Finite Group Databasebeta

← #28312657

Subgroups — #28312657

Order 13568.

182 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 50 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#20291118C211yesmaximal, characteristic
2#20291112C211yesmaximal, characteristic, complemented
3#20289880C211yesmaximal, characteristic, complemented
4#20289871C211yesmaximal, characteristic, complemented
5#20289869C211yesmaximal, characteristic
6#20289824C211yesmaximal, characteristic, complemented
7#20277034C211yesmaximal, characteristic
8#16811667—21no—
9Q16×#6157—21no—
10#16811556C2211yescharacteristic
11#16811552C2211yescharacteristic
12#16811550C2211yescharacteristic
13#16811538—21no—
14#16811536—21no—
15#16811510C2211yescharacteristic
16#16811056C2211yescharacteristic
17Q32×D53C2211yescharacteristic
18#16810463C411yescharacteristic, complemented
19#16810462C411yescharacteristic, complemented
20#16810460C411yescharacteristic, complemented
21#16810456C2211yescharacteristic
22C53×Q64C411yescharacteristic, complemented
23#13133305—41no—
24Q8×#6157—41no—
25D53⋊C16D411yescharacteristic
26#13133288—21no—
27#13133286—21no—
28#13133282—41no—
29#13133280—41no—
30C8×#6157D411yescharacteristic
31#13133267—21no—
32#13133240—21no—
33Q16×D53—21no—
34Dic424C2×C411yescharacteristic
35#13133163—21no—
36#13133160—21no—
37D848C2×C411yescharacteristic
38C16×D53C2311yescharacteristic
39#13133123C2×C411yescharacteristic
40Q32×C53C2×C411yescharacteristic
41Q32×C53C2×C411yescharacteristic
42C32×C53C2×C411yescharacteristic
43#11719949—41no—
44Q8×D53—41no—
45C4⋊#6157—41no—
46C4⋊#6157—81no—
47C4×#6157—81no—
48C4×#6157D811yescharacteristic
49#11719936—41no—
50#11719936—81no—
51D53⋊C8—81no—
52D53⋊C8D811yescharacteristic
53Q16×C53—21no—
54Q16×C53—21no—
55#11719925—41no—
56#11719924—41no—
57Dic212—21no—
58D424—21no—
59C8×D53C2×D411yescharacteristic
60#11719910—21no—
61C16×C53C22×C411yescharacteristic
62#11719908#4111yescharacteristic
63C2×#6157—21no—
64C2×#6157—81no—
65Q8×C53—41no—
66Q8×C53—41no—
67D212—41no—
68D212—81no—
69C4×D53—81no—
70C4×D53—81no—
71C4×D53C2×D811yescharacteristic
72Dic106—41no—
73Dic106—81no—
74#88324—21no—
75#88324—81no—
76C8×C53C4×D411yescharacteristic
77#88322#13511yescharacteristic
78#13657—531nomaximal, Sylow
79C2×D53—81no—
80C2×D53#50911yescharacteristic
81#6157—41no—
82#6157—161no—
83C4×C53—81no—
84C4×C53—81no—
85C4×C53C4×D811yescharacteristic
86Dic53—81no—
87Dic53#50911yescharacteristic
88#2264—531no—
89#2179—531no—
90#2176—531no—
91#1426—531no—
92#1420—531no—
93#1418—531no—
94#1398—531no—
95C2×C53#217711yescharacteristic
96D53—21no—
97D53—161no—
98#508—531no—
99C2×Q32—531no—
100#443—1061no—
101C4×Q16—1061no—
102Q64—531no—
103Q64#615711yescharacteristic, complemented
104SD64—531no—
105SD64—531no—
106M64—531no—
107M64—531no—
108#368—531no—
109#367—531no—
110#359—1061no—
111#358—1061no—
112#346—531no—
113C53#1365711yescharacteristic, complemented, Sylow
114#135—1061no—
115C2×Q16—1061no—
116#131—2121no—
117C4×Q8—2121no—
118Q32—531no—
119Q32—1061no—
120Q32C2×#615711yescharacteristic
121Q32C2×#615711yescharacteristic
122SD32—1061no—
123D16—531no—
124C2×C16—531no—
125C2×C16—531no—
126#108—1061no—
127#107—1061no—
128#104—2121no—
129#103—2121no—
130C4×C8—531no—
131C32—531no—
132C32—1061no—
133C32C2×#615711yescharacteristic
134#41—2121no—
135C2×Q8—2121no—
136Q16—21no—
137Q16—21no—
138Q16—1061no—
139Q16—2121no—
140SD16—2121no—
141D8—1061no—
142M16—4241no—
143M16—2121no—
144C2×C8—531no—
145C2×C8—531no—
146C2×C8—4241no—
147C4⋊C4—4241no—
148C4⋊C4—2121no—
149C42—531no—
150C42—4241no—
151C16—531no—
152C16—1061no—
153C16C22×#615711yescharacteristic
154Q8—41no—
155Q8—41no—
156Q8—4241no—
157Q8—2121no—
158D4—4241no—
159D4—2121no—
160C2×C4—531no—
161C2×C4—1061no—
162C2×C4—4241no—
163C2×C4—4241no—
164C2×C4—4241no—
165C8—531no—
166C8—1061no—
167C8—4241no—
168C8D4×#615711yescharacteristic
169C22—531no—
170C22—4241no—
171C4—81no—
172C4—81no—
173C4—531no—
174C4—8481no—
175C4—4241no—
176C4—2121no—
177C4D8×#615711yescharacteristic
178C2—8481no—
179C2—1061no—
180C2#2029111311yescharacteristic
subgroupG/H
centerC2#20291113
commutatorC16×C53C22×C4
FrattiniC16C22×#6157
FittingC53×Q64C4
socleC2×C53#2177
prime radical#16811056C22
cosocleC16×D53C23
hypercenterQ64#6157
hypocenterC53#13657
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q64

Complements

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

NH ≅ G/Ntype
2. #20291112C2semidirect
3. #20289880C2semidirect
4. #20289871C2semidirect
6. #20289824C2semidirect
18. #16810463C4semidirect
19. #16810462C4semidirect
20. #16810460C4semidirect
22. C53×Q64C4semidirect
103. Q64#6157semidirect
113. C53#13657semidirect

Maximal subgroups

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

subgroupindex
#202911182
#202911122
#202898802
#202898712
#202898692
#202898242
#202770342
#1365753