Finite Group Databasebeta

← #38770876

Subgroups — #38770876

Order 25472.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#27164200C211yesmaximal, characteristic, complemented
2#27164190C212yesmaximal, complemented
3#27163817C211yesmaximal, characteristic, complemented
4#27163815C212yesmaximal, complemented
5#555×C199C211yesmaximal, characteristic, complemented
6#20264003C2211yescharacteristic
7#20263986—21no—
8#20263981—22no—
9#20263883—22no—
10#20263883—21no—
11#20263874C2212yes—
12#20263874C2211yescharacteristic
13C4⋊Q8×C199C2211yescharacteristic
14#124×C199—21no—
15#124×C199C2212yes—
16#121×C199—22no—
17C4×D4×C199—21no—
18#117×C199—22no—
19#15691887—41no—
20#15691886—22no—
21#15691886—21no—
22Q8×D199—41no—
23#15691876—41no—
24#15691874—42no—
25#15691873—41no—
26#15691871—42no—
27#15691870—22no—
28#15691870—21no—
29C2×Q8×C199D411yescharacteristic
30C2×Q8×C199C2311yescharacteristic
31C2×D4×C199—21no—
32C2×D4×C199—21no—
33C2×D4×C199D412yes—
34C22×C4×C199—22no—
35C4⋊C4×C199—21no—
36C4⋊C4×C199—22no—
37C22⋊C4×C199—22no—
38C22⋊C4×C199—22no—
39C22⋊C4×C199—22no—
40C42×C199—21no—
41C42×C199D412yes—
42C42×C199D411yescharacteristic
43Dic398—22no—
44Dic398—21no—
45D796—22no—
46D796—21no—
47C22⋊D199—41no—
48C4×D199—42no—
49C4×D199—41no—
50C199⋊C8—42no—
51C199⋊C8—41no—
52C23×C199—22no—
53Q8×C199—41no—
54Q8×C199—41no—
55D4×C199—42no—
56D4×C199—42no—
57D4×C199—81no—
58C2×C4×C199—21no—
59C2×C4×C199—22no—
60C2×C4×C199—21no—
61C2×C4×C199—44no—
62C2×C4×C199C2×D411yescharacteristic
63C2×C4×C199C2×D412yes—
64C22×C199—42no—
65C22×C199—42no—
66C22×C199—42no—
67C22×C199#12011yescharacteristic
68C2×D199—41no—
69C4×C199—22no—
70C4×C199—22no—
71C4×C199—21no—
72C4×C199—42no—
73C4×C199—41no—
74Dic199—41no—
75C2×C199—21no—
76C2×C199—82no—
77C2×C199B(4,𝔽2)11yescharacteristic
78D199—81no—
79C199#220411yescharacteristic, complemented, Sylow
80#2204—1991nomaximal, Sylow
81#555D19911yescharacteristic, complemented
82#456—1991no—
83#454—1992no—
84#356—1991no—
85#355—1992no—
862−1+4—1991no—
87#137—3981no—
88AGL(1,ℤ/8)—3982no—
89C4⋊Q8C2×D19911yescharacteristic
90#124—21no—
91#124C2×D19912yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3982no—
96#104—3981no—
97#101—1991no—
98#101—1992no—
99#41—7961no—
100#41—3982no—
101#41—3981no—
102C2×Q8—7961no—
103C2×Q8C22⋊D19911yescharacteristic
104C2×Q8C22×D19911yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D19912yes—
108C22×C4—22no—
109Q16—7961no—
110SD16—7962no—
111SD16—7961no—
112D8—7962no—
113M16—3982no—
114M16—3981no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D19912yes—
122C42C22⋊D19911yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3982no—
127Q8—3981no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—7961no—
132D4—3982no—
133D4—3981no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—7962no—
139C2×C4—7961no—
140C2×C4C2×C22⋊D19911yescharacteristic
141C2×C4C2×C22⋊D19912yes—
142C8—7962no—
143C8—7961no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—7961no—
148C22#2026392411yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—7961no—
155C2—21no—
156C2—82no—
157C2—15921no—
158C2#2716420511yescharacteristic
subgroupG/H
centerC2#27164205
commutator, cosocleC2×Q8×C199C23
FrattiniC2×Q8C22×D199
Fitting#555×C199C2
socleC2×C199B(4,𝔽2)
hypercenter#555D199
hypocenterC199#2204
LCSG ▷ C2×Q8×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #27164200C2semidirect
2. #27164190C2semidirect
3. #27163817C2semidirect
4. #27163815C2semidirect
5. #555×C199C2semidirect
79. C199#2204semidirect
81. #555D199semidirect

Maximal subgroups

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

subgroupindex
#271642002
#271641902
#271638172
#271638152
#555×C1992
#2204199