Finite Group Databasebeta

← #28312662

Subgroups — #28312662

Order 13568.

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

normal subgroups only

This group has 194 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, complemented
2#20291113C211yesmaximal, characteristic, complemented
3#20289880C211yesmaximal, characteristic, complemented
4#20289870C211yesmaximal, characteristic, complemented
5#20289869C211yesmaximal, characteristic, complemented
6#20289825C211yesmaximal, characteristic, complemented
7#20277029C211yesmaximal, characteristic
8#16811667—21no—
9D8×#6157—21no—
10#16811556C2211yescharacteristic
11#16811552C2211yescharacteristic
12#16811550C2211yescharacteristic
13#16811538—21no—
14#16811535—21no—
15#16811510C2211yescharacteristic
16#16811056C2211yescharacteristic
17D16×D53C2211yescharacteristic
18#16810460C411yescharacteristic, complemented
19D1696C411yescharacteristic, complemented
20#16810458C411yescharacteristic, complemented
21#16810456C2211yescharacteristic
22C53×SD64C411yescharacteristic, complemented
23#13133305—41no—
24D4×#6157—41no—
25D53⋊C16D411yescharacteristic
26#13133288—21no—
27#13133286—21no—
28#13133282—41no—
29#13133279—41no—
30C8×#6157D411yescharacteristic
31#13133267—21no—
32#13133240—21no—
33D8×D53—21no—
34#13133160—21no—
35D848C2×C411yescharacteristic
36D848C2×C411yescharacteristic, complemented
37#13133158—21no—
38C16×D53C2311yescharacteristic
39#13133123C2×C411yescharacteristic, complemented
40Q32×C53C2×C411yescharacteristic, complemented
41D16×C53C2×C411yescharacteristic
42C32×C53C2×C411yescharacteristic, complemented
43C22×#6157—81no—
44#11719949—41no—
45D4×D53—41no—
46C22⋊#6157—81no—
47C4⋊#6157—41no—
48C4×#6157D811yescharacteristic
49#11719936—41no—
50#11719936—81no—
51D53⋊C8—81no—
52D53⋊C8D811yescharacteristic
53Q16×C53—21no—
54D8×C53—21no—
55#11719924—41no—
56#11719922—41no—
57D424—21no—
58D424—21no—
59C8×D53C2×D411yescharacteristic
60#11719910—21no—
61C16×C53C22×C411yescharacteristic
62#11719908#4111yescharacteristic
63C22×D53—81no—
64C2×#6157—21no—
65C2×#6157—81no—
66C2×#6157—162no—
67Q8×C53—41no—
68D4×C53—41no—
69C22⋊D53—81no—
70D212—41no—
71D212—41no—
72D212—81no—
73C4×D53—81no—
74C4×D53C2×D811yescharacteristic
75#88324—21no—
76#88324—81no—
77C8×C53C4×D411yescharacteristic
78#88322#13511yescharacteristic
79#13658—531nomaximal, Sylow
80C22×C53—81no—
81C2×D53—81no—
82C2×D53—81no—
83C2×D53—161no—
84C2×D53—161no—
85C2×D53#50911yescharacteristic
86#6157—41no—
87#6157—161no—
88C4×C53—81no—
89C4×C53C4×D811yescharacteristic
90Dic53#50911yescharacteristic
91#2263—531no—
92#2179—531no—
93#2177—531no—
94#1426—531no—
95#1419—531no—
96#1418—531no—
97#1398—531no—
98C2×C53—161no—
99C2×C53#217711yescharacteristic
100D53—21no—
101D53—161no—
102D53—161no—
103#508—531no—
104C2×D16—531no—
105#443—1061no—
106C4×D8—1061no—
107SD64—531no—
108SD64#615711yescharacteristic, complemented
109D32—531no—
110D32—531no—
111M64—531no—
112M64—531no—
113#368—531no—
114#367—531no—
115#359—1061no—
116#357—1061no—
117#346—531no—
118C53#1365811yescharacteristic, complemented, Sylow
119#135—1061no—
120C2×D8—1061no—
121#131—2121no—
122C4×D4—2121no—
123Q32C2×#615711yescharacteristic, complemented
124SD32—1061no—
125D16—531no—
126D16—531no—
127D16—1061no—
128D16C2×#615711yescharacteristic
129C2×C16—531no—
130C2×C16—531no—
131#108—1061no—
132#107—1061no—
133#104—2121no—
134#102—2121no—
135C4×C8—531no—
136C32—531no—
137C32—1061no—
138C32C2×#615711yescharacteristic, complemented
139#41—2121no—
140C2×D4—2121no—
141C22×C4—4241no—
142Q16—21no—
143SD16—2121no—
144D8—21no—
145D8—1061no—
146D8—1061no—
147D8—2121no—
148M16—4241no—
149M16—2121no—
150C2×C8—531no—
151C2×C8—531no—
152C2×C8—4241no—
153C4⋊C4—2121no—
154C22⋊C4—4241no—
155C42—531no—
156C16—531no—
157C16—1061no—
158C16C22×#615711yescharacteristic
159C23—4241no—
160Q8—41no—
161D4—41no—
162D4—4241no—
163D4—4241no—
164D4—2121no—
165D4—2121no—
166C2×C4—531no—
167C2×C4—8482no—
168C2×C4—1061no—
169C2×C4—4241no—
170C2×C4—4241no—
171C8—531no—
172C8—1061no—
173C8—4241no—
174C8D4×#615711yescharacteristic
175C22—81no—
176C22—531no—
177C22—8481no—
178C22—8481no—
179C22—4241no—
180C22—4241no—
181C4—81no—
182C4—531no—
183C4—8481no—
184C4—2121no—
185C4D8×#615711yescharacteristic
186C2—161no—
187C2—8481no—
188C2—8481no—
189C2—1061no—
190C2#2029111311yescharacteristic
subgroupG/H
centerC2#20291113
commutatorC16×C53C22×C4
FrattiniC16C22×#6157
FittingC53×SD64C4
socleC2×C53#2177
prime radical#20277029C2
cosocleC16×D53C23
hypercenterSD64#6157
hypocenterC53#13658
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. #20291118C2semidirect
2. #20291113C2semidirect
3. #20289880C2semidirect
4. #20289870C2semidirect
5. #20289869C2semidirect
6. #20289825C2semidirect
18. #16810460C4semidirect
19. D1696C4semidirect
20. #16810458C4semidirect
22. C53×SD64C4semidirect
36. D848C2×C4semidirect
39. #13133123C2×C4semidirect
40. Q32×C53C2×C4semidirect
42. C32×C53C2×C4semidirect
108. SD64#6157semidirect
118. C53#13658semidirect
123. Q32C2×#6157semidirect
138. C32C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202911182
#202911132
#202898802
#202898702
#202898692
#202898252
#202770292
#1365853