Finite Group Databasebeta

← #28392603

Subgroups — #28392603

Order 14464.

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#20299328C211yesmaximal, characteristic, complemented
2#20298781C212yesmaximal
3#20298397C211yesmaximal, characteristic, complemented
4#20298396C212yesmaximal, complemented
5#456×C113C211yesmaximal, characteristic
6#16813652—22no—
7#16813621—22no—
8#16813620—21no—
9#16813620C2212yes—
10C4×Dic226—21no—
11#16813587—22no—
12#16813549—22no—
13#16813541C2212yes—
142−1+4×C113C2211yescharacteristic
15#137×C113—21no—
16C4⋊Q8×C113C2211yescharacteristic
17#104×C113—21no—
18#101×C113C2211yescharacteristic
19C2×Dic226—21no—
20C2×Dic226—21no—
21C2×Dic226D412yes—
22#14268637—42no—
23#14268635—42no—
24#14268631—22no—
25C4⋊Dic113—21no—
26#14268626—22no—
27#14268626—22no—
28#14268626—22no—
29C4×Dic113—21no—
30C4×Dic113—22no—
31C4×Dic113D412yes—
32#41×C113—22no—
33#41×C113—21no—
34#41×C113—41no—
35C2×Q8×C113—41no—
36C2×Q8×C113D411yescharacteristic
37C2×Q8×C113C2311yescharacteristic
38Q16×C113—41no—
39SD16×C113—41no—
40M16×C113—21no—
41C4⋊C4×C113—22no—
42C42×C113D411yescharacteristic
43Q8×C113—21no—
44Q8×C113—22no—
45Q8×C113—41no—
46Q8×C113—41no—
47D4×C113—21no—
48D4×C113—22no—
49D4×C113—41no—
50C2×C4×C113—21no—
51C2×C4×C113—21no—
52C2×C4×C113—42no—
53C2×C4×C113—41no—
54C2×C4×C113C2×D412yes—
55C2×C4×C113C2×D411yescharacteristic
56C2×Dic113—22no—
57C2×Dic113—22no—
58Dic226—24no—
59Dic226—42no—
60Dic226—81no—
61C8×C113—41no—
62#11744751—42no—
63C22×C113—41no—
64C22×C113#12011yescharacteristic
65C4×C113—22no—
66C4×C113—22no—
67C4×C113—21no—
68C4×C113—41no—
69C4×C113—41no—
70Dic113—24no—
71Dic113—42no—
72C2×C113—21no—
73C2×C113—81no—
74C2×C113B(4,𝔽2)11yescharacteristic
75#2205—1131nomaximal, Sylow
76C113#220511yescharacteristic, complemented, Sylow
77Q82—1131no—
78#456—1132no—
79#456D11311yescharacteristic
80#356—1131no—
81#356—1132no—
822−1+4C2×D11311yescharacteristic
83#137—21no—
84#137—2262no—
85C4⋊Q8—1132no—
86C4⋊Q8—2262no—
87C4⋊Q8—2261no—
88C4⋊Q8C2×D11311yescharacteristic
89C4×Q8—2262no—
90C4×Q8—2261no—
91#104—21no—
92#104—2262no—
93#101—1132no—
94#101C2×D11311yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—1132no—
100C2×Q8—2261no—
101C2×Q8—2261no—
102C2×Q8C22⋊D11311yescharacteristic
103C2×Q8C22×D11311yescharacteristic
104Q16—41no—
105Q16—4522no—
106SD16—41no—
107SD16—4522no—
108M16—21no—
109M16—2262no—
110C4⋊C4—22no—
111C4⋊C4—2262no—
112C4⋊C4—2261no—
113C4⋊C4—2262no—
114C4⋊C4—2262no—
115C42—1132no—
116C42—2262no—
117C42—2261no—
118C42C22⋊D11311yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—9041no—
124Q8—4522no—
125Q8—2264no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—2262no—
134C2×C4—2262no—
135C2×C4D4×D11312yes—
136C2×C4C2×C22⋊D11311yescharacteristic
137C8—41no—
138C8—4522no—
139C22—41no—
140C22#1681359011yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—4522no—
147C4—2264no—
148C2—21no—
149C2—81no—
150C2#2029878411yescharacteristic
subgroupG/H
centerC2#20298784
commutator, cosocleC2×Q8×C113C23
FrattiniC2×Q8C22×D113
Fitting#456×C113C2
socleC2×C113B(4,𝔽2)
prime radical2−1+4×C113C22
hypercenter#456D113
hypocenterC113#2205
LCSG ▷ C2×Q8×C113 ▷ C22×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20299328C2semidirect
3. #20298397C2semidirect
4. #20298396C2semidirect
76. C113#2205semidirect

Maximal subgroups

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

subgroupindex
#202993282
#202987812
#202983972
#202983962
#456×C1132
#2205113