Finite Group Databasebeta

← #24915499

Subgroups — #24915499

Order 10624.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581772C211yesmaximal, characteristic
2C2×#14581770C211yesmaximal, characteristic
3#19127643C212yesmaximal, complemented
4#19127639C211yesmaximal, characteristic, complemented
5#19127630C212yesmaximal
6#19127628C212yesmaximal
7#19127622C211yesmaximal, characteristic, complemented
8#19127622C212yesmaximal, complemented
9#19127621C211yesmaximal, characteristic, complemented
10#19127563C211yesmaximal, characteristic
11#400×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581772—24no—
16#14581770—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C83⋊#95C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2211yescharacteristic
35#95×C83C2212yes—
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—22no—
40C2×Dic166—42no—
41C2×Dic166—42no—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C2×C4×D83—42no—
48C2×C4×D83D414yes—
49C4⋊Dic83—24no—
50C4⋊Dic83—42no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—42no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—42no—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C23×C83C2411yescharacteristic
66C22×D83—22no—
67C22×D83—21no—
68C22×D83—22no—
69C22×D83—21no—
70C22×D83C2×D412yes—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—24no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83C2×D412yes—
83C2×Dic83—22no—
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83C2×D412yes—
96C4×D83—44no—
97Dic166—44no—
98Dic166—44no—
99C22×C83C22×D411yescharacteristic
100C22×C83H(5,𝔽2)12yes—
101C22×C832−1+411yescharacteristic
102C22×C832−1+412yes—
103C22×C832−1+411yescharacteristic
104C2×D83—22no—
105C2×D83—22no—
106C2×D83—22no—
107C2×D83—42no—
108C2×D83—42no—
109C4×C83—22no—
110C4×C83—41no—
111C4×C83—42no—
112C4×C83—41no—
113C4×C83—42no—
114Dic83—22no—
115Dic83—41no—
116Dic83—42no—
117Dic83—41no—
118Dic83—42no—
119C2×C83#53411yescharacteristic
120C2×C83#53611yescharacteristic
121C2×C83#53611yescharacteristic
122C2×C83#56312yes—
123C2×C83#56312yes—
124D83—42no—
125#2817—831nomaximal, Sylow
126C83#281711yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—831no—
128C2×C22⋊Q8—831no—
129#400—831no—
130#400—832no—
131#400D8311yescharacteristic, complemented
132#397—832no—
133#397—832no—
134#395—831no—
135#395—832no—
136#395—831no—
137#381—831no—
138C22×Q8—831no—
139C22×Q8—831no—
140C23×C4—831no—
141C22⋊Q8—1664no—
142C22⋊Q8—1664no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4—831no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4C2×D8311yescharacteristic
149C2×C4⋊C4C2×D8311yescharacteristic
150C2×C22⋊C4—832no—
151C2×C22⋊C4—831no—
152C2×C22⋊C4—831no—
153C2×C22⋊C4—832no—
154#95—832no—
155#95—831no—
156#95—832no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95C2×D8311yescharacteristic
161#95C2×D8312yes—
162#95C2×D8312yes—
163C24—831no—
164C2×Q8—3322no—
165C2×Q8—3322no—
166C2×Q8—1662no—
167C2×Q8—1662no—
168C22×C4—3322no—
169C22×C4—834no—
170C22×C4—831no—
171C22×C4—832no—
172C22×C4—831no—
173C22×C4—832no—
174C22×C4—831no—
175C22×C4C22×D8312yes—
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—3322no—
183C4⋊C4—3322no—
184C4⋊C4—1662no—
185C4⋊C4—1664no—
186C4⋊C4—1662no—
187C22⋊C4—3322no—
188C22⋊C4—1662no—
189C22⋊C4—1664no—
190C22⋊C4—1662no—
191C23—832no—
192C23—1662no—
193C23—1661no—
194C23—1662no—
195C23—1661no—
196C23C23×D8311yescharacteristic
197Q8—3324no—
198Q8—3324no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—3324no—
211C2×C4—832no—
212C2×C4—1661no—
213C2×C4—1662no—
214C2×C4—1661no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1661no—
221C2×C4—1662no—
222C2×C4—1661no—
223C2×C4—1662no—
224C2×C4D4×D8312yes—
225C22—3322no—
226C22—3322no—
227C22—1662no—
228C22—1662no—
229C22—1662no—
230C22C2×D4×D8311yescharacteristic
231C22#1458183812yes—
232C22#1458183911yescharacteristic
233C22#1458183912yes—
234C22#1458184011yescharacteristic
235C4—22no—
236C4—42no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—3322no—
241C4—3322no—
242C4—3321no—
243C4—3321no—
244C4—1662no—
245C2—3322no—
246C2#1912826211yescharacteristic
247C2#1912827111yescharacteristic
248C2#1912827411yescharacteristic
249C2#1912845312yes—
250C2#1912845512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC23×D83C23
hypercenter#400D83
hypocenterC83#2817
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #19127643C2semidirect
4. #19127639C2semidirect
7. #19127622C2semidirect
8. #19127622C2semidirect
9. #19127621C2semidirect
11. #400×C83C2semidirect
126. C83#2817semidirect
131. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
C2×#145817702
#191276432
#191276392
#191276302
#191276282
#191276222
#191276212
#191275632
#400×C832
#281783