Finite Group Databasebeta

← #28309134

Subgroups — #28309134

Order 13568.

482 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13133278C211yesmaximal, characteristic, complemented
2#20289977C211yesmaximal, characteristic, complemented
3#20289697C214yesmaximal, complemented
4#20279718C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133278—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133204—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810537C412yes—
14#16810537C2212yes—
15#16810526—21no—
16#16810526—21no—
17#16810526—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×#127×C53C411yescharacteristic
20C2×#11719940—22no—
21C4⋊#88324—41no—
22#13133278—24no—
23C22⋊#88324—22no—
24#13133272—41no—
25#13133272—41no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28#13133204—44no—
29C4×Dic106—41no—
30C4×Dic106—41no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919C2×C411yescharacteristic
34C2×#11719919C2311yescharacteristic
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42#13133124C2×C412yes—
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53C2×C412yes—
46#127×C53—41no—
47#127×C53—41no—
48C2×C42×C53C2×C411yescharacteristic
49C24×C53—22no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—42no—
58C2×D4×C53—82no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×Dic106—22no—
68C2×Dic106—22no—
69C2×Dic106—41no—
70#11719940—24no—
71#11719940—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C42×C53—21no—
75C42×C53—21no—
76C22⋊Dic53—44no—
77C22⋊Dic53—44no—
78C22⋊Dic53—82no—
79C22⋊Dic53—82no—
80C4⋊Dic53—41no—
81C4⋊Dic53—42no—
82#11719919—41no—
83#11719919—41no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—82no—
98D4×C53—84no—
99D4×C53—84no—
100D4×C53—82no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#10412yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53AGL(1,ℤ/8)12yes—
122Dic106—44no—
123#88324—44no—
124#88324—82no—
125#11781—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—82no—
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53C2×#9911yescharacteristic
136C22×C53#41811yescharacteristic
137C22×C53C2×#10411yescharacteristic
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—41no—
141C4×C53#35512yes—
142Dic53—22no—
143Dic53—41no—
144Dic53—41no—
145Dic53—41no—
146Dic53—82no—
147#2548—531no—
148C4×#101—531no—
149#1467—531no—
150#1345—534no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—162no—
154C2×C53#150711yescharacteristic
155C2×C53#212211yescharacteristic
156C2×C53C2×#35511yescharacteristic
157C2×#127#615711yescharacteristic
158C2×#124—1061no—
159C2×C4×Q8—531no—
160#432—1061no—
161C2×#101—1061no—
162C4×M16—1061no—
163#394—532no—
164#394—532no—
165#390—1061no—
166#390—1061no—
167#390—1061no—
168C4×C4⋊C4—531no—
169#343—532no—
170#324—532no—
171#324—1061no—
172C53#1178111yescharacteristic, complemented, Sylow
173C22×Q8—531no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×#615712yes—
177C2×M16—1062no—
178#127—41no—
179#127—41no—
180#124—2124no—
181C4×Q8—2121no—
182C4×Q8—2121no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—1061no—
194C2×C42C2×#615711yescharacteristic
195C4⋊C8—2121no—
196#101—1064no—
197C22⋊C8—1062no—
198#97—2121no—
199#97—2121no—
200C4×C8—2121no—
201#95—532no—
202C24—22no—
203C2×Q8—1062no—
204C2×Q8—1062no—
205C2×Q8—2121no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1062no—
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223M16—1064no—
224M16—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—2121no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C4⋊C4—2122no—
231C22⋊C4—4242no—
232C22⋊C4—4242no—
233C22⋊C4—2124no—
234C22⋊C4—2124no—
235C42—21no—
236C42—21no—
237C42—1061no—
238C42—1061no—
239C42—1061no—
240C42—1061no—
241C42—2122no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊#615711yescharacteristic
250Q8—2124no—
251D4—84no—
252D4—82no—
253D4—84no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—532no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4#1313327511yescharacteristic
274C2×C4#1313327511yescharacteristic
275C2×C4#1313328112yes—
276C8—4242no—
277C8—2124no—
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#1313327511yescharacteristic
288C22#1681161511yescharacteristic
289C22C2×#1313328111yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—1062no—
294C4—4242no—
295C4—2121no—
296C4—2121no—
297C4—2121no—
298C4#1681153112yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2#2029011111yescharacteristic
303C2#2029097411yescharacteristic
304C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC22×C53C2×#99
cosocleC2×#11719919C23
hypercenterC2×#127#6157
hypocenterC53#11781
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C4×#13133278C2semidirect
2. #20289977C2semidirect
3. #20289697C2semidirect
172. C53#11781semidirect

Maximal subgroups

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

subgroupindex
C4×#131332782
#202899772
#202896972
#202797182
#1178153