Finite Group Databasebeta

← #24918204

Subgroups — #24918204

Order 10624.

358 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128097C211yesmaximal, characteristic, complemented
3#19128080C211yesmaximal, characteristic, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128051C211yesmaximal, characteristic, complemented
6#19127957C214yesmaximal
7#19127806C214yesmaximal, complemented
8C4×#13050574C211yesmaximal, characteristic
9#533×C83C211yesmaximal, characteristic
10C2×#13050580C2214yes—
11C2×#13050574C2212yes—
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581784—21no—
15#14581771—21no—
16#14581771C2212yes—
17C4×Dic166—21no—
18C4×Dic166C2212yes—
19#14581751—21no—
20#14581723C2214yes—
21#14581716C2214yes—
22#14581715C2214yes—
23#14581703C2212yes—
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
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37#13050580—28no—
38#13050574—24no—
39C2×C83⋊C8C2312yes—
40C2×C83⋊C8C2312yes—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83C2312yes—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47C22⋊Dic83—22no—
48C4×Dic83—22no—
49C2×Q8×C83—21no—
50C2×Q8×C83C2312yes—
51C22×C4×C83D411yescharacteristic
52C22×C4×C83D411yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—21no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83—21no—
57C4⋊C4×C83—22no—
58C4⋊C4×C83—22no—
59C4⋊C4×C83C2312yes—
60C22⋊C4×C83—22no—
61C42×C83—22no—
62C42×C83D411yescharacteristic
63C42×C83D411yescharacteristic
64C42×C83C2311yescharacteristic
65C42×C83C2311yescharacteristic
66C23×C83C2×D411yescharacteristic
67Q8×C83—24no—
68Q8×C83—42no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C2×Dic83—22no—
81C2×Dic83—22no—
82Dic166—24no—
83Dic166—42no—
84C83⋊C8—22no—
85C83⋊C8#4114yes—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83—21no—
89C22×C83C22×D411yescharacteristic
90C4×C83—21no—
91C4×C83—21no—
92C4×C83—21no—
93C4×C83—42no—
94C4×C83—42no—
95C4×C83#13712yes—
96C4×C83#13712yes—
97C4×C83C2×#4112yes—
98Dic83—42no—
99Dic83—42no—
100C2×C83—41no—
101C2×C83#53211yescharacteristic
102C2×C83C2×#13711yescharacteristic
103C2×C83C2×#13711yescharacteristic
104#3185—831nomaximal, Sylow
105C83#318511yescharacteristic, complemented, Sylow
106#533—831no—
107#533D8311yescharacteristic
108#498—831no—
109#494—831no—
110#487—831no—
111#487—831no—
112#470—834no—
113#441—834no—
114C4×M16—831no—
115C2×Q16—834no—
116C2×M16—832no—
117C4⋊Q8—831no—
118C4⋊Q8—831no—
119C4⋊Q8C2×D8311yescharacteristic
120C4⋊Q8C2×D8311yescharacteristic
121#125—21no—
122#125—1661no—
123C22⋊Q8—21no—
124C22⋊Q8—832no—
125C22⋊Q8—1661no—
126C22⋊Q8C2×D8312yes—
127C4×Q8—21no—
128C4×Q8—832no—
129C4×Q8—1661no—
130C4×Q8C2×D8312yes—
131#117—21no—
132#117—1661no—
133C2×C42C2×D8311yescharacteristic
134#106—834no—
135#103—834no—
136#103—834no—
137#97—832no—
138C4×C8—831no—
139C4×C8—831no—
140C2×Q8—21no—
141C2×Q8—832no—
142C2×Q8—1661no—
143C2×Q8C22×D8312yes—
144C22×C4C22⋊D8311yescharacteristic
145C22×C4C22⋊D8311yescharacteristic
146C22×C4C22×D8311yescharacteristic
147Q16—1668no—
148M16—1664no—
149C2×C8—832no—
150C2×C8—832no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—832no—
157C4⋊C4—1661no—
158C4⋊C4—1661no—
159C4⋊C4—1662no—
160C4⋊C4—1662no—
161C4⋊C4—1661no—
162C4⋊C4C22×D8312yes—
163C22⋊C4—22no—
164C22⋊C4—1662no—
165C42—22no—
166C42—1662no—
167C42C22⋊D8311yescharacteristic
168C42C22⋊D8311yescharacteristic
169C42C22×D8311yescharacteristic
170C42C22×D8311yescharacteristic
171C23C2×C22⋊D8311yescharacteristic
172Q8—24no—
173Q8—42no—
174Q8—3322no—
175Q8—1664no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4C2×C22⋊D8311yescharacteristic
184C2×C4C2×C22⋊D8311yescharacteristic
185C2×C4C2×C22⋊D8311yescharacteristic
186C2×C4C2×C22⋊D8311yescharacteristic
187C2×C4C2×C22⋊D8311yescharacteristic
188C2×C4C23×D8311yescharacteristic
189C8—834no—
190C8—1662no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22C22×C22⋊D8311yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4—3322no—
201C4—3322no—
202C4#1458182212yes—
203C4#1458182312yes—
204C4#41×D8312yes—
205C2—41no—
206C2#1912824711yescharacteristic
207C2C2×#1458182211yescharacteristic
208C2C2×#1458182311yescharacteristic
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#3185
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. #19128097C2semidirect
3. #19128080C2semidirect
4. #19128053C2semidirect
5. #19128051C2semidirect
7. #19127806C2semidirect
105. C83#3185semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191280972
#191280802
#191280532
#191280512
#191279572
#191278062
C4×#130505742
#533×C832
#318583