Finite Group Databasebeta

← #24918415

Subgroups — #24918415

Order 10624.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128080C211yesmaximal, characteristic, complemented
3#19128075C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19127923C214yesmaximal, complemented
7#19127861C214yesmaximal, complemented
8#19127754C211yesmaximal, characteristic, complemented
9#533×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2212yes—
11C2×#13050580C2214yes—
12C2×#13050577C2214yes—
13#14581792C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581780C2212yes—
16#14581766—21no—
17#14581766—21no—
18C4×D332—21no—
19C4×D332—21no—
20C4⋊(C83⋊C8)C2212yes—
21#14581715C2214yes—
22#14581712C2214yes—
23C22⋊(C83⋊C8)C2212yes—
24C4×C83⋊C8C2211yescharacteristic
25C4×C83⋊C8C2211yescharacteristic
26C4⋊Q8×C83C2211yescharacteristic
27C4⋊Q8×C83C2211yescharacteristic
28#125×C83—21no—
29C22⋊Q8×C83—21no—
30C22⋊Q8×C83C2212yes—
31C4×Q8×C83—21no—
32C4×Q8×C83C2212yes—
33#117×C83—21no—
34C2×C42×C83C2211yescharacteristic
35#13050590—44no—
36C2×Dic166C2312yes—
37C2×D332—21no—
38C2×D332—21no—
39C2×D332C2312yes—
40C2×C22⋊D83—22no—
41C2×C4×D83—22no—
42#13050580—44no—
43#13050577—44no—
44C2×C83⋊C8C2312yes—
45C2×C83⋊C8C2312yes—
46C4⋊Dic83—21no—
47C4⋊Dic83—21no—
48(C2×D83)⋊C4—22no—
49(C2×D83)⋊C4—22no—
50C2×Q8×C83—21no—
51C2×Q8×C83C2312yes—
52C22×C4×C83D411yescharacteristic
53C22×C4×C83D411yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—21no—
56C4⋊C4×C83—21no—
57C4⋊C4×C83—21no—
58C4⋊C4×C83—22no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83C2312yes—
61C22⋊C4×C83—22no—
62C42×C83—22no—
63C42×C83D411yescharacteristic
64C42×C83D411yescharacteristic
65C42×C83C2311yescharacteristic
66C42×C83C2311yescharacteristic
67C23×C83C2×D411yescharacteristic
68C22×D83—22no—
69Q8×C83—24no—
70Q8×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C22⋊D83—44no—
83C2×Dic83—22no—
84D332—24no—
85D332—42no—
86D332—42no—
87C4×D83—44no—
88Dic166—24no—
89C83⋊C8—22no—
90C83⋊C8—22no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83C22×D411yescharacteristic
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83—42no—
103C4×C83#13512yes—
104C4×C83#13712yes—
105C4×C83H(5,𝔽2)12yes—
106Dic83—42no—
107C2×C83—41no—
108C2×C83#53511yescharacteristic
109C2×C83C2×#13511yescharacteristic
110C2×C83C2×#13711yescharacteristic
111D83—82no—
112#3210—831nomaximal, Sylow
113C83#321011yescharacteristic, complemented, Sylow
114#533D8311yescharacteristic, complemented
115#532—831no—
116#494—831no—
117#492—831no—
118#487—831no—
119#486—831no—
120#464—834no—
121#452—834no—
122#431—831no—
123C2×#41—832no—
124C2×Q16—834no—
125C2×SD16—834no—
126C4⋊Q8—831no—
127C4⋊Q8C2×D8311yescharacteristic
128C4⋊Q8C2×D8311yescharacteristic
129#127—831no—
130#125—21no—
131#124—832no—
132C22⋊Q8—21no—
133C22⋊Q8C2×D8312yes—
134#121—1661no—
135#121—1661no—
136C4×Q8—21no—
137C4×Q8C2×D8312yes—
138C4×D4—1661no—
139C4×D4—1661no—
140#117—21no—
141C2×C42C2×D8311yescharacteristic
142C4⋊C8—832no—
143#103—834no—
144#102—834no—
145C22⋊C8—832no—
146C4×C8—831no—
147C4×C8—831no—
148#41—3324no—
149C2×Q8—21no—
150C2×Q8—832no—
151C2×Q8C22×D8312yes—
152C2×D4—832no—
153C2×D4—1662no—
154C2×D4—1661no—
155C2×D4—1661no—
156C22×C4—1662no—
157C22×C4C22⋊D8311yescharacteristic
158C22×C4C22⋊D8311yescharacteristic
159C22×C4C22×D8311yescharacteristic
160Q16—3324no—
161SD16—3324no—
162C2×C8—832no—
163C2×C8—832no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—1661no—
170C4⋊C4—1661no—
171C4⋊C4C22×D8312yes—
172C22⋊C4—22no—
173C22⋊C4—1662no—
174C22⋊C4—1662no—
175C42—22no—
176C42C22⋊D8311yescharacteristic
177C42C22⋊D8311yescharacteristic
178C42C22×D8311yescharacteristic
179C42C22×D8311yescharacteristic
180C23—1662no—
181C23C2×C22⋊D8311yescharacteristic
182Q8—24no—
183Q8—42no—
184Q8—1664no—
185D4—3324no—
186D4—3322no—
187D4—3322no—
188D4—1664no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—3324no—
195C2×C4—1662no—
196C2×C4C2×C22⋊D8311yescharacteristic
197C2×C4C2×C22⋊D8311yescharacteristic
198C2×C4C2×C22⋊D8311yescharacteristic
199C2×C4C2×C22⋊D8311yescharacteristic
200C2×C4C2×C22⋊D8311yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—1662no—
203C8—1662no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—3322no—
208C22—3322no—
209C22—3322no—
210C22C22×C22⋊D8311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—3322no—
217C4#1458181212yes—
218C4#1458182312yes—
219C4#1458183812yes—
220C2—41no—
221C2—6642no—
222C2#1912826511yescharacteristic
223C2C2×#1458181211yescharacteristic
224C2C2×#1458182311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#533D83
hypocenterC83#3210
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. #19128080C2semidirect
3. #19128075C2semidirect
4. #19128053C2semidirect
5. #19128049C2semidirect
6. #19127923C2semidirect
7. #19127861C2semidirect
8. #19127754C2semidirect
9. #533×C83C2semidirect
113. C83#3210semidirect
114. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
#191280802
#191280752
#191280532
#191280492
#191279232
#191278612
#191277542
#533×C832
#321083