Finite Group Databasebeta

← #28299677

Subgroups — #28299677

Order 13568.

520 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290500C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289531C212yesmaximal
4#124×Dic53C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×#13133272—22no—
7#16811489C2214yes—
8C2×Q8×Dic53C411yescharacteristic
9C2×D4×Dic53C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810526C2211yescharacteristic
12#16810516—22no—
13#16810489C411yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C2211yescharacteristic
16C2×#124×C53C411yescharacteristic
17C22×#88324—24no—
18#13133284—28no—
19#13133272—44no—
20C23×Dic53—21no—
21Q8×Dic53—42no—
22D4×Dic53—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D412yes—
31C2×C4×Dic53D412yes—
32C2×C4×Dic53C2311yescharacteristic
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53C2×C411yescharacteristic
37#124×C53—22no—
38#124×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—21no—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53M1612yes—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53M1612yes—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53—42no—
56C22×Dic53—44no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D412yes—
60C2×#88324—28no—
61C2×#88324—44no—
62C2×#88324—44no—
63C22⋊C4×C53—28no—
64C42×C53—21no—
65C42×C53—21no—
66C22⋊Dic53—42no—
67C22⋊Dic53—42no—
68C22⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4×Dic53—22no—
73C4×Dic53—24no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—41no—
77C4×Dic53—42no—
78C23×C53—22no—
79C23×C53—41no—
80C23×C53—41no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84Q8×C53—42no—
85Q8×C53—42no—
86D4×C53—42no—
87D4×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53C2×M1611yescharacteristic
97C2×C4×C53C2×M1611yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—84no—
107C2×Dic53#10012yes—
108C2×Dic53#10414yes—
109C2×Dic53AGL(1,ℤ/8)12yes—
110C2×Dic53#13712yes—
111#88324—48no—
112#7635—531nomaximal, Sylow
113C22×C53—42no—
114C22×C53—44no—
115C22×C53—81no—
116C22×C53—82no—
117C22×C53—81no—
118C22×C53#40711yescharacteristic
119C22×C53#41011yescharacteristic
120C22×C53C2×#10011yescharacteristic
121C22×C53#41712yes—
122C22×C53C2×#10412yes—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—42no—
127Dic53—22no—
128Dic53—22no—
129Dic53—41no—
130Dic53—81no—
131Dic53—81no—
132Dic53#32914yes—
133C4×#124—531no—
134#1767—532no—
135#1272—532no—
136#1270—532no—
137C2×C53—82no—
138C2×C53#146611yescharacteristic
139C2×C53#148912yes—
140C2×C53#150912yes—
141C2×C53C2×#32911yescharacteristic
142C2×C53#156011yescharacteristic
143C2×#124#615711yescharacteristic
144C2×C4×Q8—531no—
145C2×C4×D4—531no—
146C2×C4⋊C8—1062no—
147C2×#97—1062no—
148#391—531no—
149#390—531no—
150#388—1062no—
151#382—531no—
152C4×C22⋊C4—1062no—
153C43—531no—
154#336—534no—
155C53#763511yescharacteristic, complemented, Sylow
156C22×Q8C2×#615711yescharacteristic
157C22×D4C2×#615711yescharacteristic
158C23×C4—1061no—
159C22×C8—1064no—
160#124—22no—
161#124—22no—
162C4×Q8—2122no—
163C4×D4—2122no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—1061no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1061no—
169C2×C22⋊C4—1062no—
170C2×C42—532no—
171C2×C42—532no—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42—1061no—
175C2×C42C2×#615711yescharacteristic
176C4⋊C8—1068no—
177#97—2124no—
178#95—1062no—
179#95—1062no—
180C24—21no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#1171994012yes—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4#1171994012yes—
189C22×C4—21no—
190C22×C4—531no—
191C22×C4—532no—
192C22×C4—531no—
193C22×C4—1061no—
194C22×C4—1061no—
195C22×C4—2122no—
196C22×C4—2124no—
197C22×C4C22⋊#615712yes—
198C22×C4C22×#615711yescharacteristic
199C2×C8—1068no—
200C2×C8—2124no—
201C2×C8—2124no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2122no—
205C22⋊C4—28no—
206C22⋊C4—2122no—
207C22⋊C4—2122no—
208C22⋊C4—2122no—
209C42—21no—
210C42—21no—
211C42—1062no—
212C42—1064no—
213C42—1062no—
214C42—1062no—
215C42—2121no—
216C42—2122no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23C2×C22⋊#615711yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—532no—
236C2×C4—534no—
237C2×C4—532no—
238C2×C4—532no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—1062no—
243C2×C4—4244no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4C2×#1171994011yescharacteristic
249C2×C4C2×#1171994011yescharacteristic
250C8—2128no—
251C22—44no—
252C22—42no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22#1681159311yescharacteristic
257C22#1681160311yescharacteristic
258C22D53⋊#10011yescharacteristic
259C22#1681161412yes—
260C22D53⋊#10412yes—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—534no—
266C4—1062no—
267C4—1062no—
268C4—4241no—
269C4—4241no—
270C4—2121no—
271C2—82no—
272C2#2028997511yescharacteristic
273C2#2029005012yes—
274C2#2029011612yes—
275C2D53⋊#32911yescharacteristic
276C2#2029025311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#7635
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20290500C2semidirect
2. #20289535C2semidirect
155. C53#7635semidirect

Maximal subgroups

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

subgroupindex
#202905002
#202895352
#202895312
#124×Dic532
#763553