Finite Group Databasebeta

← #27293601

Subgroups — #27293601

Order 13568.

262 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 71 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#20283652C211yesmaximal, characteristic
2#20276528C212yesmaximal
3#20272667C212yesmaximal
4#20272666C211yesmaximal, characteristic
5C53×#1327C211yesmaximal, characteristic
6#16811180C411yescharacteristic
7#16810752—22no—
8#16810752—22no—
9#16810651C411yescharacteristic
10#16810651C2211yescharacteristic
11#16810581C411yescharacteristic
12#16810581C2211yescharacteristic
13C2×#13133128C411yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#423C2211yescharacteristic
18C53×#349C2212yes—
19C22×#88322C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21#13133217—21no—
22#13133209—21no—
23#13133196—22no—
24C4×Dic106—22no—
25#13133176C2×C411yescharacteristic
26#13133176C2×C411yescharacteristic
27#13133157—22no—
28#13133146—22no—
29#13133146—22no—
30#13133128—21no—
31#13133128—21no—
32#13133127—21no—
33#13133127D412yes—
34C8×Dic53—21no—
35C8×Dic53D412yes—
36C2×M16×C53C2×C411yescharacteristic
37C2×M16×C53C2311yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39M32×C53—22no—
40C4⋊C8×C53—22no—
41C22×C4×C53—21no—
42C22×C4×C53C22×C411yescharacteristic
43C2×Dic106—22no—
44M16×C53—41no—
45M16×C53—41no—
46C2×C8×C53—21no—
47C2×C8×C53C2×D412yes—
48C42×C53—21no—
49C42×C53M1611yescharacteristic
50C42×C53M1611yescharacteristic
51C22⋊Dic53—21no—
52C22⋊Dic53M1612yes—
53C4⋊Dic53—22no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56#11719919—24no—
57C4×Dic53—21no—
58C4×Dic53C22⋊C412yes—
59C2×#88322—22no—
60C2×#88322—21no—
61C2×#88322—41no—
62C2×#88322C22⋊C411yescharacteristic
63C2×#88322C22⋊C411yescharacteristic
64C16×C53—42no—
65#11719908—42no—
66C23×C53C2×M1611yescharacteristic
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—81no—
72C2×C4×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53C2×M1611yescharacteristic
74C2×Dic53—22no—
75C2×Dic53—22no—
76Dic106—82no—
77C8×C53—22no—
78C8×C53—41no—
79#88322—21no—
80#88322—41no—
81#88322#10112yes—
82#11540—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—41no—
85C22×C53#40711yescharacteristic
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—41no—
89C4×C53#41011yescharacteristic
90C4×C53C2×#10111yescharacteristic
91Dic53—42no—
92Dic53—42no—
93#2961—531no—
94#2166—532no—
95#1328—532no—
96#1327—531no—
97#1327D5311yescharacteristic
98C2×C53—21no—
99C2×C53—41no—
100C2×C53#146711yescharacteristic
101#533—531no—
102#446—1062no—
103#446—1062no—
104#424—531no—
105#424—531no—
106#423C2×D5311yescharacteristic
107#405—531no—
108#405—531no—
109C2×#97—531no—
110#364—1062no—
111#349—532no—
112#349C2×D5312yes—
113#347—1062no—
114C53#1154011yescharacteristic, complemented, Sylow
115C2×M16C4×D5311yescharacteristic
116C2×M16C22×D5311yescharacteristic
117C22×C8—531no—
118C22×C8—531no—
119C4⋊Q8—1061no—
120#125—1061no—
121C22⋊Q8—1062no—
122C4×Q8—1062no—
123#117—531no—
124#117—531no—
125C2×C42C4×D5311yescharacteristic
126M32—22no—
127M32—1062no—
128C4⋊C8—22no—
129C4⋊C8—1062no—
130C4⋊C8—1062no—
131#97—532no—
132#97—1061no—
133#97—1061no—
134#97—1061no—
135C4×C8—532no—
136C4×C8—1061no—
137C2×Q8—1062no—
138C22×C4—21no—
139C22×C4C2×C4×D5311yescharacteristic
140M16—41no—
141M16—41no—
142C2×C8—21no—
143C2×C8—531no—
144C2×C8—531no—
145C2×C8—1062no—
146C2×C8—1061no—
147C2×C8—2121no—
148C2×C8D4×D5312yes—
149C4⋊C4—1064no—
150C4⋊C4—1062no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C22⋊C4—532no—
154C22⋊C4—1061no—
155C42—21no—
156C42—532no—
157C42—1061no—
158C42#1171991211yescharacteristic
159C42#1171991211yescharacteristic
160C16—42no—
161C16—2122no—
162C23M16×D5311yescharacteristic
163Q8—4242no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—81no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4C22⋊C4×D5311yescharacteristic
172C2×C4C2×#1171991211yescharacteristic
173C8—22no—
174C8—41no—
175C8—532no—
176C8—1061no—
177C8—2121no—
178C22—21no—
179C22—41no—
180C22#1681058511yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—2122no—
185C4—2122no—
186C4#1681059511yescharacteristic
187C4#101×D5311yescharacteristic
188C2—21no—
189C2—41no—
190C2#2027305711yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1327C2
socleC2×C53#1467
prime radicalC23×C53C2×M16
cosocleC2×M16×C53C23
hypercenter#1327D53
hypocenterC53#11540
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
114. C53#11540semidirect

Maximal subgroups

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

subgroupindex
#202836522
#202765282
#202726672
#202726662
C53×#13272
#1154053