Finite Group Databasebeta

← #38744738

Subgroups — #38744738

Order 25216.

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#27161261C211yesmaximal, characteristic, complemented
2#27161251C212yesmaximal, complemented
3#27160878C211yesmaximal, characteristic, complemented
4#27160876C212yesmaximal, complemented
5#555×C197C211yesmaximal, characteristic, complemented
6#20263374C2211yescharacteristic
7#20263357—21no—
8#20263352—22no—
9#20263254—22no—
10#20263254—21no—
11#20263245C2211yescharacteristic
12#20263245C2212yes—
13C4⋊Q8×C197C2211yescharacteristic
14#124×C197—21no—
15#124×C197C2212yes—
16#121×C197—22no—
17C4×D4×C197—21no—
18#117×C197—22no—
19#15691359—41no—
20#15691358—22no—
21#15691358—21no—
22Q8×D197—41no—
23#15691348—41no—
24#15691346—42no—
25#15691345—41no—
26#15691343—42no—
27#15691342—22no—
28#15691342—21no—
29C2×Q8×C197D411yescharacteristic
30C2×Q8×C197C2311yescharacteristic
31C2×D4×C197—21no—
32C2×D4×C197—21no—
33C2×D4×C197D412yes—
34C22×C4×C197—22no—
35C4⋊C4×C197—21no—
36C4⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—22no—
40C42×C197—21no—
41C42×C197D412yes—
42C42×C197D411yescharacteristic
43Dic394—22no—
44Dic394—21no—
45D788—22no—
46D788—21no—
47C22⋊D197—41no—
48C4×D197—42no—
49C4×D197—41no—
50#13096802—42no—
51#13096802—41no—
52C23×C197—22no—
53Q8×C197—41no—
54Q8×C197—41no—
55D4×C197—42no—
56D4×C197—42no—
57D4×C197—81no—
58C2×C4×C197—21no—
59C2×C4×C197—22no—
60C2×C4×C197—21no—
61C2×C4×C197—44no—
62C2×C4×C197C2×D411yescharacteristic
63C2×C4×C197C2×D412yes—
64C22×C197—42no—
65C22×C197—42no—
66C22×C197—42no—
67C22×C197#12011yescharacteristic
68C2×D197—41no—
69C4×C197—22no—
70C4×C197—22no—
71C4×C197—21no—
72C4×C197—42no—
73C4×C197—41no—
74Dic197—41no—
75C2×C197—21no—
76C2×C197—82no—
77C2×C197B(4,𝔽2)11yescharacteristic
78D197—81no—
79C197#220411yescharacteristic, complemented, Sylow
80#2204—1971nomaximal, Sylow
81#555D19711yescharacteristic, complemented
82#456—1971no—
83#454—1972no—
84#356—1971no—
85#355—1972no—
862−1+4—1971no—
87#137—3941no—
88AGL(1,ℤ/8)—3942no—
89C4⋊Q8C2×D19711yescharacteristic
90#124—21no—
91#124C2×D19712yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3942no—
96#104—3941no—
97#101—1971no—
98#101—1972no—
99#41—7881no—
100#41—3942no—
101#41—3941no—
102C2×Q8—7881no—
103C2×Q8C22⋊D19711yescharacteristic
104C2×Q8C22×D19711yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D19712yes—
108C22×C4—22no—
109Q16—7881no—
110SD16—7882no—
111SD16—7881no—
112D8—7882no—
113M16—3942no—
114M16—3941no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D19712yes—
122C42C22⋊D19711yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3942no—
127Q8—3941no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—7881no—
132D4—3942no—
133D4—3941no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—7882no—
139C2×C4—7881no—
140C2×C4C2×C22⋊D19711yescharacteristic
141C2×C4C2×C22⋊D19712yes—
142C8—7882no—
143C8—7881no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—7881no—
148C22#2026329511yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—7881no—
155C2—21no—
156C2—82no—
157C2—15761no—
158C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#555×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#555D197
hypocenterC197#2204
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #27161261C2semidirect
2. #27161251C2semidirect
3. #27160878C2semidirect
4. #27160876C2semidirect
5. #555×C197C2semidirect
79. C197#2204semidirect
81. #555D197semidirect

Maximal subgroups

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

subgroupindex
#271612612
#271612512
#271608782
#271608762
#555×C1972
#2204197