Finite Group Databasebeta

← #24908010

Subgroups — #24908010

Order 10624.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#19128265C211yesmaximal, characteristic
2#19127685C211yesmaximal, characteristic, complemented
3C2×#14581708C212yesmaximal, complemented
4#19127473C212yesmaximal, complemented
5#568×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7C2×#13050574C2212yes—
8#14581788—21no—
9#14581781—21no—
10#14581766—22no—
11#14581760C414yes—
12#14581751C2211yescharacteristic
13#14581711—21no—
14#14581709—21no—
15#14581708—24no—
16C2×#41×C83C2211yescharacteristic
17#131×C83—24no—
18C2×M16×C83—22no—
19C2×M16×C83C2212yes—
20C23×D83C2×C412yes—
21C2×D332—21no—
22C2×D332—21no—
23C2×D332—22no—
24C2×D332—22no—
25C2×C22⋊D83—24no—
26#13050574—22no—
27#13050574—22no—
28C2×C83⋊C8—22no—
29C4⋊Dic83—21no—
30C22⋊Dic83C2×C412yes—
31(C2×D83)⋊C4—24no—
32C4×Dic83—21no—
33#41×C83—22no—
34#41×C83—22no—
35C2×Q8×C83D411yescharacteristic
36C2×D4×C83C2×C412yes—
37C2×D4×C83D411yescharacteristic
38C22×C4×C83D412yes—
39C22×C4×C83C2311yescharacteristic
40M16×C83—24no—
41M16×C83—22no—
42M16×C83D414yescomplemented
43C2×C8×C83—24no—
44C2×C8×C83—22no—
45C23×C83C22⋊C412yes—
46C23×C83C22×C411yescharacteristic
47C22×D83—22no—
48C22×D83—24no—
49C22×D83—82no—
50Q8×C83—22no—
51D4×C83—24no—
52D4×C83—22no—
53C2×C4×C83—22no—
54C2×C4×C83—22no—
55C2×C4×C83C22⋊C412yes—
56C2×C4×C83C2×D412yes—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83#4112yes—
60C22⋊D83—84no—
61C2×Dic83—22no—
62D332—41no—
63D332—42no—
64D332—41no—
65C8×C83—24no—
66C83⋊C8—42no—
67C22×C83—22no—
68C22×C83—41no—
69C22×C83—42no—
70C22×C83C2×C22⋊C411yescharacteristic
71C22×C83C4×D412yes—
72C2×D83—44no—
73C2×D83—42no—
74C2×D83—84no—
75C4×C83—22no—
76C4×C83#12011yescharacteristic
77C4×C83#12112yes—
78C4×C83#12311yescharacteristic
79Dic83—42no—
80C2×C83—21no—
81C2×C83—22no—
82C2×C83—42no—
83C2×C83#38611yescharacteristic
84D83—82no—
85#1914—831nomaximal, Sylow
86C83#191411yescharacteristic, complemented, Sylow
87#568D8311yescharacteristic, complemented
88#535—831no—
89#413—831no—
90C2×#100—832no—
91#344—832no—
92C2×#41C2×D8311yescharacteristic
93C22×D4—831no—
94#131—24no—
95C2×M16—22no—
96C2×M16—832no—
97C2×M16C2×D8312yes—
98#127—1661no—
99#124—1661no—
100#121—1662no—
101#120—834no—
102#117—831no—
103#101—1661no—
104#100—1661no—
105#100—1664no—
106C24—832no—
107#41—22no—
108#41—22no—
109C2×Q8D33211yescharacteristic
110C2×D4—1664no—
111C2×D4—1662no—
112C2×D4—1661no—
113C2×D4—1661no—
114C2×D4—1662no—
115C2×D4C4×D8312yes—
116C2×D4D33211yescharacteristic
117C22×C4C22⋊D8312yes—
118C22×C4C22×D8311yescharacteristic
119M16—24no—
120M16—22no—
121M16—1662no—
122M16—1662no—
123M16C22⋊D8314yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—1662no—
127C4⋊C4—1661no—
128C22⋊C4—832no—
129C22⋊C4—1664no—
130C42—1661no—
131C23—6642no—
132C23—1662no—
133C23—1664no—
134C23(C2×D83)⋊C412yes—
135C23C2×C4×D8311yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—3321no—
140D4—3322no—
141D4—3321no—
142D4—6644no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—1662no—
146C2×C4(C2×D83)⋊C412yes—
147C2×C4C2×C22⋊D8312yes—
148C2×C4C2×C22⋊D8311yescharacteristic
149C2×C4C2×D33211yescharacteristic
150C2×C4#1305059012yes—
151C8—24no—
152C8—3322no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—3324no—
157C22—3322no—
158C22—6644no—
159C22C2×(C2×D83)⋊C411yescharacteristic
160C22C4×C22⋊D8312yes—
161C4—22no—
162C4—3322no—
163C4#1458176111yescharacteristic
164C4#1458176612yes—
165C4#1458177511yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—6642no—
170C2#1912758011yescharacteristic
subgroupG/H
centerC2#19127580
commutatorC23×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#568×C83C2
socleC2×C83#386
prime radical#19128265C2
cosocleC22×C4×C83C23
hypercenter#568D83
hypocenterC83#1914
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #19127685C2semidirect
3. C2×#14581708C2semidirect
4. #19127473C2semidirect
5. #568×C83C2semidirect
42. M16×C83D4semidirect
86. C83#1914semidirect
87. #568D83semidirect
123. M16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191276852
C2×#145817082
#191274732
#568×C832
#191483