Finite Group Databasebeta

← #24913383

Subgroups — #24913383

Order 10624.

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

normal subgroups only

This group has 520 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×#14581771C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D83C211yesmaximal, characteristic
3#19127633C211yesmaximal, characteristic, complemented
4#19127628C211yesmaximal, characteristic, complemented
5#19127626C212yesmaximal, complemented
6#19127622C211yesmaximal, characteristic, complemented
7#19127617C212yesmaximal
8#19127614C211yesmaximal, characteristic, complemented
9#19127586C212yesmaximal, complemented
10#19127575C212yesmaximal, complemented
11#381×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581771—24no—
16C4×C22⋊D83—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C23×C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2212yes—
34#95×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—22no—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—42no—
50C4⋊Dic83—22no—
51C4⋊Dic83—24no—
52C4⋊Dic83—42no—
53Dic83⋊C4—24no—
54Dic83⋊C4—22no—
55C22⋊Dic83—24no—
56C22⋊Dic83—22no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—42no—
60C4×Dic83—22no—
61C24×C83C2311yescharacteristic
62C22×C4×C83—42no—
63C22×C4×C83D414yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22⋊C4×C83—42no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2×D412yes—
74C23×C83C2411yescharacteristic
75C22×D83—21no—
76C22×D83—21no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83#4112yes—
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—22no—
87C2×C4×C83—22no—
88C2×C4×C83—44no—
89C2×C4×C83C2×D412yes—
90C2×C4×C83C2×D412yes—
91C22⋊D83—28no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83#4112yes—
105C4×D83—44no—
106Dic166—44no—
107C22×C83—22no—
108C22×C83—22no—
109C22×C83—22no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83C22×D411yescharacteristic
113C22×C83C2×#4111yescharacteristic
114C22×C83C2×#4111yescharacteristic
115C22×C83H(5,𝔽2)12yes—
116C22×C832−1+412yes—
117C2×D83—22no—
118C2×D83—24no—
119C2×D83—41no—
120C2×D83—41no—
121C2×D83—42no—
122C4×C83—22no—
123C4×C83—22no—
124C4×C83—42no—
125C4×C83—42no—
126Dic83—22no—
127Dic83—42no—
128Dic83—42no—
129Dic83—41no—
130Dic83—41no—
131C2×C83—42no—
132C2×C83#52511yescharacteristic
133C2×C83#53411yescharacteristic
134C2×C83#53611yescharacteristic
135C2×C83#53912yes—
136C2×C83#54212yes—
137D83—42no—
138#2638—831nomaximal, Sylow
139C83#263811yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—831no—
141C2×C4×D4—831no—
142#398—831no—
143#397—832no—
144#397—831no—
145#395—831no—
146#394—832no—
147#393—831no—
148#387—832no—
149#385—832no—
150#381D8311yescharacteristic, complemented
151C22×Q8—831no—
152C22×D4—831no—
153C23×C4—831no—
154C23×C4C2×D8311yescharacteristic
155C22⋊Q8—1664no—
156C4×D4—1664no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4—831no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—831no—
161C2×C4⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—831no—
166C2×C22⋊C4—832no—
167C2×C22⋊C4C2×D8312yes—
168C2×C42—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95C2×D8312yes—
174#95C2×D8311yescharacteristic
175#95C2×D8311yescharacteristic
176C24—831no—
177C24C22×D8311yescharacteristic
178C2×Q8—3322no—
179C2×Q8—1662no—
180C2×D4—1662no—
181C2×D4—1662no—
182C2×D4—1662no—
183C22×C4—42no—
184C22×C4—3322no—
185C22×C4—831no—
186C22×C4—831no—
187C22×C4—832no—
188C22×C4—832no—
189C22×C4—831no—
190C22×C4—1662no—
191C22×C4D33214yes—
192C22×C4C22×D8312yes—
193C22×C4C22×D8312yes—
194C22×C4C22×D8311yescharacteristic
195C22×C4C22×D8311yescharacteristic
196C4⋊C4—3322no—
197C4⋊C4—1664no—
198C4⋊C4—1662no—
199C4⋊C4—1664no—
200C4⋊C4—1662no—
201C22⋊C4—42no—
202C22⋊C4—3322no—
203C22⋊C4—1664no—
204C22⋊C4—1662no—
205C22⋊C4—1664no—
206C22⋊C4—1662no—
207C42—1662no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—832no—
213C23—1661no—
214C23—1661no—
215C23—1662no—
216C23—1662no—
217C23C2×D33212yes—
218C23C23×D8311yescharacteristic
219Q8—3324no—
220D4—1668no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—3324no—
231C2×C4—832no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4C2×D33212yes—
245C2×C4C2×D33212yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—3321no—
252C22—3321no—
253C22—3322no—
254C22—1662no—
255C22—1664no—
256C22C22×D33211yescharacteristic
257C22#41×D8311yescharacteristic
258C22#41×D8311yescharacteristic
259C22#1458183712yes—
260C22#1458183912yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—3322no—
266C4—3321no—
267C4—3322no—
268C4—3321no—
269C4—1662no—
270C2—42no—
271C2—3322no—
272C2#1912821311yescharacteristic
273C2#1912826311yescharacteristic
274C2#1912827711yescharacteristic
275C2#1912829412yes—
276C2#1912831912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#381×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#381D83
hypocenterC83#2638
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
3. #19127633C2semidirect
4. #19127628C2semidirect
5. #19127626C2semidirect
6. #19127622C2semidirect
8. #19127614C2semidirect
9. #19127586C2semidirect
10. #19127575C2semidirect
11. #381×C83C2semidirect
139. C83#2638semidirect
150. #381D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
C2×C4×C22⋊D832
#191276332
#191276282
#191276262
#191276222
#191276172
#191276142
#191275862
#191275752
#381×C832
#263883