Finite Group Databasebeta

← #24913611

Subgroups — #24913611

Order 10624.

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

normal subgroups only

This group has 498 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×#14581775C211yesmaximal, characteristic
2C2×#14581771C211yesmaximal, characteristic, complemented
3#19127641C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic, complemented
5#19127626C212yesmaximal, complemented
6#19127622C211yesmaximal, characteristic, complemented
7#19127617C212yesmaximal
8#19127614C211yesmaximal, characteristic, complemented
9#19127599C212yesmaximal, complemented
10#19127592C212yesmaximal, complemented
11C4×C22⋊C4×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14#14581775—24no—
15#14581771—24no—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C23×C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2212yes—
34C2×C42×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—42no—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2312yes—
47C4⋊Dic83—22no—
48C4⋊Dic83—24no—
49C4⋊Dic83—42no—
50Dic83⋊C4—24no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53C22⋊Dic83—24no—
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—42no—
59C24×C83C2311yescharacteristic
60C22×C4×C83—42no—
61C22×C4×C83D414yes—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C22⋊C4×C83—24no—
67C42×C83—42no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83C2×D412yes—
73C23×C83C2411yescharacteristic
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—21no—
78C22×D83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—44no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C2×C4×C83#4114yes—
90C22⋊D83—44no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104Dic166—44no—
105C22×C83—22no—
106C22×C83—22no—
107C22×C83—22no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)12yes—
113C22×C832−1+411yescharacteristic
114C22×C832−1+411yescharacteristic
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—22no—
122C4×C83—24no—
123C4×C83—42no—
124Dic83—41no—
125Dic83—42no—
126Dic83—42no—
127Dic83—41no—
128Dic83—41no—
129C2×C83—42no—
130C2×C83#53211yescharacteristic
131C2×C83#53411yescharacteristic
132C2×C83#53611yescharacteristic
133C2×C83#53912yes—
134C2×C83#54212yes—
135D83—81no—
136#2663—831nomaximal, Sylow
137C83#266311yescharacteristic, complemented, Sylow
138C2×#123—831no—
139C2×C22⋊Q8—831no—
140#400—831no—
141#399—831no—
142#397—832no—
143#395—831no—
144#394—832no—
145#393—831no—
146#389—832no—
147#388—832no—
148C4×C22⋊C4D8311yescharacteristic, complemented
149C22×Q8—831no—
150C22×D4—831no—
151C23×C4C2×D8311yescharacteristic
152#123—1664no—
153C22⋊Q8—1664no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4—831no—
159C2×C4⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—831no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4C2×D8312yes—
167C2×C42C2×D8312yes—
168#95—832no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95C2×D8311yescharacteristic
173#95C2×D8311yescharacteristic
174C24—831no—
175C24C22×D8311yescharacteristic
176C2×Q8—3322no—
177C2×Q8—1662no—
178C2×D4—3322no—
179C2×D4—1662no—
180C22×C4—42no—
181C22×C4—831no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—832no—
185C22×C4—831no—
186C22×C4D33214yes—
187C22×C4C22×D8312yes—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8312yes—
191C4⋊C4—3322no—
192C4⋊C4—1664no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C4⋊C4—1664no—
197C22⋊C4—24no—
198C22⋊C4—3322no—
199C22⋊C4—1664no—
200C22⋊C4—1662no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1664no—
204C42—42no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—1661no—
210C23—1662no—
211C23—1662no—
212C23—1661no—
213C23—1661no—
214C23C2×D33212yes—
215C23C23×D8311yescharacteristic
216Q8—3324no—
217D4—3324no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—1661no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1661no—
238C2×C4—1661no—
239C2×C4C2×D33212yes—
240C2×C4C2×D33212yes—
241C2×C4#1305059014yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—3321no—
248C22—3321no—
249C22—3321no—
250C22—3322no—
251C22—3322no—
252C22C22×D33211yescharacteristic
253C22C2×#1305059012yes—
254C22#1458183712yes—
255C22#1458183911yescharacteristic
256C22#1458183911yescharacteristic
257C4—22no—
258C4—22no—
259C4—24no—
260C4—42no—
261C4—3321no—
262C4—3322no—
263C4—3321no—
264C4—3322no—
265C4—3321no—
266C2—42no—
267C2—6641no—
268C2#1912825111yescharacteristic
269C2#1912826311yescharacteristic
270C2#1912827711yescharacteristic
271C2#1912829312yes—
272C2#1912831612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC4×C22⋊C4×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC4×C22⋊C4D83
hypocenterC83#2663
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14581771C2semidirect
3. #19127641C2semidirect
4. #19127638C2semidirect
5. #19127626C2semidirect
6. #19127622C2semidirect
8. #19127614C2semidirect
9. #19127599C2semidirect
10. #19127592C2semidirect
11. C4×C22⋊C4×C83C2semidirect
137. C83#2663semidirect
148. C4×C22⋊C4D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817712
#191276412
#191276382
#191276262
#191276222
#191276172
#191276142
#191275992
#191275922
C4×C22⋊C4×C832
#266383