Finite Group Databasebeta

← #28312652

Subgroups — #28312652

Order 13568.

184 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 52 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#20291116C211yesmaximal, characteristic, complemented
2#20291114C211yesmaximal, characteristic, complemented
3#20289878C211yesmaximal, characteristic, complemented
4#20289867C211yesmaximal, characteristic, complemented
5#20289866C211yesmaximal, characteristic, complemented
6#20289821C211yesmaximal, characteristic, complemented
7#20277023C211yesmaximal, characteristic
8#16811667—21no—
9#16811665—21no—
10D53⋊C32C2211yescharacteristic
11#16811551C2211yescharacteristic
12#16811551C2211yescharacteristic
13#16811538—21no—
14#16811537—21no—
15C16×#6157C2211yescharacteristic
16#16811056C2211yescharacteristic
17#16811052C2211yescharacteristic
18#16810463C411yescharacteristic, complemented
19#16810462C411yescharacteristic, complemented
20#16810458C411yescharacteristic, complemented
21C32×D53C2211yescharacteristic
22C53×SD64C411yescharacteristic, complemented
23#13133305—41no—
24#13133304—41no—
25D53⋊C16D411yescharacteristic
26#13133288—21no—
27#13133288—21no—
28#13133282—41no—
29#13133281—41no—
30C8×#6157D411yescharacteristic
31#13133267—21no—
32#13133240—21no—
33#13133236—21no—
34Dic424C2×C411yescharacteristic
35#13133161—21no—
36#13133160—21no—
37D848C2×C411yescharacteristic
38C16×D53C2311yescharacteristic
39#13133123C2×C411yescharacteristic
40Q32×C53C2×C411yescharacteristic
41D16×C53C2×C411yescharacteristic
42C32×C53C2×C411yescharacteristic
43#11719949—41no—
44#11719947—41no—
45#11719940—81no—
46C2×#88324—81no—
47C4×#6157D811yescharacteristic
48#11719936—41no—
49#11719936—41no—
50#11719936—81no—
51D53⋊C8—81no—
52D53⋊C8D811yescharacteristic
53Q16×C53—21no—
54D8×C53—21no—
55#11719924—41no—
56#11719923—41no—
57Dic212—21no—
58D424—21no—
59C8×D53C2×D411yescharacteristic
60#11719910—21no—
61C16×C53C22×C411yescharacteristic
62#11719908#4111yescharacteristic
63C2×#6157—21no—
64Q8×C53—41no—
65D4×C53—41no—
66C22⋊D53—81no—
67C2×Dic53—81no—
68D212—41no—
69D212—81no—
70C4×D53—81no—
71C4×D53C2×D811yescharacteristic
72Dic106—41no—
73#88324—81no—
74#88324—81no—
75#88324D1611yescharacteristic
76#88324D1611yescharacteristic
77C8×C53C4×D411yescharacteristic
78#88322#13511yescharacteristic
79#13656—531nomaximal, Sylow
80C22×C53—81no—
81C2×D53—81no—
82C2×D53#50811yescharacteristic
83#6157—41no—
84C4×C53—81no—
85C4×C53C4×D811yescharacteristic
86Dic53—81no—
87Dic53C2×D1611yescharacteristic
88#2262—531no—
89#2178—531no—
90#2178—531no—
91#1425—531no—
92#1417—531no—
93#1417—531no—
94#1397—531no—
95C2×C53—161no—
96C2×C53C4×D1611yescharacteristic
97D53—21no—
98D53—161no—
99#508—531no—
100#508—531no—
101#443—1061no—
102#443—1061no—
103Q64—531no—
104SD64—531no—
105SD64#615711yescharacteristic, complemented
106D32—531no—
107C2×C32—531no—
108C2×C32—531no—
109#368—531no—
110#368—531no—
111#359—1061no—
112#359—1061no—
113C4×C16—531no—
114C53#1365611yescharacteristic, complemented, Sylow
115#135—1061no—
116#135—1061no—
117#131—2121no—
118#131—2121no—
119Q32—531no—
120Q32C2×#615711yescharacteristic
121SD32—1061no—
122SD32—1061no—
123D16—531no—
124D16C2×#615711yescharacteristic
125C2×C16—531no—
126C2×C16—531no—
127#108—1061no—
128#108—1061no—
129#104—2121no—
130#104—2121no—
131C4×C8—531no—
132C32—531no—
133C32—1061no—
134C32C2×#615711yescharacteristic
135#41—2121no—
136#41—2121no—
137Q16—21no—
138Q16—1061no—
139SD16—2121no—
140SD16—2121no—
141D8—21no—
142D8—1061no—
143M16—4241no—
144M16—4241no—
145M16—2121no—
146M16—2121no—
147C2×C8—531no—
148C2×C8—531no—
149C2×C8—4241no—
150C2×C8—4241no—
151C42—531no—
152C16—531no—
153C16—1061no—
154C16C22×#615711yescharacteristic
155Q8—41no—
156Q8—2121no—
157D4—41no—
158D4—4241no—
159D4—4241no—
160D4—2121no—
161C2×C4—531no—
162C2×C4—1061no—
163C2×C4—4241no—
164C2×C4—4241no—
165C8—531no—
166C8—531no—
167C8—531no—
168C8—4241no—
169C8—4241no—
170C8D4×#615711yescharacteristic
171C22—81no—
172C22—531no—
173C22—4241no—
174C4—81no—
175C4—531no—
176C4—4241no—
177C4—2121no—
178C4D8×#615711yescharacteristic
179C2—161no—
180C2—8481no—
181C2—1061no—
182C2D16×#615711yescharacteristic
subgroupG/H
centerC2D16×#6157
commutatorC16×C53C22×C4
FrattiniC16C22×#6157
FittingC53×SD64C4
socleC2×C53C4×D16
prime radical#20277023C2
cosocleC16×D53C23
hypercenterSD64#6157
hypocenterC53#13656
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64

Complements

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

NH ≅ G/Ntype
1. #20291116C2semidirect
2. #20291114C2semidirect
3. #20289878C2semidirect
4. #20289867C2semidirect
5. #20289866C2semidirect
6. #20289821C2semidirect
18. #16810463C4semidirect
19. #16810462C4semidirect
20. #16810458C4semidirect
22. C53×SD64C4semidirect
105. SD64#6157semidirect
114. C53#13656semidirect

Maximal subgroups

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

subgroupindex
#202911162
#202911142
#202898782
#202898672
#202898662
#202898212
#202770232
#1365653