Finite Group Databasebeta

← #38770803

Subgroups — #38770803

Order 25472.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#27164818C211yesmaximal, characteristic
2#27164461C212yesmaximal, complemented
3#27164276C211yesmaximal, characteristic, complemented
4#27163829C212yesmaximal, complemented
5#349×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7#20263988—22no—
8#20263983—22no—
9#20263977—21no—
10#20263977C2212yes—
11C2×Dic796C2211yescharacteristic, complemented
12#20263934—21no—
13Dic1592—22no—
14#20263900—22no—
15#20263887C2211yescharacteristic, complemented
16#20263880—21no—
17C2×M16×C199C2211yescharacteristic
18M32×C199C2212yescomplemented
19#15691886—22no—
20#15691886—22no—
21#15691886—41no—
22#15691886—41no—
23C2×Dic398—21no—
24C2×Dic398—21no—
25C2×D796—21no—
26C2×C22⋊D199—41no—
27C2×C4×D199—41no—
28Dic796—21no—
29Dic796—22no—
30Dic796D412yes—
31#15691875—22no—
32#15691875—22no—
33D1592—21no—
34D1592D412yes—
35C4⋊Dic199—21no—
36Dic199⋊C4—41no—
37C22⋊Dic199—41no—
38C22×C4×C199D411yescharacteristic
39M16×C199—22no—
40C2×C8×C199D411yescharacteristic
41C2×C8×C199C2311yescharacteristic
42C16×C199—22no—
43C22×D199—41no—
44Dic398—22no—
45Dic398—41no—
46Dic398—41no—
47Dic398—41no—
48D796—22no—
49D796—41no—
50C22⋊D199—42no—
51C22⋊D199—42no—
52C2×Dic199—41no—
53C2×Dic199—41no—
54C2×Dic199—41no—
55C4×D199—42no—
56C4×D199—42no—
57C23×C199D811yescharacteristic
58C2×C4×C199—21no—
59C2×C4×C199—21no—
60C2×C4×C199D811yescharacteristic
61C2×C4×C199C2×D411yescharacteristic
62C8×C199—21no—
63C8×C199C2×D412yes—
64C22×C199—21no—
65C22×C199—41no—
66C22×C199C2×D811yescharacteristic
67C2×D199—42no—
68C2×D199—82no—
69C4×C199—21no—
70C4×C199#12011yescharacteristic
71C4×C199AGL(1,ℤ/8)11yescharacteristic
72Dic199—42no—
73Dic199—81no—
74Dic199—81no—
75C2×C199—21no—
76C2×C199—41no—
77C2×C199#44711yescharacteristic
78D199—82no—
79C199#219111yescharacteristic, complemented, Sylow
80#2191—1991nomaximal, Sylow
81#575—1991no—
82#510—1992no—
83#470—1991no—
84#360—1992no—
85#349D19911yescharacteristic, complemented
86C2×#41—3981no—
87#137—3982no—
88AGL(1,ℤ/8)—3982no—
89#135—3981no—
90#135—1992no—
91C2×Q16—1991no—
92C2×M16C2×D19911yescharacteristic
93C22⋊Q8—3981no—
94Q32—3982no—
95SD32—3982no—
96M32C2×D19912yescomplemented
97#106—1991no—
98#103—3981no—
99#41—7961no—
100#41—7961no—
101#41—3982no—
102#41—3982no—
103C2×Q8—3981no—
104C2×Q8—3981no—
105C2×D4—7961no—
106C2×D4—3981no—
107C22×C4—7961no—
108C22×C4D79611yescharacteristic
109Q16—3981no—
110Q16—3982no—
111Q16—1992no—
112SD16—3982no—
113SD16—3982no—
114D8—3981no—
115D8—1992no—
116M16—22no—
117C2×C8D79611yescharacteristic
118C2×C8C22×D19911yescharacteristic
119C4⋊C4—7961no—
120C4⋊C4—3981no—
121C22⋊C4—7961no—
122C16—22no—
123C23—7961no—
124C23D159211yescharacteristic
125Q8—7961no—
126Q8—7961no—
127Q8—7961no—
128Q8—3982no—
129D4—7961no—
130D4—7962no—
131D4—7962no—
132D4—3982no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—7962no—
136C2×C4—7961no—
137C2×C4—7962no—
138C2×C4—7961no—
139C2×C4—7961no—
140C2×C4D159211yescharacteristic
141C2×C4C2×D79611yescharacteristic
142C8—21no—
143C8D4×D19912yes—
144C22—21no—
145C22—41no—
146C22—7962no—
147C22—15922no—
148C22C2×D159211yescharacteristic
149C4—21no—
150C4—7962no—
151C4—15921no—
152C4—15921no—
153C4#2026392511yescharacteristic
154C4#2026398311yescharacteristic
155C2—21no—
156C2—41no—
157C2—15922no—
158C2#2716415411yescharacteristic
subgroupG/H
centerC2#27164154
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#349×C199C2
socleC2×C199#447
prime radical#27164818C2
hypercenter#349D199
hypocenterC199#2191
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #27164461C2semidirect
3. #27164276C2semidirect
4. #27163829C2semidirect
5. #349×C199C2semidirect
11. C2×Dic796C22semidirect
15. #20263887C22semidirect
18. M32×C199C22semidirect
79. C199#2191semidirect
85. #349D199semidirect
96. M32C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648182
#271644612
#271642762
#271638292
#349×C1992
#2191199