Finite Group Databasebeta

← #24914190

Subgroups — #24914190

Order 10624.

612 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581773C211yesmaximal, characteristic, complemented
2#19127615C212yesmaximal, complemented
3#19127615C212yesmaximal, complemented
4#19127615C212yesmaximal, complemented
5#19127585C212yesmaximal, complemented
6#19127581C212yesmaximal, complemented
7#19127581C212yesmaximal, complemented
8#19127581C211yesmaximal, characteristic, complemented
9C2×#127×C83C211yesmaximal, characteristic
10C23×Dic83C2211yescharacteristic
11C23×Dic83C2211yescharacteristic
12#14581773—24no—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2211yescharacteristic
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C22×D4×C83C2212yes—
28C22×D4×C83C2212yes—
29C22×D4×C83C2211yescharacteristic
30C22×D4×C83C2211yescharacteristic
31#127×C83—24no—
32C2×C42×C83C2211yescharacteristic
33C22×Dic83—22no—
34C22×Dic83—41no—
35C22×Dic83—41no—
36C22×Dic83—41no—
37C22×Dic83—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42Dic83⋊C4—24no—
43Dic83⋊C4—24no—
44C22⋊Dic83—22no—
45C22⋊Dic83—24no—
46C22⋊Dic83—41no—
47C22⋊Dic83—42no—
48C22⋊Dic83—42no—
49C22⋊Dic83—42no—
50C22⋊Dic83—42no—
51C22⋊Dic83—42no—
52C24×C83C2312yes—
53C24×C83C2311yescharacteristic
54C24×C83C2311yescharacteristic
55C2×D4×C83—24no—
56C2×D4×C83—24no—
57C2×D4×C83—22no—
58C2×D4×C83—24no—
59C2×D4×C83—24no—
60C2×D4×C83—41no—
61C2×D4×C83—41no—
62C2×D4×C83—42no—
63C2×D4×C83—42no—
64C2×D4×C83—41no—
65C2×D4×C83—41no—
66C2×D4×C83D414yes—
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2312yes—
69C42×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—21no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83C2×D412yes—
86C23×C83C2×D412yes—
87C23×C83C2411yescharacteristic
88D4×C83—44no—
89D4×C83—48no—
90D4×C83—48no—
91D4×C83—44no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83C2×D412yes—
97C2×C4×C83#4114yes—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—44no—
111C2×Dic83—44no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—24no—
115C22×C83—24no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—41no—
119C22×C83—41no—
120C22×C83—41no—
121C22×C83—41no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83C22×D411yescharacteristic
130C22×C83C2×#4112yes—
131C22×C83H(5,𝔽2)11yescharacteristic
132C22×C83H(5,𝔽2)11yescharacteristic
133C22×C83H(5,𝔽2)12yes—
134C4×C83—22no—
135C4×C83—24no—
136Dic83—42no—
137Dic83—42no—
138Dic83—42no—
139Dic83—42no—
140C2×C83—42no—
141C2×C83—42no—
142C2×C83—82no—
143C2×C83#53411yescharacteristic
144C2×C83#54012yes—
145C2×C83#54612yes—
146C2×C83#55911yescharacteristic
147C2×C83#56011yescharacteristic
148#2709—831nomaximal, Sylow
149C83#270911yescharacteristic, complemented, Sylow
150C2×#127D8311yescharacteristic
151C2×#123—831no—
152#394—832no—
153#394—832no—
154#394—832no—
155#387—832no—
156#386—832no—
157#386—832no—
158#386—831no—
159C22×D4C2×D8312yes—
160C22×D4C2×D8312yes—
161C22×D4C2×D8311yescharacteristic
162C22×D4C2×D8311yescharacteristic
163C23×C4—831no—
164C23×C4—831no—
165#127—24no—
166#123—1664no—
167C2×C4⋊C4—832no—
168C2×C4⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4—831no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—832no—
176C2×C22⋊C4—831no—
177C2×C42C2×D8311yescharacteristic
178#95—832no—
179#95—832no—
180#95—832no—
181#95—832no—
182C24C22×D8312yes—
183C24C22×D8311yescharacteristic
184C24C22×D8311yescharacteristic
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—22no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C22⋊D8314yes—
197C22×C4—3321no—
198C22×C4—3321no—
199C22×C4—3322no—
200C22×C4—3321no—
201C22×C4—832no—
202C22×C4—832no—
203C22×C4—832no—
204C22×C4—832no—
205C22×C4—1662no—
206C22×C4C22×D8311yescharacteristic
207C22×C4C22×D8312yes—
208C4⋊C4—1664no—
209C4⋊C4—1664no—
210C22⋊C4—3321no—
211C22⋊C4—3322no—
212C22⋊C4—3322no—
213C22⋊C4—3322no—
214C22⋊C4—3322no—
215C22⋊C4—3322no—
216C22⋊C4—1662no—
217C22⋊C4—1664no—
218C42—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23C2×C22⋊D8312yes—
235C23C2×C22⋊D8312yes—
236C23C23×D8311yescharacteristic
237D4—44no—
238D4—48no—
239D4—48no—
240D4—44no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—3324no—
246C2×C4—3324no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4C2×C22⋊D8312yes—
260C2×C4#1305058914yes—
261C22—24no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22C22×C22⋊D8311yescharacteristic
279C22C2×#1305058912yes—
280C22#1458183712yes—
281C22#1458183711yescharacteristic
282C22#1458183711yescharacteristic
283C4—24no—
284C4—22no—
285C4—3322no—
286C4—3322no—
287C4—3322no—
288C4—3322no—
289C2—42no—
290C2—42no—
291C2—82no—
292C2#1912826011yescharacteristic
293C2#1912830512yes—
294C2#1912834512yes—
295C2#1912843711yescharacteristic
296C2#1912843811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#127×C83C2
hypercenterC2×#127D83
hypocenterC83#2709
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#14581773C2semidirect
2. #19127615C2semidirect
3. #19127615C2semidirect
4. #19127615C2semidirect
5. #19127585C2semidirect
6. #19127581C2semidirect
7. #19127581C2semidirect
8. #19127581C2semidirect
149. C83#2709semidirect

Maximal subgroups

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

subgroupindex
C2×#145817732
#191276152
#191275852
#191275812
C2×#127×C832
#270983