Finite Group Databasebeta

← #38744737

Subgroups — #38744737

Order 25216.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#27161795C211yesmaximal, characteristic, complemented
2#27161252C212yesmaximal, complemented
3#27160878C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#456×C197C211yesmaximal, characteristic, complemented
6#20263351—22no—
7#20263315—21no—
8#20263315C2212yes—
9#20263301—22no—
10C4×D788—21no—
11#20263284—22no—
12#20263253—22no—
13#20263245C2212yes—
142−1+4×C197C2211yescharacteristic
15#137×C197—21no—
16C4⋊Q8×C197C2211yescharacteristic
17#104×C197—21no—
18#101×C197C2211yescharacteristic
19C2×D788—21no—
20C2×D788—21no—
21C2×D788D412yes—
22C2×C4×D197—22no—
23#15691345—42no—
24#15691343—42no—
25#15691342—22no—
26C4⋊Dic197—21no—
27#15691335—22no—
28#15691335—22no—
29#15691335—22no—
30C4×Dic197—21no—
31C4×Dic197D412yes—
32#41×C197—21no—
33#41×C197—22no—
34#41×C197—41no—
35C2×Q8×C197—41no—
36C2×Q8×C197D411yescharacteristic
37C2×Q8×C197C2311yescharacteristic
38Q16×C197—41no—
39SD16×C197—41no—
40M16×C197—21no—
41C4⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43C22×D197—22no—
44D788—42no—
45D788—42no—
46D788—81no—
47C2×Dic197—22no—
48C4×D197—44no—
49#13096802—42no—
50Q8×C197—21no—
51Q8×C197—22no—
52Q8×C197—41no—
53Q8×C197—41no—
54D4×C197—21no—
55D4×C197—22no—
56D4×C197—41no—
57C2×C4×C197—21no—
58C2×C4×C197—21no—
59C2×C4×C197—41no—
60C2×C4×C197—42no—
61C2×C4×C197C2×D412yes—
62C2×C4×C197C2×D411yescharacteristic
63C8×C197—41no—
64C22×C197—41no—
65C22×C197#12011yescharacteristic
66C2×D197—42no—
67C2×D197—42no—
68C2×D197—42no—
69C4×C197—22no—
70C4×C197—21no—
71C4×C197—22no—
72C4×C197—41no—
73C4×C197—41no—
74Dic197—42no—
75C2×C197—21no—
76C2×C197—81no—
77C2×C197B(4,𝔽2)11yescharacteristic
78D197—82no—
79C197#220411yescharacteristic, complemented, Sylow
80#2204—1971nomaximal, Sylow
81#555—1971no—
82#456D19711yescharacteristic, complemented
83#454—1972no—
84#356—1971no—
85#355—1972no—
862−1+4C2×D19711yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—3942no—
89C4⋊Q8C2×D19711yescharacteristic
90#124—3941no—
91#124—1972no—
92#121—3942no—
93C4×D4—3941no—
94#117—3942no—
95#104—21no—
96#104—3942no—
97#101—1972no—
98#101C2×D19711yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D19711yescharacteristic
104C2×Q8C22×D19711yescharacteristic
105C2×D4—3941no—
106C2×D4—3941no—
107C2×D4—1972no—
108C22×C4—3942no—
109Q16—41no—
110SD16—41no—
111SD16—7882no—
112D8—7882no—
113M16—21no—
114M16—3942no—
115C4⋊C4—22no—
116C4⋊C4—3941no—
117C22⋊C4—3942no—
118C22⋊C4—3942no—
119C22⋊C4—3942no—
120C42—3941no—
121C42—1972no—
122C42C22⋊D19711yescharacteristic
123C23—3942no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—7882no—
132D4—7882no—
133D4—15761no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—7884no—
139C2×C4—3942no—
140C2×C4D4×D19712yes—
141C2×C4C2×C22⋊D19711yescharacteristic
142C8—41no—
143C8—7882no—
144C22—41no—
145C22—7882no—
146C22—7882no—
147C22—7882no—
148C22#2026329411yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—7882no—
155C2—21no—
156C2—81no—
157C2—15762no—
158C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#456×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#456D197
hypocenterC197#2204
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27161795C2semidirect
2. #27161252C2semidirect
3. #27160878C2semidirect
4. #27160875C2semidirect
5. #456×C197C2semidirect
79. C197#2204semidirect
82. #456D197semidirect

Maximal subgroups

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

subgroupindex
#271617952
#271612522
#271608782
#271608752
#456×C1972
#2204197