Finite Group Databasebeta

← #24912403

Subgroups — #24912403

Order 10624.

760 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14581768C212yesmaximal, complemented
2C2×#14581763C212yesmaximal, complemented
3C2×C4×D332C211yesmaximal, characteristic, complemented
4#19127637C212yesmaximal, complemented
5#19127610C212yesmaximal, complemented
6#19127601C214yesmaximal, complemented
7C4×C4⋊Dic83C211yesmaximal, characteristic, complemented
8C2×#124×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic
10C22×C22⋊D83C2214yes—
11C22×C4×D83C2212yes—
12#14581768—28no—
13#14581763—28no—
14C4×D332—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2214yes—
18C2×C22⋊Dic83C2214yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2214yes—
21C2×C4×Dic83C2212yes—
22C83⋊#95C2212yes—
23C22×Q8×C83C2211yescharacteristic
24C22×D4×C83C2211yescharacteristic
25#124×C83—24no—
26C2×C22⋊C4×C83C2212yes—
27C2×C22⋊C4×C83C2212yes—
28C2×C42×C83C2211yescharacteristic
29C23×D83C2312yes—
30C2×D332—24no—
31C2×D332—22no—
32C2×C22⋊D83—28no—
33C2×C22⋊D83—28no—
34C2×C22⋊D83—44no—
35C22×Dic83C2312yes—
36C22×Dic83C2314yes—
37C2×C4×D83—24no—
38C2×C4×D83—24no—
39C2×C4×D83—24no—
40C4⋊Dic83—22no—
41C4⋊Dic83D418yescomplemented
42Dic83⋊C4—28no—
43C22⋊Dic83—28no—
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—28no—
46C4×Dic83—42no—
47C24×C83C2312yes—
48C2×Q8×C83—24no—
49C2×Q8×C83—22no—
50C2×D4×C83—24no—
51C2×D4×C83—22no—
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22⋊C4×C83—24no—
57C22⋊C4×C83—24no—
58C42×C83D414yescomplemented
59C23×C83—22no—
60C23×C83—24no—
61C23×C83—24no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83#4114yes—
72Q8×C83—44no—
73D4×C83—44no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—24no—
79C2×C4×C83C2×D414yes—
80C2×C4×C83C2×D412yes—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D412yes—
83C22⋊D83—416no—
84C2×Dic83—24no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83C2×D418yes—
90D332—28no—
91C4×D83—48no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—44no—
96C22×C83—44no—
97C22×C83C22×D412yes—
98C22×C83C22×D411yescharacteristic
99C22×C83C2×#4112yes—
100C22×C83H(5,𝔽2)11yescharacteristic
101C22×C832−1+411yescharacteristic
102C2×D83—24no—
103C2×D83—24no—
104C2×D83—24no—
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—42no—
109C4×C83—22no—
110C4×C83—22no—
111C4×C83—42no—
112C4×C83#12714yes—
113Dic83—28no—
114Dic83—42no—
115C2×C83—82no—
116C2×C83#52511yescharacteristic
117C2×C83C2×#12711yescharacteristic
118C2×C83#53711yescharacteristic
119C2×C83#54612yes—
120C2×C83#54812yes—
121D83—44no—
122#2540—831nomaximal, Sylow
123C83#254011yescharacteristic, complemented, Sylow
124C2×#124D8311yescharacteristic, complemented
125C2×C22⋊Q8—832no—
126C2×#121—832no—
127C2×C4×D4—831no—
128#399—832no—
129#392—832no—
130#390—834no—
131C4×C4⋊C4—831no—
132C22×Q8C2×D8311yescharacteristic
133C22×D4—834no—
134C22×D4—831no—
135C22×D4C2×D8311yescharacteristic
136C23×C4—832no—
137#124—24no—
138C22⋊Q8—1668no—
139#121—1668no—
140C4×D4—1664no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4—834no—
144C2×C22⋊C4—834no—
145C2×C22⋊C4—832no—
146C2×C22⋊C4—834no—
147C2×C22⋊C4C2×D8312yes—
148C2×C22⋊C4C2×D8312yes—
149C2×C42—832no—
150C2×C42C2×D8311yescharacteristic
151#95—832no—
152C24—832no—
153C24C22×D8312yes—
154C2×Q8—24no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—3324no—
159C2×D4—1664no—
160C2×D4—1668no—
161C2×D4—1668no—
162C2×D4—1662no—
163C22×C4—834no—
164C22×C4—832no—
165C22×C4—1664no—
166C22×C4—1664no—
167C22×C4—1664no—
168C22×C4C22×D8312yes—
169C22×C4C22×D8311yescharacteristic
170C22×C4C22×D8311yescharacteristic
171C22×C4C22×D8311yescharacteristic
172C4⋊C4—838no—
173C4⋊C4—1668no—
174C4⋊C4—1662no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—1668no—
178C22⋊C4—1668no—
179C22⋊C4—1664no—
180C42—3322no—
181C42C22⋊D8314yescomplemented
182C23—24no—
183C23—22no—
184C23—24no—
185C23—22no—
186C23—22no—
187C23—834no—
188C23—1662no—
189C23—1662no—
190C23—1662no—
191C23—1662no—
192C23—1662no—
193C23—1662no—
194C23C23×D8311yescharacteristic
195Q8—44no—
196D4—44no—
197D4—33216no—
198D4—1668no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—3328no—
205C2×C4—838no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1664no—
210C2×C4—1664no—
211C2×C4D4×D8314yes—
212C2×C4C2×C22⋊D8312yes—
213C2×C4C2×C22⋊D8312yes—
214C2×C4C2×C22⋊D8312yes—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—3322no—
221C22—3322no—
222C22—3322no—
223C22—3322no—
224C22—1664no—
225C22—1664no—
226C22—1664no—
227C22C2×D4×D8312yes—
228C22C22×C22⋊D8311yescharacteristic
229C22#41×D8312yes—
230C22#1458183811yescharacteristic
231C22#1458183911yescharacteristic
232C4—22no—
233C4—22no—
234C4—42no—
235C4—3322no—
236C4—1668no—
237C4#1458178814yes—
238C2—82no—
239C2—3324no—
240C2#1912820811yescharacteristic
241C2C2×#1458178811yescharacteristic
242C2#1912827911yescharacteristic
243C2#1912834412yes—
244C2#1912835912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
hypercenterC2×#124D83
hypocenterC83#2540
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
2. C2×#14581763C2semidirect
3. C2×C4×D332C2semidirect
4. #19127637C2semidirect
5. #19127610C2semidirect
6. #19127601C2semidirect
7. C4×C4⋊Dic83C2semidirect
8. C2×#124×C83C2semidirect
41. C4⋊Dic83D4semidirect
58. C42×C83D4semidirect
123. C83#2540semidirect
124. C2×#124D83semidirect
181. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
C2×#145817632
C2×C4×D3322
#191276372
#191276102
#191276012
C4×C4⋊Dic832
C2×#124×C832
#254083