Finite Group Databasebeta

← #24907921

Subgroups — #24907921

Order 10624.

378 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14581775C211yesmaximal, characteristic
2C2×#14581706C212yesmaximal
3#19127564C211yesmaximal, characteristic, complemented
4#19127467C212yesmaximal, complemented
5C2×#117×C83C211yesmaximal, characteristic, complemented
6C22×C22⋊D83C2211yescharacteristic
7#14581775—22no—
8#14581775—21no—
9#14581775—21no—
10C2×Dic83⋊C4C412yescomplemented
11C2×C22⋊Dic83C2212yes—
12C2×C22⋊Dic83C2211yescharacteristic
13C2×(C2×D83)⋊C4C412yescomplemented
14#14581706—24no—
15C83⋊#95—22no—
16C23×C4×C83C2211yescharacteristic
17#117×C83—24no—
18C2×C4⋊C4×C83—21no—
19C2×C22⋊C4×C83C2212yes—
20C2×C42×C83—21no—
21C23×D83C2×C411yescharacteristic
22C2×C22⋊D83—21no—
23C2×C22⋊D83—21no—
24C2×C22⋊D83—22no—
25C2×C22⋊D83—22no—
26C22×Dic83—22no—
27C22×Dic83C2×C411yescharacteristic
28C22×Dic83C2×C412yes—
29Dic83⋊C4—22no—
30Dic83⋊C4—22no—
31C22⋊Dic83—21no—
32C22⋊Dic83—21no—
33C22⋊Dic83—22no—
34C22⋊Dic83—22no—
35(C2×D83)⋊C4—22no—
36(C2×D83)⋊C4—22no—
37C24×C83C2311yescharacteristic
38C22×C4×C83—22no—
39C22×C4×C83—22no—
40C22×C4×C83—22no—
41C22×C4×C83C2×C412yes—
42C22×C4×C83D412yes—
43C22×C4×C83D411yescharacteristic
44C22×C4×C83D411yescharacteristic
45C4⋊C4×C83—24no—
46C22⋊C4×C83—22no—
47C22⋊C4×C83D414yes—
48C42×C83—24no—
49C23×C83—42no—
50C23×C83C22×C411yescharacteristic
51C23×C83C2×D411yescharacteristic
52C23×C83C2×D412yes—
53C23×C83C2×D411yescharacteristic
54C23×C83#4112yes—
55C22×D83—21no—
56C22×D83—22no—
57C22×D83—42no—
58C2×C4×C83—24no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—24no—
62C2×C4×C83—21no—
63C2×C4×C83—21no—
64C2×C4×C83—24no—
65C2×C4×C83—44no—
66C2×C4×C83C22⋊C412yes—
67C2×C4×C83C22⋊C412yes—
68C22⋊D83—41no—
69C22⋊D83—42no—
70C22⋊D83—41no—
71C2×Dic83—22no—
72C2×Dic83—21no—
73C2×Dic83—22no—
74C2×Dic83—42no—
75C2×Dic83—44no—
76C2×Dic83—42no—
77C22×C83—22no—
78C22×C83—22no—
79C22×C83—22no—
80C22×C83—42no—
81C22×C83—42no—
82C22×C83C2×C22⋊C411yescharacteristic
83C22×C83C4×D412yes—
84C22×C83#12011yescharacteristic
85C22×C83#12112yes—
86C22×C83#12311yescharacteristic
87C2×D83—41no—
88C2×D83—42no—
89C2×D83—82no—
90C4×C83—22no—
91C4×C83—22no—
92C4×C83—44no—
93Dic83—41no—
94Dic83—42no—
95Dic83—82no—
96C2×C83—22no—
97C2×C83—21no—
98C2×C83—22no—
99C2×C83—21no—
100C2×C83#38611yescharacteristic
101C2×C83#41012yes—
102D83—81no—
103#1899—831nomaximal, Sylow
104C83#189911yescharacteristic, complemented, Sylow
105C2×#123—831no—
106C2×#117D8311yescharacteristic, complemented
107C2×#99—832no—
108#381—831no—
109#342—832no—
110C22×D4—831no—
111C23×C4C2×D8311yescharacteristic
112#123—1662no—
113#123—1661no—
114#123—1661no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—832no—
118C2×C22⋊C4—831no—
119C2×C22⋊C4—832no—
120C2×C22⋊C4—832no—
121C2×C22⋊C4C2×D8312yes—
122C2×C42—21no—
123#99—1664no—
124#95—1662no—
125C24—831no—
126C24C22×D8311yescharacteristic
127C2×D4—1662no—
128C2×D4—1661no—
129C2×D4—1661no—
130C2×D4—1662no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—832no—
135C22×C4—831no—
136C22×C4—1662no—
137C22×C4C4×D8312yes—
138C22×C4D33212yes—
139C22×C4C22⋊D8311yescharacteristic
140C22×C4C22⋊D8311yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—1662no—
143C4⋊C4—1662no—
144C22⋊C4—22no—
145C22⋊C4—1662no—
146C22⋊C4—1662no—
147C22⋊C4—1661no—
148C22⋊C4—1661no—
149C22⋊C4—1662no—
150C22⋊C4—1662no—
151C22⋊C4C22⋊D8314yes—
152C42—24no—
153C23—42no—
154C23—3322no—
155C23—1661no—
156C23—1662no—
157C23C2×C4×D8311yescharacteristic
158C23C2×C22⋊D8311yescharacteristic
159C23C2×C22⋊D8312yes—
160C23C2×D33211yescharacteristic
161C23#1305059012yes—
162D4—3321no—
163D4—3322no—
164D4—3321no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—44no—
173C2×C4—3322no—
174C2×C4—3324no—
175C2×C4—3322no—
176C2×C4—1662no—
177C2×C4—1661no—
178C2×C4—1662no—
179C2×C4(C2×D83)⋊C412yes—
180C2×C4(C2×D83)⋊C412yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—3321no—
187C22—3322no—
188C22—6642no—
189C22C2×(C2×D83)⋊C411yescharacteristic
190C22C4×C22⋊D8312yes—
191C22#1458176111yescharacteristic
192C22#1458176612yes—
193C22#1458177511yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—3321no—
198C4—3322no—
199C4—6642no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—6641no—
205C2#1912758011yescharacteristic
206C2#1912767412yes—
subgroupG/H
centerC22C2×(C2×D83)⋊C4
commutatorC23×C83C22×C4
FrattiniC24C22×D83
FittingC2×#117×C83C2
socleC22×C83C2×C22⋊C4
prime radicalC22×C22⋊D83C22
cosocleC24×C83C23
hypercenterC2×#117D83
hypocenterC83#1899
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #19127564C2semidirect
4. #19127467C2semidirect
5. C2×#117×C83C2semidirect
10. C2×Dic83⋊C4C4semidirect
13. C2×(C2×D83)⋊C4C4semidirect
104. C83#1899semidirect
106. C2×#117D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817062
#191275642
#191274672
C2×#117×C832
#189983