Finite Group Databasebeta

← #28299526

Subgroups — #28299526

Order 13568.

846 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290605C212yesmaximal, complemented
2#20289532C214yesmaximal, complemented
3C2×#16810524C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811489—22no—
6#16811489C2214yescomplemented
7C23×D212C411yescharacteristic
8C22×#11719921—22no—
9C22×C4×Dic53C2211yescharacteristic
10#16810524—22no—
11#16810524C412yes—
12#16810524C2212yes—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—44no—
16C24×D53—21no—
17C22×D212—21no—
18C22×D212—22no—
19C22×D212—42no—
20C22×D212—42no—
21C22×D212C814yescomplemented
22C22×D212C2×C412yes—
23C23×Dic53D412yes—
24C2×#11719921—24no—
25C2×#11719921—24no—
26C2×#11719921—48no—
27C2×C4×Dic53—42no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—42no—
35C2×C4⋊C4×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C23×D53—24no—
38C23×D53—22no—
39C23×D53—44no—
40C23×D53—44no—
41C23×D53—42no—
42C23×D53—42no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—44no—
47C22×C4×C53—42no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C2×C812yes—
50C22×C4×C53M1612yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—24no—
53C22×Dic53—24no—
54C22×Dic53—42no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C2×D412yes—
57C2×D212—24no—
58C2×D212—22no—
59C2×D212—48no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—48no—
63C2×D212—42no—
64C2×D212—42no—
65C2×D212C2×C814yes—
66C2×#88324—44no—
67C2×#88324—44no—
68C2×#88324—44no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71#11719921—48no—
72#11719921—48no—
73C4×Dic53—41no—
74C4×Dic53—42no—
75C4×Dic53—41no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53C22⋊C812yes—
79C23×C53#9912yes—
80C23×C53#10012yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C22×D53—24no—
83C22×D53—416no—
84C22×D53—48no—
85C22×D53—42no—
86C22×D53—42no—
87C22×D53—44no—
88C22×D53—44no—
89C22×D53—84no—
90C22×D53—88no—
91C22×D53—84no—
92C2×C4×C53—24no—
93C2×C4×C53—24no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—44no—
100C2×C4×C53—84no—
101C2×C4×C53C22⋊C812yes—
102C2×C4×C53C22×C811yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—48no—
110C2×Dic53AGL(1,ℤ/8)14yes—
111D212—44no—
112D212—48no—
113D212—44no—
114#88324—84no—
115#7592—531nomaximal, Sylow
116C22×C53—22no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—22no—
120C22×C53#32314yes—
121C22×C53C2×C22⋊C811yescharacteristic
122C22×C53C2×#9911yescharacteristic
123C22×C53C2×#10011yescharacteristic
124C22×C53#41812yes—
125C22×C53#42112yes—
126C2×D53—44no—
127C2×D53—48no—
128C2×D53—816no—
129C2×D53—82no—
130C2×D53—84no—
131C2×D53—82no—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—82no—
136Dic53—24no—
137Dic53—42no—
138C2×#390—531no—
139#1817—532no—
140#1270—534no—
141C2×C53—24no—
142C2×C53C2×#32311yescharacteristic
143C2×C53#147812yes—
144C2×C53#151012yes—
145C2×C53#153311yescharacteristic
146C2×C53#155511yescharacteristic
147D53—84no—
148C23×D4—531no—
149C22×C4⋊C4#615711yescharacteristic
150C22×C22⋊C4—1062no—
151C22×C42—531no—
152C2×C22⋊C8—1062no—
153#390—532no—
154#390—532no—
155#390—1062no—
156#336—534no—
157#336—1062no—
158C53#759211yescharacteristic, complemented, Sylow
159C25—1061no—
160C22×D4—532no—
161C22×D4—534no—
162C22×D4—1062no—
163C22×D4—1061no—
164C22×D4—2122no—
165C22×D4—2122no—
166C23×C4—21no—
167C23×C4—532no—
168C23×C4C2×#615711yescharacteristic
169C22×C8—1064no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×#615712yes—
173C2×C22⋊C4—1064no—
174C2×C22⋊C4—1064no—
175C2×C22⋊C4—2128no—
176C2×C42—532no—
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—2122no—
180C22⋊C8—2124no—
181C24—1064no—
182C24—1062no—
183C24—2124no—
184C24—2124no—
185C24—2122no—
186C24—2122no—
187C24C22⋊#615711yescharacteristic
188C2×D4—534no—
189C2×D4—1064no—
190C2×D4—1062no—
191C2×D4—2122no—
192C2×D4—2122no—
193C2×D4—2128no—
194C2×D4—2122no—
195C2×D4—2122no—
196C2×D4—2128no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—532no—
203C22×C4—532no—
204C22×C4—1064no—
205C22×C4—1064no—
206C22×C4—2122no—
207C22×C4D53⋊C812yes—
208C22×C4#1171993612yes—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22×#615711yescharacteristic
211C2×C8—2124no—
212C2×C8—2124no—
213C2×C8—2124no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—2128no—
217C22⋊C4—2128no—
218C42—2122no—
219C42—2121no—
220C42—2121no—
221C23—22no—
222C23—22no—
223C23—1064no—
224C23—4248no—
225C23—4244no—
226C23—4244no—
227C23—21216no—
228C23—2122no—
229C23—2122no—
230C23—2124no—
231C23—2124no—
232C23—2128no—
233C23#1313327412yes—
234C23#1313327612yes—
235C23#1313327712yes—
236C23C2×C22⋊#615711yescharacteristic
237D4—2124no—
238D4—2128no—
239D4—2124no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—84no—
249C2×C4—534no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—2128no—
256C2×C4#1313327412yes—
257C2×C4C2×D53⋊C811yescharacteristic
258C2×C4C2×#1171993611yescharacteristic
259C8—4244no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—42416no—
265C22—4244no—
266C22—4242no—
267C22—4242no—
268C22—2124no—
269C22—2128no—
270C22#1681146714yes—
271C22C2×#1313327411yescharacteristic
272C22C2×#1313327611yescharacteristic
273C22C2×#1313327711yescharacteristic
274C22#1681161512yes—
275C22#1681162212yes—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—82no—
280C4—1064no—
281C4—2122no—
282C2—24no—
283C2—4244no—
284C2C2×#1681146711yescharacteristic
285C2#2029001612yes—
286C2#2029012212yes—
287C2#2029019611yescharacteristic
288C2#2029024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D212C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7592
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290605C2semidirect
2. #20289532C2semidirect
6. #16811489C22semidirect
21. C22×D212C8semidirect
158. C53#7592semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202895322
C2×#168105242
#759253