Finite Group Databasebeta

← #27293599

Subgroups — #27293599

Order 13568.

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

normal subgroups only

This group has 270 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#20283648C211yesmaximal, characteristic
2#20276528C212yesmaximal
3#20272668C211yesmaximal, characteristic
4#20272665C212yesmaximal
5C53×#1328C211yesmaximal, characteristic
6#16811176C411yescharacteristic
7#16810752—22no—
8#16810747—22no—
9#16810649C411yescharacteristic
10#16810646C2211yescharacteristic
11#16810581C2211yescharacteristic
12#16810579C411yescharacteristic
13C2×C8×Dic53C411yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#424C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719917C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21#13133216—21no—
22#13133208—21no—
23#13133194—22no—
24C4×Dic106—22no—
25#13133176C2×C411yescharacteristic
26C2×C4×Dic53C2×C411yescharacteristic
27#13133157—22no—
28#13133146—22no—
29#13133146—22no—
30#13133128—21no—
31#13133127D412yes—
32C4×#88322—21no—
33C8×Dic53—21no—
34C8×Dic53—21no—
35C8×Dic53D412yes—
36C2×M16×C53C2311yescharacteristic
37C22×C8×C53C2×C411yescharacteristic
38#117×C53C2×C411yescharacteristic
39M32×C53—22no—
40C4⋊C8×C53—22no—
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—21no—
43C2×Dic106—22no—
44M16×C53—41no—
45C2×C8×C53—21no—
46C2×C8×C53—41no—
47C2×C8×C53C2×D412yes—
48C4⋊C4×C53—21no—
49C22⋊C4×C53M1612yes—
50C42×C53—21no—
51C22⋊Dic53—21no—
52C4⋊Dic53—22no—
53C4⋊Dic53—21no—
54#11719919—22no—
55#11719919—22no—
56C4×Dic53—21no—
57C4×Dic53C22⋊C412yes—
58C4×Dic53M1611yescharacteristic
59C4×Dic53M1611yescharacteristic
60#11719917—41no—
61C2×#88322—22no—
62C2×#88322—21no—
63C2×#88322C22⋊C411yescharacteristic
64C2×#88322C22⋊C411yescharacteristic
65C16×C53—42no—
66#11719908—42no—
67C23×C53C2×M1611yescharacteristic
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53C2×C22⋊C411yescharacteristic
72C2×C4×C53C2×M1611yescharacteristic
73C2×Dic53—21no—
74C2×Dic53—22no—
75C2×Dic53—21no—
76C2×Dic53—42no—
77Dic106—44no—
78C8×C53—22no—
79C8×C53—41no—
80#88322—21no—
81#88322—41no—
82#88322#10112yes—
83#11539—531nomaximal, Sylow
84C22×C53—21no—
85C22×C53—41no—
86C22×C53#40711yescharacteristic
87C4×C53—21no—
88C4×C53—42no—
89C4×C53#41011yescharacteristic
90C4×C53C2×#10111yescharacteristic
91Dic53—22no—
92Dic53—42no—
93Dic53—41no—
94#2960—531no—
95#2166—532no—
96#1328—531no—
97#1328D5311yescharacteristic
98#1327—532no—
99C2×C53—21no—
100C2×C53—41no—
101C2×C53#146711yescharacteristic
102#533—531no—
103#446—1062no—
104C8×Q8—1062no—
105#424—531no—
106#424C2×D5311yescharacteristic
107#423—531no—
108#405—531no—
109#405—531no—
110C2×C4×C8—531no—
111#364—1062no—
112#349—532no—
113#349C2×D5312yes—
114#347—1062no—
115C53#1153911yescharacteristic, complemented, Sylow
116C2×M16—531no—
117C2×M16C22×D5311yescharacteristic
118C22×C8—531no—
119C22×C8C4×D5311yescharacteristic
120C4⋊Q8—1061no—
121#125—1061no—
122C22⋊Q8—1062no—
123C4×Q8—1062no—
124#117—531no—
125#117C4×D5311yescharacteristic
126C2×C42—531no—
127M32—22no—
128M32—1062no—
129C4⋊C8—22no—
130C4⋊C8—1062no—
131C4⋊C8—1062no—
132#97—532no—
133#97—1061no—
134C4×C8—532no—
135C4×C8—1061no—
136C4×C8—1061no—
137C4×C8—1061no—
138C2×Q8—1062no—
139C22×C4—1061no—
140C22×C4C2×C4×D5311yescharacteristic
141M16—41no—
142M16—2121no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—531no—
146C2×C8—531no—
147C2×C8—1062no—
148C2×C8—1061no—
149C2×C8D4×D5312yes—
150C4⋊C4—21no—
151C4⋊C4—1062no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4#1171991212yes—
157C42—21no—
158C42—532no—
159C42—531no—
160C42—531no—
161C42—1061no—
162C16—42no—
163C16—2122no—
164C23C2×#1171991211yescharacteristic
165Q8—2124no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—1062no—
172C2×C4—2122no—
173C2×C4C22⋊C4×D5311yescharacteristic
174C2×C4M16×D5311yescharacteristic
175C8—22no—
176C8—41no—
177C8—532no—
178C8—1061no—
179C8—2121no—
180C22—21no—
181C22—41no—
182C22#1681058511yescharacteristic
183C4—21no—
184C4—42no—
185C4—1062no—
186C4—2121no—
187C4—2122no—
188C4#1681059511yescharacteristic
189C4#101×D5311yescharacteristic
190C2—21no—
191C2—41no—
192C2#2027305711yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1328C2
socleC2×C53#1467
prime radicalC23×C53C2×M16
cosocleC2×M16×C53C23
hypercenter#1328D53
hypocenterC53#11539
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
115. C53#11539semidirect

Maximal subgroups

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

subgroupindex
#202836482
#202765282
#202726682
#202726652
C53×#13282
#1153953