Finite Group Databasebeta

← #24910264

Subgroups — #24910264

Order 10624.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×#14581784C211yesmaximal, characteristic
2#19127598C212yesmaximal
3#19127588C212yesmaximal
4#19127586C212yesmaximal
5#19127584C212yesmaximal
6#19127583C212yesmaximal
7C4×C4⋊Dic83C211yesmaximal, characteristic
8#19127552C212yesmaximal
9#383×C83C211yesmaximal, characteristic
10#14581784C418yescomplemented
11C2×C4⋊Dic83C2212yes—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C4×Dic83C2212yes—
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C2×C42×C83C2211yescharacteristic
28#95×C83C2212yes—
29#95×C83C2212yes—
30#95×C83C2211yescharacteristic
31#95×C83C2211yescharacteristic
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C4⋊Dic83—24no—
37C4⋊Dic83C2×C414yes—
38C4⋊Dic83C2×C414yes—
39Dic83⋊C4—24no—
40Dic83⋊C4—42no—
41Dic83⋊C4C2×C418yes—
42Dic83⋊C4C2×C418yes—
43C4×Dic83—24no—
44C4×Dic83—41no—
45C4×Dic83—42no—
46C4×Dic83—41no—
47C22×C4×C83C2312yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C42×C83C2×C414yes—
53C23×C83C2411yescharacteristic
54C2×C4×C83—22no—
55C2×C4×C83—21no—
56C2×C4×C83—22no—
57C2×C4×C83—21no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83C22×C412yes—
65C2×C4×C83C22×C412yes—
66C2×C4×C83C22×C412yes—
67C2×Dic83—22no—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83C22×C414yes—
77C2×Dic83C22×C414yes—
78C2×Dic83#4114yes—
79C22×C83C23×C411yescharacteristic
80C22×C83C2×#4112yes—
81C22×C83H(5,𝔽2)11yescharacteristic
82C22×C832−1+412yes—
83C22×C832−1+411yescharacteristic
84C4×C83—22no—
85C4×C83—22no—
86C4×C83—22no—
87C4×C83—42no—
88C4×C83—42no—
89Dic83—24no—
90Dic83—24no—
91Dic83—24no—
92Dic83—42no—
93C2×C83C4×#4111yescharacteristic
94C2×C83#51911yescharacteristic
95C2×C83#52011yescharacteristic
96C2×C83#53912yes—
97C2×C83#54112yes—
98#2336—831nomaximal, Sylow
99C83#233611yescharacteristic, complemented, Sylow
100C2×#125—831no—
101#389—832no—
102#387—832no—
103#387—832no—
104#387—832no—
105#387—832no—
106#383D8311yescharacteristic
107C4×C4⋊C4—831no—
108C4×C4⋊C4—832no—
109#125—838no—
110C2×C4⋊C4—832no—
111C2×C4⋊C4—832no—
112C2×C4⋊C4—831no—
113C2×C4⋊C4—831no—
114C2×C4⋊C4—832no—
115C2×C4⋊C4—832no—
116C2×C4⋊C4—832no—
117C2×C42—831no—
118C2×C42—831no—
119C2×C42—832no—
120C2×C42—832no—
121C2×C42C2×D8311yescharacteristic
122#95—832no—
123#95—832no—
124#95—832no—
125#95—832no—
126#95—832no—
127#95C2×D8312yes—
128#95C2×D8311yescharacteristic
129#95C2×D8312yes—
130#95C2×D8311yescharacteristic
131C22×C4—832no—
132C22×C4—832no—
133C22×C4—832no—
134C22×C4—832no—
135C22×C4C22×D8312yes—
136C22×C4C22×D8312yes—
137C22×C4C22×D8311yescharacteristic
138C22×C4C22×D8311yescharacteristic
139C22×C4C22×D8311yescharacteristic
140C4⋊C4—3322no—
141C4⋊C4—838no—
142C4⋊C4—838no—
143C4⋊C4—834no—
144C4⋊C4—834no—
145C4⋊C4—1664no—
146C4⋊C4—1664no—
147C42—3321no—
148C42—3321no—
149C42—3322no—
150C42—1664no—
151C42C4×D8314yes—
152C23C23×D8311yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—834no—
164C2×C4—834no—
165C2×C4—834no—
166C2×C4—1662no—
167C2×C4—1662no—
168C2×C4—1662no—
169C2×C4—1662no—
170C2×C4—1662no—
171C2×C4—1662no—
172C2×C4—1662no—
173C2×C4—1662no—
174C2×C4—1662no—
175C2×C4C2×C4×D8312yes—
176C2×C4C2×C4×D8312yes—
177C2×C4C2×C4×D8312yes—
178C22C22×C4×D8311yescharacteristic
179C22#41×D8312yes—
180C22#1458183811yescharacteristic
181C22#1458183912yes—
182C22#1458183911yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—3322no—
189C4—1664no—
190C4—1664no—
191C4—1664no—
192C2#1912816911yescharacteristic
193C2#1912817811yescharacteristic
194C2#1912818111yescharacteristic
195C2#1912829212yes—
196C2#1912830812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutatorC22×C83C23×C4
Fitting#383×C83C2
socle, prime radical, cosocleC23×C83C24
hypercenter#383D83
hypocenterC83#2336
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #14581784C4semidirect
99. C83#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#145817842
#191275982
#191275882
#191275862
#191275842
#191275832
C4×C4⋊Dic832
#191275522
#383×C832
#233683