Finite Group Databasebeta

← #28301877

Subgroups — #28301877

Order 13568.

708 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C4×#13133284C212yesmaximal, complemented
2C2×#16811481C212yesmaximal
3#20289593C212yesmaximal, complemented
4#20277279C211yesmaximal, characteristic
5C2×#13133284C2214yes—
6C2×C4×#88324—22no—
7#16811489—22no—
8#16811481C418yescomplemented
9C2×#13133213C2211yescharacteristic
10C2×D4×Dic53C411yescharacteristic
11C2×#13133179C411yescharacteristic
12C2×#13133177—22no—
13#16810525C411yescharacteristic
14#16810523—22no—
15#16810492C2211yescharacteristic
16C42×Dic53C2211yescharacteristic
17C53×#390C411yescharacteristic
18C22×#88324—24no—
19#13133284—24no—
20#13133284C2×C418yescomplemented
21C4×#88324—44no—
22C22×D212C2×C411yescharacteristic
23C22×C22⋊D53—22no—
24C23×Dic53—21no—
25C22×C4×D53—21no—
26#13133213—22no—
27#13133213C2×C414yes—
28D4×Dic53—42no—
29#13133179—42no—
30#13133177—44no—
31#13133177—44no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—22no—
34C2×C22⋊Dic53—21no—
35C2×#11719921—22no—
36C2×#11719921—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D412yes—
39C2×C4×Dic53D412yes—
40C2×C4×Dic53C2311yescharacteristic
41C22×D4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C42×C53C2×C411yescharacteristic
45C24×C53—21no—
46C23×D53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53C4212yes—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—28no—
55C2×C22⋊D53—42no—
56C2×C22⋊D53—42no—
57C2×C22⋊D53—44no—
58C22×Dic53—21no—
59C22×Dic53—42no—
60C22×Dic53—44no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D412yes—
64C2×D212—21no—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212C4212yes—
68C2×C4×D53—42no—
69C2×C4×D53—44no—
70C2×#88324—28no—
71C2×#88324—44no—
72C2×#88324—44no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—44no—
75C42×C53—41no—
76C22⋊Dic53—42no—
77#11719921—42no—
78#11719921—44no—
79C4⋊Dic53—41no—
80#11719919—42no—
81#11719919—42no—
82C4×Dic53—22no—
83C4×Dic53—24no—
84C4×Dic53D814yes—
85C4×Dic53SD1614yes—
86C4×Dic53C22×C412yes—
87C4×Dic53#4112yes—
88C23×C53—22no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—22no—
95C22×D53—41no—
96C22×D53—41no—
97C22×D53—42no—
98C22×D53—42no—
99D4×C53—42no—
100D4×C53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53#10214yes—
108C2×C4×C53C2×C4211yescharacteristic
109C2×C4×C53#11711yescharacteristic
110C22⋊D53—48no—
111C22⋊D53—48no—
112C2×Dic53—21no—
113C2×Dic53—22no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—84no—
117C2×Dic53#10012yes—
118C2×Dic53#10214yes—
119C2×Dic53C4×D414yes—
120C2×Dic53C2×D812yes—
121C2×Dic53C2×SD1612yes—
122C2×Dic53#13514yes—
123D212—42no—
124D212—42no—
125C4×D53—84no—
126#88324—48no—
127C4×#331—531nomaximal, Sylow
128C22×C53—42no—
129C22×C53—44no—
130C22×C53—81no—
131C22×C53—82no—
132C22×C53—81no—
133C22×C53C4×C22⋊C411yescharacteristic
134C22×C53C2×#10011yescharacteristic
135C22×C53#41311yescharacteristic
136C22×C53C2×#10212yes—
137C22×C53#41612yes—
138C2×D53—42no—
139C2×D53—44no—
140C2×D53—81no—
141C2×D53—82no—
142C2×D53—81no—
143C4×C53—24no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—81no—
147Dic53—41no—
148Dic53—81no—
149Dic53#33114yescomplemented
150Dic53C4×D814yes—
151Dic53C4×SD1614yes—
152C4×#127—531no—
153C4×C4⋊C8—532no—
154C2×#331—532no—
155#1296—532no—
156C2×C53—82no—
157C2×C53C2×#33111yescharacteristic
158C2×C53#154012yes—
