Finite Group Databasebeta

← #24918622

Subgroups — #24918622

Order 10624.

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

normal subgroups only

This group has 326 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#19128258C211yesmaximal, characteristic
2#19128104C211yesmaximal, characteristic, complemented
3#19128097C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128053C211yesmaximal, characteristic, complemented
6#19128042C214yesmaximal
7#19128005C214yesmaximal, complemented
8#19127754C211yesmaximal, characteristic
9#533×C83C211yesmaximal, characteristic
10#14581792C2211yescharacteristic
11#14581792C2211yescharacteristic
12#14581784—21no—
13#14581771—21no—
14#14581771C2212yes—
15C4×Dic166—21no—
16C4×Dic166C2212yes—
17#14581751—21no—
18#14581725C2214yes—
19#14581723C2214yes—
20C4⋊(C83⋊C8)C2212yes—
21#14581715C2214yes—
22#14581715C2214yes—
23C22⋊(C83⋊C8)C2212yes—
24C4×C83⋊C8C2211yescharacteristic
25C4×C83⋊C8C2211yescharacteristic
26C4⋊Q8×C83C2211yescharacteristic
27C4⋊Q8×C83C2211yescharacteristic
28#125×C83C2212yes—
29C22⋊Q8×C83—21no—
30C22⋊Q8×C83—21no—
31C4×Q8×C83—21no—
32C4×Q8×C83—21no—
33#117×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37C2×C83⋊C8C2312yes—
38C2×C83⋊C8C2312yes—
39C4⋊Dic83—21no—
40C4⋊Dic83—21no—
41C4⋊Dic83—21no—
42C4⋊Dic83C2312yes—
43Dic83⋊C4—22no—
44Dic83⋊C4—22no—
45C22⋊Dic83—22no—
46C4×Dic83—22no—
47C2×Q8×C83—21no—
48C2×Q8×C83—21no—
49C22×C4×C83D411yescharacteristic
50C22×C4×C83D411yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—22no—
53C4⋊C4×C83—22no—
54C4⋊C4×C83—21no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83C2312yes—
57C4⋊C4×C83C2312yes—
58C22⋊C4×C83—22no—
59C42×C83—22no—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83C2×D411yescharacteristic
65Q8×C83—42no—
66Q8×C83—42no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C2×Dic83—22no—
79C2×Dic83—22no—
80Dic166—24no—
81Dic166—42no—
82C83⋊C8—22no—
83C83⋊C8—22no—
84C22×C83—21no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83C22×D411yescharacteristic
88C4×C83—21no—
89C4×C83—21no—
90C4×C83—21no—
91C4×C83—42no—
92C4×C83—42no—
93C4×C83#13512yes—
94C4×C83#13712yes—
95C4×C832−1+412yes—
96Dic83—42no—
97Dic83—42no—
98C2×C83—41no—
99C2×C83#53611yescharacteristic
100C2×C83C2×#13511yescharacteristic
101C2×C83C2×#13711yescharacteristic
102#3231—831nomaximal, Sylow
103C83#323111yescharacteristic, complemented, Sylow
104#533—831no—
105#533D8311yescharacteristic
106#500—831no—
107#498—831no—
108#487—831no—
109#487—831no—
110#485—834no—
111#479—834no—
112#431—831no—
113C4⋊Q8—831no—
114C4⋊Q8—831no—
115C4⋊Q8C2×D8311yescharacteristic
116C4⋊Q8C2×D8311yescharacteristic
117#125—1661no—
118#125C2×D8312yes—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—832no—
122C22⋊Q8—1661no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—832no—
126C4×Q8—1661no—
127#117—1661no—
128#117C2×D8312yes—
129C2×C42C2×D8311yescharacteristic
130#107—834no—
131#106—834no—
132C4⋊C8—832no—
133#103—834no—
134#103—834no—
135C22⋊C8—832no—
136C4×C8—831no—
137C4×C8—831no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—832no—
141C2×Q8—1661no—
142C22×C4C22⋊D8311yescharacteristic
143C22×C4C22⋊D8311yescharacteristic
144C22×C4C22×D8311yescharacteristic
145C2×C8—832no—
146C2×C8—832no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—832no—
152C4⋊C4—1661no—
153C4⋊C4—1661no—
154C4⋊C4—1662no—
155C4⋊C4—1662no—
156C4⋊C4—1661no—
157C4⋊C4C22×D8312yes—
158C4⋊C4C22×D8312yes—
159C22⋊C4—22no—
160C22⋊C4—1662no—
161C42—22no—
162C42—1662no—
163C42C22⋊D8311yescharacteristic
164C42C22⋊D8311yescharacteristic
165C42C22×D8311yescharacteristic
166C42C22×D8311yescharacteristic
167C23C2×C22⋊D8311yescharacteristic
168Q8—42no—
169Q8—42no—
170Q8—3322no—
171Q8—1664no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—1662no—
178C2×C4—1662no—
179C2×C4C2×C22⋊D8311yescharacteristic
180C2×C4C2×C22⋊D8311yescharacteristic
181C2×C4C2×C22⋊D8311yescharacteristic
182C2×C4C2×C22⋊D8311yescharacteristic
183C2×C4C2×C22⋊D8311yescharacteristic
184C2×C4C23×D8311yescharacteristic
185C8—1662no—
186C8—1662no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22C22×C22⋊D8311yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—3322no—
197C4—3322no—
198C4#1458181212yes—
199C4#1458182212yes—
200C4#1458184012yes—
201C2—41no—
202C2#1912827511yescharacteristic
203C2C2×#1458181211yescharacteristic
204C2C2×#1458182211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenter#533D83
hypocenterC83#3231
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. #19128104C2semidirect
3. #19128097C2semidirect
4. #19128053C2semidirect
5. #19128053C2semidirect
7. #19128005C2semidirect
103. C83#3231semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191281042
#191280972
#191280532
#191280422
#191280052
#191277542
#533×C832
#323183