Finite Group Databasebeta

← #27277680

Subgroups — #27277680

Order 13568.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277680C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272616C212yesmaximal
4#20272575C212yesmaximal
5C53×#1801C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810530—21no—
8#16810509—22no—
9#16810491C412yes—
10#16810489C2211yescharacteristic
11C4⋊C4×Dic53—21no—
12C4×C4⋊Dic53C2211yescharacteristic
13#16810469C412yes—
14#16810349—22no—
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#423C2212yes—
17C2×#101×C53—21no—
18C53×#343—21no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C4×Dic53—22no—
22C2×C4×Dic53—21no—
23C4⋊#88322—22no—
24C22⋊#88322—22no—
25C4×#88322—22no—
26#13133124—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124C2×C411yescharacteristic
31#13133124C2×C412yes—
32#13133124C2×C411yescharacteristic
33C22×Q8×C53D411yescharacteristic
34C2×M16×C53—22no—
35C4×Q8×C53—22no—
36C4×Q8×C53—22no—
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C42×C53C2×C412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—22no—
42C4⋊C8×C53—22no—
43#101×C53—42no—
44C2×Q8×C53—22no—
45C2×Q8×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—21no—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D411yescharacteristic
52C22×C4×C53C2×D411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—22no—
55C22×Dic53—21no—
56M16×C53—42no—
57M16×C53—42no—
58C2×C8×C53—42no—
59C4⋊C4×C53—24no—
60C4⋊C4×C53—22no—
61C42×C53—22no—
62C42×C53—22no—
63C42×C53#4112yes—
64C42×C53#4112yes—
65C4⋊Dic53—42no—
66C4⋊Dic53—41no—
67C4⋊Dic53—41no—
68C4×Dic53—44no—
69C4×Dic53—42no—
70C2×#88322—42no—
71C23×C53C2×C22⋊C411yescharacteristic
72Q8×C53—42no—
73Q8×C53—42no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—42no—
81C2×C4×C53#11711yescharacteristic
82C2×C4×C53#11711yescharacteristic
83C2×C4×C53#12312yes—
84C2×C4×C53#12312yes—
85C2×C4×C53#13512yes—
86C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C53#13711yescharacteristic
88C2×Dic53—22no—
89C2×Dic53—41no—
90C2×Dic53—41no—
91C2×Dic53—41no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—42no—
97C8×C53—82no—
98#88322—82no—
99#8769—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—21no—
102C22×C53#38111yescharacteristic
103C22×C53#41611yescharacteristic
104C22×C53#41711yescharacteristic
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—42no—
109C4×C53—42no—
110C4×C53#48212yes—
111C4×C53#48512yes—
112Dic53—42no—
113Dic53—81no—
114Dic53—82no—
115#2356—531no—
116#1801D5311yescharacteristic
117#1461—531no—
118#1303—532no—
119#1286—532no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#182511yescharacteristic
123C2×C53#182711yescharacteristic
124C2×C53#183011yescharacteristic
125C2×C4×Q8C2×D5311yescharacteristic
126#431—532no—
127#423C2×D5312yes—
128C2×#101—21no—
129#391—1061no—
130#387—1062no—
131#383—532no—
132#382—531no—
133C4×C4⋊C4—531no—
134C4×C4⋊C4—1061no—
135#376—532no—
136#343—21no—
137#324—1062no—
138C53#876911yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5311yescharacteristic
140C2×M16—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—1061no—
145C2×C4⋊C4D21212yes—
146C2×C4⋊C4C22⋊D5311yescharacteristic
147C2×C42—1061no—
148C2×C42—1062no—
149C2×C42C4×D5312yes—
150C2×C42C22×D5311yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—1062no—
154#101—42no—
155C22⋊C8—1062no—
156C4×C8—1062no—
157#95—531no—
158#95—532no—
159#95—531no—
160#95—1061no—
161#95—1062no—
162#95—1062no—
163#95—1062no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—1061no—
169C22×C4—1062no—
170C22×C4—1061no—
171C22×C4#1171992112yes—
172C22×C4#1171992112yes—
173C22×C4C2×C4×D5311yescharacteristic
174C22×C4C2×D21211yescharacteristic
175C22×C4C2×C22⋊D5311yescharacteristic
176M16—42no—
177M16—42no—
178C2×C8—42no—
179C2×C8—2122no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—2122no—
183C4⋊C4—2121no—
184C4⋊C4—2121no—
185C42—22no—
186C42—22no—
187C42—2124no—
188C42—2122no—
189C42#1171994712yes—
190C42#1171994712yes—
191C23C2×#1171992111yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—1062no—
202C2×C4—2122no—
203C2×C4—2121no—
204C2×C4—2122no—
205C2×C4—2121no—
206C2×C4—2121no—
207C2×C4—2122no—
208C2×C4—2121no—
209C2×C4—2121no—
210C2×C4#1313317311yescharacteristic
211C2×C4#1313317311yescharacteristic
212C2×C4#1313319812yes—
213C2×C4#1313320112yes—
214C2×C4#1313323712yes—
215C2×C4#1313324311yescharacteristic
216C2×C4#1313324811yescharacteristic
217C8—82no—
218C8—4242no—
219C22—21no—
220C22—21no—
221C22#1681048411yescharacteristic
222C22#1681061711yescharacteristic
223C22#1681062211yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—4242no—
230C4—4241no—
231C4—2122no—
232C4#1681094412yes—
233C4#1681096112yes—
234C2—21no—
235C2—21no—
236C2#2027480711yescharacteristic
237C2#2027482011yescharacteristic
238C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#381
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8769
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
138. C53#8769semidirect

Maximal subgroups

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

subgroupindex
#202776802
#202730392
#202726162
#202725752
C53×#18012
#876953