Finite Group Databasebeta

← #38767826

Subgroups — #38767826

Order 25472.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164649C211yesmaximal, characteristic
2#27164176C212yesmaximal
3#27163938C211yesmaximal, characteristic
4#27163756C212yesmaximal
5#324×C199C211yesmaximal, characteristic
6C22×Dic398C2211yescharacteristic
7C2×Dic796—22no—
8#20263954—22no—
9#20263946—22no—
10#20263934—21no—
11#20263934C2212yes—
12#20263914—21no—
13C2×Dic199⋊C4—21no—
14#20263880—22no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C2×Dic398—21no—
20C2×Dic398—22no—
21C2×Dic398—42no—
22C2×Dic398D412yes—
23C22×Dic199—21no—
24C22×Dic199—21no—
25Dic796—44no—
26C4⋊Dic199—21no—
27C4⋊Dic199—21no—
28C4⋊Dic199—21no—
29C4⋊Dic199D412yes—
30Dic199⋊C4—22no—
31Dic199⋊C4—22no—
32Dic199⋊C4—42no—
33C22⋊Dic199—22no—
34C4×Dic199—22no—
35C22×C4×C199D411yescharacteristic
36C22×C4×C199D411yescharacteristic
37C22×C4×C199C2311yescharacteristic
38C2×C8×C199—22no—
39C4⋊C4×C199—22no—
40Dic398—24no—
41Dic398—42no—
42Dic398—42no—
43Dic398—81no—
44C2×Dic199—22no—
45C2×Dic199—22no—
46C2×Dic199—22no—
47C2×Dic199—22no—
48C2×Dic199—41no—
49C2×Dic199—41no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C23×C199C2×D411yescharacteristic
53C2×C4×C199—21no—
54C2×C4×C199—21no—
55C2×C4×C199—21no—
56C2×C4×C199—41no—
57C2×C4×C199—41no—
58C2×C4×C199Q1612yescomplemented
59C2×C4×C199C2×D412yes—
60C8×C199—42no—
61C22×C199—21no—
62C22×C199—21no—
63C22×C199#12011yescharacteristic
64C22×C199C2×Q1611yescharacteristic
65C22×C199#13711yescharacteristic
66C4×C199—22no—
67C4×C199—22no—
68C4×C199—41no—
69Dic199—42no—
70Dic199—42no—
71Dic199—42no—
72Dic199—42no—
73C2×C199—21no—
74C2×C199—21no—
75C2×C199#45111yescharacteristic
76C2×C199#45611yescharacteristic
77C2×C199B(4,𝔽2)11yescharacteristic
78C199#160511yescharacteristic, complemented, Sylow
79#1605—1991nomaximal, Sylow
80#544—1991no—
81#451—1992no—
82#395—1991no—
83#328—1992no—
84#324D19911yescharacteristic
85C22×Q8—1991no—
86C2×Q16—3982no—
87C4⋊Q8—3982no—
88#125—3982no—
89C22⋊Q8—3981no—
90C22⋊Q8—1992no—
91#117—3981no—
92C2×C4⋊C4—3981no—
93C2×C4⋊C4C2×D19911yescharacteristic
94#103—3982no—
95C22⋊C8C2×D19912yes—
96#95—3981no—
97#95—1991no—
98C2×Q8—7962no—
99C2×Q8—3981no—
100C2×Q8—3982no—
101C2×Q8—1992no—
102C22×C4—3981no—
103C22×C4—3981no—
104C22×C4D79611yescharacteristic
105C22×C4D79611yescharacteristic
106C22×C4C22×D19911yescharacteristic
107Q16—7964no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—7962no—
111C4⋊C4—3982no—
112C4⋊C4—3981no—
113C4⋊C4—3981no—
114C4⋊C4—3981no—
115C4⋊C4—3982no—
116C4⋊C4—1992no—
117C22⋊C4—3982no—
118C42—3982no—
119C23C2×D79611yescharacteristic
120Q8—7962no—
121Q8—7962no—
122Q8—15921no—
123Q8—3984no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—7961no—
130C2×C4—7961no—
131C2×C4—7961no—
132C2×C4—7961no—
133C2×C4—3982no—
134C2×C4—3982no—
135C2×C4—3982no—
136C2×C4—3982no—
137C2×C4Dic79612yescomplemented
138C2×C4D4×D19912yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2026392511yescharacteristic
143C22C2×Dic79611yescharacteristic
144C22#2026398811yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—7962no—
149C4—7962no—
150C4—7962no—
151C4—7962no—
152C2—21no—
153C2—21no—
154C2#2716417611yescharacteristic
155C2#2716420311yescharacteristic
156C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radicalC23×C199C2×D4
hypercenter#324D199
hypocenterC199#1605
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C199Q16semidirect
78. C199#1605semidirect
137. C2×C4Dic796semidirect

Maximal subgroups

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

subgroupindex
#271646492
#271641762
#271639382
#271637562
#324×C1992
#1605199