Finite Group Databasebeta

← #35216484

Subgroups — #35216484

Order 20864.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782465C211yesmaximal, characteristic, complemented
2#23782103C212yesmaximal, complemented
3#23781918C211yesmaximal, characteristic, complemented
4#23781476C212yesmaximal, complemented
5#349×C163C211yesmaximal, characteristic, complemented
6C2×#14579699—21no—
7#19104768—22no—
8#19104763—22no—
9#19104757—21no—
10#19104757C2212yes—
11C2×D1304C2211yescharacteristic, complemented
12#19104709—21no—
13#19104680—22no—
14D2608—22no—
15#19104667C2211yescharacteristic, complemented
16#19104657—21no—
17C2×M16×C163C2211yescharacteristic
18M32×C163C2212yescomplemented
19#14579699—22no—
20#14579699—22no—
21#14579699—41no—
22#14579699—41no—
23C2×Dic326—21no—
24C2×D652—21no—
25C2×D652—21no—
26C2×C22⋊D163—41no—
27C2×C22⋊D163—41no—
28C2×C4×D163—41no—
29Dic652—21no—
30Dic652D412yescomplemented
31#14579688—22no—
32#14579688—22no—
33D1304—21no—
34D1304—22no—
35D1304D412yescomplemented
36C4⋊Dic163—21no—
37(C2×D163)⋊C4—41no—
38C22×C4×C163D411yescharacteristic
39M16×C163—22no—
40C2×C8×C163D411yescharacteristic
41C2×C8×C163C2311yescharacteristic
42C16×C163—22no—
43C22×D163—41no—
44C22×D163—41no—
45Dic326—22no—
46Dic326—41no—
47D652—22no—
48D652—41no—
49D652—41no—
50D652—41no—
51C22⋊D163—42no—
52C22⋊D163—42no—
53C22⋊D163—82no—
54C2×Dic163—41no—
55C2×Dic163—41no—
56C4×D163—42no—
57C4×D163—42no—
58C23×C163D811yescharacteristic
59C2×C4×C163—21no—
60C2×C4×C163—21no—
61C2×C4×C163D811yescharacteristic
62C2×C4×C163C2×D411yescharacteristic
63C8×C163—21no—
64C8×C163C2×D412yes—
65C22×C163—21no—
66C22×C163—41no—
67C22×C163C2×D811yescharacteristic
68C2×D163—42no—
69C2×D163—82no—
70C2×D163—81no—
71C2×D163—81no—
72C2×D163—81no—
73C4×C163—21no—
74C4×C163#12011yescharacteristic
75C4×C163AGL(1,ℤ/8)11yescharacteristic
76Dic163—42no—
77Dic163—81no—
78C2×C163—21no—
79C2×C163—41no—
80C2×C163#44711yescharacteristic
81D163—82no—
82D163—161no—
83C163#219011yescharacteristic, complemented, Sylow
84#2190—1631nomaximal, Sylow
85#575—1631no—
86#509—1632no—
87#469—1631no—
88#360—1632no—
89#349D16311yescharacteristic, complemented
90C2×#41—3261no—
91#137—3262no—
92AGL(1,ℤ/8)—3262no—
93#135—3261no—
94#135—1632no—
95C2×D8—1631no—
96C2×M16C2×D16311yescharacteristic
97#121—3261no—
98SD32—3262no—
99D16—3262no—
100M32C2×D16312yescomplemented
101#106—1631no—
102#102—3261no—
103#41—3262no—
104#41—3262no—
105#41—6521no—
106#41—6521no—
107C2×Q8—3261no—
108C2×D4—3261no—
109C2×D4—3261no—
110C2×D4—6521no—
111C2×D4—6521no—
112C22×C4—6521no—
113C22×C4D65211yescharacteristic
114Q16—3261no—
115Q16—1632no—
116SD16—3262no—
117SD16—3262no—
118D8—3261no—
119D8—3262no—
120D8—1632no—
121M16—22no—
122C2×C8D65211yescharacteristic
123C2×C8C22×D16311yescharacteristic
124C4⋊C4—3261no—
125C22⋊C4—6521no—
126C16—22no—
127C23—6521no—
128C23—6521no—
129C23D130411yescharacteristic
130Q8—3262no—
131Q8—6521no—
132D4—13042no—
133D4—3262no—
134D4—6521no—
135D4—6522no—
136D4—6522no—
137D4—6521no—
138D4—6521no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—6522no—
142C2×C4—6521no—
143C2×C4—6521no—
144C2×C4—6522no—
145C2×C4D130411yescharacteristic
146C2×C4C2×D65211yescharacteristic
147C8—21no—
148C8D4×D16312yes—
149C22—21no—
150C22—41no—
151C22—13041no—
152C22—13041no—
153C22—13042no—
154C22—13041no—
155C22—6522no—
156C22C2×D130411yescharacteristic
157C4—21no—
158C4—13041no—
159C4—6522no—
160C4#1910470511yescharacteristic
161C4#1910476311yescharacteristic
162C2—21no—
163C2—41no—
164C2—13042no—
165C2—26081no—
166C2#2378180111yescharacteristic
subgroupG/H
centerC2#23781801
commutator, cosocleC2×C8×C163C23
FrattiniC2×C8C22×D163
Fitting#349×C163C2
socleC2×C163#447
hypercenter#349D163
hypocenterC163#2190
LCSG ▷ C2×C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #23782465C2semidirect
2. #23782103C2semidirect
3. #23781918C2semidirect
4. #23781476C2semidirect
5. #349×C163C2semidirect
11. C2×D1304C22semidirect
15. #19104667C22semidirect
18. M32×C163C22semidirect
30. Dic652D4semidirect
35. D1304D4semidirect
83. C163#2190semidirect
89. #349D163semidirect
100. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824652
#237821032
#237819182
#237814762
#349×C1632
#2190163