Finite Group Databasebeta

← #35213541

Subgroups — #35213541

Order 20864.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#23782246C211yesmaximal, characteristic
2#23781979C212yesmaximal, complemented
3#23781595C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#324×C163C211yesmaximal, characteristic, complemented
6C2×#14579699—21no—
7#19104715—22no—
8#19104710—22no—
9#19104709—21no—
10#19104709C2212yes—
11C2×C4⋊Dic163C2211yescharacteristic
12C2×Dic163⋊C4—21no—
13#19104669—22no—
14#19104657—22no—
15C163⋊#95—21no—
16C163⋊#95C2211yescharacteristic
17C2×C4⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yes—
19#14579699—42no—
20#14579699—42no—
21C2×Dic326—21no—
22C2×D652—21no—
23C2×D652—21no—
24C2×D652D412yes—
25C2×C22⋊D163—22no—
26C22×Dic163—21no—
27C22×Dic163—21no—
28C2×C4×D163—22no—
29C4⋊Dic163—21no—
30C4⋊Dic163—22no—
31C4⋊Dic163D412yes—
32Dic163⋊C4—42no—
33(C2×D163)⋊C4—22no—
34(C2×D163)⋊C4—22no—
35C22×C4×C163D411yescharacteristic
36C22×C4×C163D411yescharacteristic
37C22×C4×C163C2311yescharacteristic
38C2×C8×C163—22no—
39C4⋊C4×C163—22no—
40C22×D163—22no—
41Dic326—42no—
42D652—42no—
43D652—42no—
44D652—81no—
45C22⋊D163—42no—
46C22⋊D163—42no—
47C2×Dic163—22no—
48C2×Dic163—22no—
49C2×Dic163—41no—
50C2×Dic163—41no—
51C2×Dic163—41no—
52C2×Dic163—41no—
53C2×Dic163—41no—
54C4×D163—44no—
55C23×C163C2×D411yescharacteristic
56C2×C4×C163—21no—
57C2×C4×C163—21no—
58C2×C4×C163—21no—
59C2×C4×C163—41no—
60C2×C4×C163—41no—
61C2×C4×C163D812yes—
62C2×C4×C163C2×D412yes—
63C8×C163—42no—
64C22×C163—21no—
65C22×C163—21no—
66C22×C163#12011yescharacteristic
67C22×C163C2×D811yescharacteristic
68C22×C163AGL(1,ℤ/8)11yescharacteristic
69C2×D163—42no—
70C2×D163—42no—
71C2×D163—42no—
72C4×C163—22no—
73C4×C163—22no—
74C4×C163—41no—
75Dic163—42no—
76Dic163—42no—
77Dic163—81no—
78C2×C163—21no—
79C2×C163—21no—
80C2×C163#44711yescharacteristic
81C2×C163#45611yescharacteristic
82C2×C163#45811yescharacteristic
83D163—82no—
84C163#161011yescharacteristic, complemented, Sylow
85#1610—1631nomaximal, Sylow
86#537—1631no—
87#480—1632no—
88#398—1631no—
89#327—1632no—
90#324D16311yescharacteristic, complemented
91C2×#41—3261no—
92C22⋊Q8—3262no—
93#121—3262no—
94#121—3261no—
95#121—1632no—
96C2×C4⋊C4—3261no—
97C2×C4⋊C4—1631no—
98C2×C4⋊C4C2×D16311yescharacteristic
99#107—3262no—
100#102—3262no—
101C22⋊C8C2×D16312yes—
102#95—3261no—
103#95—1631no—
104#41—6522no—
105#41—6522no—
106C2×Q8—3261no—
107C2×D4—3261no—
108C2×D4—3261no—
109C2×D4—3262no—
110C2×D4—1632no—
111C22×C4—3262no—
112C22×C4—3261no—
113C22×C4—3261no—
114C22×C4D65211yescharacteristic
115C22×C4D65211yescharacteristic
116C22×C4C22×D16311yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—3261no—
120C4⋊C4—3262no—
121C4⋊C4—6522no—
122C4⋊C4—1632no—
123C22⋊C4—3262no—
124C22⋊C4—3262no—
125C23—3262no—
126C23C2×D65211yescharacteristic
127Q8—6522no—
128D4—13041no—
129D4—6522no—
130D4—6522no—
131D4—6522no—
132D4—6522no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—3262no—
139C2×C4—3262no—
140C2×C4—6524no—
141C2×C4—6521no—
142C2×C4—6521no—
143C2×C4—6521no—
144C2×C4—6521no—
145C2×C4—6521no—
146C2×C4D130412yes—
147C2×C4D4×D16312yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—6522no—
152C22—6522no—
153C22—6522no—
154C22#1910470511yescharacteristic
155C22C2×D130411yescharacteristic
156C22#1910476311yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—13041no—
161C4—6522no—
162C4—6522no—
163C2—21no—
164C2—21no—
165C2—13042no—
166C2#2378180111yescharacteristic
167C2#2378185011yescharacteristic
168C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
prime radical#23782246C2
hypercenter#324D163
hypocenterC163#1610
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #23781979C2semidirect
3. #23781595C2semidirect
4. #23781400C2semidirect
5. #324×C163C2semidirect
84. C163#1610semidirect
90. #324D163semidirect

Maximal subgroups

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

subgroupindex
#237822462
#237819792
#237815952
#237814002
#324×C1632
#1610163