Finite Group Databasebeta

← #24909150

Subgroups — #24909150

Order 10624.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#19128275C211yesmaximal, characteristic
2C2×#14581711C211yesmaximal, characteristic, complemented
3#19127499C212yesmaximal, complemented
4#19127497C212yesmaximal, complemented
5C2×#101×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C411yescharacteristic
7C2×#13050574—21no—
8#14581790C411yescharacteristic
9#14581790C2211yescharacteristic
10#14581781C411yescharacteristic
11#14581781C2211yescharacteristic
12#14581775—21no—
13#14581775—21no—
14#14581768—21no—
15#14581768—21no—
16#14581751C411yescharacteristic
17#14581711—21no—
18#14581711C2212yes—
19C22×Q8×C83C2211yescharacteristic
20C2×M16×C83—21no—
21#101×C83—21no—
22#101×C83C2212yes—
23#13050590—41no—
24#13050590—41no—
25C2×Dic166C2×C411yescharacteristic
26C2×D332C2×C411yescharacteristic
27C2×C22⋊D83—21no—
28C2×C4×D83—21no—
29#13050574—22no—
30#13050574—41no—
31C2×C83⋊C8—41no—
32C4⋊Dic83—21no—
33Dic83⋊C4—21no—
34Dic83⋊C4—21no—
35Dic83⋊C4—21no—
36Dic83⋊C4—21no—
37C22⋊Dic83—21no—
38(C2×D83)⋊C4—21no—
39(C2×D83)⋊C4—21no—
40(C2×D83)⋊C4—21no—
41(C2×D83)⋊C4—21no—
42C4×Dic83C2×C411yescharacteristic
43C4×Dic83C2×C411yescharacteristic
44C2×Q8×C83—41no—
45C2×Q8×C83—41no—
46C2×Q8×C83C2×C411yescharacteristic
47C2×Q8×C83D411yescharacteristic
48C2×Q8×C83D411yescharacteristic
49C2×Q8×C83C2311yescharacteristic
50C22×C4×C83C2×C411yescharacteristic
51C22×C4×C83D411yescharacteristic
52C22×C4×C83D411yescharacteristic
53M16×C83—22no—
54M16×C83—41no—
55C2×C8×C83—41no—
56C23×C83C22⋊C411yescharacteristic
57C22×D83—21no—
58Q8×C83—41no—
59Q8×C83—41no—
60Q8×C83—41no—
61Q8×C83—41no—
62C2×C4×C83—21no—
63C2×C4×C83—21no—
64C2×C4×C83—21no—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83C22⋊C411yescharacteristic
69C2×C4×C83C22⋊C411yescharacteristic
70C2×C4×C83C22⋊C411yescharacteristic
71C2×C4×C83C22×C411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C22⋊D83—41no—
75C22⋊D83—41no—
76C2×Dic83—21no—
77C2×Dic83—21no—
78C2×Dic83—21no—
79D332—41no—
80C4×D83—42no—
81Dic166—41no—
82C8×C83—42no—
83C83⋊C8—42no—
84C22×C83—41no—
85C22×C83#9911yescharacteristic
86C22×C83#9911yescharacteristic
87C22×C83C2×C22⋊C411yescharacteristic
88C2×D83—42no—
89C2×D83—41no—
90C4×C83—21no—
91C4×C83—21no—
92C4×C83—21no—
93C4×C83—21no—
94C4×C83—21no—
95C4×C83—21no—
96Dic83—41no—
97Dic83—41no—
98Dic83—41no—
99C2×C83—21no—
100C2×C83—21no—
101C2×C83—21no—
102C2×C83C2×#9911yescharacteristic
103D83—81no—
104#2133—831nomaximal, Sylow
105C83#213311yescharacteristic, complemented, Sylow
106#536—831no—
107C2×#101—831no—
108C2×#101D8311yescharacteristic, complemented
109#356—832no—
110#355—832no—
111C2×#41—831no—
112C22×Q8C2×D8311yescharacteristic
113C2×M16—21no—
114C2×M16—1661no—
115C4⋊Q8—831no—
116C4⋊Q8—831no—
117#124—831no—
118#124—831no—
119#123—1661no—
120#123—1661no—
121C22⋊Q8—1661no—
122C22⋊Q8—1661no—
123#117—831no—
124#101—21no—
125#101—832no—
126#101—1661no—
127#101C2×D8312yes—
128#41—3321no—
129#41—3321no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—831no—
133C2×Q8C4×D8311yescharacteristic
134C2×Q8D33211yescharacteristic
135C2×Q8C22⋊D8311yescharacteristic
136C2×Q8C22×D8311yescharacteristic
137C2×D4—831no—
138C2×D4—1661no—
139C22×C4—1661no—
140C22×C4C4×D8311yescharacteristic
141C22×C4D33211yescharacteristic
142C22×C4C22⋊D8311yescharacteristic
143M16—22no—
144M16—41no—
145M16—3321no—
146M16—1662no—
147C2×C8—41no—
148C2×C8—3321no—
149C4⋊C4—1661no—
150C4⋊C4—1661no—
151C4⋊C4—1661no—
152C4⋊C4—1661no—
153C4⋊C4—1661no—
154C22⋊C4—1661no—
155C22⋊C4—1661no—
156C22⋊C4—1661no—
157C22⋊C4—1661no—
158C22⋊C4—1661no—
159C42—831no—
160C42—831no—
161C23—1661no—
162C23(C2×D83)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—3321no—
168D4—3321no—
169D4—3321no—
170D4—3321no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—3322no—
178C2×C4—1661no—
179C2×C4—1661no—
180C2×C4—1661no—
181C2×C4(C2×D83)⋊C411yescharacteristic
182C2×C4(C2×D83)⋊C411yescharacteristic
183C2×C4(C2×D83)⋊C411yescharacteristic
184C2×C4C2×C4×D8311yescharacteristic
185C2×C4C2×C22⋊D8311yescharacteristic
186C2×C4C2×D33211yescharacteristic
187C8—42no—
188C8—3322no—
189C22—41no—
190C22—3322no—
191C22—3321no—
192C22#1458170611yescharacteristic
193C22#1458170611yescharacteristic
194C22C2×(C2×D83)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—3321no—
202C4—3321no—
203C4—3321no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—6641no—
208C2C2×#1458170611yescharacteristic
subgroupG/H
centerC2C2×#14581706
commutatorC2×C4×C83C22×C4
FrattiniC2×Q8C22×D83
FittingC2×#101×C83C2
socleC2×C83C2×#99
prime radicalC2×#13050590C4
cosocleC2×Q8×C83C23
hypercenterC2×#101D83
hypocenterC83#2133
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#14581711C2semidirect
3. #19127499C2semidirect
4. #19127497C2semidirect
5. C2×#101×C83C2semidirect
105. C83#2133semidirect
108. C2×#101D83semidirect

Maximal subgroups

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

subgroupindex
#191282752
C2×#145817112
#191274992
#191274972
C2×#101×C832
#213383