Finite Group Databasebeta

← #38744735

Subgroups — #38744735

Order 25216.

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

normal subgroups only

This group has 216 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#27161829C211yesmaximal, characteristic, complemented
2#27161264C212yesmaximal, complemented
3#27161251C211yesmaximal, characteristic
4#27160877C212yesmaximal, complemented
5#355×C197C211yesmaximal, characteristic, complemented
6#20263374C2211yescharacteristic
7#20263359—22no—
8#20263352—21no—
9#20263324C2212yes—
10#20263319—22no—
11#20263318—21no—
12#20263307—21no—
13#20263303—22no—
14#20263254—21no—
15#20263252—22no—
16#20263245C2211yescharacteristic
17#124×C197C2211yescharacteristic
18#101×C197C2212yes—
19#15691359—41no—
20#15691358—22no—
21#15691358—21no—
22C2×Dic394D412yes—
23Q8×D197—41no—
24C22×Dic197—21no—
25Dic788—42no—
26#15691347—42no—
27#15691346—41no—
28#15691343—41no—
29#15691342—21no—
30C4⋊Dic197—21no—
31C4⋊Dic197—21no—
32#15691337—22no—
33#15691337—22no—
34#15691337—22no—
35C22⋊Dic197—22no—
36C22⋊Dic197—21no—
37C4×Dic197D412yes—
38C2×Q8×C197C2311yescharacteristic
39C2×D4×C197D411yescharacteristic
40M16×C197—22no—
41C22⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43Dic394—21no—
44Dic394—22no—
45Dic394—42no—
46D788—21no—
47D788—22no—
48C22⋊D197—41no—
49C2×Dic197—22no—
50C2×Dic197—22no—
51C2×Dic197—81no—
52C4×D197—42no—
53C4×D197—41no—
54#13096802—41no—
55C23×C197—21no—
56Q8×C197—41no—
57D4×C197—41no—
58C2×C4×C197—21no—
59C2×C4×C197C2×D412yes—
60C2×C4×C197C2×D411yescharacteristic
61C8×C197—42no—
62C22×C197—42no—
63C22×C197—41no—
64C22×C197#12011yescharacteristic
65C2×D197—41no—
66C4×C197—22no—
67C4×C197—21no—
68C4×C197—41no—
69Dic197—42no—
70Dic197—41no—
71Dic197—42no—
72C2×C197—21no—
73C2×C197—81no—
74C2×C197B(4,𝔽2)11yescharacteristic
75D197—81no—
76C197#220311yescharacteristic, complemented, Sylow
77#2203—1971nomaximal, Sylow
78#563—1971no—
79#456—1972no—
80#454—1971no—
81#356—1972no—
82#355D19711yescharacteristic, complemented
832−1+4—1971no—
84#137—3942no—
85AGL(1,ℤ/8)—3941no—
86C4⋊Q8—1972no—
87#126—3942no—
88#125—3941no—
89#124C2×D19711yescharacteristic
90#123—3941no—
91C22⋊Q8—3942no—
92#104—3942no—
93#104—3941no—
94#101—1971no—
95#101C2×D19712yes—
96#41—7881no—
97#41—3942no—
98#41—3941no—
99C2×Q8—7881no—
100C2×Q8—1972no—
101C2×Q8C22×D19711yescharacteristic
102C2×D4D78811yescharacteristic
103C22×C4—3941no—
104Q16—7882no—
105SD16—7882no—
106SD16—7881no—
107D8—7881no—
108M16—22no—
109M16—3941no—
110C4⋊C4—3942no—
111C4⋊C4—3941no—
112C4⋊C4—3941no—
113C4⋊C4—3942no—
114C4⋊C4—3942no—
115C22⋊C4—22no—
116C22⋊C4—3942no—
117C22⋊C4—3941no—
118C42—1972no—
119C42D78811yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—7882no—
123Q8—3941no—
124Q8—3942no—
125D4—41no—
126D4—7881no—
127D4—3941no—
128D4—3942no—
129C2×C4—21no—
130C2×C4—7882no—
131C2×C4—7881no—
132C2×C4—15761no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4D4×D19712yes—
136C2×C4C2×D78811yescharacteristic
137C8—42no—
138C8—7881no—
139C22—42no—
140C22—41no—
141C22—7881no—
142C22#2026329611yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—7882no—
147C4—7881no—
148C4—7882no—
149C2—21no—
150C2—81no—
151C2—15761no—
152C2#2716126711yescharacteristic
subgroupG/H
centerC2#27161267
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#355×C197C2
socleC2×C197B(4,𝔽2)
prime radical#27161251C2
hypercenter#355D197
hypocenterC197#2203
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #27161829C2semidirect
2. #27161264C2semidirect
4. #27160877C2semidirect
5. #355×C197C2semidirect
76. C197#2203semidirect
82. #355D197semidirect

Maximal subgroups

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

subgroupindex
#271618292
#271612642
#271612512
#271608772
#355×C1972
#2203197