Finite Group Databasebeta

← #24913426

Subgroups — #24913426

Order 10624.

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

normal subgroups only

This group has 420 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
1#19127630C212yesmaximal, complemented
2#19127628C212yesmaximal, complemented
3#19127627C212yesmaximal, complemented
4#19127623C212yesmaximal, complemented
5#19127603C211yesmaximal, characteristic
6#19127602C211yesmaximal, characteristic
7#19127602C211yesmaximal, characteristic
8#19127600C211yesmaximal, characteristic, complemented
9#19127597C211yesmaximal, characteristic, complemented
10#19127547C211yesmaximal, characteristic, complemented
11#389×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2211yescharacteristic
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C4⋊C4×C83C2212yes—
33C2×C42×C83C2211yescharacteristic
34#95×C83C2212yes—
35C23×D83C2312yes—
36C2×D332—22no—
37C2×D332—22no—
38C2×D332—22no—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C4⋊Dic83—22no—
45C4⋊Dic83—42no—
46Dic83⋊C4—42no—
47(C2×D83)⋊C4—42no—
48(C2×D83)⋊C4—42no—
49(C2×D83)⋊C4—42no—
50(C2×D83)⋊C4—42no—
51(C2×D83)⋊C4—42no—
52(C2×D83)⋊C4—42no—
53C4×Dic83—22no—
54C4×Dic83—22no—
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C4⋊C4×C83—22no—
61C4⋊C4×C83—22no—
62C4⋊C4×C83—42no—
63C42×C83—22no—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83#4112yes—
85C2×C4×C83#4112yes—
86C2×C4×C83#4112yes—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102D332—44no—
103C22×C83C2×#4111yescharacteristic
104C22×C83C2×#4111yescharacteristic
105C22×C83C2×#4111yescharacteristic
106C22×C83H(5,𝔽2)11yescharacteristic
107C22×C83H(5,𝔽2)11yescharacteristic
108C22×C83H(5,𝔽2)11yescharacteristic
109C22×C832−1+411yescharacteristic
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—42no—
116C2×D83—42no—
117C4×C83—42no—
118C4×C83—41no—
119C4×C83—42no—
120C4×C83—41no—
121C4×C83#12614yes—
122Dic83—41no—
123Dic83—41no—
124Dic83—42no—
125Dic83—41no—
126Dic83—41no—
127C2×C83C2×#12611yescharacteristic
128C2×C83#54011yescharacteristic
129C2×C83#54011yescharacteristic
130C2×C83#54011yescharacteristic
131C2×C83#54211yescharacteristic
132C2×C83#54211yescharacteristic
133C2×C83#54211yescharacteristic
134D83—82no—
135#2644—831nomaximal, Sylow
136C83#264411yescharacteristic, complemented, Sylow
137#397—832no—
138#397—832no—
139#397—832no—
140#396—832no—
141#390—831no—
142#390—831no—
143#390—831no—
144#389—831no—
145#389—831no—
146#389D8311yescharacteristic, complemented
147#376—831no—
148C22×D4—831no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4C2×D8311yescharacteristic
153C2×C4⋊C4C2×D8311yescharacteristic
154C2×C4⋊C4C2×D8312yes—
155C2×C22⋊C4—832no—
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C42—831no—
162C2×C42—831no—
163C2×C42C2×D8311yescharacteristic
164#95—831no—
165#95—831no—
166#95—831no—
167#95—831no—
168#95—832no—
169#95—832no—
170#95C2×D8312yes—
171C24—832no—
172C2×D4—1662no—
173C2×D4—1662no—
174C2×D4—1662no—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—831no—
178C22×C4—831no—
179C22×C4—832no—
180C22×C4C22×D8311yescharacteristic
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8312yes—
183C22×C4C22×D8312yes—
184C22×C4C22×D8311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—3322no—
189C4⋊C4—3322no—
190C4⋊C4—1662no—
191C22⋊C4—3322no—
192C22⋊C4—3322no—
193C22⋊C4—3322no—
194C22⋊C4—3322no—
195C22⋊C4—3322no—
196C22⋊C4—3322no—
197C42—22no—
198C42—1662no—
199C42—1662no—
200C23—1662no—
201C23—1662no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23—1662no—
207C23C23×D8311yescharacteristic
208D4—3324no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—1661no—
222C2×C4—1661no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1661no—
228C2×C4—1661no—
229C2×C4—1661no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1661no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4—1661no—
236C2×C4#1305058912yes—
237C2×C4#1305059012yes—
238C2×C4#1305059112yes—
239C22—3322no—
240C22—3322no—
241C22—3322no—
242C22—3322no—
243C22—3322no—
244C22—3322no—
245C22—3322no—
246C22C2×#1305058911yescharacteristic
247C22C2×#1305059011yescharacteristic
248C22C2×#1305059111yescharacteristic
249C22#1458183711yescharacteristic
250C22#1458183811yescharacteristic
251C22#1458183811yescharacteristic
252C22#1458184011yescharacteristic
253C4—41no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—3321no—
258C4—3321no—
259C4—3321no—
260C4—3321no—
261C4—3322no—
262C4#1458178714yes—
263C2—6642no—
264C2C2×#1458178711yescharacteristic
265C2#1912830211yescharacteristic
266C2#1912830411yescharacteristic
267C2#1912830711yescharacteristic
268C2#1912832311yescharacteristic
269C2#1912832411yescharacteristic
270C2#1912832511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#389×C83C2
prime radicalC22×D332C22
hypercenter#389D83
hypocenterC83#2644
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #19127630C2semidirect
2. #19127628C2semidirect
3. #19127627C2semidirect
4. #19127623C2semidirect
8. #19127600C2semidirect
9. #19127597C2semidirect
10. #19127547C2semidirect
11. #389×C83C2semidirect
136. C83#2644semidirect
146. #389D83semidirect

Maximal subgroups

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

subgroupindex
#191276302
#191276282
#191276272
#191276232
#191276032
#191276022
#191276002
#191275972
#191275472
#389×C832
#264483