Finite Group Databasebeta

← #38770017

Subgroups — #38770017

Order 25472.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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#27164644C211yesmaximal, characteristic
2#27163881C212yesmaximal
3#27163785C211yesmaximal, characteristic
4#27163784C212yesmaximal
5#381×C199C211yesmaximal, characteristic
6#20263934—22no—
7#20263932—24no—
8C2×C22⋊Dic199C2212yes—
9C2×C22⋊Dic199C2212yes—
10C199⋊#95—24no—
11C23×C4×C199C2211yescharacteristic
12C2×C22⋊C4×C199C2211yescharacteristic
13C2×C22⋊C4×C199C2211yescharacteristic
14#95×C199—22no—
15C2×Dic398—22no—
16C22×Dic199—22no—
17C22×Dic199D414yes—
18C4⋊Dic199—22no—
19Dic199⋊C4—24no—
20C22⋊Dic199—22no—
21C22⋊Dic199—22no—
22C22⋊Dic199—22no—
23C22⋊Dic199—22no—
24C24×C199C2311yescharacteristic
25C22×C4×C199—21no—
26C22×C4×C199—22no—
27C22×C4×C199—22no—
28C22×C4×C199—21no—
29C22×C4×C199—21no—
30C22×C4×C199D412yes—
31C22×C4×C199Q812yescomplemented
32C22⋊C4×C199—22no—
33C22⋊C4×C199—22no—
34Dic398—82no—
35C2×Dic199—24no—
36C2×Dic199—44no—
37C2×Dic199—44no—
38C2×Dic199—42no—
39C23×C199—42no—
40C23×C199C2×D412yes—
41C23×C199C2×D411yescharacteristic
42C23×C199C2×Q811yescharacteristic
43C23×C199#4112yes—
44C23×C199#4111yescharacteristic
45C2×C4×C199—22no—
46C2×C4×C199—24no—
47C2×C4×C199—22no—
48C2×C4×C199—41no—
49C2×C4×C199—42no—
50C2×C4×C199—41no—
51C2×C4×C199—42no—
52C2×C4×C199—42no—
53C2×C4×C199—41no—
54C22×C199—22no—
55C22×C199—24no—
56C22×C199—41no—
57C22×C199—42no—
58C22×C199—41no—
59C22×C199#12011yescharacteristic
60C22×C199C22⋊Q812yes—
61C22×C199C22⋊Q811yescharacteristic
62C22×C199#12412yes—
63C22×C199#12411yescharacteristic
64C4×C199—42no—
65C4×C199—42no—
66C4×C199—81no—
67Dic199—44no—
68Dic199—82no—
69C2×C199—22no—
70C2×C199—21no—
71C2×C199—22no—
72C2×C199—21no—
73C2×C199#39311yescharacteristic
74C2×C199#45812yes—
75C199#203011yescharacteristic, complemented, Sylow
76#2030—1991nomaximal, Sylow
77#543—1991no—
78#381—1992no—
79#381D19911yescharacteristic
80#342—1992no—
81#342—1991no—
82C23×C4C2×D19911yescharacteristic
83C22⋊Q8—3984no—
84C22⋊Q8—3982no—
85C2×C22⋊C4—1992no—
86C2×C22⋊C4—1992no—
87C2×C22⋊C4C2×D19911yescharacteristic
88C2×C22⋊C4C2×D19911yescharacteristic
89#95—22no—
90#95—3984no—
91C24C22×D19911yescharacteristic
92C2×Q8—3982no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—3982no—
99C22×C4—1994no—
100C22×C4C22⋊D19912yes—
101C22×C4Dic39812yescomplemented
102C4⋊C4—3984no—
103C4⋊C4—3982no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—3982no—
107C22⋊C4—3982no—
108C22⋊C4—3982no—
109C22⋊C4—3982no—
110C23—42no—
111C23D4×D19912yes—
112C23C2×C22⋊D19911yescharacteristic
113C23C2×Dic39811yescharacteristic
114C23#1569188512yes—
115C23#1569188611yescharacteristic
116Q8—15922no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—7964no—
127C2×C4—7964no—
128C2×C4—7962no—
129C2×C4—3984no—
130C22—22no—
131C22—24no—
132C22—41no—
133C22—42no—
134C22—41no—
135C22#2026392311yescharacteristic
136C22#2026393212yes—
137C22#2026393411yescharacteristic
138C22#2026394112yes—
139C22#2026394211yescharacteristic
140C4—42no—
141C4—42no—
142C4—81no—
143C4—7964no—
144C4—15922no—
145C2—21no—
146C2—22no—
147C2—22no—
148C2—21no—
149C2#2716393011yescharacteristic
150C2#2716420712yes—
subgroupG/H
centerC22#20263923
commutator, prime radical, cosocleC24×C199C23
FrattiniC24C22×D199
Fitting#381×C199C2
socleC22×C199#120
hypercenter#381D199
hypocenterC199#2030
LCSG ▷ C24×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
31. C22×C4×C199Q8semidirect
75. C199#2030semidirect
101. C22×C4Dic398semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271638812
#271637852
#271637842
#381×C1992
#2030199