Finite Group Databasebeta

← #28311609

Subgroups — #28311609

Order 13568.

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 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#20291684C211yesmaximal, characteristic, complemented
2#20290953C211yesmaximal, characteristic, complemented
3C2×#16811519C212yesmaximal, complemented
4#20289766C212yesmaximal, complemented
5#20283414C211yesmaximal, characteristic
6#16811682—24no—
7D53⋊M32—22no—
8C2×#13133291C2212yes—
9C2×#13133291C2212yes—
10#16811520—22no—
11#16811519—24no—
12C2×#13133225C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811134C411yescharacteristic
15#16810698—22no—
16#16810697—22no—
17#16810651C411yescharacteristic
18#16810591C411yescharacteristic
19#16810581C2211yescharacteristic
20C53×#407C411yescharacteristic
21#13133292—24no—
22#13133291—22no—
23#13133291—22no—
24#13133291—22no—
25#13133291D414yescomplemented
26D53⋊C16—24no—
27C2×#11719910—22no—
28C2×#11719910—22no—
29C2×#11719945C2×C411yescharacteristic
30#13133225—22no—
31#13133225—22no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719917—22no—
35C2×#11719917—22no—
36C2×#11719912D412yes—
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2311yescharacteristic
39C2×C8×D53D412yes—
40#13133200C812yes—
41#13133196—21no—
42#13133191—21no—
43#13133186C812yes—
44C4×C22⋊D53—22no—
45#13133176C2×C411yescharacteristic
46#13133146—22no—
47#13133129—22no—
48#13133127—21no—
49C8×Dic53—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×M16×C53C2×C411yescharacteristic
52C2×M16×C53C2×C411yescharacteristic
53C22⋊C8×C53—22no—
54C24×C53C2×C811yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53C2×C811yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C2×C22⋊D53—22no—
62C2×C22⋊D53C2×C812yes—
63#11719945—22no—
64#11719945—22no—
65C2×D212M1611yescharacteristic
66C2×C4×D53C22⋊C412yes—
67C2×Dic106M1611yescharacteristic
68M16×C53—22no—
69M16×C53—22no—
70C2×C8×C53—21no—
71C2×C8×C53C22⋊C412yes—
72C22⋊Dic53—22no—
73C22⋊Dic53C2×C812yes—
74#11719921—22no—
75C4⋊Dic53—21no—
76#11719919—22no—
77C4×Dic53—21no—
78#11719917—41no—
79#11719917—42no—
80#11719917—41no—
81C2×#88322—21no—
82C2×#88322—22no—
83C2×#88322C2×D412yes—
84C2×#88322C2×D411yescharacteristic
85C2×#88322C2×D411yescharacteristic
86C2×#88322#4112yes—
87#11719912—24no—
88C8×D53—22no—
89C8×D53—22no—
90#11719910—24no—
91#11719910—42no—
92C23×C53—21no—
93C23×C53—22no—
94C23×C53—81no—
95C23×C53C22×C811yescharacteristic
96C22×D53C22⋊C812yes—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53C4×D412yes—
104C2×C4×C53C2×M1611yescharacteristic
105C2×C4×C53#13112yes—
106C22⋊D53—24no—
107C22⋊D53—82no—
108C2×Dic53—22no—
109C2×Dic53C22⋊C812yes—
110D212—22no—
111C4×D53—22no—
112C4×D53—22no—
113Dic106—22no—
114C8×C53—22no—
115C8×C53—41no—
116#88322—41no—
117#88322#12011yescharacteristic
118#88322#12112yes—
119#88322#12311yescharacteristic
120#12989—531nomaximal, Sylow
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—82no—
126C22×C53C2×C22⋊C811yescharacteristic
127C22×C53C8×D412yes—
128C2×D53—22no—
129C2×D53—42no—
130C4×C53—41no—
131C4×C53#38611yescharacteristic
132C4×C53#40811yescharacteristic
133C4×C53#43512yes—
