Finite Group Databasebeta

← #24912809

Subgroups — #24912809

Order 10624.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×Dic83C211yesmaximal, characteristic
2C22×Dic83⋊C4C211yesmaximal, characteristic, complemented
3#19127635C212yesmaximal, complemented
4#19127624C212yesmaximal, complemented
5#19127584C212yesmaximal, complemented
6#19127581C211yesmaximal, characteristic
7#19127576C211yesmaximal, characteristic, complemented
8#19127574C211yesmaximal, characteristic, complemented
9#19127573C212yesmaximal, complemented
10#19127559C211yesmaximal, characteristic, complemented
11#386×C83C211yesmaximal, characteristic
12C23×Dic83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15D4×Dic83—24no—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4—24no—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C23×C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2212yes—
37C22×Dic83—22no—
38C22×Dic83—24no—
39C22×Dic83—22no—
40C22×Dic83—42no—
41C22×Dic83—42no—
42C22×Dic83D414yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2312yes—
49C4⋊Dic83—22no—
50C4⋊Dic83—24no—
51Dic83⋊C4—24no—
52Dic83⋊C4—24no—
53Dic83⋊C4—42no—
54Dic83⋊C4—42no—
55C22⋊Dic83—22no—
56C22⋊Dic83—22no—
57C22⋊Dic83—22no—
58C22⋊Dic83—24no—
59C4×Dic83—22no—
60C24×C83C2311yescharacteristic
61C24×C83C2311yescharacteristic
62C2×D4×C83—22no—
63C2×D4×C83—22no—
64C2×D4×C83Q814yes—
65C22×C4×C83—22no—
66C22×C4×C83—42no—
67C22×C4×C83C2312yes—
68C22×C4×C83C2312yes—
69C22×C4×C83C2311yescharacteristic
70C22⋊C4×C83—22no—
71C22⋊C4×C83—42no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83C2×D412yes—
79C23×C83C2×Q812yes—
80C23×C83C2×Q812yes—
81C23×C83#4112yes—
82C23×C83C2411yescharacteristic
83D4×C83—28no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—44no—
93C2×C4×C83C2×Q812yes—
94C2×Dic83—28no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—44no—
107C2×Dic83—44no—
108C2×Dic83C2×D412yes—
109C2×Dic83C2×D412yes—
110C2×Dic83#4114yes—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—24no—
114C22×C83—24no—
115C22×C83—22no—
116C22×C83—24no—
117C22×C83—41no—
118C22×C83—41no—
119C22×C83—42no—
120C22×C83C22⋊Q814yes—
121C22×C83C22×D411yescharacteristic
122C22×C83C22×Q811yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83C2×#4111yescharacteristic
125C22×C83H(5,𝔽2)12yes—
126C4×C83—22no—
127C4×C83—42no—
128C4×C83—42no—
129Dic83—24no—
130Dic83—22no—
131Dic83—22no—
132Dic83—41no—
133Dic83—41no—
134Dic83—42no—
135C2×C83—24no—
136C2×C83—42no—
137C2×C83C2×C22⋊Q811yescharacteristic
138C2×C83#52511yescharacteristic
139C2×C83#53411yescharacteristic
140C2×C83#55312yes—
141C2×C83#55412yes—
142#2579—831nomaximal, Sylow
143C83#257911yescharacteristic, complemented, Sylow
144C2×C4×D4—831no—
145C22×C4⋊C4—831no—
146#398—832no—
147#396—832no—
148#387—832no—
149#386—831no—
150#386D8311yescharacteristic
151#385—831no—
152#385—832no—
153#385—831no—
154#380—831no—
155C22×D4C2×D8311yescharacteristic
156C23×C4—831no—
157C23×C4—831no—
158C23×C4—831no—
159C23×C4C2×D8311yescharacteristic
160C4×D4—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—832no—
164C2×C4⋊C4—832no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—832no—
167C2×C4⋊C4—1664no—
168C2×C22⋊C4—831no—
169C2×C22⋊C4—831no—
170C2×C22⋊C4—831no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4C2×D8311yescharacteristic
173C2×C22⋊C4C2×D8312yes—
174C2×C42—831no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95—832no—
179#95C2×D8312yes—
180C24C22×D8311yescharacteristic
181C24C22×D8311yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4Dic16614yes—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—3322no—
188C22×C4—3322no—
189C22×C4—834no—
190C22×C4—831no—
191C22×C4—832no—
192C22×C4—831no—
193C22×C4—832no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—1662no—
197C22×C4—1664no—
198C22×C4—1662no—
199C22×C4C22×D8312yes—
200C22×C4C22×D8312yes—
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—3322no—
203C4⋊C4—3322no—
204C4⋊C4—1664no—
205C4⋊C4—1664no—
206C4⋊C4—1662no—
207C4⋊C4—1664no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—1662no—
211C22⋊C4—1664no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C42—1662no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—22no—
219C23—21no—
220C23—22no—
221C23D4×D8312yes—
222C23C2×Dic16612yes—
223C23C2×Dic16612yes—
224C23#1305058912yes—
225C23C23×D8311yescharacteristic
226D4—28no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—3324no—
237C2×C4—3324no—
238C2×C4—832no—
239C2×C4—832no—
240C2×C4—834no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1668no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4C2×Dic16612yes—
254C22—24no—
255C22—24no—
256C22—22no—
257C22—22no—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22#1458176914yes—
264C22C2×D4×D8311yescharacteristic
265C22C22×Dic16611yescharacteristic
266C22#41×D8312yes—
267C22C2×#1305058911yescharacteristic
268C22#1458183712yes—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—3322no—
273C4—3321no—
274C4—3321no—
275C4—1664no—
276C4—1662no—
277C4—1662no—
278C2—24no—
279C2—42no—
280C2C2×#1458176911yescharacteristic
281C2#1912820911yescharacteristic
282C2#1912826111yescharacteristic
283C2#1912839412yes—
284C2#1912841012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2579
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic83⋊C4C2semidirect
3. #19127635C2semidirect
4. #19127624C2semidirect
5. #19127584C2semidirect
7. #19127576C2semidirect
8. #19127574C2semidirect
9. #19127573C2semidirect
10. #19127559C2semidirect
143. C83#2579semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic832
C22×Dic83⋊C42
#191276352
#191276242
#191275842
#191275812
#191275762
#191275742
#191275732
#191275592
#386×C832
#257983