Finite Group Databasebeta

← #27299278

Subgroups — #27299278

Order 13568.

464 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16810422C212yesmaximal
2C4×#13133136C212yesmaximal
3#20272842C212yesmaximal
4C53×#2327C211yesmaximal, characteristic
5C2×#13133136C2214yes—
6C4×#11719917—22no—
7#16810422C418yescomplemented
8#16810392—22no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×C4×Q8×C53C411yescharacteristic
11C2×C4×Q8×C53C2211yescharacteristic
12C53×#391C411yescharacteristic
13C53×#391C2211yescharacteristic
14C53×#387—22no—
15C53×#387—22no—
16C53×#382C411yescharacteristic
17C53×#376C411yescharacteristic
18C2×#11719917—24no—
19#13133136—24no—
20#13133136C2×C418yes—
21#13133128—42no—
22C4×#88322—42no—
23C22×Q8×C53C2×C411yescharacteristic
24C22×Q8×C53C2311yescharacteristic
25C4⋊Q8×C53—22no—
26C4⋊Q8×C53C2×C414yes—
27C4×Q8×C53—42no—
28C4×Q8×C53—42no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53D412yes—
34C2×C42×C53—21no—
35C2×C42×C53C2×C411yescharacteristic
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53D412yes—
39#95×C53—21no—
40#95×C53—21no—
41#95×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C4212yes—
50C2×Q8×C53C22×C412yes—
51C2×Q8×C53#4112yes—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D412yes—
61C4⋊C4×C53—28no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C42×C53—22no—
67C42×C53—22no—
68C42×C53—22no—
69C42×C53—21no—
70C42×C53—42no—
71C42×C53—41no—
72C42×C53C4212yes—
73#11719917—28no—
74#11719917—44no—
75C2×#88322—44no—
76C23×C53C2×C22⋊C411yescharacteristic
77Q8×C53—42no—
78Q8×C53—42no—
79Q8×C53—42no—
80Q8×C53—42no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53#9912yes—
102C2×C4×C53C2×C4211yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×C4×C53C4×D414yes—
105#88322—48no—
106#12631—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53C4×C22⋊C411yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53#41011yescharacteristic
112C4×C53—22no—
113C4×C53—24no—
114C4×C53—21no—
115C4×C53—21no—
116C4×C53—42no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53—81no—
121C4×C53—81no—
122#2327D5311yescharacteristic
123C2×#356—532no—
124C4×#101—532no—
125#1394—532no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53C4×#9911yescharacteristic
129C2×C53#213311yescharacteristic
130C2×C53#213511yescharacteristic
131C2×C4⋊Q8C2×D5311yescharacteristic
132C2×C4×Q8Dic5311yescharacteristic
133C2×C4×Q8C2×D5311yescharacteristic
134C2×#101—534no—
135C4×M16—1062no—
136#391Dic5311yescharacteristic
137#391C2×D5311yescharacteristic
138#387—22no—
139#387—22no—
140#382Dic5311yescharacteristic
141#376Dic5311yescharacteristic
142#356—538no—
143#343—1062no—
144C53#1263111yescharacteristic, complemented, Sylow
145C22×Q8C2×Dic5311yescharacteristic
146C22×Q8C22×D5311yescharacteristic
147C2×M16—1064no—
148C4⋊Q8—22no—
149C4⋊Q8C4×D5314yes—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×Dic5311yescharacteristic
155C2×C4⋊C4C2×Dic5311yescharacteristic
156C2×C4⋊C4C22⋊D5312yes—
157C2×C42—21no—
158C2×C42C2×Dic5311yescharacteristic
159C2×C42C2×Dic5311yescharacteristic
160C2×C42C2×Dic5311yescharacteristic
161C2×C42C22⋊D5312yes—
162#101—538no—
163#101—1064no—
164#97—2122no—
165C4×C8—2122no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—21no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C4×Dic5312yes—
177C2×Q8C2×C4×D5312yes—
178C2×Q8#1171994512yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4C22⋊Dic5312yes—
184C22×C4C22⋊Dic5311yescharacteristic
185C22×C4C22⋊Dic5311yescharacteristic
186C22×C4C22×Dic5311yescharacteristic
187C22×C4C2×C22⋊D5312yes—
188M16—1068no—
189M16—2124no—
190C2×C8—2124no—
191C4⋊C4—28no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—41no—
201C42—42no—
202C42C4×Dic5312yes—
203C23C2×C22⋊Dic5311yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4#1313313212yes—
229C2×C4C2×C4×Dic5311yescharacteristic
230C2×C4#1313317611yescharacteristic
231C2×C4C4×C22⋊D5314yes—
232C8—2128no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊Dic5311yescharacteristic
236C22C2×#1313313211yescharacteristic
237C22#1681059711yescharacteristic
238C4—24no—
239C4—21no—
240C4—21no—
241C4—22no—
242C4—42no—
243C4—42no—
244C4—41no—
245C4—42no—
246C4—81no—
247C4—81no—
248C2—21no—
249C2—21no—
250C2C4×#1313313211yescharacteristic
251C2#2027640411yescharacteristic
252C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC53×#2327C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2327D53
hypocenterC53#12631
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
7. #16810422C4semidirect
144. C53#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
C4×#131331362
#202728422
C53×#23272
#1263153