Finite Group Databasebeta

← #24913360

Subgroups — #24913360

Order 10624.

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

normal subgroups only

This group has 562 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×C4×D332C211yesmaximal, characteristic, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127623C212yesmaximal, complemented
5#19127618C211yesmaximal, characteristic, complemented
6#19127617C212yesmaximal, complemented
7#19127610C211yesmaximal, characteristic
8#19127577C212yesmaximal, complemented
9#19127572C212yesmaximal, complemented
10#19127567C211yesmaximal, characteristic, complemented
11C2×#126×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C4×D83C2212yes—
15C4×D332—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C83⋊#95C2212yes—
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×D332—24no—
39C2×D332—22no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C2×C4×D83—24no—
47C2×C4×D83—42no—
48C2×C4×D83—42no—
49C4⋊Dic83—22no—
50C4⋊Dic83—42no—
51Dic83⋊C4—42no—
52C22⋊Dic83—42no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—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#4114yes—
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—22no—
108D332—28no—
109C4×D83—48no—
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—24no—
121C2×D83—24no—
122C2×D83—24no—
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—42no—
134Dic83—42no—
135Dic83—42no—
136C2×C83—81no—
137C2×C83#52511yescharacteristic
138C2×C83#53511yescharacteristic
139C2×C83#53711yescharacteristic
140C2×C83#53812yes—
141C2×C83#53912yes—
142D83—44no—
143#2636—831nomaximal, Sylow
144C83#263611yescharacteristic, complemented, Sylow
145C2×#126D8311yescharacteristic, complemented
146C2×C4×D4—831no—
147#400—831no—
148#399—831no—
149#396—832no—
150#394—831no—
151#394—832no—
152#392—831no—
153#386—832no—
154#385—832no—
155#382—831no—
156C22×D4—832no—
157C22×D4—831no—
158C23×C4—832no—
159#126—24no—
160C4×D4—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4C2×D8311yescharacteristic
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C4⋊C4C2×D8311yescharacteristic
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×C42C2×D8311yescharacteristic
176#95—832no—
177#95—831no—
178#95—831no—
179#95—832no—
180#95—832no—
181C24—832no—
182C24C22×D8311yescharacteristic
183C2×D4—3322no—
184C2×D4—3322no—
185C2×D4—1664no—
186C2×D4—1664no—
187C2×D4—1662no—
188C22×C4—3322no—
189C22×C4—3322no—
190C22×C4—832no—
191C22×C4—832no—
192C22×C4—832no—
193C22×C4—1664no—
194C22×C4C22×D8311yescharacteristic
195C22×C4C22×D8311yescharacteristic
196C22×C4C22×D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199C22×C4C22×D8311yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—3322no—
204C4⋊C4—3322no—
205C4⋊C4—1662no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—3322no—
210C22⋊C4—3322no—
211C22⋊C4—3322no—
212C22⋊C4—1664no—
213C22⋊C4—1664no—
214C42C22⋊D8314yescomplemented
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—834no—
221C23—1662no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23—1662no—
227C23C23×D8311yescharacteristic
228D4—3328no—
229D4—1668no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—3328no—
242C2×C4—1662no—
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—1664no—
264C22—1664no—
265C22—1664no—
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—3322no—
280C2—81no—
281C2—3324no—
282C2#1912820811yescharacteristic
283C2#1912826511yescharacteristic
284C2#1912827911yescharacteristic
285C2#1912829012yes—
286C2#1912829212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#126×C83C2
prime radical#19127610C2
hypercenterC2×#126D83
hypocenterC83#2636
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C4×D332C2semidirect
2. #19127640C2semidirect
3. #19127637C2semidirect
4. #19127623C2semidirect
5. #19127618C2semidirect
6. #19127617C2semidirect
8. #19127577C2semidirect
9. #19127572C2semidirect
10. #19127567C2semidirect
11. C2×#126×C83C2semidirect
70. C42×C83D4semidirect
144. C83#2636semidirect
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×C4×D3322
#191276402
#191276372
#191276232
#191276182
#191276172
#191276102
#191275772
#191275722
#191275672
C2×#126×C832
#263683