Finite Group Databasebeta

← C4×#16811528 #28310610

Subgroups — C4×#16811528 #28310610

Order 13568.

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

normal subgroups only

This group has 720 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
1C2×#16811528C212yesmaximal
2C4×#13133276C212yesmaximal, complemented
3#20289809C212yesmaximal, complemented
4C4×#13133214C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6C4×C22⋊#6157—22no—
7#16811528C418yesdirect product factor
8#16811501—22no—
9C2×#13133214C2211yescharacteristic
10C2×C4×D212—22no—
11C2×C4×D212C411yescharacteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810525—22no—
14#16810525C411yescharacteristic, complemented
15#16810525C2211yescharacteristic
16#16810493C411yescharacteristic, complemented
17C43×C53C411yescharacteristic, complemented
18C2×C22⋊#6157—24no—
19C2×C4×#6157—42no—
20#13133276—24no—
21#13133276C2×C418yescomplemented
22(C2×#6157)⋊C4—42no—
23C22×D212—22no—
24C22×D212C2×C411yescharacteristic
25C22×D212C2311yescharacteristic
26C22×C4×D53—21no—
27C22×C4×D53D412yes—
28#13133214—22no—
29#13133214C2×C414yes—
30C4×D212—41no—
31C4×D212—41no—
32C4×D212—48no—
33C4×D212—41no—
34C4×D212—41no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—22no—
39C2×#11719921—21no—
40C2×#11719921—22no—
41C2×#11719921D412yes—
42C2×C42×C53—21no—
43C2×C42×C53—21no—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47C23×D53—21no—
48C23×D53C2×D412yes—
49C22×#6157—44no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53C22⋊C412yes—
57C2×D212—28no—
58C2×D212—21no—
59C2×D212—41no—
60C2×D212—41no—
61C2×D212—44no—
62C2×D212—44no—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212C4212yescomplemented
66C2×D212C22×C412yes—
67C2×D212#4112yes—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—44no—
71C2×C4×D53—44no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C22⋊#6157—28no—
75C22⋊#6157—44no—
76C4×#6157—84no—
77C42×C53—24no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—42no—
82C42×C53—41no—
83C42×C53C4212yescomplemented
84#11719921—42no—
85#11719921—44no—
86#11719921—44no—
87#11719921—42no—
88C4⋊Dic53—41no—
89C4⋊Dic53—44no—
90C4⋊Dic53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53C4×D414yes—
102C2×#6157—48no—
103C2×#6157—84no—
104C2×#6157—84no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53C2×C4211yescharacteristic
119C2×C4×C53#11711yescharacteristic
120C2×Dic53—22no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—41no—
124D212—42no—
125D212—42no—
126D212—42no—
127D212—416no—
128D212—42no—
129C4×D53—44no—
130C4×D53—82no—
131C4×D53—82no—
132C4×D53—82no—
133C4×#353—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53C4×C22⋊C411yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53#41011yescharacteristic
139C2×D53—24no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—44no—
143C2×D53—44no—
144C2×D53—42no—
145C2×D53—82no—
146C2×D53—82no—
147C2×D53—81no—
148C2×D53—81no—
149#6157—88no—
150C4×C53—24no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—42no—
155C4×C53—41no—
156C4×C53#35312yescomplemented
157Dic53—42no—
158Dic53—81no—
159C4×#127—531no—
160C2×#353—532no—
161C4×#99—532no—
162#1393—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53C4×#9911yescharacteristic
166C2×C53C2×#35311yescharacteristic
167C2×C53#212611yescharacteristic
168D53—44no—
169D53—82no—
170C2×#127—531no—
171C2×C4×D4—531no—
172C2×C4×D4—531no—
173C2×C4×D4—1062no—
174C2×#99—534no—
175#390—531no—
176#390—531no—
177#390—1062no—
178#384—531no—
