Finite Group Databasebeta

← #28305695

Subgroups — #28305695

Order 13568.

582 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 134 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#20290648C211yesmaximal, characteristic, complemented
2#20290429C211yesmaximal, characteristic, complemented
3C2×Q8⋊#88324C212yesmaximal, complemented
4C2×#16811471C212yesmaximal
5#20277258C211yesmaximal, characteristic
6C2×#13133284—22no—
7C2×C4⋊#88324C2212yes—
8C2×C4×#88324C2212yes—
9#16811489—22no—
10Q8⋊#88324—24no—
11#16811471—24no—
12C2×#13133206C411yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15#16810525C411yescharacteristic
16#16810515—22no—
17#16810488C411yescharacteristic
18#16810470—22no—
19C42×Dic53C2211yescharacteristic
20C53×#391C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23#13133284—44no—
24C4⋊#88324—22no—
25C4⋊#88324—22no—
26C4×#88324—22no—
27C4×#88324D414yes—
28C22×D212C2×C411yescharacteristic
29C22×C4×D53—21no—
30#13133206—22no—
31#13133206C814yescomplemented
32Q8×Dic53—22no—
33Q8×Dic53—22no—
34#13133179—22no—
35#13133179—22no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D412yes—
44C2×C4×Dic53D412yes—
45C2×C4×Dic53C2311yescharacteristic
46#13133124—22no—
47C22×Q8×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—22no—
51C23×D53—21no—
52C2×Q8×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—22no—
55C2×Q8×C53C2×C812yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53C22⋊C412yes—
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×D212—22no—
65C2×D212—22no—
66C2×D212—21no—
67C2×D212C2×C812yes—
68C2×C4×D53—41no—
69C2×C4×D53—41no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72C2×#88324—24no—
73C2×#88324—44no—
74C2×#88324—42no—
75C2×#88324—44no—
76C4⋊C4×C53—22no—
77C42×C53—22no—
78#11719921—28no—
79#11719921—44no—
80#11719921—42no—
81C4⋊Dic53—22no—
82C4⋊Dic53—42no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—42no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53C2×C812yes—
89C4×Dic53M1612yes—
90C4×Dic53C2×D412yes—
91C4×Dic53#4112yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—22no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53—42no—
97C22×D53—42no—
98Q8×C53—24no—
99Q8×C53—42no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53C4×D412yes—
109C2×C4×C53C22×C811yescharacteristic
110C2×C4×C53C2×M1611yescharacteristic
111C2×C4×C53#13112yes—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53C22⋊C814yes—
118C2×Dic53#12112yes—
119C2×Dic53#12411yescharacteristic
120C2×Dic53#12711yescharacteristic
121C2×Dic53AGL(1,ℤ/8)12yes—
122C2×Dic53#13712yes—
123D212—24no—
124D212—42no—
125C4×D53—82no—
126C4×D53—81no—
127C4×D53—81no—
128#88324—44no—
129#88324—82no—
130#9640—531nomaximal, Sylow
131C22×C53#39011yescharacteristic
132C22×C53C2×C22⋊C811yescharacteristic
133C22×C53#40811yescharacteristic
134C22×C53#41712yes—
135C22×C53#42112yes—
136C2×D53—42no—
137C2×D53—44no—
138C2×D53—81no—
139C2×D53—81no—
140C2×D53—82no—
141C4×C53—42no—
142C4×C53—42no—
143C4×C53—81no—
144C4×C53C8×D412yes—
145C4×C53#43612yes—
146Dic53—22no—
147Dic53—24no—
148Dic53—81no—
149Dic53#48912yes—
150Dic53#49611yescharacteristic
151Dic53#49711yescharacteristic
152C4×#124—531no—
153#1842—531no—
154#1744—531no—
155C2×Q8⋊C8—532no—
156C2×D4⋊C8—532no—
