Finite Group Databasebeta

← #27271127

Subgroups — #27271127

Order 13568.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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#20274798C211yesmaximal, characteristic
2#20274481C211yesmaximal, characteristic
3C53×#1464C211yesmaximal, characteristic
4#16810587—21no—
5C2×C22⋊#88322—21no—
6C2×#13133128C411yescharacteristic
7C2×C4×#88322C411yescharacteristic
8#16810392—21no—
9#16810392C412yes—
10#16810378—21no—
11C2×#117×C53C2211yescharacteristic
12C53×#432—21no—
13C53×#324—21no—
14C53×#323C412yes—
15C2×#11719917—22no—
16C22×#88322—21no—
17C22×#88322—21no—
18C22⋊#88322—24no—
19C22⋊#88322—41no—
20C22⋊#88322—41no—
21#13133128—41no—
22C4×#88322—41no—
23C23×C4×C53D411yescharacteristic
24#117×C53—41no—
25#117×C53—41no—
26C2×C4⋊C4×C53C2×C411yescharacteristic
27C2×C4⋊C4×C53D411yescharacteristic
28C2×C22⋊C4×C53C2×C411yescharacteristic
29C2×C22⋊C4×C53Q811yescharacteristic
30C2×C42×C53C2×C411yescharacteristic
31C2×C42×C53D411yescharacteristic
32C4⋊C8×C53—41no—
33C22⋊C8×C53—22no—
34#97×C53—41no—
35C24×C53C4⋊C411yescharacteristic
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53—22no—
39C22×C4×C53—41no—
40C22×C4×C53C4211yescharacteristic
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C22×C4×C53C4⋊C411yescharacteristic
45C22×C4×C53C4⋊C411yescharacteristic
46C2×C8×C53—42no—
47C4⋊C4×C53—41no—
48C4⋊C4×C53—41no—
49C22⋊C4×C53—41no—
50C22⋊C4×C53—41no—
51C42×C53—21no—
52C42×C53—21no—
53C42×C53—21no—
54C42×C53—21no—
55#11719917—42no—
56#11719917—42no—
57C2×#88322—22no—
58C2×#88322—22no—
59C2×#88322—41no—
60C2×#88322—41no—
61C2×#88322—41no—
62C2×#88322—41no—
63C2×#88322—42no—
64C23×C53—21no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53#9511yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53#9911yescharacteristic
85C2×C4×C53#9911yescharacteristic
86C2×C4×C53#10412yes—
87C8×C53—82no—
88#88322—42no—
89#88322—42no—
90#88322—82no—
91#7216—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—42no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—81no—
99C22×C53H(3,ℤ/4)11yescharacteristic
100C22×C53#33911yescharacteristic
101C22×C53#34211yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—22no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109#1822—531no—
110#1750—531no—
111#1464D5311yescharacteristic
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—81no—
115C2×C53#128411yescharacteristic
116C2×C53#139211yescharacteristic
117C2×C53#139511yescharacteristic
118C2×#117C2×D5311yescharacteristic
119#432—21no—
120#407—1061no—
121C2×C22⋊C8—1061no—
122C2×#97—531no—
123C2×C4×C8—531no—
124#343—532no—
125#343—1061no—
126#336—1061no—
127#324—21no—
128#323Dic5312yes—
129C53#721611yescharacteristic, complemented, Sylow
130C23×C4D21211yescharacteristic
131C2×M16—1062no—
132C22×C8—1061no—
133C22×C8—1061no—
134#117—41no—
135#117—41no—
136C2×C4⋊C4C4×D5311yescharacteristic
137C2×C4⋊C4C22⋊D5311yescharacteristic
138C2×C22⋊C4Dic10611yescharacteristic
139C2×C22⋊C4C2×Dic5311yescharacteristic
140C2×C42C4×D5311yescharacteristic
141C2×C42C22⋊D5311yescharacteristic
142C4⋊C8—41no—
143C22⋊C8—22no—
144C22⋊C8—1064no—
145C22⋊C8—2121no—
146C22⋊C8—2121no—
147#97—41no—
148#97—2121no—
149C4×C8—2121no—
150C24C4⋊Dic5311yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4C4×Dic5311yescharacteristic
156C22×C4#1171991911yescharacteristic
157C22×C4#1171991911yescharacteristic
158C22×C4#1171992111yescharacteristic
159C22×C4#1171992111yescharacteristic
160C22×C4C22⋊Dic5311yescharacteristic
161M16—2122no—
162M16—2122no—
163C2×C8—42no—
164C2×C8—1062no—
165C2×C8—1062no—
166C2×C8—2122no—
167C2×C8—2121no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8—2121no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23#1313312411yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4#1313313111yescharacteristic
200C2×C4#1313313111yescharacteristic
201C2×C4#1313314312yes—
202C8—82no—
203C8—4242no—
204C8—2122no—
205C8—2122no—
206C22—21no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—81no—
213C22#1681038011yescharacteristic
214C22#1681038411yescharacteristic
215C22#1681038911yescharacteristic
216C4—22no—
217C4—21no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—41no—
223C2—21no—
224C2—21no—
225C2—81no—
226C2#2027256911yescharacteristic
227C2#2027283711yescharacteristic
228C2#2027284311yescharacteristic
subgroupG/H
centerC22#16810389
commutatorC22×C4×C53C42
FrattiniC2×#117C2×D53
FittingC53×#1464C2
socleC22×C53#342
prime radicalC24×C53C4⋊C4
cosocleC2×#117×C53C22
hypercenter#1464D53
hypocenterC53#7216
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
129. C53#7216semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202744812
C53×#14642
#721653