Finite Group Databasebeta

← #24918655

Subgroups — #24918655

Order 10624.

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

normal subgroups only

This group has 368 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
2C2×#14581784C211yesmaximal, characteristic, complemented
3#19128110C212yesmaximal, complemented
4#19128066C212yesmaximal, complemented
5#19128020C212yesmaximal, complemented
6#19128015C212yesmaximal, complemented
7#19128002C214yesmaximal, complemented
8#432×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—21no—
10#14581792C2211yescharacteristic
11#14581789C2211yescharacteristic
12#14581784—21no—
13#14581784—21no—
14#14581784C2212yes—
15#14581784C2212yes—
16#14581780—21no—
17#14581766—21no—
18#14581766C2212yes—
19C4×D332—21no—
20C4×D332C2212yes—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊Dic83C2211yescharacteristic
23C2×Dic83⋊C4—21no—
24C2×Dic83⋊C4—21no—
25#14581726C2214yes—
26#14581724C2214yes—
27#14581714C2214yes—
28#14581714C2214yes—
29C2×C42×C83C2211yescharacteristic
30C4⋊C8×C83C2212yes—
31C22⋊C8×C83C2212yes—
32#97×C83C2212yes—
33#13050590—44no—
34C2×Dic166—21no—
35C2×D332—21no—
36C2×D332—21no—
37C2×D332C2312yes—
38C2×C22⋊D83—22no—
39C22×Dic83—21no—
40C22×Dic83—21no—
41C2×C4×D83—22no—
42C4⋊Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83—21no—
45C4⋊Dic83C2312yes—
46C4⋊Dic83C2312yes—
47C4⋊Dic83C2312yes—
48Dic83⋊C4—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54C22×C4×C83D411yescharacteristic
55C22×C4×C83D411yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C2×C8×C83C2314yes—
58C42×C83D411yescharacteristic
59C42×C83D411yescharacteristic
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83C2×D411yescharacteristic
63C22×D83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—41no—
67C2×C4×C83C2×D411yescharacteristic
68C2×C4×C83C2×D411yescharacteristic
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2411yescharacteristic
73C22⋊D83—44no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—42no—
78C2×Dic83—42no—
79D332—24no—
80D332—42no—
81D332—42no—
82C4×D83—44no—
83Dic166—42no—
84C8×C83—24no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83AGL(1,ℤ/8)12yes—
88C22×C83C22×D411yescharacteristic
89C2×D83—42no—
90C2×D83—42no—
91C2×D83—42no—
92C4×C83—21no—
93C4×C83—21no—
94C4×C83—21no—
95C4×C83—21no—
96C4×C83—21no—
97C4×C832−1+412yes—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101C2×C83—22no—
102C2×C83#53611yescharacteristic
103C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C83#57511yescharacteristic
105D83—82no—
106#3234—831nomaximal, Sylow
107C83#323411yescharacteristic, complemented, Sylow
108#532—831no—
109C2×#125—831no—
110#501—832no—
111#490—832no—
112#481—832no—
113#480—832no—
114#478—834no—
115#432D8311yescharacteristic, complemented
116C2×#41—1661no—
117C4⋊Q8—831no—
118#127—831no—
119#125—832no—
120#125—832no—
121#125—1661no—
122#125—1661no—
123#124—1661no—
124#121—832no—
125#121—1661no—
126C4×D4—832no—
127C4×D4—1661no—
128C2×C4⋊C4—831no—
129C2×C4⋊C4—831no—
130C2×C4⋊C4—1661no—
131C2×C4⋊C4—1661no—
132C2×C42C2×D8311yescharacteristic
133#107—834no—
134#106—834no—
135C4⋊C8C2×D8312yes—
136#102—834no—
137#102—834no—
138C22⋊C8C2×D8312yes—
139#97C2×D8312yes—
140#41—3324no—
141C2×Q8—1661no—
142C2×D4—832no—
143C2×D4—1662no—
144C2×D4—1661no—
145C2×D4—1661no—
146C22×C4—1661no—
147C22×C4—1661no—
148C22×C4—1662no—
149C22×C4D33211yescharacteristic
150C22×C4D33211yescharacteristic
151C22×C4C22×D8311yescharacteristic
152C2×C8C22×D8314yes—
153C4⋊C4—832no—
154C4⋊C4—832no—
155C4⋊C4—832no—
156C4⋊C4—1661no—
157C4⋊C4—1661no—
158C4⋊C4—1662no—
159C4⋊C4—1662no—
160C4⋊C4—1662no—
161C4⋊C4—1662no—
162C4⋊C4—1661no—
163C22⋊C4—1662no—
164C22⋊C4—1662no—
165C42D33211yescharacteristic
166C42D33211yescharacteristic
167C42C22×D8311yescharacteristic
168C42C22×D8311yescharacteristic
169C23—1662no—
170C23C2×D33211yescharacteristic
171Q8—3322no—
172D4—3322no—
173D4—3322no—
174D4—3324no—
175D4—1664no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—3324no—
180C2×C4—3322no—
181C2×C4—3322no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4C2×D33211yescharacteristic
186C2×C4C2×D33211yescharacteristic
187C2×C4C2×D33211yescharacteristic
188C2×C4C2×D33211yescharacteristic
189C2×C4C2×D33211yescharacteristic
190C2×C4C23×D8311yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—3322no—
195C22—3322no—
196C22—3322no—
197C22#1458182012yes—
198C22C22×D33211yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—3322no—
205C4—3322no—
206C4—3322no—
207C4#1458183912yes—
208C2—22no—
209C2—6642no—
210C2#1912827711yescharacteristic
211C2C2×#1458182011yescharacteristic
212C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#432×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#432D83
hypocenterC83#3234
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×#14581784C2semidirect
3. #19128110C2semidirect
4. #19128066C2semidirect
5. #19128020C2semidirect
6. #19128015C2semidirect
7. #19128002C2semidirect
8. #432×C83C2semidirect
107. C83#3234semidirect
115. #432D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
C2×#145817842
#191281102
#191280662
#191280202
#191280152
#191280022
#432×C832
#323483