Finite Group Databasebeta

← #35216567

Subgroups — #35216567

Order 20864.

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#23782395C211yesmaximal, characteristic, complemented
2#23781850C212yesmaximal
3#23781847C211yesmaximal, characteristic
4#23781463C212yesmaximal, complemented
5#356×C163C211yesmaximal, characteristic, complemented
6#19104783C2211yescharacteristic
7#19104768—22no—
8#19104766—21no—
9#19104734—22no—
10#19104733—21no—
11#19104733C2212yes—
12C4×Dic326—21no—
13#19104700—22no—
14#19104663—21no—
15#19104661—22no—
16#19104654C2211yescharacteristic
17C4⋊Q8×C163C2211yescharacteristic
18#101×C163C2212yes—
19#14579700—41no—
20#14579699—22no—
21#14579699—21no—
22C2×Dic326—21no—
23C2×Dic326—21no—
24C2×Dic326D412yes—
25Q8×D163—41no—
26Dic652—42no—
27#14579689—41no—
28#14579688—42no—
29#14579686—41no—
30#14579683—21no—
31C4⋊Dic163—21no—
32Dic163⋊C4—22no—
33Dic163⋊C4—22no—
34Dic163⋊C4—22no—
35C4×Dic163—21no—
36C4×Dic163—22no—
37C4×Dic163D412yes—
38C2×Q8×C163D411yescharacteristic
39C2×Q8×C163C2311yescharacteristic
40M16×C163—22no—
41C4⋊C4×C163—22no—
42C42×C163D411yescharacteristic
43Dic326—24no—
44Dic326—21no—
45Dic326—22no—
46Dic326—42no—
47Dic326—81no—
48D652—21no—
49D652—22no—
50C22⋊D163—41no—
51C2×Dic163—22no—
52C2×Dic163—22no—
53C4×D163—42no—
54C4×D163—41no—
55C163⋊C8—41no—
56Q8×C163—41no—
57Q8×C163—41no—
58C2×C4×C163—21no—
59C2×C4×C163—21no—
60C2×C4×C163C2×D412yes—
61C2×C4×C163C2×D411yescharacteristic
62C8×C163—42no—
63C22×C163#12011yescharacteristic
64C2×D163—41no—
65C4×C163—22no—
66C4×C163—22no—
67C4×C163—21no—
68C4×C163—41no—
69Dic163—24no—
70Dic163—41no—
71Dic163—42no—
72C2×C163—21no—
73C2×C163B(4,𝔽2)11yescharacteristic
74D163—81no—
75C163#220511yescharacteristic, complemented, Sylow
76#2205—1631nomaximal, Sylow
77Q82—1631no—
78#456—1631no—
79#456—1632no—
80#356—1632no—
81#356D16311yescharacteristic, complemented
822−1+4—1631no—
83#137—3262no—
84#137—3261no—
85C4⋊Q8—3262no—
86C4⋊Q8—3261no—
87C4⋊Q8—1632no—
88C4⋊Q8C2×D16311yescharacteristic
89C4×Q8—3262no—
90C4×Q8—3261no—
91#104—3262no—
92#104—3261no—
93#101—1631no—
94#101C2×D16312yes—
95#41—3262no—
96#41—3261no—
97#41—6521no—
98C2×Q8—3261no—
99C2×Q8—3261no—
100C2×Q8—6521no—
101C2×Q8—1632no—
102C2×Q8D65211yescharacteristic
103C2×Q8C22×D16311yescharacteristic
104Q16—6522no—
105Q16—6521no—
106SD16—6522no—
107SD16—6521no—
108M16—22no—
109M16—3261no—
110C4⋊C4—22no—
111C4⋊C4—3262no—
112C4⋊C4—3261no—
113C4⋊C4—3262no—
114C4⋊C4—3262no—
115C42—3262no—
116C42—3261no—
117C42—1632no—
118C42D65211yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—13041no—
122Q8—3264no—
123Q8—3262no—
124Q8—3261no—
125Q8—6522no—
126D4—3262no—
127D4—3261no—
128D4—6521no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—3262no—
132C2×C4—3262no—
133C2×C4—6522no—
134C2×C4—6521no—
135C2×C4D4×D16312yes—
136C2×C4C2×D65211yescharacteristic
137C8—42no—
138C8—6521no—
139C22—6521no—
140C22#1910470511yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—3264no—
146C4—6522no—
147C4—6521no—
148C2—21no—
149C2—13041no—
150C2#2378185311yescharacteristic
subgroupG/H
centerC2#23781853
commutator, cosocleC2×Q8×C163C23
FrattiniC2×Q8C22×D163
Fitting#356×C163C2
socleC2×C163B(4,𝔽2)
prime radical#19104783C22
hypercenter#356D163
hypocenterC163#2205
LCSG ▷ C2×Q8×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #23782395C2semidirect
4. #23781463C2semidirect
5. #356×C163C2semidirect
75. C163#2205semidirect
81. #356D163semidirect

Maximal subgroups

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

subgroupindex
#237823952
#237818502
#237818472
#237814632
#356×C1632
#2205163