Finite Group Databasebeta

← #24907262

Subgroups — #24907262

Order 10624.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14581800C211yesmaximal, characteristic
2#19127683C211yesmaximal, characteristic
3#19127674C211yesmaximal, characteristic
4#19127659C211yesmaximal, characteristic, complemented
5#19127657C211yesmaximal, characteristic, complemented
6#19127472C211yesmaximal, characteristic, complemented
7#344×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581800—21no—
10#14581800—21no—
11#14581800—21no—
12#14581800—21no—
13C2×#13050574C2211yescharacteristic
14C2×#13050573C412yescomplemented
15C22×C83⋊C8C2211yescharacteristic
16C2×C8×D83C412yescomplemented
17#14581751C2211yescharacteristic
18#14581710C412yescomplemented
19#14581708C412yescomplemented
20#14581706C414yes—
21#14581703—21no—
22#14581702—21no—
23C4×C83⋊C8—21no—
24C8×Dic83—21no—
25C2×M16×C83C2211yescharacteristic
26C2×M16×C83C2211yescharacteristic
27#117×C83C2211yescharacteristic
28#13050590—21no—
29#13050590—21no—
30#13050590—21no—
31#13050590—21no—
32C2×Dic166C2×C411yescharacteristic
33C2×D332C2×C411yescharacteristic
34C2×C22⋊D83C2×C412yes—
35C2×C4×D83C2×C412yes—
36#13050574—21no—
37#13050574C2×C412yes—
38#13050573—22no—
39#13050573—22no—
40C2×C83⋊C8—21no—
41C2×C83⋊C8C2×C411yescharacteristic
42C2×C83⋊C8C2×C411yescharacteristic
43C2×C83⋊C8D411yescharacteristic
44C2×C83⋊C8D411yescharacteristic
45C2×C83⋊C8D411yescharacteristic
46C2×C83⋊C8D411yescharacteristic
47C8×D83—22no—
48C8×D83—22no—
49C4⋊Dic83—21no—
50C22⋊Dic83C2×C412yes—
51C4×Dic83—21no—
52C22×C4×C83C2311yescharacteristic
53M16×C83—21no—
54M16×C83—21no—
55M16×C83—21no—
56M16×C83C2×C412yes—
57C2×C8×C83—21no—
58C2×C8×C83C2×C412yes—
59C4⋊C4×C83—21no—
60C22⋊C4×C83C2×C412yes—
61C42×C83—21no—
62C23×C83C22×C411yescharacteristic
63C22×D83C4212yes—
64C2×C4×C83—22no—
65C2×C4×C83C22×C411yescharacteristic
66C2×C4×C83C22×C411yescharacteristic
67C2×C4×C83C2×D411yescharacteristic
68C2×C4×C83C2×D411yescharacteristic
69C2×C4×C83#4111yescharacteristic
70C2×C4×C83#4111yescharacteristic
71C22⋊D83—22no—
72C22⋊D83—22no—
73C2×Dic83—22no—
74C2×Dic83C4212yes—
75D332—21no—
76D332—21no—
77C4×D83—22no—
78C4×D83—22no—
79Dic166—21no—
80Dic166—21no—
81C8×C83—22no—
82C8×C83—22no—
83C83⋊C8—22no—
84C83⋊C8C22⋊C411yescharacteristic
85C83⋊C8C22⋊C411yescharacteristic
86C83⋊C8C22⋊C411yescharacteristic
87C83⋊C8C22⋊C411yescharacteristic
88C22×C83—41no—
89C22×C83C2×C4211yescharacteristic
90C22×C83C4×D411yescharacteristic
91C22×C83C4×D411yescharacteristic
92C2×D83—22no—
93C2×D83—42no—
94C4×C83—42no—
95C4×C83C2×C22⋊C411yescharacteristic
96C4×C83#11711yescharacteristic
97C4×C83C4×D411yescharacteristic
98C4×C83C4×D411yescharacteristic
99Dic83—22no—
100Dic83—42no—
101C2×C83—21no—
102C2×C83—21no—
103C2×C83—21no—
