Finite Group Databasebeta

← #24913396

Subgroups — #24913396

Order 10624.

576 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581771C211yesmaximal, characteristic, complemented
2#19127639C212yesmaximal, complemented
3#19127626C212yesmaximal, complemented
4#19127615C212yesmaximal, complemented
5#19127613C212yesmaximal, complemented
6#19127581C214yesmaximal, complemented
7C2×C83⋊#95C211yesmaximal, characteristic, complemented
8C2×#121×C83C211yesmaximal, characteristic
9C22×Dic166C2211yescharacteristic
10C23×Dic83C2212yes—
11#14581771—24no—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2214yes—
16C2×C22⋊Dic83C2214yes—
17C83⋊#95C2214yes—
18C83⋊#95C2214yes—
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C22×D4×C83C2212yes—
22C22×D4×C83C2211yescharacteristic
23C23×C4×C83C2211yescharacteristic
24#121×C83—24no—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2212yes—
27C2×Dic166—22no—
28C2×Dic166—42no—
29C22×Dic83—24no—
30C22×Dic83—42no—
31C22×Dic83—42no—
32C22×Dic83C2314yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C4⋊Dic83—22no—
36Dic83⋊C4—24no—
37C22⋊Dic83—24no—
38C22⋊Dic83—44no—
39C22⋊Dic83—44no—
40C24×C83C2312yes—
41C24×C83C2311yescharacteristic
42C2×D4×C83—24no—
43C2×D4×C83—28no—
44C2×D4×C83—22no—
45C2×D4×C83—42no—
46C22×C4×C83—42no—
47C22×C4×C83D414yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C22⋊C4×C83—24no—
53C23×C83—22no—
54C23×C83—21no—
55C23×C83—21no—
56C23×C83—22no—
57C23×C83—24no—
58C23×C83—24no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83C2×D412yes—
62C23×C83#4114yes—
63C23×C83C2411yescharacteristic
64D4×C83—44no—
65D4×C83—48no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—24no—
70C2×C4×C83—44no—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×D412yes—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—24no—
76C2×Dic83—22no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—22no—
81C2×Dic83—48no—
82Dic166—44no—
83C22×C83—22no—
84C22×C83—22no—
85C22×C83—24no—
86C22×C83—24no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83—44no—
91C22×C83—44no—
92C22×C83#12414yes—
93C22×C83C22×D411yescharacteristic
94C22×C83C2×#4112yes—
95C22×C83H(5,𝔽2)12yes—
96C22×C83H(5,𝔽2)11yescharacteristic
97C22×C832−1+411yescharacteristic
98C4×C83—22no—
99C4×C83—22no—
100C4×C83—42no—
101Dic83—42no—
102Dic83—42no—
103Dic83—44no—
104C2×C83—24no—
105C2×C83—82no—
106C2×C83C2×#12411yescharacteristic
107C2×C83#53411yescharacteristic
108C2×C83#53711yescharacteristic
109C2×C83#53812yes—
110C2×C83#53912yes—
111#2639—831nomaximal, Sylow
112C83#263911yescharacteristic, complemented, Sylow
113C2×C22⋊Q8—831no—
114C2×#121D8311yescharacteristic
115#400—832no—
116#397—832no—
117#394—832no—
118#393—832no—
119#386—834no—
120C2×#95—831no—
121C22×Q8—831no—
122C22×D4C2×D8312yes—
123C22×D4C2×D8311yescharacteristic
124C23×C4—832no—
125C23×C4C2×D8311yescharacteristic
126C22⋊Q8—1664no—
127#121—24no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—831no—
130C2×C4⋊C4C2×D8311yescharacteristic
131C2×C22⋊C4—834no—
132C2×C22⋊C4—834no—
133C2×C22⋊C4—832no—
134C2×C22⋊C4C2×D8312yes—
135#95—832no—
136#95—832no—
137#95—834no—
138#95—834no—
139C24C22×D8312yes—
140C24C22×D8311yescharacteristic
141C2×Q8—3322no—
142C2×Q8—1662no—
143C2×D4—24no—
144C2×D4—28no—
145C2×D4—22no—
146C2×D4—42no—
147C22×C4—42no—
148C22×C4—3322no—
149C22×C4—3322no—
150C22×C4—834no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—1664no—
154C22×C4C22⋊D8314yes—
155C22×C4C22×D8312yes—
156C22×C4C22×D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—1664no—
160C4⋊C4—1662no—
161C22⋊C4—24no—
162C22⋊C4—3324no—
163C22⋊C4—3324no—
164C22⋊C4—1664no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—24no—
170C23—24no—
171C23—22no—
172C23—22no—
173C23C2×C22⋊D8312yes—
174C23#1305058914yes—
175C23C23×D8311yescharacteristic
176Q8—3324no—
177D4—44no—
178D4—48no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—44no—
184C2×C4—3328no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1664no—
188C2×C4—1664no—
189C2×C4—1664no—
190C2×C4—1662no—
191C2×C4—1664no—
192C2×C4—1662no—
193C2×C4C2×C22⋊D8312yes—
194C2×C4C2×C22⋊D8312yes—
195C22—22no—
196C22—22no—
197C22—24no—
198C22—24no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—44no—
203C22—44no—
204C22#1458177914yes—
205C22C22×C22⋊D8311yescharacteristic
206C22C2×#1305058912yes—
207C22#1458183712yes—
208C22#1458183811yescharacteristic
209C22#1458183911yescharacteristic
210C4—22no—
211C4—22no—
212C4—42no—
213C4—3322no—
214C4—3322no—
215C4—3324no—
216C2—24no—
217C2—82no—
218C2C2×#1458177911yescharacteristic
219C2#1912826011yescharacteristic
220C2#1912827911yescharacteristic
221C2#1912828812yes—
222C2#1912829612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2639
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. #19127639C2semidirect
3. #19127626C2semidirect
4. #19127615C2semidirect
5. #19127613C2semidirect
6. #19127581C2semidirect
7. C2×C83⋊#95C2semidirect
112. C83#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
#191276392
#191276262
#191276152
#191276132
#191275812
C2×C83⋊#952
C2×#121×C832
#263983