Finite Group Databasebeta

← #24918910

Subgroups — #24918910

Order 10624.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128313C211yesmaximal, characteristic, complemented
2#19128306C211yesmaximal, characteristic
3#19128009C212yesmaximal, complemented
4#19128002C212yesmaximal, complemented
5#19127947C212yesmaximal, complemented
6#19127930C212yesmaximal, complemented
7#19127907C212yesmaximal, complemented
8#19127902C212yesmaximal, complemented
9#424×C83C211yesmaximal, characteristic, complemented
10C2×#13050579C2212yes—
11C2×#13050579C2212yes—
12#14581792C2211yescharacteristic
13#14581789C2211yescharacteristic
14#14581785—22no—
15#14581784C2211yescharacteristic
16#14581784C2211yescharacteristic
17#14581782—22no—
18#14581777—22no—
19#14581771C2212yes—
20#14581766C2212yes—
21#14581765—22no—
22C4×Dic166C2212yes—
23C4×D332C2212yes—
24#14581726C2212yes—
25#14581724C2212yes—
26#14581717C2212yes—
27#14581717C2212yes—
28#14581714C2212yes—
29#14581714C2212yes—
30C2×M16×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic
32#117×C83C2211yescharacteristic
33C4⋊C8×C83C2212yes—
34C4⋊C8×C83C2212yes—
35C2×Dic166C2312yes—
36C2×D332—22no—
37C2×D332C2312yes—
38C2×C22⋊D83—22no—
39C2×C4×D83—22no—
40#13050579—42no—
41#13050579—42no—
42C4⋊Dic83—22no—
43C4⋊Dic83C2312yes—
44C4⋊Dic83C2312yes—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48Dic83⋊C4—22no—
49C22⋊Dic83—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52C4×Dic83—22no—
53C22×C4×C83C2311yescharacteristic
54M16×C83—41no—
55C2×C8×C83—41no—
56C2×C8×C83C2312yes—
57C2×C8×C83C2312yes—
58C4⋊C4×C83D412yes—
59C22⋊C4×C83D412yes—
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83C2×D411yescharacteristic
63C22×D83—22no—
64C2×C4×C83—21no—
65C2×C4×C83—21no—
66C2×C4×C83C2×D412yes—
67C2×C4×C83C2×D412yes—
68C2×C4×C83C2×D411yescharacteristic
69C2×C4×C83C2411yescharacteristic
70C22⋊D83—82no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74D332—22no—
75D332—22no—
76D332—82no—
77C4×D83—82no—
78Dic166—22no—
79Dic166—22no—
80C8×C83—22no—
81C8×C83—22no—
82C22×C83—21no—
83C22×C83—22no—
84C22×C83C22×D411yescharacteristic
85C2×D83—42no—
86C2×D83—42no—
87C2×D83—42no—
88C4×C83—21no—
89C4×C83—22no—
90C4×C83—22no—
91C4×C83H(5,𝔽2)11yescharacteristic
92C4×C832−1+411yescharacteristic
93Dic83—42no—
94Dic83—42no—
95Dic83—42no—
96C2×C83—41no—
97C2×C83#53711yescharacteristic
98C2×C83#57712yes—
99D83—82no—
100#3260—831nomaximal, Sylow
101C83#326011yescharacteristic, complemented, Sylow
102#541—831no—
103#540—831no—
104#479—832no—
105#478—832no—
106#468—832no—
107#465—832no—
108#461—832no—
109#460—832no—
110#424D8311yescharacteristic, complemented
111C2×SD16—832no—
112C2×SD16—832no—
113C2×M16C2×D8311yescharacteristic
114C22×C8C2×D8311yescharacteristic
115C4⋊Q8—831no—
116#127—831no—
117#126—1662no—
118#125—831no—
119#125—831no—
120#125—1662no—
121#123—1662no—
122C22⋊Q8—832no—
123#121—832no—
124#121—1662no—
125C4×Q8—832no—
126C4×D4—832no—
127#117C2×D8311yescharacteristic
128#107—832no—
129#106—832no—
130C4⋊C8C2×D8312yes—
131C4⋊C8C2×D8312yes—
132#103—832no—
133#103—832no—
134#102—832no—
135#102—832no—
136C2×Q8—832no—
137C2×D4—832no—
138C2×D4—1662no—
139C2×D4—1662no—
140C22×C4—1662no—
141C22×C4C22×D8311yescharacteristic
142SD16—3322no—
143SD16—3322no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D8312yes—
147C2×C8C22×D8312yes—
148C4⋊C4—832no—
149C4⋊C4—832no—
150C4⋊C4—1662no—
151C4⋊C4—1662no—
152C4⋊C4—1662no—
153C4⋊C4—1662no—
154C4⋊C4—1662no—
155C4⋊C4D33212yes—
156C22⋊C4—1662no—
157C22⋊C4—1662no—
158C22⋊C4—1662no—
159C22⋊C4D33212yes—
160C42—1662no—
161C42C22×D8311yescharacteristic
162C42C22×D8311yescharacteristic
163C23—1662no—
164C23C2×D33211yescharacteristic
165Q8—1662no—
166Q8—1662no—
167D4—6642no—
168D4—6642no—
169D4—1662no—
170D4—1662no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—6642no—
174C2×C4—1662no—
175C2×C4—1662no—
176C2×C4—1662no—
177C2×C4C2×D33212yes—
178C2×C4C2×D33212yes—
179C2×C4C2×D33211yescharacteristic
180C2×C4C23×D8311yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—3322no—
186C22—3322no—
187C22—3322no—
188C22C22×D33211yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—3322no—
193C4—3322no—
194C4—3322no—
195C4#1458183711yescharacteristic
196C4#1458184011yescharacteristic
197C2—41no—
198C2—6642no—
199C2#1912828211yescharacteristic
200C2#1912851312yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#424×C83C2
socleC22×C83C22×D4
prime radical#19128306C2
hypercenter#424D83
hypocenterC83#3260
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #19128313C2semidirect
3. #19128009C2semidirect
4. #19128002C2semidirect
5. #19127947C2semidirect
6. #19127930C2semidirect
7. #19127907C2semidirect
8. #19127902C2semidirect
9. #424×C83C2semidirect
101. C83#3260semidirect
110. #424D83semidirect

Maximal subgroups

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

subgroupindex
#191283132
#191283062
#191280092
#191280022
#191279472
#191279302
#191279072
#191279022
#424×C832
#326083