179C4×C22⋊C4—1062no—
180C43#615711yescharacteristic, complemented
181#353—538no—
182#342—1062no—
183C53C4×#35311yescharacteristic, complemented, Sylow
184C22×D4—531no—
185C22×D4—531no—
186C22×D4—1062no—
187C23×C4—532no—
188C23×C4—1061no—
189#127—534no—
190#127—1062no—
191C4×D4—2121no—
192C4×D4—2121no—
193C4×D4—2128no—
194C4×D4—2121no—
195C4×D4—2121no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—1062no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—1064no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1061no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—2122no—
207C2×C42C2×#615711yescharacteristic
208C2×C42C2×#615711yescharacteristic
209C2×C42C2×#615711yescharacteristic
210#99—538no—
211#99—1064no—
212#95—2122no—
213C24—532no—
214C24—1061no—
215C2×D4—532no—
216C2×D4—532no—
217C2×D4—532no—
218C2×D4—1068no—
219C2×D4—1061no—
220C2×D4—2121no—
221C2×D4—2121no—
222C2×D4—2121no—
223C2×D4—2121no—
224C2×D4—2124no—
225C2×D4—2124no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—532no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1061no—
232C22×C4—2124no—
233C22×C4—2124no—
234C22×C4—2124no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4C22⋊#615711yescharacteristic
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22×#615711yescharacteristic
240C4⋊C4—2121no—
241C4⋊C4—2121no—
242C4⋊C4—2124no—
243C22⋊C4—1068no—
244C22⋊C4—2122no—
245C22⋊C4—2124no—
246C22⋊C4—2124no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—42no—
255C42—4244no—
256C42C4×#615712yescomplemented
257C23—534no—
258C23—1062no—
259C23—1062no—
260C23—1062no—
261C23—2122no—
262C23—2122no—
263C23—2122no—
264C23—2122no—
265C23—2121no—
266C23—2121no—
267C23C2×C22⋊#615711yescharacteristic
268D4—2122no—
269D4—2122no—
270D4—2122no—
271D4—21216no—
272D4—2122no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—1062no—
285C2×C4—4242no—
286C2×C4—4242no—
287C2×C4—4244no—
288C2×C4—4244no—
289C2×C4—4242no—
290C2×C4—2122no—
291C2×C4—2128no—
292C2×C4—2121no—
293C2×C4—2124no—
294C2×C4—2122no—
295C2×C4#1313327611yescharacteristic
296C2×C4#1313327611yescharacteristic
297C2×C4C2×C4×#615711yescharacteristic
298C2×C4#1313329611yescharacteristic
299C22—21no—
300C22—21no—
301C22—1064no—
302C22—4242no—
303C22—4242no—
304C22—4241no—
305C22—4241no—
306C22—2124no—
307C22—2124no—
308C22—2122no—
309C22—2122no—
310C22—2122no—
311C22C4×C22⋊#615711yescharacteristic
312C22C2×#1313327611yescharacteristic
313C22#1681160211yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—41no—
320C4—4248no—
321C4—4241no—
322C4—2122no—
323C4#1681152812yesdirect product factor
324C2—21no—
325C2—21no—
326C2—4242no—
327C2—2124no—
328C2C4×#1313327611yescharacteristic
329C2C2×#1681152811yescharacteristic
330C2#2029098811yescharacteristic
subgroupG/H
centerC2×C4#13133276
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC43×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133214C22
cosocleC22×D212C23
hypercenterC43#6157
hypocenterC53C4×#353
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
2. C4×#13133276C2semidirect
3. #20289809C2semidirect
7. #16811528C4direct (N × H)
11. C2×C4×D212C4semidirect
14. #16810525C4semidirect
16. #16810493C4semidirect
17. C43×C53C4semidirect
21. #13133276C2×C4semidirect
65. C2×D212C42semidirect
83. C42×C53C42semidirect
156. C4×C53#353semidirect
180. C43#6157semidirect
183. C53C4×#353semidirect
256. C42C4×#6157semidirect
323. C4#16811528direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115282
C4×#131332762
#202898092
C4×#131332142
C4×#35353