Finite Group Databasebeta

← #24913618

Subgroups — #24913618

Order 10624.

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

normal subgroups only

This group has 498 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×#14581777C211yesmaximal, characteristic
2C2×#14581768C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic, complemented
5#19127628C212yesmaximal, complemented
6#19127618C212yesmaximal
7#19127612C211yesmaximal, characteristic, complemented
8#19127597C212yesmaximal, complemented
9#19127590C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#395×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14#14581777—24no—
15#14581768—24no—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×(C2×D83)⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C22×Q8×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×D332—22no—
40C2×D332—42no—
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—42no—
46C2×C4×D83D414yes—
47C4⋊Dic83—22no—
48C4⋊Dic83—24no—
49Dic83⋊C4—24no—
50Dic83⋊C4—22no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—42no—
58C4×Dic83—42no—
59C2×Q8×C83—22no—
60C2×Q8×C83—42no—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C23×C83C2411yescharacteristic
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—21no—
72C22×D83—21no—
73C22×D83—21no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—21no—
77C22×D83—21no—
78C22×D83C2×D412yes—
79Q8×C83—44no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83C2×D412yes—
92C2×C4×C83#4114yes—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83C2×D412yes—
103D332—44no—
104C4×D83—44no—
105C22×C83C22×D411yescharacteristic
106C22×C83C2×#4112yes—
107C22×C83H(5,𝔽2)12yes—
108C22×C832−1+411yescharacteristic
109C22×C832−1+411yescharacteristic
110C2×D83—22no—
111C2×D83—22no—
112C2×D83—22no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—24no—
122C4×C83—41no—
123C4×C83—42no—
124C4×C83—41no—
125Dic83—22no—
126Dic83—41no—
127Dic83—42no—
128Dic83—42no—
129C2×C83#53211yescharacteristic
130C2×C83#53411yescharacteristic
131C2×C83#53611yescharacteristic
132C2×C83#53912yes—
133C2×C83#54212yes—
134D83—42no—
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#395D8311yescharacteristic, complemented
144#394—832no—
145#393—831no—
146#389—832no—
147#388—832no—
148C4×C22⋊C4—831no—
149C22×Q8C2×D8311yescharacteristic
150C22×D4—831no—
151C23×C4—831no—
152#123—1664no—
153C22⋊Q8—1664no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8312yes—
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⋊C4—832no—
167C2×C42—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95C2×D8311yescharacteristic
173#95C2×D8312yes—
174C24—831no—
175C24—831no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—3322no—
179C2×D4—1662no—
180C22×C4—3322no—
181C22×C4—834no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—831no—
185C22×C4—832no—
186C22×C4C22×D8311yescharacteristic
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8312yes—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—1664no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C4⋊C4—1664no—
197C22⋊C4—3322no—
198C22⋊C4—1662no—
199C22⋊C4—1664no—
200C22⋊C4—1662no—
201C22⋊C4—1662no—
202C22⋊C4—1664no—
203C22⋊C4—1664no—
204C42—3322no—
205C23—832no—
206C23—1661no—
207C23—1662no—
208C23—1662no—
209C23—1661no—
210C23—1661no—
211C23—1662no—
212C23—1662no—
213C23—1661no—
214C23—1661no—
215C23C23×D8311yescharacteristic
216Q8—44no—
217D4—3324no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—3324no—
230C2×C4—832no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4D4×D8312yes—
241C2×C4#1305059114yes—
242C22—3322no—
243C22—3322no—
244C22—3322no—
245C22—3321no—
246C22—3321no—
247C22—3321no—
248C22—3322no—
249C22—1662no—
250C22—1662no—
251C22—1662no—
252C22C2×D4×D8311yescharacteristic
253C22C2×#1305059112yes—
254C22#1458183712yes—
255C22#1458183911yescharacteristic
256C22#1458184011yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—41no—
262C4—3322no—
263C4—3321no—
264C4—3322no—
265C4—1662no—
266C2—3322no—
267C2—6641no—
268C2#1912825211yescharacteristic
269C2#1912826111yescharacteristic
270C2#1912827411yescharacteristic
271C2#1912830012yes—
272C2#1912832512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC22×D332C22
hypercenter#395D83
hypocenterC83#2663
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#14581768C2semidirect
3. #19127640C2semidirect
4. #19127638C2semidirect
5. #19127628C2semidirect
7. #19127612C2semidirect
8. #19127597C2semidirect
9. #19127590C2semidirect
10. #19127548C2semidirect
11. #395×C83C2semidirect
137. C83#2663semidirect
143. #395D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817682
#191276402
#191276382
#191276282
#191276182
#191276122
#191275972
#191275902
#191275482
#395×C832
#266383