159C2×C53#154311yescharacteristic
160C2×C53C4×#10011yescharacteristic
161C2×C53C4×#10212yes—
162D53—82no—
163C2×#127—531no—
164C2×C4×D4—531no—
165C2×C4×D4—531no—
166C2×C4×D4—1062no—
167C2×C4⋊C8—534no—
168C2×C4×C8—1062no—
169#390—531no—
170#390—1062no—
171#390#615711yescharacteristic
172#384—531no—
173C43—531no—
174#336—1062no—
175#331—538no—
176C53C4×#33111yescharacteristic, complemented, Sylow
177C22×D4—531no—
178C22×D4—1062no—
179C22×D4C2×#615711yescharacteristic
180C23×C4—1061no—
181C23×C4—1061no—
182C22×C8—1064no—
183#127—534no—
184#127—1062no—
185C4×D4—2124no—
186C4×D4—2124no—
187C4×D4—2122no—
188C4×D4—2122no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1062no—
191C2×C4⋊C4C2×#615711yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—1061no—
194C2×C22⋊C4—1061no—
195C2×C22⋊C4—1062no—
196C2×C42—532no—
197C2×C42—531no—
198C2×C42—532no—
199C2×C42—531no—
200C2×C42C2×#615711yescharacteristic
201C4⋊C8—538no—
202C4⋊C8—1064no—
203C4×C8—2124no—
204C24—21no—
205C24—1061no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—532no—
210C2×D4—1068no—
211C2×D4—1061no—
212C2×D4—2121no—
213C2×D4—2121no—
214C2×D4—2122no—
215C2×D4—2122no—
216C2×D4—2124no—
217C2×D4C4×#615712yes—
218C22×C4—21no—
219C22×C4—531no—
220C22×C4—532no—
221C22×C4—531no—
222C22×C4—1061no—
223C22×C4—2124no—
224C22×C4—2122no—
225C22×C4—2124no—
226C22×C4—2122no—
227C22×C4C22⋊#615712yes—
228C22×C4C22×#615711yescharacteristic
229C2×C8—1068no—
230C2×C8—2124no—
231C2×C8—2124no—
232C4⋊C4—41no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C22⋊C4—44no—
237C22⋊C4—2122no—
238C22⋊C4—2122no—
239C22⋊C4—2124no—
240C42—41no—
241C42—534no—
242C42—534no—
243C42—532no—
244C42—532no—
245C42—1062no—
246C42—1064no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—1062no—
253C23—2121no—
254C23—2121no—
255C23—2122no—
256C23—2122no—
257C23C2×C22⋊#615711yescharacteristic
258D4—42no—
259D4—42no—
260D4—2122no—
261D4—2122no—
262D4—2128no—
263D4—2128no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—534no—
271C2×C4—532no—
272C2×C4—534no—
273C2×C4—532no—
274C2×C4—532no—
275C2×C4—534no—
276C2×C4—1061no—
277C2×C4—1062no—
278C2×C4—4244no—
279C2×C4—4244no—
280C2×C4—2121no—
281C2×C4—2122no—
282C2×C4#1313327914yes—
283C2×C4C2×C4×#615711yescharacteristic
284C2×C4#1313329611yescharacteristic
285C8—2128no—
286C22—42no—
287C22—44no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22—4241no—
292C22—4242no—
293C22—4241no—
294C22—2122no—
295C22—2124no—
296C22C4×C22⋊#615711yescharacteristic
297C22D53⋊#10011yescharacteristic
298C22#1681161011yescharacteristic
299C22C2×#1313327912yes—
300C22#1681161312yes—
301C4—21no—
302C4—21no—
303C4—24no—
304C4—81no—
305C4—534no—
306C4—534no—
307C4—534no—
308C4—4241no—
309C4—2121no—
310C2—82no—
311C2—4242no—
312C2D53⋊#33111yescharacteristic
313C2#2029021312yes—
314C2#2029021911yescharacteristic
315C2#2029046511yescharacteristic
316C2C4×#1313327912yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C22
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53C4×#331
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C4×#13133284C2semidirect
3. #20289593C2semidirect
8. #16811481C4semidirect
20. #13133284C2×C4semidirect
149. Dic53#331semidirect
176. C53C4×#331semidirect

Maximal subgroups

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

subgroupindex
C4×#131332842
C2×#168114812
#202895932
#202772792
C4×#33153