Finite Group Databasebeta

← #38770018

Subgroups — #38770018

Order 25472.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#27163879C212yesmaximal
3#27163879C211yesmaximal, characteristic
4#27163784C212yesmaximal
5#342×C199C211yesmaximal, characteristic
6C23×Dic199C2211yescharacteristic
7#20263934—22no—
8#20263932—24no—
9C2×C22⋊Dic199C2211yescharacteristic
10C2×C22⋊Dic199C2212yes—
11C199⋊#95—24no—
12C199⋊#95—22no—
13C2×C22⋊C4×C199C2212yes—
14C2×C22⋊C4×C199C2211yescharacteristic
15C2×Dic398—22no—
16C22×Dic199—22no—
17C22×Dic199—21no—
18C22×Dic199—21no—
19C22×Dic199—21no—
20C22×Dic199—22no—
21C22×Dic199D414yes—
22C22×Dic199Q812yescomplemented
23C4⋊Dic199—22no—
24Dic199⋊C4—24no—
25C22⋊Dic199—22no—
26C22⋊Dic199—22no—
27C22⋊Dic199—22no—
28C24×C199C2311yescharacteristic
29C22×C4×C199—22no—
30C22×C4×C199D412yes—
31C22⋊C4×C199—22no—
32C22⋊C4×C199—22no—
33C22⋊C4×C199—22no—
34Dic398—82no—
35C2×Dic199—24no—
36C2×Dic199—24no—
37C2×Dic199—22no—
38C2×Dic199—41no—
39C2×Dic199—42no—
40C2×Dic199—41no—
41C2×Dic199—44no—
42C2×Dic199—42no—
43C2×Dic199—41no—
44C23×C199—42no—
45C23×C199C2×D412yes—
46C23×C199C2×D411yescharacteristic
47C23×C199C2×Q811yescharacteristic
48C23×C199#4111yescharacteristic
49C23×C199#4112yes—
50C2×C4×C199—22no—
51C2×C4×C199—42no—
52C2×C4×C199—44no—
53C2×C4×C199—42no—
54C22×C199—24no—
55C22×C199—22no—
56C22×C199—41no—
57C22×C199—41no—
58C22×C199—42no—
59C22×C199#12011yescharacteristic
60C22×C199C22⋊Q812yes—
61C22×C199C22⋊Q811yescharacteristic
62C22×C199#12412yes—
63C22×C199#12411yescharacteristic
64C4×C199—42no—
65C4×C199—82no—
66Dic199—42no—
67Dic199—44no—
68Dic199—81no—
69C2×C199—21no—
70C2×C199—22no—
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#381—1991no—
80#342—1992no—
81#342D19911yescharacteristic
82C23×C4—1991no—
83C22⋊Q8—3984no—
84C22⋊Q8—3982no—
85C2×C22⋊C4—1991no—
86C2×C22⋊C4—1992no—
87C2×C22⋊C4C2×D19912yes—
88C2×C22⋊C4C2×D19911yescharacteristic
89#95—3984no—
90#95—3982no—
91C24C22×D19911yescharacteristic
92C2×Q8—3982no—
93C22×C4—22no—
94C22×C4—3981no—
95C22×C4—3982no—
96C22×C4—3981no—
97C22×C4—3981no—
98C22×C4—3982no—
99C22×C4—1994no—
100C22×C4—1992no—
101C22×C4D79612yes—
102C4⋊C4—3984no—
103C4⋊C4—3982no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—3982no—
108C22⋊C4—3982no—
109C22⋊C4—3982no—
110C23—42no—
111C23D4×D19912yes—
112C23C2×D79611yescharacteristic
113C23Q8×D19911yescharacteristic
114C23#1569188511yescharacteristic
115C23#1569188612yes—
116Q8—15922no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—7962no—
122C2×C4—7961no—
123C2×C4—7961no—
124C2×C4—7964no—
125C2×C4—7962no—
126C2×C4—7961no—
127C2×C4—3984no—
128C2×C4—3984no—
129C2×C4—3982no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#2026392511yescharacteristic
136C22#2026393312yes—
137C22#2026393511yescharacteristic
138C22#2026394112yes—
139C22#2026394311yescharacteristic
140C4—42no—
141C4—82no—
142C4—7962no—
143C4—7964no—
144C4—15921no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2716393111yescharacteristic
150C2#2716420912yes—
subgroupG/H
centerC22#20263925
commutator, prime radical, cosocleC24×C199C23
FrattiniC24C22×D199
Fitting#342×C199C2
socleC22×C199#120
hypercenter#342D199
hypocenterC199#2030
LCSG ▷ C24×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic199Q8semidirect
75. C199#2030semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271638792
#271637842
#342×C1992
#2030199