Finite Group Databasebeta

← #24915547

Subgroups — #24915547

Order 10624.

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

normal subgroups only

This group has 472 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#19127643C212yesmaximal, complemented
2#19127628C212yesmaximal, complemented
3#19127619C212yesmaximal, complemented
4#19127617C212yesmaximal, complemented
5#19127612C212yesmaximal, complemented
6#19127610C211yesmaximal, characteristic
7#19127578C212yesmaximal, complemented
8#19127573C211yesmaximal, characteristic, complemented
9#393×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C23×Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2211yescharacteristic
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2212yes—
26C22×Q8×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29#95×C83C2212yes—
30#95×C83C2211yescharacteristic
31C23×D83C2312yes—
32C2×D332—41no—
33C2×D332—42no—
34C2×C22⋊D83—24no—
35C2×C22⋊D83—42no—
36C2×C22⋊D83—42no—
37C22×Dic83—41no—
38C22×Dic83—42no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—42no—
43Dic83⋊C4—42no—
44C22⋊Dic83—22no—
45(C2×D83)⋊C4—42no—
46(C2×D83)⋊C4—42no—
47(C2×D83)⋊C4—42no—
48(C2×D83)⋊C4—42no—
49(C2×D83)⋊C4—42no—
50C24×C83C2311yescharacteristic
51C2×Q8×C83—42no—
52C2×Q8×C83—41no—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57C22⋊C4×C83—42no—
58C22⋊C4×C83—41no—
59C23×C83—21no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—21no—
63C23×C83#4112yes—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72Q8×C83—44no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C22⋊D83—48no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—44no—
94D332—44no—
95C22×C83—24no—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—41no—
99C22×C83—42no—
100C22×C83C2×#4111yescharacteristic
101C22×C83H(5,𝔽2)12yes—
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C83H(5,𝔽2)12yes—
104C22×C832−1+411yescharacteristic
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C4×C83—42no—
113C4×C83—41no—
114C4×C83—41no—
115C4×C83—42no—
116Dic83—42no—
117Dic83—42no—
118Dic83—42no—
119C2×C83—42no—
120C2×C83#53812yes—
121C2×C83#53911yescharacteristic
122C2×C83#56011yescharacteristic
123C2×C83H(3,𝔽4)11yescharacteristic
124C2×C83#56212yes—
125D83—82no—
126#2824—831nomaximal, Sylow
127C83#282411yescharacteristic, complemented, Sylow
128#400—832no—
129#397—832no—
130#394—832no—
131#394—832no—
132#393—832no—
133#393D8311yescharacteristic, complemented
134#392—831no—
135#386—832no—
136#385—831no—
137C22×Q8C2×D8311yescharacteristic
138C22×D4—832no—
139C22×D4—831no—
140C23×C4—831no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—832no—
143C2×C22⋊C4—832no—
144C2×C22⋊C4—832no—
145C2×C22⋊C4—832no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—831no—
149C2×C22⋊C4C2×D8311yescharacteristic
150C2×C22⋊C4C2×D8312yes—
151#95—832no—
152#95—832no—
153#95—832no—
154#95—831no—
155#95—832no—
156#95C2×D8312yes—
157#95C2×D8311yescharacteristic
158C24—832no—
159C24C22×D8311yescharacteristic
160C2×Q8—41no—
161C2×Q8—42no—
162C2×D4—3321no—
163C2×D4—3322no—
164C2×D4—3322no—
165C2×D4—3322no—
166C2×D4—1664no—
167C22×C4—3321no—
168C22×C4—3322no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4—832no—
172C22×C4C22×D8311yescharacteristic
173C22×C4C22×D8312yes—
174C22×C4C22×D8312yes—
175C22×C4C22×D8311yescharacteristic
176C4⋊C4—3322no—
177C4⋊C4—3322no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C22⋊C4—3322no—
181C22⋊C4—3322no—
182C22⋊C4—3322no—
183C22⋊C4—3322no—
184C22⋊C4—3322no—
185C22⋊C4—1662no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—1662no—
191C23—1662no—
192C23—1662no—
193C23—1662no—
194C23—1662no—
195C23—1662no—
196C23—1662no—
197C23#1305059112yes—
198C23C23×D8311yescharacteristic
199Q8—44no—
200D4—3324no—
201D4—3328no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—3324no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—3322no—
228C22—3322no—
229C22—3322no—
230C22—3322no—
231C22—3322no—
232C22—3322no—
233C22—3322no—
234C22C2×#1305059111yescharacteristic
235C22#1458183711yescharacteristic
236C22#1458183712yes—
237C22#1458183812yes—
238C22#1458183911yescharacteristic
239C4—41no—
240C4—42no—
241C4—41no—
242C4—42no—
243C4—3322no—
244C4—3322no—
245C4—3322no—
246C2—42no—
247C2—6642no—
248C2#1912829112yes—
249C2#1912830011yescharacteristic
250C2#1912844011yescharacteristic
251C2#1912844311yescharacteristic
252C2#1912844512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radical#19127610C2
hypercenter#393D83
hypocenterC83#2824
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #19127643C2semidirect
2. #19127628C2semidirect
3. #19127619C2semidirect
4. #19127617C2semidirect
5. #19127612C2semidirect
7. #19127578C2semidirect
8. #19127573C2semidirect
9. #393×C83C2semidirect
127. C83#2824semidirect
133. #393D83semidirect

Maximal subgroups

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

subgroupindex
#191276432
#191276282
#191276192
#191276172
#191276122
#191276102
#191275782
#191275732
#393×C832
#282483