Finite Group Databasebeta

← #38770801

Subgroups — #38770801

Order 25472.

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

normal subgroups only

This group has 216 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#27164458C212yesmaximal, complemented
2#27164276C211yesmaximal, characteristic, complemented
3#27163828C212yesmaximal, complemented
4#27163800C211yesmaximal, characteristic, complemented
5#575×C199C211yesmaximal, characteristic
6C2×Dic796C2211yescharacteristic, complemented
7#20263934—21no—
8#20263901—22no—
9#20263899—22no—
10#20263895C2212yescomplemented
11#20263887C2211yescharacteristic, complemented
12#20263880—21no—
13C2×#41×C199—21no—
14#137×C199—22no—
15AGL(1,ℤ/8)×C199—22no—
16#135×C199—21no—
17#135×C199C2212yes—
18C2×M16×C199C2211yescharacteristic
19C2×Dic398—21no—
20Dic796—22no—
21C4⋊Dic199—21no—
22Dic199⋊C4—41no—
23C22⋊Dic199—41no—
24C199⋊C16—22no—
25#41×C199—22no—
26#41×C199—22no—
27#41×C199—41no—
28#41×C199—41no—
29C2×Q8×C199—21no—
30C2×D4×C199—21no—
31C2×D4×C199—41no—
32C22×C4×C199—41no—
33C22×C4×C199D411yescharacteristic
34Q16×C199—21no—
35Q16×C199D412yes—
36SD16×C199—22no—
37SD16×C199—22no—
38D8×C199—21no—
39D8×C199D412yes—
40M16×C199—22no—
41C2×C8×C199D411yescharacteristic
42C2×C8×C199C2311yescharacteristic
43Dic398—41no—
44Dic398—41no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C23×C199—41no—
48C23×C199D811yescharacteristic
49Q8×C199—22no—
50Q8×C199—41no—
51D4×C199—22no—
52D4×C199—41no—
53D4×C199—42no—
54D4×C199—42no—
55C2×C4×C199—21no—
56C2×C4×C199—21no—
57C2×C4×C199—42no—
58C2×C4×C199—41no—
59C2×C4×C199—42no—
60C2×C4×C199D811yescharacteristic
61C2×C4×C199C2×D411yescharacteristic
62C8×C199—21no—
63C8×C199C2×D412yes—
64C22×C199—21no—
65C22×C199—41no—
66C22×C199—42no—
67C22×C199—82no—
68C22×C199C2×D811yescharacteristic
69C4×C199—21no—
70C4×C199—42no—
71C4×C199#12011yescharacteristic
72C4×C199AGL(1,ℤ/8)11yescharacteristic
73Dic199—81no—
74Dic199—81no—
75C2×C199—21no—
76C2×C199—41no—
77C2×C199—82no—
78C2×C199#44711yescharacteristic
79C199#219111yescharacteristic, complemented, Sylow
80#2191—1991nomaximal, Sylow
81#575D19911yescharacteristic
82#510—1992no—
83#470—1991no—
84#360—1992no—
85#349—1991no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D19912yes—
91C2×Q16—1991no—
92C2×M16C2×D19911yescharacteristic
93C22⋊Q8—3981no—
94Q32—3982no—
95SD32—3982no—
96M32—1992no—
97#106—1991no—
98#103—3981no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—3981no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D19911yescharacteristic
109Q16—21no—
110Q16—3982no—
111Q16C22⋊D19912yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D19912yes—
116M16—22no—
117C2×C8C22⋊D19911yescharacteristic
118C2×C8C22×D19911yescharacteristic
119C4⋊C4—7961no—
120C4⋊C4—3981no—
121C22⋊C4—7961no—
122C16—3982no—
123C23—41no—
124C23#1569187111yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—7961no—
128Q8—7961no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—7961no—
139C2×C4—7961no—
140C2×C4#1569187111yescharacteristic
141C2×C4C2×C22⋊D19911yescharacteristic
142C8—21no—
143C8C2×C22⋊D19912yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1569187111yescharacteristic
149C4—21no—
150C4—42no—
151C4—15921no—
152C4—15921no—
153C4#2026392411yescharacteristic
154C4#2026398111yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2716415211yescharacteristic
subgroupG/H
centerC2#27164152
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting, prime radical#575×C199C2
socleC2×C199#447
hypercenter#575D199
hypocenterC199#2191
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27164458C2semidirect
2. #27164276C2semidirect
3. #27163828C2semidirect
4. #27163800C2semidirect
6. C2×Dic796C22semidirect
10. #20263895C22semidirect
11. #20263887C22semidirect
79. C199#2191semidirect

Maximal subgroups

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

subgroupindex
#271644582
#271642762
#271638282
#271638002
#575×C1992
#2191199