Finite Group Databasebeta

← #28312486

Subgroups — #28312486

Order 13568.

312 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#16811534C212yesmaximal
2#20289859C214yesmaximal, complemented
3#20285096C211yesmaximal, characteristic
4C2×#13133278—22no—
5#16811534—22no—
6#16811534C2214yes—
7#16811180C411yescharacteristic
8C2×#13133217C2211yescharacteristic
9#16810992C412yes—
10#16810992C2212yes—
11#16810974—22no—
12#16810633—22no—
13C4×M16×C53C411yescharacteristic
14C2×#11719940—42no—
15#13133278—24no—
16#13133278—42no—
17C22×Dic106D412yes—
18#13133217—21no—
19#13133217—41no—
20#13133217C2×C411yescharacteristic
21#13133217D412yes—
22#13133217C2311yescharacteristic
23#13133209C2×C412yes—
24#13133196—41no—
25C4×Dic106—41no—
26#13133176—21no—
27C2×C4⋊Dic53—41no—
28#13133142—24no—
29#13133142—24no—
30C2×M16×C53—21no—
31C2×C42×C53C2×C411yescharacteristic
32#97×C53C2×C412yes—
33C4×C8×C53—21no—
34C22×C4×C53—21no—
35C22×C4×C53C22⋊C411yescharacteristic
36C22×Dic53—22no—
37C2×Dic106—22no—
38C2×Dic106—42no—
39C2×Dic106—42no—
40C2×Dic106—41no—
41C2×Dic106C22⋊C412yes—
42C2×Dic106C2×D412yes—
43#11719940—44no—
44#11719940—82no—
45C2×#88324—82no—
46M16×C53—42no—
47C2×C8×C53—22no—
48C42×C53—21no—
49C42×C53C22⋊C411yescharacteristic
50C42×C53C22×C411yescharacteristic
51C22⋊Dic53—41no—
52C4⋊Dic53—22no—
53C4⋊Dic53—41no—
54C4⋊Dic53—42no—
55C4⋊Dic53—41no—
56C4⋊Dic53—41no—
57#11719919—42no—
58C4×Dic53—41no—
59C23×C53#9911yescharacteristic
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—42no—
64C2×C4×C53#9911yescharacteristic
65C2×C4×C53C2×C22⋊C411yescharacteristic
66C2×Dic53—22no—
67C2×Dic53—22no—
68C2×Dic53—42no—
69C2×Dic53—42no—
70C2×Dic53—42no—
71Dic106—42no—
72Dic106—42no—
73Dic106—82no—
74Dic106—82no—
75#88324—84no—
76C8×C53—42no—
77#13599—531nomaximal, Sylow
78C22×C53—41no—
79C22×C53#35412yes—
80C22×C53C2×#9911yescharacteristic
81C4×C53—22no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—41no—
85Dic53—42no—
86Dic53—42no—
87Dic53—82no—
88#3139—531no—
89C2×#356—532no—
90#1413—534no—
91C2×C53—21no—
92C2×C53—22no—
93C2×C53C2×#35411yescharacteristic
94#533—531no—
95C2×C4⋊Q8—531no—
96#491—532no—
97#491—532no—
98#487—1062no—
99#419—1062no—
100C2×#101—1062no—
101C4×M16#615711yescharacteristic
102#356—534no—
103#356—1062no—
104C53#1359911yescharacteristic, complemented, Sylow
105C22×Q8—532no—
106C2×M16—21no—
107C2×M16—2122no—
108C4⋊Q8—532no—
109C4⋊Q8—531no—
110C4⋊Q8—531no—
111C4⋊Q8—1061no—
112C4⋊Q8—2121no—
113#125—532no—
114C22⋊Q8—2121no—
115C4×Q8—2121no—
116#117—1061no—
117C2×C4⋊C4—2121no—
118C2×C42C2×#615711yescharacteristic
119#103—1064no—
120#103—1064no—
121#101—1064no—
122#101—2122no—
123#97C2×#615712yes—
124C4×C8—21no—
125C2×Q8—532no—
126C2×Q8—532no—
127C2×Q8—1062no—
128C2×Q8—2121no—
129C2×Q8—2122no—
130C2×Q8—2122no—
131C22×C4—21no—
132C22×C4—1062no—
133C22×C4C22⋊#615711yescharacteristic
134M16—42no—
135M16—4242no—
136M16—2124no—
137C2×C8—22no—
138C2×C8—4242no—
139C4⋊C4—1062no—
140C4⋊C4—2122no—
141C4⋊C4—2121no—
142C4⋊C4—2121no—
143C4⋊C4—2122no—
144C4⋊C4—2121no—
145C22⋊C4—2121no—
146C42—21no—
147C42—2121no—
148C42C22⋊#615711yescharacteristic
149C42C22×#615711yescharacteristic
150C23#1313327611yescharacteristic
151Q8—4242no—
152Q8—4242no—
153Q8—2122no—
154Q8—2122no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—1062no—
160C2×C4—1062no—
161C2×C4—2122no—
162C2×C4—2122no—
163C2×C4—2122no—
164C2×C4#1313327611yescharacteristic
165C2×C4C2×C22⋊#615711yescharacteristic
166C8—42no—
167C8—4244no—
168C22—41no—
169C22#1681153012yes—
170C22C2×#1313327611yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—4242no—
176C4—2122no—
177C4—2122no—
178C2—21no—
179C2—22no—
180C2C2×#1681153011yescharacteristic
subgroupG/H
centerC2C2×#16811530
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC4×M16×C53C4
socleC2×C53C2×#354
prime radicalC23×C53#99
cosocle#13133217C23
hypercenterC4×M16#6157
hypocenterC53#13599
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #20289859C2semidirect
104. C53#13599semidirect

Maximal subgroups

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

subgroupindex
C2×#168115342
#202898592
#202850962
#1359953