Finite Group Databasebeta

← #24907656

Subgroups — #24907656

Order 10624.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581723C212yesmaximal, complemented
2C2×C22⋊(C83⋊C8)C211yesmaximal, characteristic
3#19127561C211yesmaximal, characteristic
4#19127463C212yesmaximal, complemented
5C22×C4⋊C4×C83C211yesmaximal, characteristic
6C22×C83⋊C8C2212yes—
7C2×C4⋊Dic83C2211yescharacteristic
8C2×C4⋊Dic83C2211yescharacteristic
9C2×C22⋊Dic83—21no—
10#14581723—24no—
11C22⋊(C83⋊C8)C414yescomplemented
12C83⋊#95—21no—
13C23×C4×C83—21no—
14C23×C4×C83C2211yescharacteristic
15C2×C4⋊C4×C83—21no—
16C2×C4⋊C4×C83—24no—
17C2×C4⋊C4×C83C2212yes—
18C22×Dic83—21no—
19C22×Dic83—21no—
20C2×C83⋊C8—22no—
21C2×C83⋊C8—22no—
22C2×C83⋊C8C2×C414yescomplemented
23C4⋊Dic83—22no—
24C4⋊Dic83—22no—
25C22⋊Dic83—42no—
26C24×C83D411yescharacteristic
27C22×C4×C83—22no—
28C22×C4×C83—22no—
29C22×C4×C83—24no—
30C22×C4×C83—22no—
31C22×C4×C83—42no—
32C22×C4×C83—42no—
33C22×C4×C83C2×C412yes—
34C22×C4×C83D411yescharacteristic
35C22×C4×C83Q812yes—
36C22×C4×C83C2311yescharacteristic
37C4⋊C4×C83—24no—
38C4⋊C4×C83—22no—
39C4⋊C4×C83D414yes—
40C23×C83—21no—
41C23×C83—21no—
42C23×C83—21no—
43C23×C83—21no—
44C23×C83C4⋊C412yes—
45C23×C83SD1612yes—
46C23×C83SD1612yes—
47C23×C83C2×D411yescharacteristic
48C2×C4×C83—21no—
49C2×C4×C83—21no—
50C2×C4×C83—24no—
51C2×C4×C83—24no—
52C2×C4×C83—24no—
53C2×C4×C83—48no—
54C2×C4×C83—42no—
55C2×C4×C83—42no—
56C2×C4×C83C4⋊C412yes—
57C2×C4×C83C22×C411yescharacteristic
58C2×C4×C83C2×D412yes—
59C2×C4×C83C2×Q811yescharacteristic
60C2×C4×C83#4112yes—
61C2×Dic83—41no—
62C2×Dic83—41no—
63C2×Dic83—41no—
64C2×Dic83—41no—
65C2×Dic83—41no—
66C2×Dic83—41no—
67C83⋊C8—24no—
68C22×C83—22no—
69C22×C83—22no—
70C22×C83—22no—
71C22×C83—22no—
72C22×C83—22no—
73C22×C83—22no—
74C22×C83#10614yescomplemented
75C22×C83C2×C4⋊C411yescharacteristic
76C22×C83#12011yescharacteristic
77C22×C83#12311yescharacteristic
78C22×C83C2×SD1611yescharacteristic
79C22×C83C2×SD1611yescharacteristic
80C22×C83AGL(1,ℤ/8)11yescharacteristic
81C22×C83#13711yescharacteristic
82C4×C83—22no—
83C4×C83—44no—
84C4×C83C4×D412yes—
85C4×C83C22⋊Q812yes—
86Dic83—81no—
87Dic83—81no—
88C2×C83—24no—
89C2×C83#38511yescharacteristic
90C2×C83C2×#10611yescharacteristic
91C2×C83#42811yescharacteristic
92C2×C83#44811yescharacteristic
93C2×C83#45011yescharacteristic
94C2×C83#48111yescharacteristic
95C2×C83#48311yescharacteristic
96#1853—831nomaximal, Sylow
97C83#185311yescharacteristic, complemented, Sylow
98C22×C4⋊C4D8311yescharacteristic
99C2×#106—832no—
100C2×C22⋊C8—831no—
101#380—831no—
102#340—832no—
103C23×C4—21no—
104C23×C4C2×D8311yescharacteristic
105C22×C8—832no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—831no—
109C2×C4⋊C4—831no—
110C2×C4⋊C4C2×D8312yes—
111C2×C22⋊C4—1661no—
112#106—1664no—
113C22⋊C8—834no—
114#95—1661no—
115C24C22⋊D8311yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—1661no—
123C22×C4—1661no—
124C22×C4Dic16612yes—
125C22×C4C4×D8312yes—
126C22×C4C22⋊D8311yescharacteristic
127C22×C4C22×D8311yescharacteristic
128C2×C8—834no—
129C2×C8—1662no—
130C2×C8—1662no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—1662no—
134C4⋊C4—1662no—
135C4⋊C4C22⋊D8314yes—
136C22⋊C4—3322no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23Dic83⋊C412yes—
142C23#1305057712yes—
143C23#1305057812yes—
144C23C2×C22⋊D8311yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—3321no—
154C2×C4—3321no—
155C2×C4—3321no—
156C2×C4—3321no—
157C2×C4—3321no—
158C2×C4—3321no—
159C2×C4Dic83⋊C412yes—
160C2×C4C2×C4×D8311yescharacteristic
161C2×C4C2×C22⋊D8312yes—
162C2×C4C2×Dic16611yescharacteristic
163C2×C4#1305059012yes—
164C8—1664no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1458172314yescomplemented
172C22C2×Dic83⋊C411yescharacteristic
173C22#1458176111yescharacteristic
174C22#1458177511yescharacteristic
175C22C2×#1305057711yescharacteristic
176C22C2×#1305057811yescharacteristic
177C22#1458181811yescharacteristic
178C22#1458182311yescharacteristic
179C4—22no—
180C4—44no—
181C4—6641no—
182C4—6641no—
183C4C4×C22⋊D8312yes—
184C4#1458177112yes—
185C2—24no—
186C2#1912757411yescharacteristic
187C2C2×#1458172311yescharacteristic
188C2#1912774111yescharacteristic
189C2#1912784011yescharacteristic
190C2#1912785211yescharacteristic
191C2#1912801911yescharacteristic
192C2#1912803111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC22×C4⋊C4×C83C2
socleC23×C83C2×D4
prime radicalC24×C83D4
cosocleC22×C4×C83C23
hypercenterC22×C4⋊C4D83
hypocenterC83#1853
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14581723C2semidirect
4. #19127463C2semidirect
11. C22⋊(C83⋊C8)C4semidirect
22. C2×C83⋊C8C2×C4semidirect
74. C22×C83#106semidirect
97. C83#1853semidirect
171. C22#14581723semidirect

Maximal subgroups

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

subgroupindex
C2×#145817232
C2×C22⋊(C83⋊C8)2
#191275612
#191274632
C22×C4⋊C4×C832
#185383