Finite Group Databasebeta

← #24913605

Subgroups — #24913605

Order 10624.

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

normal subgroups only

This group has 534 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
1C2×#14581785C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic, complemented
3#19127624C212yesmaximal, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127615C212yesmaximal, complemented
6#19127615C211yesmaximal, characteristic, complemented
7#19127613C211yesmaximal, characteristic, complemented
8#19127604C212yesmaximal, complemented
9#19127594C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#392×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13#14581785—24no—
14#14581779—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C22⋊Dic83C2212yes—
27C2×C4×Dic83C2212yes—
28C2×C4×Dic83C2211yescharacteristic
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2212yes—
33C22×D4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—42no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C4⋊Dic83—22no—
46C4⋊Dic83—42no—
47Dic83⋊C4—22no—
48Dic83⋊C4—24no—
49Dic83⋊C4—22no—
50C22⋊Dic83—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—42no—
55C22⋊Dic83—42no—
56C22⋊Dic83—42no—
57C4×Dic83—42no—
58C4×Dic83D414yescomplemented
59C24×C83C2311yescharacteristic
60C24×C83C2312yes—
61C2×D4×C83—22no—
62C2×D4×C83—24no—
63C2×D4×C83—42no—
64C2×D4×C83—42no—
65C2×D4×C83—42no—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2312yes—
69C22⋊C4×C83—22no—
70C22⋊C4×C83—24no—
71C23×C83—21no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—21no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—21no—
82C23×C83—22no—
83C23×C83C2411yescharacteristic
84D4×C83—44no—
85D4×C83—48no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83C2×D412yes—
92C2×C4×C83#4114yes—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83C2×D412yes—
108C2×Dic83C2×D412yes—
109Dic166—44no—
110C22×C83—41no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83C22×D411yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83H(5,𝔽2)12yes—
125C22×C83H(5,𝔽2)11yescharacteristic
126C22×C832−1+411yescharacteristic
127C4×C83—22no—
128C4×C83—24no—
129C4×C83—41no—
130Dic83—22no—
131Dic83—22no—
132Dic83—42no—
133Dic83—41no—
134Dic83—41no—
135Dic83—42no—
136C2×C83—82no—
137C2×C83—81no—
138C2×C83#53211yescharacteristic
139C2×C83#53511yescharacteristic
140C2×C83#53711yescharacteristic
141C2×C83#53812yes—
142C2×C83#54212yes—
143#2662—831nomaximal, Sylow
144C83#266211yescharacteristic, complemented, Sylow
145C2×#126—831no—
146C2×#124—831no—
147#396—832no—
148#395—831no—
149#394—831no—
150#394—832no—
151#393—831no—
152#392D8311yescharacteristic
153#390—832no—
154#388—832no—
155C4×C4⋊C4—831no—
156C22×Q8—831no—
157C22×D4C2×D8311yescharacteristic
158C22×D4C2×D8312yes—
159#126—1664no—
160#124—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—831no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4—831no—
165C2×C4⋊C4—832no—
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—831no—
172C2×C22⋊C4—831no—
173C2×C22⋊C4C2×D8311yescharacteristic
174C2×C22⋊C4C2×D8312yes—
175C2×C42—832no—
176C2×C42—831no—
177#95—831no—
178#95—832no—
179#95—832no—
180#95C2×D8311yescharacteristic
181C24C22×D8311yescharacteristic
182C24C22×D8312yes—
183C2×Q8—3322no—
184C2×Q8—1662no—
185C2×D4—22no—
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—832no—
193C22×C4—831no—
194C22×C4—832no—
195C22×C4—831no—
196C22×C4C22×D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8312yes—
199C4⋊C4—3322no—
200C4⋊C4—1662no—
201C4⋊C4—1662no—
202C4⋊C4—1662no—
203C4⋊C4—1664no—
204C22⋊C4—22no—
205C22⋊C4—24no—
206C22⋊C4—3322no—
207C22⋊C4—3322no—
208C22⋊C4—3322no—
209C22⋊C4—1664no—
210C22⋊C4—1664no—
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C42—3322no—
214C42—834no—
215C23—21no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23C23×D8311yescharacteristic
228Q8—3324no—
229D4—44no—
230D4—48no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—832no—
237C2×C4—832no—
238C2×C4—1662no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1661no—
249C2×C4—1662no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4D4×D8312yes—
253C2×C4#1305058914yes—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22C2×D4×D8311yescharacteristic
267C22C2×#1305058912yes—
268C22#1458183712yes—
269C22#1458183711yescharacteristic
270C22#1458183911yescharacteristic
271C4—22no—
272C4—24no—
273C4—41no—
274C4—3322no—
275C4—3321no—
276C4—3321no—
277C4—3322no—
278C4—1662no—
279C4—1662no—
280C2—82no—
281C2—81no—
282C2#1912824911yescharacteristic
283C2#1912826611yescharacteristic
284C2#1912828011yescharacteristic
285C2#1912828612yes—
286C2#1912831812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radical#392×C83C2
hypercenter#392D83
hypocenterC83#2662
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581785C2semidirect
2. C2×#14581779C2semidirect
3. #19127624C2semidirect
4. #19127622C2semidirect
5. #19127615C2semidirect
6. #19127615C2semidirect
7. #19127613C2semidirect
8. #19127604C2semidirect
9. #19127594C2semidirect
10. #19127552C2semidirect
58. C4×Dic83D4semidirect
144. C83#2662semidirect

Maximal subgroups

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

subgroupindex
C2×#145817852
C2×#145817792
#191276242
#191276222
#191276152
#191276132
#191276042
#191275942
#191275522
#392×C832
#266283