Finite Group Databasebeta

← #24918470

Subgroups — #24918470

Order 10624.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128251C211yesmaximal, characteristic
2#19128093C212yesmaximal, complemented
3#19128063C212yesmaximal, complemented
4#19127911C212yesmaximal, complemented
5#19127901C212yesmaximal, complemented
6#19127861C214yesmaximal, complemented
7#19127758C211yesmaximal, characteristic, complemented
8#533×C83C211yesmaximal, characteristic, complemented
9C2×#13050590C2212yes—
10C2×#13050580C2214yes—
11C2×#13050577C2214yes—
12#14581792C2211yescharacteristic
13#14581789C2211yescharacteristic
14#14581780C2212yes—
15#14581766—21no—
16#14581766—21no—
17C4×D332—21no—
18C4×D332—21no—
19C4⋊(C83⋊C8)C2212yes—
20#14581715C2214yes—
21#14581712C2214yes—
22C22⋊(C83⋊C8)C2212yes—
23#14581703C2212yes—
24C4⋊Q8×C83C2211yescharacteristic
25C4⋊Q8×C83C2211yescharacteristic
26#125×C83—21no—
27C22⋊Q8×C83—21no—
28C22⋊Q8×C83C2212yes—
29C4×Q8×C83—21no—
30C4×Q8×C83C2212yes—
31#117×C83—21no—
32C2×C42×C83C2211yescharacteristic
33#13050590—44no—
34C2×Dic166C2312yes—
35C2×D332—21no—
36C2×D332—21no—
37C2×D332C2312yes—
38C2×C22⋊D83—22no—
39C2×C4×D83—22no—
40#13050580—44no—
41#13050577—44no—
42C2×C83⋊C8C2314yes—
43C4⋊Dic83—21no—
44C4⋊Dic83—21no—
45(C2×D83)⋊C4—22no—
46(C2×D83)⋊C4—22no—
47C2×Q8×C83—21no—
48C2×Q8×C83C2312yes—
49C22×C4×C83D411yescharacteristic
50C22×C4×C83D411yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C4⋊C4×C83—21no—
55C4⋊C4×C83—22no—
56C4⋊C4×C83—22no—
57C4⋊C4×C83C2312yes—
58C22⋊C4×C83—22no—
59C42×C83—22no—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66Q8×C83—22no—
67Q8×C83—22no—
68Q8×C83—42no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—41no—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—44no—
81C2×Dic83—22no—
82D332—24no—
83D332—42no—
84D332—42no—
85C4×D83—44no—
86Dic166—24no—
87C83⋊C8—24no—
88C22×C83—21no—
89C22×C83—21no—
90C22×C83—21no—
91C22×C83C22×D411yescharacteristic
92C2×D83—42no—
93C2×D83—42no—
94C2×D83—42no—
95C4×C83—21no—
96C4×C83—21no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—42no—
100C4×C83—42no—
101C4×C83#13712yes—
102C4×C83H(5,𝔽2)12yes—
103Dic83—42no—
104C2×C83—41no—
105C2×C83#53511yescharacteristic
106C2×C83C2×#13711yescharacteristic
107C2×C83#57511yescharacteristic
108D83—82no—
109#3216—831nomaximal, Sylow
110C83#321611yescharacteristic, complemented, Sylow
111#533D8311yescharacteristic, complemented
112#532—831no—
113#497—832no—
114#489—832no—
115#462—832no—
116#460—832no—
117#452—834no—
118#432—831no—
119C2×#41—832no—
120C2×Q16—834no—
121C2×SD16—834no—
122C4⋊Q8—831no—
123C4⋊Q8C2×D8311yescharacteristic
124C4⋊Q8C2×D8311yescharacteristic
125#127—831no—
126#125—21no—
127#124—832no—
128C22⋊Q8—21no—
129C22⋊Q8C2×D8312yes—
130#121—1661no—
131#121—1661no—
132C4×Q8—21no—
133C4×Q8C2×D8312yes—
134C4×D4—1661no—
135C4×D4—1661no—
136#117—21no—
137C2×C42C2×D8311yescharacteristic
138C4⋊C8—832no—
139#103—834no—
140#102—834no—
141C22⋊C8—832no—
142#97—832no—
143#41—3324no—
144C2×Q8—21no—
145C2×Q8—832no—
146C2×Q8C22×D8312yes—
147C2×D4—832no—
148C2×D4—1662no—
149C2×D4—1661no—
150C2×D4—1661no—
151C22×C4—1662no—
152C22×C4C22⋊D8311yescharacteristic
153C22×C4C22⋊D8311yescharacteristic
154C22×C4C22×D8311yescharacteristic
155Q16—3324no—
156SD16—3324no—
157C2×C8—834no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—1661no—
164C4⋊C4—1661no—
165C4⋊C4C22×D8312yes—
166C22⋊C4—22no—
167C22⋊C4—1662no—
168C22⋊C4—1662no—
169C42—22no—
170C42C22⋊D8311yescharacteristic
171C42C22⋊D8311yescharacteristic
172C42C22×D8311yescharacteristic
173C42C22×D8311yescharacteristic
174C23—1662no—
175C23C2×C22⋊D8311yescharacteristic
176Q8—22no—
177Q8—22no—
178Q8—42no—
179Q8—1664no—
180D4—3322no—
181D4—3324no—
182D4—3322no—
183D4—1664no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—3324no—
190C2×C4—1662no—
191C2×C4C2×C22⋊D8311yescharacteristic
192C2×C4C2×C22⋊D8311yescharacteristic
193C2×C4C2×C22⋊D8311yescharacteristic
194C2×C4C2×C22⋊D8311yescharacteristic
195C2×C4C2×C22⋊D8311yescharacteristic
196C2×C4C23×D8311yescharacteristic
197C8—1664no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—3322no—
202C22—3322no—
203C22—3322no—
204C22C22×C22⋊D8311yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—3322no—
212C4#1458182312yes—
213C4#1458183812yes—
214C2—41no—
215C2—6642no—
216C2#1912826511yescharacteristic
217C2C2×#1458182311yescharacteristic
218C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#533D83
hypocenterC83#3216
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #19128093C2semidirect
3. #19128063C2semidirect
4. #19127911C2semidirect
5. #19127901C2semidirect
6. #19127861C2semidirect
7. #19127758C2semidirect
8. #533×C83C2semidirect
110. C83#3216semidirect
111. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
#191280932
#191280632
#191279112
#191279012
#191278612
#191277582
#533×C832
#321683