Finite Group Databasebeta

← C83×#1507 #24903192

Subgroups — C83×#1507 #24903192

Order 10624.

346 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 96 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×#121×C83C211yesmaximal, characteristic
2#407×C83C211yesmaximal, characteristic, complemented
3C4×C22⋊C4×C83C211yesmaximal, characteristic, complemented
4#327×C83C214yesmaximal, complemented
5C22×D4×C83—21no—
6C22×D4×C83C411yescharacteristic, complemented
7C23×C4×C83C2211yescharacteristic
8C2×M16×C83—21no—
9#121×C83—21no—
10#121×C83—21no—
11#121×C83C412yescomplemented
12#121×C83C2212yes—
13C2×C4⋊C4×C83C411yescharacteristic, complemented
14C2×C22⋊C4×C83—21no—
15C2×C22⋊C4×C83—21no—
16C2×C42×C83—21no—
17C22⋊C8×C83—21no—
18C22⋊C8×C83C2212yescomplemented
19#95×C83C2212yescomplemented
20C24×C83—21no—
21C24×C83D411yescharacteristic
22C2×D4×C83—22no—
23C2×D4×C83—22no—
24C2×D4×C83—21no—
25C2×D4×C83—42no—
26C2×D4×C83—44no—
27C2×D4×C83C2×C412yes—
28C22×C4×C83—21no—
29C22×C4×C83—22no—
30C22×C4×C83—42no—
31C22×C4×C83C2×C411yescharacteristic
32C22×C4×C83C2×C411yescharacteristic
33C22×C4×C83D411yescharacteristic
34C22×C4×C83D411yescharacteristic
35C22×C4×C83D411yescharacteristic
36C22×C4×C83C2311yescharacteristic
37M16×C83—42no—
38C2×C8×C83—22no—
39C4⋊C4×C83—21no—
40C4⋊C4×C83C2×C412yes—
41C22⋊C4×C83—24no—
42C22⋊C4×C83—22no—
43C22⋊C4×C83—22no—
44C42×C83—42no—
45C23×C83—22no—
46C23×C83—22no—
47C23×C83—22no—
48C23×C83—41no—
49C23×C83—42no—
50C23×C83—42no—
51C23×C83—42no—
52C23×C83C22⋊C411yescharacteristic
53C23×C83C22⋊C411yescharacteristic
54C23×C83C2×D411yescharacteristic
55D4×C83—42no—
56D4×C83—44no—
57D4×C83—44no—
58D4×C83—42no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83—24no—
63C2×C4×C83—22no—
64C2×C4×C83—41no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—42no—
68C2×C4×C83—42no—
69C2×C4×C83—42no—
70C2×C4×C83C22⋊C411yescharacteristic
71C2×C4×C83C22⋊C411yescharacteristic
72C2×C4×C83C22×C411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C8×C83—42no—
75C22×C83—21no—
76C22×C83—21no—
77C22×C83—21no—
78C22×C83—21no—
79C22×C83—21no—
80C22×C83—21no—
81C22×C83—21no—
82C22×C83—44no—
83C22×C83—42no—
84C22×C83—42no—
85C22×C83—42no—
86C22×C83—84no—
87C22×C83#10412yes—
88C22×C83C2×C22⋊C411yescharacteristic
89C22×C83AGL(1,ℤ/8)12yes—
90C4×C83—21no—
91C4×C83—21no—
92C4×C83—21no—
93C4×C83—44no—
94C4×C83—42no—
95C4×C83#9912yes—
96C2×C83—22no—
97C2×C83—22no—
98C2×C83—41no—
99C2×C83—82no—
100C2×C83C2×#9911yescharacteristic
101C2×C83#41811yescharacteristic
102C2×C83C2×#10411yescharacteristic
103#1507C8311yesmaximal, characteristic, direct product factor, Sylow
104C83#150711yescharacteristic, direct product factor, Sylow
105C2×#121C2×C8311yescharacteristic
106#407C2×C8311yescharacteristic, complemented
107C4×C22⋊C4C2×C8311yescharacteristic, complemented
108#327C2×C8314yescomplemented
109C22×D4—21no—
110C22×D4C4×C8311yescharacteristic, complemented
111C23×C4C22×C8311yescharacteristic
112C2×M16—21no—
113#121—21no—
114#121—21no—
115#121C4×C8312yescomplemented
116#121C22×C8312yes—
117C2×C4⋊C4C4×C8311yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C22⋊C8—21no—
122C22⋊C8C22×C8312yescomplemented
123#95C22×C8312yescomplemented
124C24—21no—
125C24D4×C8311yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×C4×C8312yes—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4C2×C4×C8311yescharacteristic
136C22×C4C2×C4×C8311yescharacteristic
137C22×C4D4×C8311yescharacteristic
138C22×C4D4×C8311yescharacteristic
139C22×C4D4×C8311yescharacteristic
140C22×C4C23×C8311yescharacteristic
141M16—42no—
142C2×C8—22no—
143C4⋊C4—21no—
144C4⋊C4C2×C4×C8312yes—
145C22⋊C4—24no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊C4×C8311yescharacteristic
157C23C22⋊C4×C8311yescharacteristic
158C23C2×D4×C8311yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4C22⋊C4×C8311yescharacteristic
175C2×C4C22⋊C4×C8311yescharacteristic
176C2×C4C22×C4×C8311yescharacteristic
177C2×C4C2×D4×C8311yescharacteristic
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22#104×C8312yes—
192C22C2×C22⋊C4×C8311yescharacteristic
193C22AGL(1,ℤ/8)×C8312yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—42no—
199C4#99×C8312yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×#99×C8311yescharacteristic
205C2#418×C8311yescharacteristic
206C2C2×#104×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C2×C22⋊C4
commutatorC2×C4C22×C4×C83
Frattini, cosocleC22×C4C23×C83
prime radicalC2×#121×C83C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. #407×C83C2semidirect
3. C4×C22⋊C4×C83C2semidirect
4. #327×C83C2semidirect
6. C22×D4×C83C4semidirect
11. #121×C83C4semidirect
13. C2×C4⋊C4×C83C4semidirect
18. C22⋊C8×C83C22semidirect
19. #95×C83C22semidirect
103. #1507C83direct (N × H)
104. C83#1507direct (N × H)
106. #407C2×C83semidirect
107. C4×C22⋊C4C2×C83semidirect
108. #327C2×C83semidirect
110. C22×D4C4×C83semidirect
115. #121C4×C83semidirect
117. C2×C4⋊C4C4×C83semidirect
122. C22⋊C8C22×C83semidirect
123. #95C22×C83semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C832
#407×C832
C4×C22⋊C4×C832
#327×C832
#150783