104C2×C83C4×C22⋊C411yescharacteristic
105D83—42no—
106#1757—831nomaximal, Sylow
107C83#175711yescharacteristic, complemented, Sylow
108C2×#131—831no—
109#413—831no—
110#410—831no—
111#405—831no—
112#405—831no—
113#344—831no—
114#344D8311yescharacteristic, complemented
115C2×#41—831no—
116#131—1661no—
117#131—1661no—
118#131—1661no—
119#131—1661no—
120C2×M16—832no—
121C2×M16—831no—
122C2×M16C2×D8311yescharacteristic
123C2×M16C2×D8311yescharacteristic
124C22×C8—831no—
125C22×C8—832no—
126#117—831no—
127#117C2×D8311yescharacteristic
128#101—832no—
129#100—832no—
130#99—834no—
131#97—1661no—
132#97—1661no—
133C4×C8—1661no—
134C4×C8—1661no—
135#41—1661no—
136#41—1661no—
137#41—1661no—
138#41—1661no—
139C2×Q8—831no—
140C2×D4—832no—
141C2×D4—831no—
142C22×C4—832no—
143C22×C4C22×D8311yescharacteristic
144M16—21no—
145M16—21no—
146M16—21no—
147M16—832no—
148M16—1662no—
149M16—1662no—
150M16—1661no—
151M16C4×D8312yes—
152C2×C8—21no—
153C2×C8—831no—
154C2×C8—831no—
155C2×C8—831no—
156C2×C8—831no—
157C2×C8—831no—
158C2×C8—831no—
159C2×C8—1662no—
160C2×C8—1662no—
161C2×C8—1661no—
162C2×C8C4×D8312yes—
163C4⋊C4—21no—
164C4⋊C4—1661no—
165C22⋊C4—832no—
166C22⋊C4C4×D8312yes—
167C42—21no—
168C42—1661no—
169C23—832no—
170C23C2×C4×D8311yescharacteristic
171Q8—1661no—
172Q8—1661no—
173D4—1662no—
174D4—1662no—
175D4—1661no—
176D4—1661no—
177C2×C4—22no—
178C2×C4—832no—
179C2×C4—1662no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4C2×C4×D8311yescharacteristic
183C2×C4C2×C4×D8311yescharacteristic
184C2×C4D4×D8311yescharacteristic
185C2×C4D4×D8311yescharacteristic
186C2×C4#1305058911yescharacteristic
187C2×C4#1305059111yescharacteristic
188C8—22no—
189C8—22no—
190C8—831no—
191C8—831no—
192C8—831no—
193C8—831no—
194C8—1662no—
195C22—41no—
196C22—3322no—
197C22—1662no—
198C22C42×D8311yescharacteristic
199C22(C22⋊D83)⋊C411yescharacteristic
200C22#1458175411yescharacteristic
201C4—42no—
202C4—3322no—
203C4—1662no—
204C4C22⋊C4×D8311yescharacteristic
205C4#1458175011yescharacteristic
206C4(C22⋊D83)⋊C411yescharacteristic
207C4#1458175411yescharacteristic
208C2—21no—
209C2—21no—
210C2—21no—
211C2—3322no—
212C2#1912754811yescharacteristic
subgroupG/H
centerC4C22⋊C4×D83
commutatorC22×C83C2×C42
FrattiniC22×C4C22×D83
Fitting#344×C83C2
socleC2×C83C4×C22⋊C4
prime radicalC2×#13050590C22
cosocleC22×C4×C83C23
hypercenter#344D83
hypocenterC83#1757
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #19127659C2semidirect
5. #19127657C2semidirect
6. #19127472C2semidirect
7. #344×C83C2semidirect
14. C2×#13050573C4semidirect
16. C2×C8×D83C4semidirect
18. #14581710C4semidirect
19. #14581708C4semidirect
107. C83#1757semidirect
114. #344D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818002
#191276832
#191276742
#191276592
#191276572
#191274722
#344×C832
#175783