Finite Group Databasebeta

← #24915519

Subgroups — #24915519

Order 10624.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581783C211yesmaximal, characteristic, complemented
2C2×#14581778C211yesmaximal, characteristic
3#19127638C211yesmaximal, characteristic, complemented
4#19127637C211yesmaximal, characteristic, complemented
5#19127627C211yesmaximal, characteristic, complemented
6#19127625C212yesmaximal, complemented
7#19127624C212yesmaximal, complemented
8#19127622C212yesmaximal, complemented
9#19127617C212yesmaximal
10#19127565C211yesmaximal, characteristic, complemented
11#397×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14#14581783—24no—
15#14581778—24no—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—42no—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C4⋊Dic83—22no—
48C4⋊Dic83—24no—
49C4⋊Dic83—42no—
50Dic83⋊C4—24no—
51Dic83⋊C4—42no—
52Dic83⋊C4—42no—
53C22⋊Dic83—22no—
54C22⋊Dic83—42no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—42no—
58(C2×D83)⋊C4—42no—
59C4×Dic83D414yes—
60C24×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—42no—
68C23×C83—21no—
69C23×C83—21no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—21no—
73C23×C83C2411yescharacteristic
74C22×D83—21no—
75C22×D83—21no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D412yes—
89C22⋊D83—44no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103Dic166—44no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83C22×D411yescharacteristic
110C22×C83H(5,𝔽2)11yescharacteristic
111C22×C83H(5,𝔽2)12yes—
112C22×C832−1+412yes—
113C22×C832−1+411yescharacteristic
114C2×D83—41no—
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—42no—
118C2×D83—42no—
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—42no—
122C4×C83—42no—
123Dic83—22no—
124Dic83—22no—
125Dic83—41no—
126Dic83—42no—
127Dic83—42no—
128C2×C83—81no—
129C2×C83#53511yescharacteristic
130C2×C83#53611yescharacteristic
131C2×C83#53711yescharacteristic
132C2×C83#56212yes—
133C2×C83#56312yes—
134D83—81no—
135#2820—831nomaximal, Sylow
136C83#282011yescharacteristic, complemented, Sylow
137C2×#125—831no—
138C2×#124—831no—
139#399—831no—
140#399—831no—
141#397—831no—
142#397D8311yescharacteristic, complemented
143#396—832no—
144#396—832no—
145#395—832no—
146#394—832no—
147#382—831no—
148C22×Q8—831no—
149C22×D4—831no—
150#125—1664no—
151#124—1664no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—831no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4C2×D8312yes—
167C2×C42—831no—
168#95—832no—
169#95—831no—
170#95—832no—
171#95C2×D8311yescharacteristic
172#95C2×D8312yes—
173C24—831no—
174C24C22×D8311yescharacteristic
175C2×Q8—3322no—
176C2×Q8—1662no—
177C2×D4—3322no—
178C2×D4—1662no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—832no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4C22×D8312yes—
185C22×C4C22×D8312yes—
186C22×C4C22×D8311yescharacteristic
187C22×C4C22×D8311yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—3322no—
190C4⋊C4—3322no—
191C4⋊C4—3322no—
192C4⋊C4—1664no—
193C4⋊C4—1664no—
194C4⋊C4—1662no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—3322no—
198C22⋊C4—3322no—
199C22⋊C4—3322no—
200C22⋊C4—1662no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C42—834no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—1661no—
210C23—1661no—
211C23—1662no—
212C23—1662no—
213C23—1661no—
214C23C23×D8311yescharacteristic
215Q8—3324no—
216D4—3324no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—832no—
227C2×C4—832no—
228C2×C4—1661no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4D4×D8312yes—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—3322no—
246C22—3322no—
247C22—3321no—
248C22—3321no—
249C22—3321no—
250C22C2×D4×D8311yescharacteristic
251C22#1458183711yescharacteristic
252C22#1458183812yes—
253C22#1458183912yes—
254C22#1458183911yescharacteristic
255C4—22no—
256C4—42no—
257C4—41no—
258C4—42no—
259C4—3322no—
260C4—3321no—
261C4—3322no—
262C4—1662no—
263C4—1662no—
264C2—81no—
265C2—6641no—
266C2#1912826911yescharacteristic
267C2#1912827311yescharacteristic
268C2#1912828011yescharacteristic
269C2#1912844812yes—
270C2#1912845312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2820
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581783C2semidirect
3. #19127638C2semidirect
4. #19127637C2semidirect
5. #19127627C2semidirect
6. #19127625C2semidirect
7. #19127624C2semidirect
8. #19127622C2semidirect
10. #19127565C2semidirect
11. #397×C83C2semidirect
136. C83#2820semidirect
142. #397D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817832
C2×#145817782
#191276382
#191276372
#191276272
#191276252
#191276242
#191276222
#191276172
#191275652
#397×C832
#282083