Finite Group Databasebeta

← #38768023

Subgroups — #38768023

Order 25472.

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

normal subgroups only

This group has 194 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#27164742C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic, complemented
3#27164314C211yesmaximal, characteristic, complemented
4#27164309C211yesmaximal, characteristic
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#332×C199C211yesmaximal, characteristic, complemented
8#20263955C2211yescharacteristic
9#20263950—22no—
10#20263944—21no—
11#20263935—21no—
12Q8×Dic199—21no—
13C4×Dic398C2211yescharacteristic
14C4×D796C2211yescharacteristic
15#20263913—21no—
16#20263889—21no—
17#20263887—21no—
18#20263880—21no—
19#20263878—22no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×D796D411yescharacteristic
28C2×C4×D199—21no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D411yescharacteristic
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—21no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—21no—
39C4×Dic199—22no—
40C4×Dic199—21no—
41C2×Q8×C199—21no—
42C2×C8×C199—21no—
43C2×C8×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D412yes—
46C42×C199C2311yescharacteristic
47C22×D199—21no—
48Dic398—21no—
49Dic398—21no—
50Dic398—42no—
51D796—21no—
52D796—21no—
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—41no—
57C4×D199—41no—
58C4×D199—41no—
59C199⋊C8—41no—
60Q8×C199—41no—
61Q8×C199—41no—
62C2×C4×C199—22no—
63C2×C4×C199C2×D411yescharacteristic
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C8×C199—41no—
67C8×C199—41no—
68C22×C199#12011yescharacteristic
69C2×D199—41no—
70C2×D199—41no—
71C2×D199—41no—
72C4×C199—21no—
73C4×C199—42no—
74C4×C199#13511yescharacteristic
75C4×C199#13511yescharacteristic
76C4×C199#13711yescharacteristic
77C4×C199#13711yescharacteristic
78Dic199—41no—
79Dic199—41no—
80Dic199—41no—
81Dic199—81no—
82C2×C199#45211yescharacteristic
83C2×C199#45211yescharacteristic
84C2×C199#45411yescharacteristic
85D199—81no—
86C199#163811yescharacteristic, complemented, Sylow
87#1638—1991nomaximal, Sylow
88#556—1991no—
89#487—1991no—
90#477—1991no—
91#476—1991no—
92#332D19911yescharacteristic, complemented
93Q8⋊C8—1991no—
94D4⋊C8—1991no—
95C4⋊Q8—1991no—
96C4⋊Q8C2×D19911yescharacteristic
97#126—3982no—
98#124—3981no—
99C22⋊Q8—3981no—
100C4×Q8—3981no—
101C4×Q8—1991no—
102C4×D4—1991no—
103#117—3981no—
104#107—3981no—
105#106—3981no—
106C4⋊C8C2×D19911yescharacteristic
107C4⋊C8C2×D19911yescharacteristic
108#103—3981no—
109#103—3982no—
110#102—3981no—
111C4×C8—1991no—
112C2×Q8—21no—
113C2×Q8—3981no—
114C2×Q8—1991no—
115C2×D4—1991no—
116C22×C4—3981no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3981no—
120C4⋊C4—21no—
121C4⋊C4—3982no—
122C4⋊C4—3981no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—1991no—
126C4⋊C4—1991no—
127C4⋊C4D79612yes—
128C22⋊C4—3982no—
129C22⋊C4—3982no—
130C22⋊C4—3981no—
131C42—3982no—
132C42—3981no—
133C42C22×D19911yescharacteristic
134C23—3981no—
135Q8—41no—
136Q8—41no—
137Q8—7962no—
138Q8—3981no—
139Q8—3981no—
140D4—3981no—
141D4—3981no—
142C2×C4—22no—
143C2×C4—7961no—
144C2×C4—7961no—
145C2×C4—7961no—
146C2×C4—3981no—
147C2×C4—3981no—
148C2×C4—3981no—
149C2×C4D4×D19911yescharacteristic
150C2×C4D4×D19911yescharacteristic
151C2×C4C2×D79611yescharacteristic
152C8—41no—
153C8—41no—
154C8—7961no—
155C22—7961no—
156C22—7961no—
157C22—7961no—
158C22#2026392511yescharacteristic
159C4—21no—
160C4—42no—
161C4—7961no—
162C4—7961no—
163C4—7961no—
164C4—15921no—
165C4#2026397411yescharacteristic
166C4#2026397611yescharacteristic
167C4#2026398411yescharacteristic
168C4#2026398711yescharacteristic
169C2—15921no—
170C2#2716418111yescharacteristic
171C2#2716418311yescharacteristic
172C2#2716419311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#332D199
hypocenterC199#1638
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27164370C2semidirect
3. #27164314C2semidirect
5. #27163748C2semidirect
7. #332×C199C2semidirect
86. C199#1638semidirect
92. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271647422
#271643702
#271643142
#271643092
#271637482
#271637452
#332×C1992
#1638199