Finite Group Databasebeta

← #24913625

Subgroups — #24913625

Order 10624.

434 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581784C211yesmaximal, characteristic, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic
4#19127634C212yesmaximal, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127626C211yesmaximal, characteristic
7#19127624C212yesmaximal
8#19127596C212yesmaximal, complemented
9#19127595C212yesmaximal
10C4×C4⋊Dic83C211yesmaximal, characteristic, complemented
11C2×#126×C83C211yesmaximal, characteristic
12#14581784—24no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C4×Dic83C2212yes—
23C83⋊#95C2211yescharacteristic
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29#126×C83—24no—
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C42×C83C2211yescharacteristic
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C4⋊Dic83—22no—
42C4⋊Dic83—22no—
43C4⋊Dic83—24no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47Dic83⋊C4—42no—
48C22⋊Dic83—24no—
49C22⋊Dic83—42no—
50C4×Dic83—42no—
51C24×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C22×C4×C83C2311yescharacteristic
58C4⋊C4×C83—22no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83—22no—
61C22⋊C4×C83—22no—
62C22⋊C4×C83—22no—
63C22⋊C4×C83—22no—
64C42×C83D414yes—
65C23×C83—21no—
66C23×C83—22no—
67C23×C83—21no—
68C23×C83—22no—
69C23×C83—21no—
70C23×C83C2411yescharacteristic
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D412yes—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83#4114yes—
97C22×C83—41no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83—42no—
101C22×C83—42no—
102C22×C83C22×D411yescharacteristic
103C22×C83C2×#4112yes—
104C22×C83H(5,𝔽2)11yescharacteristic
105C22×C832−1+412yes—
106C22×C832−1+411yescharacteristic
107C4×C83—22no—
108C4×C83—22no—
109C4×C83—22no—
110C4×C83—41no—
111C4×C83—41no—
112C4×C83—41no—
113Dic83—24no—
114Dic83—42no—
115Dic83—42no—
116Dic83—42no—
117C2×C83—81no—
118C2×C83#53211yescharacteristic
119C2×C83#53611yescharacteristic
120C2×C83#53711yescharacteristic
121C2×C83#53912yes—
122C2×C83#54112yes—
123#2664—831nomaximal, Sylow
124C83#266411yescharacteristic, complemented, Sylow
125C2×#126D8311yescharacteristic
126C2×#125—831no—
127#400—831no—
128#399—831no—
129#398—831no—
130#398—832no—
131#397—831no—
132#396—832no—
133#389—832no—
134#388—832no—
135C4×C4⋊C4—831no—
136#126—24no—
137#125—1664no—
138C2×C4⋊C4—832no—
139C2×C4⋊C4—832no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C22⋊C4—832no—
149C2×C22⋊C4—832no—
150C2×C22⋊C4C2×D8311yescharacteristic
151C2×C22⋊C4C2×D8311yescharacteristic
152C2×C22⋊C4C2×D8311yescharacteristic
153C2×C42—832no—
154C2×C42C2×D8311yescharacteristic
155#95—831no—
156#95—831no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95—832no—
161C24C22×D8311yescharacteristic
162C22×C4—832no—
163C22×C4—832no—
164C22×C4—832no—
165C22×C4—832no—
166C22×C4C22×D8311yescharacteristic
167C22×C4C22×D8311yescharacteristic
168C22×C4C22×D8311yescharacteristic
169C22×C4C22×D8311yescharacteristic
170C22×C4C22×D8311yescharacteristic
171C22×C4C22×D8311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—3322no—
176C4⋊C4—1664no—
177C4⋊C4—1664no—
178C4⋊C4—1664no—
179C4⋊C4—1664no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—3322no—
186C22⋊C4—1664no—
187C42—3322no—
188C42C22⋊D8314yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23C23×D8311yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—834no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1662no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4C2×C22⋊D8312yes—
219C2×C4C2×C22⋊D8312yes—
220C2×C4C2×C22⋊D8312yes—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22C22×C22⋊D8311yescharacteristic
227C22#41×D8312yes—
228C22#1458183811yescharacteristic
229C22#1458183912yes—
230C22#1458183911yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—3322no—
238C4—3322no—
239C4—3322no—
240C4—1664no—
241C2—81no—
242C2#1912824711yescharacteristic
243C2#1912827211yescharacteristic
244C2#1912827911yescharacteristic
245C2#1912829212yes—
246C2#1912830812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#126×C83C2
prime radicalC24×C83C23
hypercenterC2×#126D83
hypocenterC83#2664
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14581784C2semidirect
2. #19127640C2semidirect
4. #19127634C2semidirect
5. #19127631C2semidirect
8. #19127596C2semidirect
10. C4×C4⋊Dic83C2semidirect
124. C83#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#145817842
#191276402
#191276372
#191276342
#191276312
#191276262
#191276242
#191275962
#191275952
C4×C4⋊Dic832
C2×#126×C832
#266483