134Dic53—22no—
135Dic53—42no—
136#3407—531no—
137#2930—531no—
138#2115—531no—
139C2×#349—532no—
140#1378—532no—
141C2×C53—21no—
142C2×C53—22no—
143C2×C53—81no—
144C2×C53#186911yescharacteristic
145D53—42no—
146C2×#131—531no—
147C22×M16—531no—
148#525—531no—
149#504—1064no—
150C2×M32—532no—
151C2×M32—532no—
152C2×M32—1062no—
153#436—1062no—
154#435—1062no—
155#424—531no—
156#408—531no—
157#407#615711yescharacteristic
158#405—531no—
159#349—1064no—
160#349—1062no—
161C53#1298911yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C23×C4C2×#615711yescharacteristic
164#131—1062no—
165#131—1062no—
166C2×M16—532no—
167C2×M16—1062no—
168C2×M16—1062no—
169C2×M16—1061no—
170C2×M16—1061no—
171C2×M16C2×#615711yescharacteristic
172C2×M16C2×#615711yescharacteristic
173C22×C8—531no—
174C22×C8—532no—
175C22×C8—531no—
176#123—532no—
177C22⋊Q8—1061no—
178#121—1061no—
179#120—532no—
180C4×D4—1062no—
181#117—531no—
182M32—534no—
183M32—1062no—
184M32—1064no—
185M32—1062no—
186M32—1062no—
187C2×C16—1062no—
188C2×C16—1064no—
189C2×C16—1062no—
190C4⋊C8—1062no—
191C22⋊C8—22no—
192C22⋊C8—1062no—
193#97—1061no—
194C4×C8—1061no—
195C24D53⋊C811yescharacteristic
196#41—1062no—
197#41—1062no—
198C2×Q8—531no—
199C2×D4—532no—
200C2×D4—531no—
201C2×D4—1062no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—532no—
207C22×C4D53⋊C811yescharacteristic
208C22×C4C22×#615711yescharacteristic
209M16—22no—
210M16—22no—
211M16—1064no—
212M16—2121no—
213M16—2122no—
214M16—2121no—
215C2×C8—21no—
216C2×C8—531no—
217C2×C8—531no—
218C2×C8—532no—
219C2×C8—532no—
220C2×C8—1062no—
221C2×C8—1062no—
222C2×C8—1061no—
223C2×C8—1062no—
224C2×C8C22⋊#615712yes—
225C4⋊C4—1062no—
226C4⋊C4—1061no—
227C22⋊C4—532no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C42—1061no—
231C16—1064no—
232C16—2122no—
233C23—21no—
234C23—22no—
235C23—81no—
236C23—532no—
237C23C2×D53⋊C811yescharacteristic
238Q8—1062no—
239D4—1064no—
240D4—1062no—
241D4—4242no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—532no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4C2×C22⋊#615711yescharacteristic
252C2×C4D4×#615712yes—
253C2×C4D53⋊M1611yescharacteristic
254C2×C4#1313330412yes—
255C8—22no—
256C8—41no—
257C8—531no—
258C8—532no—
259C8—531no—
260C8—2121no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—82no—
266C22—1062no—
267C22—2122no—
268C22D53⋊(C22⋊C8)11yescharacteristic
269C22#1681165012yes—
270C4—41no—
271C4—1062no—
272C4—2122no—
273C4#1681157111yescharacteristic
274C4#1681159711yescharacteristic
275C4#1681165512yes—
276C2—21no—
277C2—22no—
278C2—81no—
279C2—2122no—
280C2#2029068911yescharacteristic
subgroupG/H
centerC4#16811571
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#407C4
socleC2×C53#1869
prime radical#16811134C4
cosocleC22×#88322C23
hypercenter#407#6157
hypocenterC53#12989
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #20291684C2semidirect
2. #20290953C2semidirect
3. C2×#16811519C2semidirect
4. #20289766C2semidirect
25. #13133291D4semidirect
161. C53#12989semidirect

Maximal subgroups

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

subgroupindex
#202916842
#202909532
C2×#168115192
#202897662
#202834142
#1298953