Finite Group Databasebeta

← #24907911

Subgroups — #24907911

Order 10624.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#19128212C211yesmaximal, characteristic
2C2×#14581720C212yesmaximal, complemented
3#19127665C211yesmaximal, characteristic, complemented
4#19127461C212yesmaximal, complemented
5C2×#117×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2211yescharacteristic
7C2×#13050574C2212yes—
8#14581775—21no—
9#14581771C412yescomplemented
10#14581766C412yescomplemented
11#14581761—21no—
12C4×C22⋊D83—22no—
13#14581751C2211yescharacteristic
14#14581720—24no—
15C22⋊(C83⋊C8)—22no—
16C23×C4×C83C2211yescharacteristic
17#117×C83—24no—
18C2×C4⋊C4×C83—21no—
19C2×C22⋊C4×C83—21no—
20C2×C42×C83C2212yes—
21#13050590—22no—
22#13050590—22no—
23C2×Dic166C2×C411yescharacteristic
24C2×D332C2×C411yescharacteristic
25C2×C22⋊D83—22no—
26C2×C22⋊D83—21no—
27C2×C4×D83—21no—
28#13050574—22no—
29#13050574—22no—
30C2×C83⋊C8—22no—
31C4⋊Dic83C2×C412yes—
32Dic83⋊C4—22no—
33C22⋊Dic83—22no—
34C22⋊Dic83—21no—
35(C2×D83)⋊C4—22no—
36C4×Dic83—21no—
37C24×C83D411yescharacteristic
38C22×C4×C83—22no—
39C22×C4×C83—22no—
40C22×C4×C83—22no—
41C22×C4×C83C2×C412yes—
42C22×C4×C83D412yes—
43C22×C4×C83D411yescharacteristic
44C22×C4×C83C2311yescharacteristic
45C4⋊C4×C83—24no—
46C22⋊C4×C83—24no—
47C42×C83—22no—
48C42×C83D414yescomplemented
49C23×C83—21no—
50C23×C83—21no—
51C23×C83—42no—
52C23×C83C22⋊C412yes—
53C23×C83C2×D411yescharacteristic
54C22×D83—21no—
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—24no—
58C2×C4×C83—24no—
59C2×C4×C83—24no—
60C2×C4×C83—44no—
61C2×C4×C83C22⋊C412yes—
62C2×C4×C83C22×C411yescharacteristic
63C2×C4×C83C2×D412yes—
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83#4112yes—
66C22⋊D83—44no—
67C22⋊D83—41no—
68C22⋊D83—41no—
69C2×Dic83—22no—
70C2×Dic83—21no—
71D332—22no—
72C4×D83—42no—
73Dic166—22no—
74C83⋊C8—42no—
75C22×C83—22no—
76C22×C83—22no—
77C22×C83—22no—
78C22×C83—22no—
79C22×C83—42no—
80C22×C83—42no—
81C22×C83C2×C22⋊C411yescharacteristic
82C22×C83#12011yescharacteristic
83C22×C83#12311yescharacteristic
84C2×D83—41no—
85C2×D83—42no—
86C4×C83—22no—
87C4×C83—44no—
88C4×C83C4×D412yes—
89C4×C83#12112yes—
90Dic83—41no—
91Dic83—42no—
92C2×C83—21no—
93C2×C83—21no—
94C2×C83—42no—
95C2×C83#38611yescharacteristic
96C2×C83#42112yes—
97D83—81no—
98#1897—831nomaximal, Sylow
99C83#189711yescharacteristic, complemented, Sylow
100#525—831no—
101C2×#117D8311yescharacteristic, complemented
102C2×#104—832no—
103#407—831no—
104#339—832no—
105C2×#41—831no—
106C23×C4C2×D8311yescharacteristic
107C2×M16—832no—
108#123—1661no—
109C22⋊Q8—832no—
110#121—832no—
111#120—1661no—
112C4×D4—1662no—
113#117—24no—
114#117—831no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D8312yes—
118#104—1664no—
119C22⋊C8—1662no—
120C24C22⋊D8311yescharacteristic
121#41—1662no—
122#41—1662no—
123C2×Q8—831no—
124C2×D4—831no—
125C2×D4—1662no—
126C2×D4—1661no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—1661no—
131C22×C4C4×D8312yes—
132C22×C4D33212yes—
133C22×C4C22⋊D8311yescharacteristic
134C22×C4C22×D8311yescharacteristic
135M16—1662no—
136M16—1662no—
137C2×C8—1662no—
138C4⋊C4—24no—
139C4⋊C4—832no—
140C4⋊C4—1662no—
141C22⋊C4—24no—
142C22⋊C4—1662no—
143C22⋊C4—1661no—
144C22⋊C4—1662no—
145C42—22no—
146C42—1661no—
147C42C22⋊D8314yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—1661no—
152C23(C2×D83)⋊C412yes—
153C23C2×C22⋊D8311yescharacteristic
154Q8—1662no—
155D4—3324no—
156D4—3321no—
157D4—3321no—
158D4—1662no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—3322no—
166C2×C4—1662no—
167C2×C4—1661no—
168C2×C4(C2×D83)⋊C412yes—
169C2×C4C2×C4×D8311yescharacteristic
170C2×C4C2×C22⋊D8312yes—
171C2×C4C2×D33211yescharacteristic
172C2×C4#1305059012yes—
173C8—3322no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—3321no—
181C22—3322no—
182C22C2×(C2×D83)⋊C411yescharacteristic
183C22#1458176111yescharacteristic
184C22#1458177511yescharacteristic
185C4—22no—
186C4—44no—
187C4—3321no—
188C4—3322no—
189C4C4×C22⋊D8312yes—
190C4#1458176612yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—6641no—
195C2#1912758011yescharacteristic
196C2#1912771912yes—
subgroupG/H
centerC2×C4C2×D332
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#117×C83C2
socleC22×C83C2×C22⋊C4
prime radical#19128212C2
cosocleC22×C4×C83C23
hypercenterC2×#117D83
hypocenterC83#1897
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#14581720C2semidirect
3. #19127665C2semidirect
4. #19127461C2semidirect
5. C2×#117×C83C2semidirect
9. #14581771C4semidirect
10. #14581766C4semidirect
48. C42×C83D4semidirect
99. C83#1897semidirect
101. C2×#117D83semidirect
147. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191282122
C2×#145817202
#191276652
#191274612
C2×#117×C832
#189783