Finite Group Databasebeta

← #27276875

Subgroups — #27276875

Order 13568.

456 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 121 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#125×Dic53C211yesmaximal, characteristic
2#20274569C212yesmaximal
3C2×#16810365C211yesmaximal, characteristic
4#20272597C212yesmaximal
5C2×C53×#330C211yesmaximal, characteristic
6C2×#13133146—22no—
7C2×#13133128C2212yes—
8#16810522C412yes—
9#16810509—22no—
10#16810489C2211yescharacteristic
11C4⋊C4×Dic53—22no—
12C4⋊C4×Dic53C412yes—
13C42×Dic53C2211yescharacteristic
14#16810377—22no—
15#16810365—22no—
16C2×#125×C53C2211yescharacteristic
17C2×#97×C53C2212yes—
18C53×#330—22no—
19C22×#88322—22no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C4×Dic53—22no—
23C2×C4×Dic53C2×C411yescharacteristic
24C2×C4×Dic53C2×C411yescharacteristic
25C2×C4×Dic53D412yes—
26C2×C4×Dic53D412yes—
27#13133146—28no—
28#13133128—22no—
29#13133128—22no—
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33C22×C8×C53—22no—
34#125×C53—22no—
35#125×C53D412yes—
36#125×C53Q812yes—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C42×C53C2311yescharacteristic
40#97×C53—22no—
41#97×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53C22×C411yescharacteristic
44C22×C4×C53C2×D412yes—
45C22×Dic53—22no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C22⋊C412yes—
49C2×C8×C53—24no—
50C2×C8×C53—42no—
51C2×C8×C53—42no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—24no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53C4⋊C414yes—
56C42×C53C2×D411yescharacteristic
57C42×C53C2×Q811yescharacteristic
58C42×C53#4112yes—
59C4⋊Dic53—44no—
60C4⋊Dic53—42no—
61C4×Dic53—22no—
62C4×Dic53—24no—
63C4×Dic53—41no—
64C4×Dic53—42no—
65C4×Dic53—44no—
66C2×#88322—24no—
67C2×#88322—42no—
68C2×#88322—42no—
69C23×C53C2×C22⋊C411yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53C2×C4⋊C411yescharacteristic
78C2×C4×C53#11711yescharacteristic
79C2×C4×C53#12112yes—
80C2×C4×C53C22⋊Q812yes—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53#10112yes—
90C2×Dic53#10414yes—
91C8×C53—44no—
92#88322—44no—
93#8584—531nomaximal, Sylow
94C22×C53#38011yescharacteristic
95C22×C53C2×#10111yescharacteristic
96C22×C53#41311yescharacteristic
97C22×C53C2×#10412yes—
98C22×C53#42112yes—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—44no—
104Dic53—24no—
105Dic53—22no—
106Dic53—41no—
107Dic53—82no—
108C4×#125—531no—
109#1767—532no—
110C2×#330—531no—
111C2×#330D5311yescharacteristic
112#1295—532no—
113C2×C53#152611yescharacteristic
114C2×C53#153212yes—
115C2×C53#153411yescharacteristic
116C2×C53#180111yescharacteristic
117C2×C53#180612yes—
118C2×#125C2×D5311yescharacteristic
119C2×C4⋊C8—1062no—
120C2×#97—532no—
121C2×#97C2×D5312yes—
122#389—532no—
123#387—1062no—
124#382—531no—
125C4×C4⋊C4—532no—
126C4×C4⋊C4—1062no—
127C43—531no—
128#336—1062no—
129#330—22no—
130#330—1062no—
131C53#858411yescharacteristic, complemented, Sylow
132C22×C8—22no—
133C22×C8—1062no—
134#125—22no—
135#125Dic10612yes—
136#125C22⋊D5312yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1062no—
140C2×C4⋊C4C4×D5312yes—
141C2×C42—531no—
142C2×C42—532no—
143C2×C42—531no—
144C2×C42—532no—
145C2×C42—1062no—
146C2×C42C22×D5311yescharacteristic
147C4⋊C8—1068no—
148#97—22no—
149#97—22no—
150#97—1062no—
151#97—1062no—
152#95—1062no—
153#95—1062no—
154#95—1062no—
155C22×C4—22no—
156C22×C4—532no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—1062no—
160C22×C4C2×C4×D5311yescharacteristic
161C22×C4D4×D5312yes—
162C2×C8—24no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—1064no—
166C2×C8—2122no—
167C2×C8—2122no—
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—2122no—
172C4⋊C4—2124no—
173C4⋊C4#1171991914yes—
174C42—1064no—
175C42—1062no—
176C42—2121no—
177C42—2122no—
178C42—2124no—
179C42C2×Dic10611yescharacteristic
180C42#1171994712yes—
181C42C2×C22⋊D5311yescharacteristic
182C23C22⋊C4×D5311yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—532no—
191C2×C4—534no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1061no—
196C2×C4—1062no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4C2×#1171991911yescharacteristic
201C2×C4#1313317311yescharacteristic
202C2×C4#1313318912yes—
203C2×C4#1313319412yes—
204C8—44no—
205C8—2124no—
206C22#1681048111yescharacteristic
207C22#101×D5311yescharacteristic
208C22#1681060611yescharacteristic
209C22#104×D5312yes—
210C22#1681063812yes—
211C4—21no—
212C4—21no—
213C4—22no—
214C4—22no—
215C4—44no—
216C4—1064no—
217C4—1062no—
218C4—4242no—
219C4—2121no—
220C2#2027330811yescharacteristic
221C2#2027333412yes—
222C2#2027334011yescharacteristic
223C2#2027470911yescharacteristic
224C2#2027473312yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#330C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#330D53
hypocenterC53#8584
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
131. C53#8584semidirect

Maximal subgroups

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

subgroupindex
#125×Dic532
#202745692
C2×#168103652
#202725972
C2×C53×#3302
#858453