Finite Group Databasebeta

← #24912062

Subgroups — #24912062

Order 10624.

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

normal subgroups only

This group has 440 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
1#19127629C212yesmaximal, complemented
2#19127629C212yesmaximal, complemented
3#19127604C211yesmaximal, characteristic
4#19127604C211yesmaximal, characteristic
5#19127597C211yesmaximal, characteristic, complemented
6#19127595C212yesmaximal, complemented
7#19127589C212yesmaximal, complemented
8#19127567C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
10C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
11#390×C83C211yesmaximal, characteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2211yescharacteristic
15C2×Dic83⋊C4C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C22×D4×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C22⋊C4×C83C2212yes—
33C2×C22⋊C4×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35C22×Dic83C2311yescharacteristic
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2311yescharacteristic
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2311yescharacteristic
41C4⋊Dic83—22no—
42C4⋊Dic83—22no—
43Dic83⋊C4—22no—
44Dic83⋊C4—22no—
45Dic83⋊C4—42no—
46C22⋊Dic83—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83—42no—
49C22⋊Dic83—42no—
50C4×Dic83—22no—
51C4×Dic83—22no—
52C4×Dic83—42no—
53C24×C83C2312yes—
54C2×D4×C83—22no—
55C2×D4×C83—22no—
56C2×D4×C83—22no—
57C22×C4×C83C2311yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C22⋊C4×C83—42no—
63C22⋊C4×C83—42no—
64C42×C83—22no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83C2411yescharacteristic
73D4×C83—44no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83#4112yes—
84C2×C4×C83#4112yes—
85C2×C4×C83#4112yes—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83#4112yes—
103C2×Dic83#4112yes—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83C2×#4111yescharacteristic
112C22×C83C2×#4111yescharacteristic
113C22×C83C2×#4111yescharacteristic
114C22×C83C2×#4111yescharacteristic
115C22×C83C2×#4111yescharacteristic
116C22×C83H(5,𝔽2)11yescharacteristic
117C22×C83H(5,𝔽2)11yescharacteristic
118C4×C83—42no—
119C4×C83—41no—
120C4×C83—41no—
121C4×C83#12614yes—
122Dic83—22no—
123Dic83—22no—
124Dic83—41no—
125Dic83—41no—
126Dic83—42no—
127Dic83—42no—
128C2×C83—82no—
129C2×C83C2×#12611yescharacteristic
130C2×C83#52911yescharacteristic
131C2×C83#54011yescharacteristic
132C2×C83#55311yescharacteristic
133C2×C83#55511yescharacteristic
134C2×C83#55911yescharacteristic
135C2×C83#55911yescharacteristic
136#2513—831nomaximal, Sylow
137C83#251311yescharacteristic, complemented, Sylow
138#397—832no—
139#397—832no—
140#390—831no—
141#390—831no—
142#390D8311yescharacteristic
143#389—831no—
144#388—832no—
145#388—832no—
146#382—831no—
147C4×C4⋊C4—831no—
148C4×C4⋊C4—831no—
149C22×D4C2×D8311yescharacteristic
150C2×C4⋊C4—832no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4C2×D8312yes—
161C2×C22⋊C4C2×D8312yes—
162C2×C42—832no—
163C2×C42—831no—
164C2×C42—831no—
165C2×C42C2×D8311yescharacteristic
166#95—831no—
167#95—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172C24C22×D8312yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C22×C4—831no—
177C22×C4—831no—
178C22×C4—832no—
179C22×C4—831no—
180C22×C4—832no—
181C22×C4—831no—
182C22×C4C22×D8311yescharacteristic
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8312yes—
185C22×C4C22×D8311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—3322no—
188C4⋊C4—1662no—
189C4⋊C4—1662no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—3322no—
195C22⋊C4—3322no—
196C22⋊C4—1664no—
197C22⋊C4—1664no—
198C42—22no—
199C42—3322no—
200C42—1662no—
201C42—1662no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23C23×D8311yescharacteristic
210D4—44no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—832no—
221C2×C4—832no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4#1305058912yes—
239C2×C4#1305058912yes—
240C2×C4#1305059012yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22#41×D8311yescharacteristic
249C22#41×D8311yescharacteristic
250C22C2×#1305058911yescharacteristic
251C22C2×#1305058911yescharacteristic
252C22C2×#1305059011yescharacteristic
253C22#1458183711yescharacteristic
254C22#1458183711yescharacteristic
255C4—41no—
256C4—41no—
257C4—42no—
258C4—3321no—
259C4—3322no—
260C4—3322no—
261C4—3321no—
262C4—1662no—
263C4—1662no—
264C4#1458178514yes—
265C2—82no—
266C2C2×#1458178511yescharacteristic
267C2#1912823011yescharacteristic
268C2#1912830511yescharacteristic
269C2#1912839811yescharacteristic
270C2#1912841511yescharacteristic
271C2#1912843411yescharacteristic
272C2#1912843411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×C83C22
hypercenter#390D83
hypocenterC83#2513
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #19127629C2semidirect
2. #19127629C2semidirect
5. #19127597C2semidirect
6. #19127595C2semidirect
7. #19127589C2semidirect
8. #19127567C2semidirect
9. C4⋊C4×Dic83C2semidirect
10. C4×Dic83⋊C4C2semidirect
137. C83#2513semidirect

Maximal subgroups

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

subgroupindex
#191276292
#191276042
#191275972
#191275952
#191275892
#191275672
C4⋊C4×Dic832
C4×Dic83⋊C42
#390×C832
#251383