Finite Group Databasebeta

← #24909153

Subgroups — #24909153

Order 10624.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#19128269C211yesmaximal, characteristic
2#19127684C212yesmaximal, complemented
3#19127498C211yesmaximal, characteristic, complemented
4#19127497C212yesmaximal, complemented
5#355×C83C211yesmaximal, characteristic, complemented
6C2×#13050591C2211yescharacteristic
7C2×D4×D83C411yescharacteristic
8M16×D83—22no—
9#14581789C411yescharacteristic
10#14581788C411yescharacteristic
11#14581781C2211yescharacteristic
12#14581765—22no—
13#14581762—22no—
14#14581749C411yescharacteristic
15#14581711C2212yescomplemented
16#14581708—22no—
17#124×C83C2211yescharacteristic
18#101×C83C2212yescomplemented
19C23×D83—21no—
20#13050591—42no—
21C2×D332—22no—
22C2×D332C2×C411yescharacteristic
23C2×D332C2×C411yescharacteristic
24C2×D332D412yes—
25C2×C22⋊D83—22no—
26C2×C22⋊D83—21no—
27D4×D83—42no—
28C2×C4×D83C2×C411yescharacteristic
29C2×C4×D83D412yes—
30#13050574—22no—
31#13050573—42no—
32C8×D83—42no—
33Dic83⋊C4—21no—
34(C2×D83)⋊C4—21no—
35(C2×D83)⋊C4—22no—
36C4×Dic83C2×C411yescharacteristic
37C2×Q8×C83C2311yescharacteristic
38C2×D4×C83C2×C411yescharacteristic
39M16×C83—22no—
40C22⋊C4×C83—22no—
41C42×C83C2×C411yescharacteristic
42C23×C83—21no—
43C22×D83—21no—
44C22×D83—42no—
45C22×D83—82no—
46C22×D83C22⋊C411yescharacteristic
47C22×D83C22⋊C412yes—
48Q8×C83—41no—
49D4×C83—41no—
50C2×C4×C83—21no—
51C2×C4×C83C22×C411yescharacteristic
52C2×C4×C83C2×D412yes—
53C22⋊D83—42no—
54C22⋊D83—82no—
55C2×Dic83—21no—
56C2×Dic83C22⋊C411yescharacteristic
57D332—41no—
58D332—42no—
59D332—41no—
60D332—82no—
61C4×D83—22no—
62C4×D83—22no—
63C4×D83—22no—
64C8×C83—42no—
65C83⋊C8—42no—
66C22×C83—42no—
67C22×C83—41no—
68C22×C83C2×C22⋊C411yescharacteristic
69C2×D83—21no—
70C2×D83—22no—
71C2×D83—42no—
72C2×D83—41no—
73C2×D83—42no—
74C2×D83—41no—
75C2×D83—84no—
76C4×C83—22no—
77C4×C83—21no—
78C4×C83—41no—
79Dic83—41no—
80Dic83#9912yes—
81C2×C83—21no—
82C2×C83—81no—
83C2×C83C2×#9911yescharacteristic
84D83—42no—
85D83—41no—
86D83—81no—
87#2134—831nomaximal, Sylow
88C83#213411yescharacteristic, complemented, Sylow
89#535—831no—
90#413—832no—
91#355—831no—
92#355—832no—
93#355D8311yescharacteristic, complemented
94C2×#41—831no—
95C22×D4—831no—
96C2×M16—1662no—
97#127—831no—
98#127—831no—
99#124—831no—
100#124C2×D8311yescharacteristic
101#121—1662no—
102#120—1662no—
103#117—831no—
104#101—832no—
105#101C2×D8312yescomplemented
106#100—1662no—
107C24—1661no—
108#41—3322no—
109C2×Q8C22×D8311yescharacteristic
110C2×D4—3322no—
111C2×D4—831no—
112C2×D4—831no—
113C2×D4—832no—
114C2×D4—1662no—
115C2×D4—1661no—
116C2×D4—1662no—
117C2×D4C4×D8311yescharacteristic
118C22×C4—831no—
119C22×C4—832no—
120M16—22no—
121M16—3322no—
122M16—1662no—
123C2×C8—3322no—
124C4⋊C4—1661no—
125C22⋊C4—22no—
126C22⋊C4—1661no—
127C22⋊C4—1662no—
128C42—831no—
129C42C4×D8311yescharacteristic
130C23—21no—
131C23—3322no—
132C23—831no—
133C23—832no—
134C23—6642no—
135C23—1661no—
136Q8—41no—
137D4—41no—
138D4—3321no—
139D4—3322no—
140D4—3321no—
141D4—3322no—
142D4—6642no—
143D4—6642no—
144C2×C4—21no—
145C2×C4—831no—
146C2×C4—1662no—
147C2×C4—1662no—
148C2×C4—1662no—
149C2×C4—1661no—
150C2×C4C2×C4×D8311yescharacteristic
151C2×C4D4×D8312yes—
152C8—42no—
153C8—3322no—
154C22—42no—
155C22—41no—
156C22—3322no—
157C22—3321no—
158C22—3322no—
159C22—3321no—
160C22—6644no—
161C22—1661no—
162C22—1662no—
163C22C22⋊C4×D8311yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—3321no—
168C4—832no—
169C2—21no—
170C2—81no—
171C2—3321no—
172C2—3322no—
173C2—6641no—
174C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC2×C4×C83C22×C4
FrattiniC2×Q8C22×D83
Fitting#355×C83C2
socleC2×C83C2×#99
prime radical#19128269C2
cosocleC2×Q8×C83C23
hypercenter#355D83
hypocenterC83#2134
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #19127684C2semidirect
3. #19127498C2semidirect
4. #19127497C2semidirect
5. #355×C83C2semidirect
15. #14581711C22semidirect
18. #101×C83C22semidirect
88. C83#2134semidirect
93. #355D83semidirect
105. #101C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282692
#191276842
#191274982
#191274972
#355×C832
#213483