Finite Group Databasebeta

← #24915587

Subgroups — #24915587

Order 10624.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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#19127619C212yesmaximal, complemented
3#19127618C211yesmaximal, characteristic, complemented
4#19127618C212yesmaximal, complemented
5#19127611C212yesmaximal, complemented
6#19127611C212yesmaximal, complemented
7#19127602C211yesmaximal, characteristic, complemented
8#19127590C212yesmaximal, complemented
9#400×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×D332C2212yes—
12C22×D332C2211yescharacteristic
13C22×D332C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×(C2×D83)⋊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×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28#95×C83C2212yes—
29#95×C83C2212yes—
30#95×C83C2211yescharacteristic
31C23×D83C2312yes—
32C23×D83C2312yes—
33C2×D332—22no—
34C2×D332—41no—
35C2×D332—41no—
36C2×D332—41no—
37C2×D332—42no—
38C2×D332—42no—
39C2×D332—42no—
40C2×D332—42no—
41C2×D332—42no—
42C2×D332—42no—
43C2×D332—42no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46Dic83⋊C4—42no—
47(C2×D83)⋊C4—24no—
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—
53(C2×D83)⋊C4—42no—
54(C2×D83)⋊C4—42no—
55C4×Dic83—41no—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—41no—
62C4⋊C4×C83—41no—
63C23×C83C2411yescharacteristic
64C22×D83—22no—
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
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—22no—
89C2×C4×C83—22no—
90C2×C4×C83#4112yes—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97D332—44no—
98D332—44no—
99D332—48no—
100D332—48no—
101C22×C83C2×#4111yescharacteristic
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C83H(5,𝔽2)11yescharacteristic
104C22×C83H(5,𝔽2)12yes—
105C22×C83H(5,𝔽2)12yes—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—42no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83—41no—
124C4×C83—42no—
125Dic83—42no—
126Dic83—42no—
127C2×C83#53811yescharacteristic
128C2×C83#54012yes—
129C2×C83#56011yescharacteristic
130C2×C83#56012yes—
131C2×C83H(3,𝔽4)11yescharacteristic
132D83—82no—
133D83—82no—
134#2828—831nomaximal, Sylow
135C83#282811yescharacteristic, complemented, Sylow
136#400D8311yescharacteristic, complemented
137#397—832no—
138#394—832no—
139#394—831no—
140#394—832no—
141#392—832no—
142#392—832no—
143#390—831no—
144#388—832no—
145C22×D4—832no—
146C22×D4—831no—
147C22×D4—832no—
148C22×D4—831no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4C2×D8311yescharacteristic
151C2×C4⋊C4C2×D8311yescharacteristic
152C2×C22⋊C4—832no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4—832no—
155C2×C22⋊C4—832no—
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C42—831no—
161#95—832no—
162#95—831no—
163#95C2×D8312yes—
164#95C2×D8312yes—
165#95C2×D8311yescharacteristic
166C24—832no—
167C24—832no—
168C2×D4—3321no—
169C2×D4—3321no—
170C2×D4—3321no—
171C2×D4—3322no—
172C2×D4—3322no—
173C2×D4—3322no—
174C2×D4—3322no—
175C2×D4—3322no—
176C2×D4—3322no—
177C2×D4—3322no—
178C2×D4—1662no—
179C22×C4—832no—
180C22×C4—832no—
181C22×C4C22×D8312yes—
182C22×C4C22×D8311yescharacteristic
183C22×C4C22×D8312yes—
184C22×C4C22×D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—3322no—
189C22⋊C4—3322no—
190C22⋊C4—3322no—
191C22⋊C4—3322no—
192C22⋊C4—3322no—
193C22⋊C4—3322no—
194C22⋊C4—3322no—
195C22⋊C4—3322no—
196C22⋊C4—1664no—
197C42—3321no—
198C23—1662no—
199C23—1662no—
200C23—1662no—
201C23—1662no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23—1662no—
207C23—1662no—
208C23—1662no—
209C23—1662no—
210C23—1662no—
211C23—1662no—
212C23C23×D8311yescharacteristic
213D4—3324no—
214D4—3324no—
215D4—3328no—
216D4—3328no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
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—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4#1305059112yes—
236C22—3322no—
237C22—3322no—
238C22—3322no—
239C22—3322no—
240C22—3322no—
241C22—3322no—
242C22—3322no—
243C22—3322no—
244C22—3322no—
245C22—3322no—
246C22—3322no—
247C22—3322no—
248C22—3322no—
249C22—3322no—
250C22C2×#1305059111yescharacteristic
251C22#1458183711yescharacteristic
252C22#1458183812yes—
253C22#1458183811yescharacteristic
254C22#1458183812yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—42no—
260C4—3322no—
261C4—3322no—
262C2—6642no—
263C2—6642no—
264C2#1912829111yescharacteristic
265C2#1912830712yes—
266C2#1912844111yescharacteristic
267C2#1912844212yes—
268C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
hypercenter#400D83
hypocenterC83#2828
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #19127630C2semidirect
2. #19127619C2semidirect
3. #19127618C2semidirect
4. #19127618C2semidirect
5. #19127611C2semidirect
6. #19127611C2semidirect
7. #19127602C2semidirect
8. #19127590C2semidirect
9. #400×C83C2semidirect
135. C83#2828semidirect
136. #400D83semidirect

Maximal subgroups

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

subgroupindex
#191276302
#191276192
#191276182
#191276112
#191276022
#191275902
#400×C832
#282883