Finite Group Databasebeta

← #27300099

Subgroups — #27300099

Order 13568.

480 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16810420C212yesmaximal
2#20274711C212yesmaximal, complemented
3#20272842C212yesmaximal, complemented
4C53×#2326C211yesmaximal, characteristic
5#16810647—22no—
6C2×#13133136C2214yes—
7#16810420—24no—
8#16810392—22no—
9C2×#124×C53C2211yescharacteristic
10C2×C4×Q8×C53C2211yescharacteristic
11C2×C4×D4×C53C411yescharacteristic
12C53×#391C2211yescharacteristic
13C53×#390C411yescharacteristic
14C53×#388—22no—
15C53×#385—22no—
16C53×#384C411yescharacteristic
17C53×#376C411yescharacteristic
18C2×#11719917—24no—
19C4⋊#88322—44no—
20#13133136—24no—
21#13133136—24no—
22C22×Q8×C53C2311yescharacteristic
23C22×D4×C53C2×C411yescharacteristic
24C23×C4×C53—21no—
25#124×C53—22no—
26#124×C53D412yes—
27#124×C53Q812yes—
28C4×Q8×C53—42no—
29C4×D4×C53—42no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53D412yes—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53D412yes—
41#95×C53—21no—
42#95×C53—21no—
43#95×C53—22no—
44C24×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C2×D411yescharacteristic
48C2×Q8×C53C2×Q811yescharacteristic
49C2×Q8×C53#4112yes—
50C2×D4×C53—21no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53C4⋊C412yes—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—42no—
58C22×C4×C53—82no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D412yes—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—82no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—82no—
72C42×C53—22no—
73C42×C53—24no—
74C42×C53—21no—
75C42×C53—41no—
76C42×C53C4⋊C412yes—
77#11719917—44no—
78#11719917—44no—
79C2×#88322—44no—
80C23×C53—22no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—41no—
87Q8×C53—41no—
88Q8×C53—42no—
89D4×C53—42no—
90D4×C53—42no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—84no—
109C2×C4×C53#9912yes—
110C2×C4×C53C2×C4⋊C411yescharacteristic
111C2×C4×C53#11711yescharacteristic
112C2×C4×C53#12112yes—
113C2×C4×C53C22⋊Q812yes—
114#88322—84no—
115#12803—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—44no—
119C22×C53—42no—
120C22×C53—82no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53#38011yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53#41011yescharacteristic
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—22no—
131C4×C53—42no—
132C4×C53—42no—
133C4×C53—41no—
134C4×C53—81no—
135C4×C53—81no—
136#2326D5311yescharacteristic
137C2×#355—532no—
138#1801—532no—
139#1394—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—82no—
143C2×C53#179911yescharacteristic
144C2×C53#213311yescharacteristic
145C2×C53#213411yescharacteristic
146C2×#124C2×D5311yescharacteristic
147C2×C4×Q8C2×D5311yescharacteristic
148C2×C4×D4Dic5311yescharacteristic
149#423—1062no—
150C2×#101—534no—
151#391C2×D5311yescharacteristic
152#390Dic5311yescharacteristic
153#388—22no—
154#385—22no—
155#384Dic5311yescharacteristic
156#376Dic5311yescharacteristic
157#355—1064no—
158#343—1062no—
159C53#1280311yescharacteristic, complemented, Sylow
160C22×Q8C22×D5311yescharacteristic
161C22×D4C2×Dic5311yescharacteristic
162C23×C4—21no—
163C2×M16—1064no—
164#124—22no—
165#124Dic10612yes—
166#124D21212yes—
167C4×Q8—42no—
168C4×D4—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic5311yescharacteristic
171C2×C4⋊C4C2×Dic5311yescharacteristic
172C2×C4⋊C4C22⋊D5312yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C42C2×Dic5311yescharacteristic
177C2×C42C2×Dic5311yescharacteristic
178C2×C42C2×Dic5311yescharacteristic
179C2×C42C22⋊D5312yes—
180C4⋊C8—2124no—
181#101—1064no—
182#101—1064no—
183#95—22no—
184#95—21no—
185#95—21no—
186C24—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×Dic10611yescharacteristic
190C2×Q8C2×D21211yescharacteristic
191C2×Q8#1171994512yes—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C4⋊Dic5312yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4C22⋊Dic5311yescharacteristic
202C22×C4C22⋊Dic5312yes—
203C22×C4C22⋊Dic5311yescharacteristic
204C22×C4C22×Dic5311yescharacteristic
205C22×C4C2×C22⋊D5312yes—
206M16—2124no—
207M16—2124no—
208C2×C8—2124no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—24no—
218C42—22no—
219C42—21no—
220C42—41no—
221C42C4⋊Dic5312yes—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23C2×C22⋊Dic5311yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4#1313313212yes—
252C2×C4C2×C4⋊Dic5311yescharacteristic
253C2×C4#1313317611yescharacteristic
254C2×C4#1313319112yes—
255C2×C4#1313319612yes—
256C8—4244no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22#1681048311yescharacteristic
265C22C2×#1313313211yescharacteristic
266C22#1681059711yescharacteristic
267C4—22no—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—81no—
276C4—81no—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#2027470311yescharacteristic
281C2#2027640411yescharacteristic
282C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2326C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Q8×C53C23
hypercenter#2326D53
hypocenterC53#12803
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #20274711C2semidirect
3. #20272842C2semidirect
159. C53#12803semidirect

Maximal subgroups

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

subgroupindex
C2×#168104202
#202747112
#202728422
C53×#23262
#1280353