Finite Group Databasebeta

← #38767855

Subgroups — #38767855

Order 25472.

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

normal subgroups only

This group has 254 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#27164599C211yesmaximal, characteristic
2#27164332C212yesmaximal, complemented
3#27163948C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7#20263935—22no—
8#20263930—22no—
9#20263929—21no—
10#20263929C2212yes—
11C2×C4⋊Dic199C2211yescharacteristic
12C2×Dic199⋊C4—21no—
13#20263889—22no—
14#20263877—22no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19#15691886—42no—
20#15691886—42no—
21C2×Dic398—21no—
22C2×D796—21no—
23C2×D796—21no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C22×Dic199—21no—
27C22×Dic199—21no—
28C2×C4×D199—22no—
29C4⋊Dic199—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199D412yes—
32Dic199⋊C4—42no—
33(C2×D199)⋊C4—22no—
34(C2×D199)⋊C4—22no—
35C22×C4×C199D411yescharacteristic
36C22×C4×C199D411yescharacteristic
37C22×C4×C199C2311yescharacteristic
38C2×C8×C199—22no—
39C4⋊C4×C199—22no—
40C22×D199—22no—
41Dic398—42no—
42D796—42no—
43D796—42no—
44D796—81no—
45C22⋊D199—42no—
46C22⋊D199—42no—
47C2×Dic199—22no—
48C2×Dic199—22no—
49C2×Dic199—41no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C2×Dic199—41no—
53C2×Dic199—41no—
54C4×D199—44no—
55C23×C199C2×D411yescharacteristic
56C2×C4×C199—21no—
57C2×C4×C199—21no—
58C2×C4×C199—21no—
59C2×C4×C199—41no—
60C2×C4×C199—41no—
61C2×C4×C199D812yes—
62C2×C4×C199C2×D412yes—
63C8×C199—42no—
64C22×C199—21no—
65C22×C199—21no—
66C22×C199#12011yescharacteristic
67C22×C199C2×D811yescharacteristic
68C22×C199AGL(1,ℤ/8)11yescharacteristic
69C2×D199—42no—
70C2×D199—42no—
71C2×D199—42no—
72C4×C199—22no—
73C4×C199—22no—
74C4×C199—41no—
75Dic199—42no—
76Dic199—42no—
77Dic199—81no—
78C2×C199—21no—
79C2×C199—21no—
80C2×C199#44711yescharacteristic
81C2×C199#45611yescharacteristic
82C2×C199#45811yescharacteristic
83D199—82no—
84C199#161011yescharacteristic, complemented, Sylow
85#1610—1991nomaximal, Sylow
86#537—1991no—
87#480—1992no—
88#398—1991no—
89#327—1992no—
90#324D19911yescharacteristic, complemented
91C2×#41—3981no—
92C22⋊Q8—3982no—
93#121—3982no—
94#121—3981no—
95#121—1992no—
96C2×C4⋊C4—3981no—
97C2×C4⋊C4—1991no—
98C2×C4⋊C4C2×D19911yescharacteristic
99#107—3982no—
100#102—3982no—
101C22⋊C8C2×D19912yes—
102#95—3981no—
103#95—1991no—
104#41—7962no—
105#41—7962no—
106C2×Q8—3981no—
107C2×D4—3981no—
108C2×D4—3981no—
109C2×D4—3982no—
110C2×D4—1992no—
111C22×C4—3982no—
112C22×C4—3981no—
113C22×C4—3981no—
114C22×C4D79611yescharacteristic
115C22×C4D79611yescharacteristic
116C22×C4C22×D19911yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—7962no—
120C4⋊C4—3981no—
121C4⋊C4—3982no—
122C4⋊C4—1992no—
123C22⋊C4—3982no—
124C22⋊C4—3982no—
125C23—3982no—
126C23C2×D79611yescharacteristic
127Q8—7962no—
128D4—7962no—
129D4—7962no—
130D4—7962no—
131D4—7962no—
132D4—15921no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—7964no—
139C2×C4—7961no—
140C2×C4—7961no—
141C2×C4—7961no—
142C2×C4—7961no—
143C2×C4—7961no—
144C2×C4—3982no—
145C2×C4—3982no—
146C2×C4D159212yes—
147C2×C4D4×D19912yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—7962no—
152C22—7962no—
153C22—7962no—
154C22#2026392511yescharacteristic
155C22C2×D159211yescharacteristic
156C22#2026398311yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—7962no—
161C4—7962no—
162C4—15921no—
163C2—21no—
164C2—21no—
165C2—15922no—
166C2#2716415411yescharacteristic
167C2#2716420311yescharacteristic
168C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#324D199
hypocenterC199#1610
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
2. #27164332C2semidirect
3. #27163948C2semidirect
4. #27163753C2semidirect
5. #324×C199C2semidirect
84. C199#1610semidirect
90. #324D199semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271643322
#271639482
#271637532
#324×C1992
#1610199