmaximal, characteristic, complemented
11C2×#126×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C22×C22⋊D83C2212yes—
15#14581780—24no—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29#126×C83—24no—
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C42×C83C2211yescharacteristic
37C23×D83C2312yes—
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×D332—22no—
41C2×D332—42no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—42no—
44C2×C22⋊D83—42no—
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C22×Dic83C2312yes—
48C4⋊Dic83—24no—
49Dic83⋊C4—42no—
50C22⋊Dic83—42no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—42no—
55(C2×D83)⋊C4—42no—
56C4×Dic83—42no—
57C24×C83C2311yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C4⋊C4×C83—22no—
65C4⋊C4×C83—22no—
66C4⋊C4×C83—22no—
67C22⋊C4×C83—22no—
68C22⋊C4×C83—22no—
69C22⋊C4×C83—22no—
70C42×C83D414yescomplemented
71C23×C83—21no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83C2411yescharacteristic
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D412yes—
98C22⋊D83—48no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83#4114yes—
108D332—44no—
109Dic166—44no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83—41no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83C22×D411yescharacteristic
116C22×C83C2×#4112yes—
117C22×C83H(5,𝔽2)12yes—
118C22×C83H(5,𝔽2)11yescharacteristic
119C22×C832−1+411yescharacteristic
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83—42no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—42no—
127C4×C83—22no—
128C4×C83—22no—
129C4×C83—22no—
130C4×C83—41no—
131C4×C83—41no—
132C4×C83—41no—
133Dic83—24no—
134Dic83—42no—
135Dic83—42no—
136C2×C83—81no—
137C2×C83#53211yescharacteristic
138C2×C83#53511yescharacteristic
139C2×C83#53711yescharacteristic
140C2×C83#53812yes—
141C2×C83#54212yes—
142D83—82no—
143#2662—831nomaximal, Sylow
144C83#266211yescharacteristic, complemented, Sylow
145C2×#126D8311yescharacteristic, complemented
146C2×#124—831no—
147#396—832no—
148#395—831no—
149#394—831no—
150#394—832no—
151#393—831no—
152#392—831no—
153#390—832no—
154#388—832no—
155C4×C4⋊C4—831no—
156C22×Q8—831no—
157C22×D4—832no—
158C22×D4—831no—
159#126—24no—
160#124—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4C2×D8311yescharacteristic
164C2×C4⋊C4C2×D8311yescharacteristic
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4C2×D8311yescharacteristic
173C2×C22⋊C4C2×D8311yescharacteristic
174C2×C22⋊C4C2×D8311yescharacteristic
175C2×C42—832no—
176C2×C42C2×D8311yescharacteristic
177#95—831no—
178#95—831no—
179#95—832no—
180#95—832no—
181C24—832no—
182C24C22×D8311yescharacteristic
183C2×Q8—3322no—
184C2×Q8—1662no—
185C2×D4—3322no—
186C2×D4—3322no—
187C2×D4—3322no—
188C2×D4—1664no—
189C2×D4—1662no—
190C22×C4—832no—
191C22×C4—832no—
192C22×C4—832no—
193C22×C4C22×D8311yescharacteristic
194C22×C4C22×D8311yescharacteristic
195C22×C4C22×D8311yescharacteristic
196C22×C4C22×D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—3322no—
203C4⋊C4—1664no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—3322no—
208C22⋊C4—3322no—
209C22⋊C4—3322no—
210C22⋊C4—1664no—
211C22⋊C4—1664no—
212C22⋊C4—1664no—
213C42—3322no—
214C42C22⋊D8314yescomplemented
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—1662no—
221C23—1662no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23—1662no—
227C23C23×D8311yescharacteristic
228Q8—3324no—
229D4—3324no—
230D4—3328no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—834no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4C2×C22⋊D8312yes—
252C2×C4C2×C22⋊D8312yes—
253C2×C4C2×C22⋊D8312yes—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—3322no—
260C22—3322no—
261C22—3322no—
262C22—3322no—
263C22—3322no—
264C22—3322no—
265C22—3322no—
266C22C22×C22⋊D8311yescharacteristic
267C22#41×D8312yes—
268C22#1458183812yes—
269C22#1458183811yescharacteristic
270C22#1458183911yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—3322no—
278C4—3322no—
279C4—1664no—
280C2—81no—
281C2—6642no—
282C2#1912824711yescharacteristic
283C2#1912826511yescharacteristic
284C2#1912827911yescharacteristic
285C2#1912829012yes—
286C2#1912831712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#126×C83C2
prime radical#19127610C2
hypercenterC2×#126D83
hypocenterC83#2662
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14581780C2semidirect
2. #19127623C2semidirect
3. #19127621C2semidirect
4. #19127618C2semidirect
5. #19127617C2semidirect
6. #19127613C2semidirect
8. #19127601C2semidirect
9. #19127591C2semidirect
10. C4×C4⋊Dic83C2semidirect
11. C2×#126×C83C2semidirect
70. C42×C83D4semidirect
144. C83#2662semidirect
145. C2×#126D83semidirect
214. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817802
#191276232
#191276212
#191276182
#191276172
#191276132
#191276102
#191276012
#191275912
C4×C4⋊Dic832
C2×#126×C832
#266283