Finite Group Databasebeta

← #24913249

Subgroups — #24913249

Order 10624.

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

normal subgroups only

This group has 526 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×#14581776C211yesmaximal, characteristic
2C22×Dic83⋊C4C211yesmaximal, characteristic, complemented
3#19127643C212yesmaximal, complemented
4#19127636C212yesmaximal, complemented
5#19127634C212yesmaximal, complemented
6#19127630C212yesmaximal, complemented
7#19127580C211yesmaximal, characteristic
8#19127578C211yesmaximal, characteristic
9#19127573C211yesmaximal, characteristic, complemented
10#19127562C211yesmaximal, characteristic, complemented
11#385×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15#14581776—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4—24no—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×C22⋊Dic83C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C23×C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2212yes—
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83—22no—
41C22×Dic83—22no—
42C22×Dic83—22no—
43C22×Dic83—22no—
44C22×Dic83—42no—
45C22×Dic83D414yes—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2312yes—
49C22×Dic83C2311yescharacteristic
50C22×Dic83C2311yescharacteristic
51C4⋊Dic83—22no—
52C4⋊Dic83—22no—
53Dic83⋊C4—24no—
54Dic83⋊C4—24no—
55Dic83⋊C4—42no—
56C22⋊Dic83—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—42no—
60(C2×D83)⋊C4—42no—
61C24×C83C2311yescharacteristic
62C22×C4×C83—22no—
63C22×C4×C83—42no—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—42no—
69C22⋊C4×C83—22no—
70C22⋊C4×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2×D412yes—
74C23×C83#4112yes—
75C23×C83#4112yes—
76C23×C83C2411yescharacteristic
77C22×D83—21no—
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
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—
93C22⋊D83—44no—
94C2×Dic83—28no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—44no—
109C2×Dic83C2×D412yes—
110C2×Dic83C2×D412yes—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—24no—
114C22×C83—24no—
115C22×C83#12314yes—
116C22×C83C22×D411yescharacteristic
117C22×C83C2×#4111yescharacteristic
118C22×C83C2×#4111yescharacteristic
119C22×C83H(5,𝔽2)12yes—
120C22×C832−1+412yes—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—42no—
125C2×D83—42no—
126C4×C83—41no—
127C4×C83—42no—
128C4×C83—41no—
129C4×C83—42no—
130Dic83—22no—
131Dic83—22no—
132Dic83—41no—
133Dic83—42no—
134Dic83—42no—
135C2×C83—24no—
136C2×C83C2×#12311yescharacteristic
137C2×C83#53511yescharacteristic
138C2×C83#53611yescharacteristic
139C2×C83#53912yes—
140C2×C83#53912yes—
141D83—81no—
142#2623—831nomaximal, Sylow
143C83#262311yescharacteristic, complemented, Sylow
144C2×#123—831no—
145C22×C4⋊C4—831no—
146#400—832no—
147#399—832no—
148#398—832no—
149#397—832no—
150#386—831no—
151#386—831no—
152#385—831no—
153#385D8311yescharacteristic, complemented
154#381—831no—
155C22×D4—831no—
156C23×C4—831no—
157C23×C4—831no—
158C23×C4C2×D8311yescharacteristic
159#123—1664no—
160C2×C4⋊C4—832no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4—831no—
165C2×C4⋊C4—1664no—
166C2×C4⋊C4C2×D8312yes—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—831no—
170C2×C22⋊C4—831no—
171C2×C22⋊C4—831no—
172C2×C22⋊C4C2×D8311yescharacteristic
173C2×C22⋊C4C2×D8311yescharacteristic
174#95—832no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95—832no—
179#95C2×D8312yes—
180C24—831no—
181C24C22×D8311yescharacteristic
182C2×D4—1662no—
183C2×D4—1662no—
184C2×D4—1662no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—3322no—
188C22×C4—834no—
189C22×C4—831no—
190C22×C4—831no—
191C22×C4—832no—
192C22×C4—831no—
193C22×C4—832no—
194C22×C4—1662no—
195C22×C4—1662no—
196C22×C4—1662no—
197C22×C4C22×D8312yes—
198C22×C4C22×D8311yescharacteristic
199C22×C4C22×D8312yes—
200C22×C4C22×D8311yescharacteristic
201C4⋊C4—42no—
202C4⋊C4—3322no—
203C4⋊C4—1664no—
204C4⋊C4—1664no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—3322no—
210C22⋊C4—3322no—
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C23—22no—
215C23—22no—
216C23—1661no—
217C23—1661no—
218C23—1661no—
219C23—1662no—
220C23—1662no—
221C23D4×D8312yes—
222C23#1305059012yes—
223C23#1305059112yes—
224C23C23×D8311yescharacteristic
225D4—3324no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—44no—
237C2×C4—3324no—
238C2×C4—832no—
239C2×C4—832no—
240C2×C4—1662no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1661no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1668no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4—1661no—
253C2×C4—1661no—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—3321no—
259C22—3321no—
260C22—3321no—
261C22—3322no—
262C22—3322no—
263C22#1458177714yes—
264C22C2×D4×D8311yescharacteristic
265C22C2×#1305059011yescharacteristic
266C22C2×#1305059111yescharacteristic
267C22#1458183712yes—
268C22#1458183912yes—
269C4—42no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—3321no—
274C4—3322no—
275C4—3322no—
276C4—1662no—
277C4—1662no—
278C2—24no—
279C2—6641no—
280C2C2×#1458177711yescharacteristic
281C2#1912826611yescharacteristic
282C2#1912827311yescharacteristic
283C2#1912829312yes—
284C2#1912830012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#385D83
hypocenterC83#2623
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×Dic83⋊C4C2semidirect
3. #19127643C2semidirect
4. #19127636C2semidirect
5. #19127634C2semidirect
6. #19127630C2semidirect
9. #19127573C2semidirect
10. #19127562C2semidirect
11. #385×C83C2semidirect
143. C83#2623semidirect
153. #385D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817762
C22×Dic83⋊C42
#191276432
#191276362
#191276342
#191276302
#191275802
#191275782
#191275732
#191275622
#385×C832
#262383