157C2×C53#147712yes—
158C2×C53#149512yes—
159C2×C53#195411yescharacteristic
160C2×C53#195911yescharacteristic
161C2×C53#196311yescharacteristic
162D53—82no—
163C2×#124—531no—
164C2×C4×Q8—531no—
165C2×C4×D4—531no—
166C2×C4⋊C8—532no—
167C2×C4⋊C8—1062no—
168C2×C4×C8—532no—
169#391#615711yescharacteristic
170#390—531no—
171#388—1062no—
172#382—531no—
173C4×C22⋊C4—1062no—
174C43—531no—
175#336—1062no—
176Q8⋊C8—1064no—
177D4⋊C8—1064no—
178C53#964011yescharacteristic, complemented, Sylow
179C22×Q8C2×#615711yescharacteristic
180C22×D4—531no—
181C23×C4—1061no—
182C22×C8—1062no—
183C22×C8—1062no—
184#124—534no—
185#124—1062no—
186C4×Q8—1062no—
187C4×Q8—1062no—
188C4×D4—1062no—
189C4×D4—1062no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1061no—
195C2×C22⋊C4—1062no—
196C2×C42—531no—
197C2×C42—531no—
198C2×C42—532no—
199C2×C42—532no—
200C2×C42—1061no—
201C2×C42C2×#615711yescharacteristic
202C4⋊C8—1062no—
203C4⋊C8—1062no—
204C4⋊C8—2124no—
205C4×C8—534no—
206C4×C8—1062no—
207#95—22no—
208#95—1062no—
209C24—1061no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8D53⋊C812yes—
214C2×D4—532no—
215C2×D4—1062no—
216C2×D4—1061no—
217C2×D4—1062no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—531no—
221C22×C4—532no—
222C22×C4—531no—
223C22×C4—1061no—
224C22×C4—2121no—
225C22×C4—2121no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4C22⋊#615712yes—
229C22×C4C22×#615711yescharacteristic
230C2×C8—1064no—
231C2×C8—2122no—
232C2×C8—2124no—
233C2×C8—2124no—
234C4⋊C4—22no—
235C4⋊C4—1062no—
236C4⋊C4—2122no—
237C22⋊C4—1068no—
238C22⋊C4—2124no—
239C22⋊C4—2122no—
240C42—22no—
241C42—532no—
242C42—532no—
243C42—532no—
244C42—532no—
245C42—1062no—
246C42—1062no—
247C42—2122no—
248C42—2121no—
249C42—2121no—
250C23—1062no—
251C23—2121no—
252C23—2121no—
253C23—2122no—
254C23—2122no—
255C23C2×C22⋊#615711yescharacteristic
256Q8—24no—
257Q8—42no—
258D4—1064no—
259D4—2122no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—532no—
271C2×C4—532no—
272C2×C4—532no—
273C2×C4—534no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—1062no—
277C2×C4—4241no—
278C2×C4—4242no—
279C2×C4—4241no—
280C2×C4—2121no—
281C2×C4—2122no—
282C2×C4D4×#615712yes—
283C2×C4C2×D53⋊C811yescharacteristic
284C2×C4D53⋊M1611yescharacteristic
285C2×C4#1313330412yes—
286C8—4242no—
287C8—2124no—
288C22—4242no—
289C22—4241no—
290C22—4241no—
291C22—2122no—
292C22—2124no—
293C22#1681157511yescharacteristic
294C22D53⋊(C22⋊C8)11yescharacteristic
295C22#1681159611yescharacteristic
296C22#1681161412yes—
297C22#1681162212yes—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—531no—
302C4—532no—
303C4—531no—
304C4—1062no—
305C4—1064no—
306C4—4241no—
307C4#1681165012yes—
308C4#1681165612yes—
309C2—4242no—
310C2#2029001312yes—
311C2#2029007212yes—
312C2#2029083811yescharacteristic
313C2#2029084511yescharacteristic
314C2#2029085111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#9640
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20290648C2semidirect
2. #20290429C2semidirect
3. C2×Q8⋊#88324C2semidirect
31. #13133206C8semidirect
178. C53#9640semidirect

Maximal subgroups

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

subgroupindex
#202906482
#202904292
C2×Q8⋊#883242
C2×#168114712
#202772582
#964053