Finite Group Databasebeta

← #38770827

Subgroups — #38770827

Order 25472.

206 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 47 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#27164831C211yesmaximal, characteristic
2#27164459C211yesmaximal, characteristic, complemented
3#27164290C211yesmaximal, characteristic, complemented
4#27163834C211yesmaximal, characteristic, complemented
5#27163826C211yesmaximal, characteristic, complemented
6#27163801C211yesmaximal, characteristic, complemented
7C2×Q32×C199C211yesmaximal, characteristic, complemented
8#20264003—21no—
9#20263987—22no—
10#20263986—21no—
11#20263986—21no—
12#20263978—21no—
13#20263977C2211yescharacteristic
14Q16×D199—21no—
15#20263961C2211yescharacteristic
16#20263901—21no—
17#20263898—21no—
18#20263895C2211yescharacteristic
19#20263891C2211yescharacteristic
20#20263874—21no—
21C2×Q16×C199C2211yescharacteristic
22C2×Q16×C199C2211yescharacteristic
23Q32×C199—22no—
24C2×C16×C199C2211yescharacteristic
25#15691887—22no—
26#15691887—41no—
27#15691886—21no—
28#15691886—42no—
29#15691886D411yescharacteristic
30Q8×D199—22no—
31Q8×D199—41no—
32Dic796D411yescharacteristic
33#15691876—22no—
34#15691876—41no—
35#15691875—21no—
36#15691873—22no—
37#15691873—41no—
38D1592D411yescharacteristic
39#15691870—21no—
40#15691870D411yescharacteristic
41#15691869—21no—
42C8×D199—21no—
43C199⋊C16—21no—
44C2×Q8×C199—21no—
45C2×Q8×C199—21no—
46Q16×C199—21no—
47Q16×C199—21no—
48Q16×C199—21no—
49Q16×C199D412yes—
50C2×C8×C199C2311yescharacteristic
51C16×C199—21no—
52Dic398—21no—
53Dic398—42no—
54Dic398D811yescharacteristic
55D796—21no—
56D796—42no—
57D796D811yescharacteristic
58C22⋊D199—21no—
59C22⋊D199—41no—
60C4×D199—21no—
61C4×D199—42no—
62C4×D199—42no—
63C4×D199—41no—
64C199⋊C8—21no—
65C199⋊C8—41no—
66Q8×C199—22no—
67Q8×C199—41no—
68Q8×C199—41no—
69Q8×C199—41no—
70C2×C4×C199—41no—
71C2×C4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C8×C199C2×D411yescharacteristic
74C8×C199C2×D411yescharacteristic
75C22×C199AGL(1,ℤ/8)11yescharacteristic
76C2×D199—21no—
77C2×D199—41no—
78C4×C199—42no—
79C4×C199—81no—
80C4×C199#12011yescharacteristic
81C4×C199C2×D811yescharacteristic
82Dic199—21no—
83Dic199—41no—
84C2×C199—21no—
85C2×C199#44711yescharacteristic
86D199—41no—
87D199—81no—
88C199#219511yescharacteristic, complemented, Sylow
89#2195—1991nomaximal, Sylow
90#578—1991no—
91#510—1991no—
92C2×Q32D19911yescharacteristic, complemented
93#473—1991no—
94#362—1991no—
95#359—1991no—
96#350—1991no—
972−1+4—3981no—
98#137—3981no—
99#137—3982no—
100#137—3981no—
101#135—3981no—
102#135—1991no—
103C2×Q16—3981no—
104C2×Q16C2×D19911yescharacteristic
105C2×Q16C2×D19911yescharacteristic
106#131—1991no—
107Q32—22no—
108Q32—3981no—
109SD32—3981no—
110M32—1991no—
111C2×C16C2×D19911yescharacteristic
112#108—1991no—
113#101—3981no—
114#41—7962no—
115#41—7961no—
116#41—3982no—
117#41—3981no—
118#41—1991no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—7961no—
122C2×Q8—3982no—
123Q16—21no—
124Q16—21no—
125Q16—21no—
126Q16—7961no—
127Q16—3982no—
128Q16—1991no—
129Q16C22⋊D19912yes—
130SD16—7961no—
131SD16—3982no—
132SD16—3981no—
133D8—1991no—
134M16—3981no—
135M16—3981no—
136M16—1991no—
137C2×C8—3981no—
138C2×C8C22×D19911yescharacteristic
139C16—21no—
140C16—3981no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—7962no—
146Q8—3981no—
147Q8—1991no—
148D4—7962no—
149D4—7961no—
150D4—3981no—
151D4—3981no—
152D4—1991no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—7962no—
156C2×C4—7962no—
157C2×C4—7961no—
158C2×C4—3981no—
159C2×C4D4×D19911yescharacteristic
160C8—7961no—
161C8—3981no—
162C8D4×D19911yescharacteristic
163C8C2×C22⋊D19911yescharacteristic
164C22—7961no—
165C22—3981no—
166C22#2026397911yescharacteristic
167C4—42no—
168C4—81no—
169C4—7961no—
170C4—3981no—
171C4#2026392311yescharacteristic
172C4D8×D19911yescharacteristic
173C2—21no—
174C2—7961no—
175C2—15921no—
176C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
FittingC2×Q32×C199C2
socleC2×C199#447
prime radical#27164831C2
hypercenterC2×Q32D199
hypocenterC199#2195
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

Complements

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

NH ≅ G/Ntype
2. #27164459C2semidirect
3. #27164290C2semidirect
4. #27163834C2semidirect
5. #27163826C2semidirect
6. #27163801C2semidirect
7. C2×Q32×C199C2semidirect
88. C199#2195semidirect
92. C2×Q32D199semidirect

Maximal subgroups

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

subgroupindex
#271648312
#271644592
#271642902
#271638342
#271638262
#271638012
C2×Q32×C1992
#2195199