Finite Group Databasebeta

← #24918853

Subgroups — #24918853

Order 10624.

342 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128253C212yesmaximal
2#19127985C212yesmaximal, complemented
3#19127984C212yesmaximal, complemented
4#19127959C211yesmaximal, characteristic
5#19127957C212yesmaximal
6#19127910C212yesmaximal, complemented
7#19127909C212yesmaximal, complemented
8#19127724C211yesmaximal, characteristic
9#470×C83C211yesmaximal, characteristic
10C2×Dic332C2211yescharacteristic
11C2×#13050580C2212yes—
12C2×#13050574C2211yescharacteristic
13#14581791C2212yes—
14#14581790C2212yes—
15#14581782—22no—
16#14581771C2212yes—
17#14581769—22no—
18Q8×Dic83C2212yes—
19#14581757—22no—
20#14581757C2212yes—
21#14581748—22no—
22C2×C4×Dic83C2211yescharacteristic
23#14581724C2211yescharacteristic
24#14581723C2212yes—
25Dic83⋊C8C2212yes—
26Dic83⋊C8C2212yes—
27#14581717C2212yes—
28#14581716C2212yes—
29#14581715C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C2×M16×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#106×C83C2211yescharacteristic
34#103×C83C2212yes—
35C2×Dic166—22no—
36C2×Dic166C2312yes—
37C22×Dic83D412yes—
38Dic332—41no—
39#13050580—42no—
40#13050574—41no—
41C2×C83⋊C8C2312yes—
42C4⋊Dic83—22no—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48Dic83⋊C4—22no—
49C22⋊Dic83—22no—
50C4×Dic83—22no—
51C4×Dic83—22no—
52C4×Dic83D411yescharacteristic
53C4×Dic83D411yescharacteristic
54C4×Dic83C2312yes—
55C2×Q8×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57Q16×C83—41no—
58M16×C83—41no—
59C2×C8×C83C2311yescharacteristic
60C2×C8×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C4⋊C4×C83C2312yes—
63C22⋊C4×C83—22no—
64C23×C83C2×D411yescharacteristic
65Q8×C83—22no—
66Q8×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2411yescharacteristic
73C2×Dic83—24no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83C2×D412yes—
77C2×Dic83C2×D412yes—
78Dic166—22no—
79Dic166—22no—
80Dic166—44no—
81C8×C83—21no—
82C8×C83—21no—
83C83⋊C8—22no—
84C22×C83—21no—
85C22×C83—22no—
86C22×C83C22×D411yescharacteristic
87C4×C83—21no—
88C4×C83—42no—
89C4×C83—42no—
90C4×C83H(5,𝔽2)11yescharacteristic
91C4×C832−1+411yescharacteristic
92Dic83—22no—
93Dic83—42no—
94Dic83—42no—
95Dic83#13714yes—
96C2×C83—41no—
97C2×C83#53711yescharacteristic
98C2×C83C2×#13712yes—
99#3255—831nomaximal, Sylow
100C83#325511yescharacteristic, complemented, Sylow
101#533—832no—
102#475—832no—
103#475—832no—
104#470—831no—
105#470—832no—
106#470D8311yescharacteristic
107#462—832no—
108#462—832no—
109#423—831no—
110C2×Q16—832no—
111C2×Q16—831no—
112C2×Q16C2×D8311yescharacteristic
113C2×M16—831no—
114C2×M16C2×D8311yescharacteristic
115C4⋊Q8—832no—
116C4⋊Q8—832no—
117#125—1662no—
118C22⋊Q8—832no—
119C22⋊Q8—1662no—
120C22⋊Q8C2×D8312yes—
121C4×Q8—832no—
122C4×Q8—832no—
123C4×Q8—1662no—
124#117—1662no—
125C2×C42—831no—
126#106—831no—
127#106—832no—
128#106C2×D8311yescharacteristic
129C4⋊C8—832no—
130C4⋊C8—832no—
131#103—832no—
132#103—832no—
133#103—832no—
134#103C2×D8312yes—
135C2×Q8—832no—
136C2×Q8—1662no—
137C2×Q8C22×D8312yes—
138C22×C4—832no—
139C22×C4C22×D8311yescharacteristic
140Q16—41no—
141Q16—3321no—
142Q16—3322no—
143M16—41no—
144M16—3321no—
145C2×C8—832no—
146C2×C8C22×D8311yescharacteristic
147C2×C8C22×D8311yescharacteristic
148C4⋊C4—22no—
149C4⋊C4—832no—
150C4⋊C4—1662no—
151C4⋊C4—1662no—
152C4⋊C4—1662no—
153C4⋊C4—1662no—
154C4⋊C4—1662no—
155C4⋊C4—1662no—
156C4⋊C4C22×D8312yes—
157C22⋊C4—22no—
158C22⋊C4—1662no—
159C42—831no—
160C42—831no—
161C42—832no—
162C42—1662no—
163C42—1662no—
164C23D4×D8311yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—3324no—
168Q8—1662no—
169Q8—1662no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—832no—
175C2×C4—832no—
176C2×C4—1662no—
177C2×C4—1662no—
178C2×C4—1664no—
179C2×C4D4×D8311yescharacteristic
180C2×C4C23×D8311yescharacteristic
181C8—21no—
182C8—21no—
183C8—1662no—
184C22—22no—
185C22—21no—
186C22C2×D4×D8311yescharacteristic
187C4—21no—
188C4—42no—
189C4—42no—
190C4—3322no—
191C4—3322no—
192C4—834no—
193C4—1662no—
194C4#1458183711yescharacteristic
195C4#1458183911yescharacteristic
196C2—41no—
197C2#1912827811yescharacteristic
198C2#137×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#470×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenter#470D83
hypocenterC83#3255
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #19127985C2semidirect
3. #19127984C2semidirect
6. #19127910C2semidirect
7. #19127909C2semidirect
100. C83#3255semidirect

Maximal subgroups

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

subgroupindex
#191282532
#191279852
#191279842
#191279592
#191279572
#191279102
#191279092
#191277242
#470×C832
#325583