Finite Group Databasebeta

← #27304964

Subgroups — #27304964

Order 13568.

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

normal subgroups only

This group has 184 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#20277008C211yesmaximal, characteristic, complemented
2#20276837C211yesmaximal, characteristic, complemented
3#20276832C211yesmaximal, characteristic
4#20272919C211yesmaximal, characteristic, complemented
5#20272908C211yesmaximal, characteristic, complemented
6#20272905C211yesmaximal, characteristic, complemented
7C2×C53×SD64C211yesmaximal, characteristic, complemented
8#16811039C2211yescharacteristic
9#16810895—21no—
10#16810890—21no—
11#16810457C2211yescharacteristic
12#16810456—21no—
13C32×D53—21no—
14#16810453C2211yescharacteristic
15#16810453C2211yescharacteristic
16#16810439—21no—
17#16810436—21no—
18C2×Q32×C53C2211yescharacteristic
19C2×D16×C53C2211yescharacteristic
20C53×SD64—21no—
21C53×SD64—21no—
22C2×C32×C53C2211yescharacteristic
23#13133248—41no—
24#13133243—41no—
25#13133223D411yescharacteristic
26#13133157D411yescharacteristic
27#13133156—21no—
28C16×D53—21no—
29#13133153—21no—
30#13133153—21no—
31#13133136—41no—
32#13133134—41no—
33#13133123—21no—
34C2×Q16×C53—21no—
35C2×D8×C53—21no—
36Q32×C53—21no—
37Q32×C53—21no—
38D16×C53—21no—
39D16×C53—21no—
40C2×C16×C53C2311yescharacteristic
41C32×C53D411yescharacteristic
42C32×C53D411yescharacteristic
43C2×Q8×C53—41no—
44C2×D4×C53—41no—
45#11719945D811yescharacteristic
46Q16×C53—41no—
47Q16×C53—41no—
48D8×C53—41no—
49D8×C53—41no—
50C2×C8×C53C2×D411yescharacteristic
51#11719925—81no—
52#11719924—81no—
53#11719923—81no—
54#11719922—81no—
55#11719917—41no—
56#11719917—41no—
57#11719917D811yescharacteristic
58#11719912—21no—
59C8×D53—21no—
60C16×C53C2×D411yescharacteristic
61C16×C53#4111yescharacteristic
62#11719908—21no—
63C23×C53—81no—
64Q8×C53—81no—
65Q8×C53—81no—
66D4×C53—81no—
67D4×C53—81no—
68C2×C4×C53—81no—
69C2×C4×C53C2×D811yescharacteristic
70C22⋊D53—21no—
71D212D1611yescharacteristic
72C4×D53—21no—
73Dic106D1611yescharacteristic
74C8×C53#12111yescharacteristic
75C8×C53#13511yescharacteristic
76#88322—21no—
77#88322—81no—
78#88322—81no—
79#13702—531nomaximal, Sylow
80C22×C53—161no—
81C22×C53—321no—
82C22×C53#50811yescharacteristic
83C2×D53—21no—
84C4×C53—161no—
85C4×C53#46611yescharacteristic
86C4×C53C2×D1611yescharacteristic
87Dic53—21no—
88C2×SD64D5311yescharacteristic, complemented
89#2258—531no—
90#2222—531no—
91#2221—531no—
92#1425—531no—
93#1420—531no—
94#1419—531no—
95C2×C53—21no—
96C2×C53—321no—
97C2×C53#221511yescharacteristic
98D53—41no—
99C2×Q32C2×D5311yescharacteristic
100C2×D16C2×D5311yescharacteristic
101#504—531no—
102#473—1061no—
103#472—1061no—
104SD64—21no—
105SD64—21no—
106M64—531no—
107M64—1061no—
108C2×C32—1061no—
109C2×C32C2×D5311yescharacteristic
110#368—531no—
111#368—531no—
112#362—1061no—
113#361—1061no—
114C53#1370211yescharacteristic, complemented, Sylow
115#137—2121no—
116AGL(1,ℤ/8)—2121no—
117C2×Q16—21no—
118C2×D8—21no—
119#131—531no—
120Q32—21no—
121Q32—21no—
122D16—21no—
123D16—21no—
124M32—531no—
125M32—1061no—
126C2×C16—1061no—
127C2×C16C22×D5311yescharacteristic
128#108—1061no—
129#108—1061no—
130#101—2121no—
131#100—2121no—
132C32—1061no—
133C32C22⋊D5311yescharacteristic
134C32C22⋊D5311yescharacteristic
135#41—531no—
136C2×Q8—41no—
137C2×D4—41no—
138Q16—41no—
139Q16—41no—
140Q16—4241no—
141SD16—4241no—
142SD16—4241no—
143D8—41no—
144D8—41no—
145D8—4241no—
146M16—531no—
147M16—1061no—
148M16—2121no—
149M16—2121no—
150C2×C8—1061no—
151C2×C8D4×D5311yescharacteristic
152C16—1061no—
153C16#1171994711yescharacteristic
154C16C2×C22⋊D5311yescharacteristic
155C23—81no—
156Q8—81no—
157Q8—81no—
158Q8—531no—
159D4—81no—
160D4—81no—
161D4—531no—
162D4—1061no—
163C2×C4—81no—
164C2×C4—1061no—
165C2×C4D8×D5311yescharacteristic
166C8—1061no—
167C8—4241no—
168C8—4241no—
169C8#1313318811yescharacteristic
170C8#1313323611yescharacteristic
171C22—161no—
172C22—321no—
173C22—1061no—
174C22#1681105211yescharacteristic
175C4—161no—
176C4—1061no—
177C4#1681085611yescharacteristic
178C4D16×D5311yescharacteristic
179C2—21no—
180C2—321no—
181C2—2121no—
182C2#2027679811yescharacteristic
subgroupG/H
centerC2#20276798
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC2×C53×SD64C2
socleC2×C53#2215
prime radical#20276832C2
hypercenterC2×SD64D53
hypocenterC53#13702
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
1. #20277008C2semidirect
2. #20276837C2semidirect
4. #20272919C2semidirect
5. #20272908C2semidirect
6. #20272905C2semidirect
7. C2×C53×SD64C2semidirect
88. C2×SD64D53semidirect
114. C53#13702semidirect

Maximal subgroups

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

subgroupindex
#202770082
#202768372
#202768322
#202729192
#202729082
#202729052
C2×C53×SD642
#1370253