Finite Group Databasebeta

← #35216549

Subgroups — #35216549

Order 20864.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#23781854C212yesmaximal
2#23781842C211yesmaximal, characteristic
3#23781460C211yesmaximal, characteristic, complemented
4#23781455C212yesmaximal, complemented
5#563×C163C211yesmaximal, characteristic, complemented
6#19104780C2211yescharacteristic
7#19104766—21no—
8#19104718—22no—
9#19104663—21no—
10#19104652C2211yescharacteristic
11#19104650C2212yes—
12#19104649—22no—
13C4⋊Q8×C163—21no—
14#126×C163—22no—
15#125×C163—21no—
16#124×C163C2211yescharacteristic
17#123×C163C2212yes—
18C22⋊Q8×C163—22no—
19#14579699—21no—
20#14579698—41no—
21C2×C22⋊D163—22no—
22D4×D163—41no—
23#14579689—41no—
24#14579686—41no—
25#14579683—21no—
26Dic163⋊C4—42no—
27C22⋊Dic163—22no—
28(C2×D163)⋊C4—42no—
29C2×Q8×C163—21no—
30C2×Q8×C163D411yescharacteristic
31C2×D4×C163C2311yescharacteristic
32C22×C4×C163D412yes—
33C4⋊C4×C163—21no—
34C4⋊C4×C163—22no—
35C4⋊C4×C163—22no—
36C4⋊C4×C163—21no—
37C4⋊C4×C163—22no—
38C22⋊C4×C163—22no—
39C22⋊C4×C163—22no—
40C22⋊C4×C163D412yes—
41C42×C163—21no—
42C42×C163D411yescharacteristic
43C22×D163—22no—
44Dic326—21no—
45D652—21no—
46C22⋊D163—42no—
47C22⋊D163—41no—
48C2×Dic163—22no—
49C2×Dic163—42no—
50C4×D163—41no—
51C163⋊C8—41no—
52C23×C163C2×D412yes—
53Q8×C163—41no—
54Q8×C163—81no—
55D4×C163—41no—
56C2×C4×C163—21no—
57C2×C4×C163—22no—
58C2×C4×C163—21no—
59C2×C4×C163—22no—
60C2×C4×C163—42no—
61C2×C4×C163C2×D411yescharacteristic
62C22×C163—22no—
63C22×C163—42no—
64C22×C163#12011yescharacteristic
65C2×D163—41no—
66C2×D163—82no—
67C4×C163—21no—
68C4×C163—42no—
69C4×C163—41no—
70C4×C163—42no—
71C4×C163—41no—
72Dic163—41no—
73Dic163—82no—
74C2×C163—21no—
75C2×C163—42no—
76C2×C163B(4,𝔽2)11yescharacteristic
77D163—81no—
78C163#220111yescharacteristic, complemented, Sylow
79#2201—1631nomaximal, Sylow
80#563D16311yescharacteristic, complemented
81#458—1632no—
82#455—1631no—
83#354—1631no—
84#352—1632no—
85H(5,𝔽2)—1631no—
86#137—3261no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D16311yescharacteristic
91#123—3262no—
92#123C2×D16312yes—
93C22⋊Q8—22no—
94#104—3261no—
95#100—1631no—
96#99—3262no—
97#99—1632no—
98#41—3261no—
99#41—6521no—
100C2×Q8—21no—
101C2×Q8C22⋊D16311yescharacteristic
102C2×D4—3262no—
103C2×D4—6521no—
104C2×D4C22×D16311yescharacteristic
105C22×C4C22⋊D16312yes—
106Q16—6521no—
107SD16—6521no—
108M16—3261no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—6522no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3262no—
118C22⋊C4—6522no—
119C22⋊C4C22⋊D16312yes—
120C42—21no—
121C42C22⋊D16311yescharacteristic
122C23—3262no—
123C23C2×C22⋊D16312yes—
124Q8—41no—
125Q8—81no—
126Q8—3261no—
127D4—41no—
128D4—3261no—
129D4—6522no—
130D4—6521no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3262no—
137C2×C4—6521no—
138C2×C4—6522no—
139C2×C4C2×C22⋊D16311yescharacteristic
140C8—6521no—
141C22—22no—
142C22—42no—
143C22—13042no—
144C22—6521no—
145C22#1910470411yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—13042no—
152C4—6521no—
153C2—21no—
154C2—42no—
155C2—13041no—
156C2#2378185211yescharacteristic
subgroupG/H
centerC2#23781852
commutator, cosocleC2×D4×C163C23
FrattiniC2×D4C22×D163
Fitting#563×C163C2
socleC2×C163B(4,𝔽2)
prime radical#19104780C22
hypercenter#563D163
hypocenterC163#2201
LCSG ▷ C2×D4×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #23781460C2semidirect
4. #23781455C2semidirect
5. #563×C163C2semidirect
78. C163#2201semidirect
80. #563D163semidirect

Maximal subgroups

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

subgroupindex
#237818542
#237818422
#237814602
#237814552
#563×C1632
#2201163