Finite Group Databasebeta

← #24913243

Subgroups — #24913243

Order 10624.

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

normal subgroups only

This group has 526 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×#14581775C211yesmaximal, characteristic
2#19127639C212yesmaximal, complemented
3#19127636C212yesmaximal, complemented
4#19127632C212yesmaximal, complemented
5#19127627C212yesmaximal, complemented
6#19127580C212yesmaximal
7#19127574C212yesmaximal, complemented
8#19127564C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C83C211yesmaximal, characteristic, complemented
10C22×C22⋊D83C2211yescharacteristic
11#14581775—24no—
12C2×Dic83⋊C4C2211yescharacteristic
13C2×Dic83⋊C4C2211yescharacteristic
14C2×Dic83⋊C4C2214yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2211yescharacteristic
17C2×(C2×D83)⋊C4C2211yescharacteristic
18C2×(C2×D83)⋊C4C2211yescharacteristic
19C2×(C2×D83)⋊C4C2214yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2214yes—
23C83⋊#95C2214yes—
24C23×C4×C83C2211yescharacteristic
25C23×C4×C83C2212yes—
26C2×C4⋊C4×C83—24no—
27C2×C4⋊C4×C83C2212yes—
28C2×C4⋊C4×C83C2212yes—
29C23×D83C2311yescharacteristic
30C2×C22⋊D83—22no—
31C2×C22⋊D83—24no—
32C22×Dic83C2311yescharacteristic
33C22×Dic83C2311yescharacteristic
34C22×Dic83C2314yes—
35C22×Dic83C2311yescharacteristic
36Dic83⋊C4—22no—
37Dic83⋊C4—22no—
38Dic83⋊C4—44no—
39C22⋊Dic83—22no—
40C22⋊Dic83—24no—
41(C2×D83)⋊C4—22no—
42(C2×D83)⋊C4—22no—
43(C2×D83)⋊C4—44no—
44C24×C83C2311yescharacteristic
45C22×C4×C83—24no—
46C22×C4×C83—24no—
47C22×C4×C83—44no—
48C22×C4×C83D414yes—
49C22×C4×C83C2314yes—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—24no—
53C4⋊C4×C83—24no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83C2×D412yes—
57C23×C83#4114yes—
58C23×C83C2411yescharacteristic
59C22×D83—21no—
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C2×C4×C83—28no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—24no—
67C2×C4×C83—24no—
68C2×C4×C83—24no—
69C2×C4×C83—48no—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C22⋊D83—44no—
73C2×Dic83—21no—
74C2×Dic83—21no—
75C2×Dic83—21no—
76C2×Dic83—24no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—24no—
82C22×C83—22no—
83C22×C83—22no—
84C22×C83—28no—
85C22×C83#12314yes—
86C22×C83C22×D411yescharacteristic
87C22×C83C2×#4112yes—
88C22×C83H(5,𝔽2)12yes—
89C22×C832−1+412yes—
90C2×D83—41no—
91C2×D83—41no—
92C2×D83—41no—
93C2×D83—44no—
94C4×C83—22no—
95C4×C83—22no—
96C4×C83—44no—
97Dic83—41no—
98Dic83—41no—
99Dic83—41no—
100Dic83—44no—
101C2×C83—24no—
102C2×C83C2×#12311yescharacteristic
103C2×C83#53511yescharacteristic
104C2×C83#53611yescharacteristic
105C2×C83#53914yes—
106D83—81no—
107#2623—831nomaximal, Sylow
108C83#262311yescharacteristic, complemented, Sylow
109C2×#123—831no—
110C22×C4⋊C4D8311yescharacteristic, complemented
111#400—832no—
112#399—832no—
113#398—832no—
114#397—832no—
115#386—832no—
116#385—832no—
117#381—831no—
118C22×D4—831no—
119C23×C4C2×D8311yescharacteristic
120C23×C4C2×D8312yes—
121#123—1664no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—834no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—831no—
126C2×C4⋊C4C2×D8312yes—
127C2×C4⋊C4C2×D8312yes—
128C2×C22⋊C4—834no—
129C2×C22⋊C4—831no—
130C2×C22⋊C4—832no—
131C2×C22⋊C4—831no—
132C2×C22⋊C4—831no—
133#95—832no—
134#95—832no—
135#95—834no—
136#95—834no—
137C24—831no—
138C24C22×D8311yescharacteristic
139C2×D4—1662no—
140C2×D4—1664no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—831no—
145C22×C4—834no—
146C22×C4—831no—
147C22×C4—831no—
148C22×C4C22⋊D8314yes—
149C22×C4C22×D8314yes—
150C22×C4C22×D8311yescharacteristic
151C22×C4C22×D8311yescharacteristic
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—3324no—
155C4⋊C4—1662no—
156C4⋊C4—1662no—
157C22⋊C4—3324no—
158C22⋊C4—1662no—
159C22⋊C4—1664no—
160C22⋊C4—1662no—
161C22⋊C4—1662no—
162C23—22no—
163C23—22no—
164C23—1661no—
165C23—1662no—
166C23—1662no—
167C23—1662no—
168C23C2×C22⋊D8312yes—
169C23#1305059014yes—
170C23C23×D8311yescharacteristic
171D4—3324no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—48no—
179C2×C4—1661no—
180C2×C4—1661no—
181C2×C4—1661no—
182C2×C4—1664no—
183C2×C4—1662no—
184C2×C4—1664no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1664no—
188C2×C4C2×C22⋊D8312yes—
189C2×C4C2×C22⋊D8312yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—3321no—
194C22—3321no—
195C22—3321no—
196C22—3324no—
197C22#1458177514yes—
198C22C22×C22⋊D8311yescharacteristic
199C22C2×#1305059012yes—
200C22#1458183712yes—
201C22#1458184012yes—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—3321no—
206C4—3321no—
207C4—3321no—
208C4—3324no—
209C2—24no—
210C2—6641no—
211C2C2×#1458177511yescharacteristic
212C2#1912826811yescharacteristic
213C2#1912827511yescharacteristic
214C2#1912829814yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C4⋊C4×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC22×C4⋊C4D83
hypocenterC83#2623
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
3. #19127636C2semidirect
4. #19127632C2semidirect
5. #19127627C2semidirect
7. #19127574C2semidirect
8. #19127564C2semidirect
9. C22×C4⋊C4×C83C2semidirect
108. C83#2623semidirect
110. C22×C4⋊C4D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
#191276392
#191276362
#191276322
#191276272
#191275802
#191275742
#191275642
C22×C4⋊C4×C832
#262383