Finite Group Databasebeta

← #34102165

Subgroups — #34102165

Order 20096.

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

normal subgroups only

This group has 224 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#23758003C211yesmaximal, characteristic, complemented
2#23757456C212yesmaximal
3#23757072C211yesmaximal, characteristic, complemented
4#23757071C212yesmaximal, complemented
5#456×C157C211yesmaximal, characteristic
6#19097801—22no—
7#19097770—22no—
8#19097769—21no—
9#19097769C2212yes—
10C4×Dic314—21no—
11#19097736—22no—
12#19097698—22no—
13#19097690C2212yes—
142−1+4×C157C2211yescharacteristic
15#137×C157—21no—
16C4⋊Q8×C157C2211yescharacteristic
17#104×C157—21no—
18#101×C157C2211yescharacteristic
19C2×Dic314—21no—
20C2×Dic314—21no—
21C2×Dic314D412yes—
22#14578944—42no—
23#14578942—42no—
24#14578938—22no—
25C4⋊Dic157—21no—
26#14578933—22no—
27#14578933—22no—
28#14578933—22no—
29C4×Dic157—21no—
30C4×Dic157—22no—
31C4×Dic157D412yes—
32#41×C157—21no—
33#41×C157—22no—
34#41×C157—41no—
35C2×Q8×C157—41no—
36C2×Q8×C157D411yescharacteristic
37C2×Q8×C157C2311yescharacteristic
38Q16×C157—41no—
39SD16×C157—41no—
40M16×C157—21no—
41C4⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43Dic314—24no—
44Dic314—42no—
45Dic314—81no—
46C2×Dic157—22no—
47C2×Dic157—22no—
48#11929434—42no—
49Q8×C157—22no—
50Q8×C157—21no—
51Q8×C157—41no—
52Q8×C157—41no—
53D4×C157—21no—
54D4×C157—22no—
55D4×C157—41no—
56C2×C4×C157—21no—
57C2×C4×C157—21no—
58C2×C4×C157—41no—
59C2×C4×C157—42no—
60C2×C4×C157C2×D412yes—
61C2×C4×C157C2×D411yescharacteristic
62C8×C157—41no—
63C22×C157—41no—
64C22×C157#12011yescharacteristic
65C4×C157—22no—
66C4×C157—21no—
67C4×C157—22no—
68C4×C157—41no—
69C4×C157—41no—
70Dic157—24no—
71Dic157—42no—
72C2×C157—21no—
73C2×C157—81no—
74C2×C157B(4,𝔽2)11yescharacteristic
75C157#220511yescharacteristic, complemented, Sylow
76#2205—1571nomaximal, Sylow
77Q82—1571no—
78#456—1572no—
79#456D15711yescharacteristic
80#356—1571no—
81#356—1572no—
822−1+4C2×D15711yescharacteristic
83#137—21no—
84#137—3142no—
85C4⋊Q8—3142no—
86C4⋊Q8—3141no—
87C4⋊Q8—1572no—
88C4⋊Q8C2×D15711yescharacteristic
89C4×Q8—3142no—
90C4×Q8—3141no—
91#104—21no—
92#104—3142no—
93#101—1572no—
94#101C2×D15711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—3141no—
100C2×Q8—3141no—
101C2×Q8—1572no—
102C2×Q8C22⋊D15711yescharacteristic
103C2×Q8C22×D15711yescharacteristic
104Q16—41no—
105Q16—6282no—
106SD16—41no—
107SD16—6282no—
108M16—21no—
109M16—3142no—
110C4⋊C4—22no—
111C4⋊C4—3142no—
112C4⋊C4—3141no—
113C4⋊C4—3142no—
114C4⋊C4—3142no—
115C42—3142no—
116C42—3141no—
117C42—1572no—
118C42C22⋊D15711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—3144no—
124Q8—6282no—
125Q8—12561no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—3142no—
134C2×C4—3142no—
135C2×C4D4×D15712yes—
136C2×C4C2×C22⋊D15711yescharacteristic
137C8—41no—
138C8—6282no—
139C22—41no—
140C22#1909773911yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—3144no—
147C4—6282no—
148C2—21no—
149C2—81no—
150C2#2375745911yescharacteristic
subgroupG/H
centerC2#23757459
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#456×C157C2
socleC2×C157B(4,𝔽2)
prime radical2−1+4×C157C22
hypercenter#456D157
hypocenterC157#2205
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23758003C2semidirect
3. #23757072C2semidirect
4. #23757071C2semidirect
75. C157#2205semidirect

Maximal subgroups

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

subgroupindex
#237580032
#237574562
#237570722
#237570712
#456×C1572
#2205157