Finite Group Databasebeta

← C4×#16810416 #27299254

Subgroups — C4×#16810416 #27299254

Order 13568.

720 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16810416C212yesmaximal
2C4×#13133132C212yesmaximal, complemented
3#20272840C212yesmaximal, complemented
4C4×#127×C53C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6C4×C22⋊Dic53—22no—
7#16810416C418yesdirect product factor
8#16810390—22no—
9C2×#127×C53C2211yescharacteristic
10C2×C4×D4×C53—22no—
11C2×C4×D4×C53C411yescharacteristic, complemented
12C2×C4×D4×C53C2211yescharacteristic
13C53×#390—22no—
14C53×#390C411yescharacteristic, complemented
15C53×#390C2211yescharacteristic
16C53×#384C411yescharacteristic, complemented
17C43×C53C411yescharacteristic, complemented
18C2×C22⋊Dic53—24no—
19C2×C4×Dic53—42no—
20#13133132—24no—
21#13133132C2×C418yescomplemented
22#13133124—42no—
23C22×D4×C53—22no—
24C22×D4×C53C2×C411yescharacteristic
25C22×D4×C53C2311yescharacteristic
26C23×C4×C53—21no—
27C23×C4×C53D412yes—
28#127×C53—22no—
29#127×C53C2×C414yes—
30C4×D4×C53—41no—
31C4×D4×C53—41no—
32C4×D4×C53—48no—
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53D412yes—
42C2×C42×C53—21no—
43C2×C42×C53—21no—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47C24×C53—21no—
48C24×C53C2×D412yes—
49C2×D4×C53—28no—
50C2×D4×C53—21no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—44no—
54C2×D4×C53—44no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C4212yescomplemented
58C2×D4×C53C22×C412yes—
59C2×D4×C53#4112yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—44no—
66C22×C4×C53—44no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×Dic53—44no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—44no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—44no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—42no—
81C42×C53—24no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—42no—
86C42×C53—41no—
87C42×C53C4212yescomplemented
88C22⋊Dic53—28no—
89C22⋊Dic53—44no—
90C4×Dic53—84no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53C4×D414yes—
102D4×C53—42no—
103D4×C53—42no—
104D4×C53—42no—
105D4×C53—416no—
106D4×C53—42no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53—82no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53C2×C4211yescharacteristic
129C2×C4×C53#11711yescharacteristic
130C2×Dic53—48no—
131C2×Dic53—84no—
132C2×Dic53—84no—
133C4×#353—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—24no—
137C22×C53—44no—
138C22×C53—44no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—82no—
143C22×C53—82no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53C4×C22⋊C411yescharacteristic
147C22×C53C2×#9911yescharacteristic
148C22×C53#41011yescharacteristic
149C4×C53—24no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—42no—
154C4×C53—42no—
155C4×C53—41no—
156C4×C53—81no—
157C4×C53#35312yescomplemented
158Dic53—88no—
159C4×#127D5311yescharacteristic
160C2×#353—532no—
161C4×#99—532no—
162#1393—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—44no—
166C2×C53—82no—
167C2×C53C4×#9911yescharacteristic
168C2×C53C2×#35311yescharacteristic
169C2×C53#212611yescharacteristic
170C2×#127C2×D5311yescharacteristic
171C2×C4×D4—22no—
172C2×C4×D4Dic5311yescharacteristic, complemented
173C2×C4×D4C2×D5311yescharacteristic
174C2×#99—534no—
175#390—22no—
176#390Dic5311yescharacteristic, complemented
177#390C2×D5311yescharacteristic
178#384Dic5311yescharacteristic, complemented
179C4×C22⋊C4—1062no—
180C43Dic5311yescharacteristic, complemented
181#353—538no—
182#342—1062no—
183C53C4×#35311yescharacteristic, complemented, Sylow
184C22×D4—22no—
185C22×D4C2×Dic5311yescharacteristic
186C22×D4C22×D5311yescharacteristic
187C23×C4—21no—
188C23×C4C22⋊D5312yes—
189#127—22no—
190#127C4×D5314yes—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—48no—
194C4×D4—41no—
195C4×D4—41no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—1064no—
203C2×C22⋊C4C22⋊D5312yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—2122no—
207C2×C42C2×Dic5311yescharacteristic
208C2×C42C2×Dic5311yescharacteristic
209C2×C42C2×Dic5311yescharacteristic
210#99—538no—
211#99—1064no—
212#95—2122no—
213C24—21no—
214C24C2×C22⋊D5312yes—
215C2×D4—28no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4C4×Dic5312yescomplemented
224C2×D4C2×C4×D5312yes—
225C2×D4#1171994512yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—2124no—
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22⋊Dic5312yes—
238C22×C4C22⋊Dic5311yescharacteristic
239C22×C4C22×Dic5311yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—1068no—
248C22⋊C4—2124no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—42no—
255C42—4244no—
256C42C4×Dic5312yescomplemented
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic5311yescharacteristic
267C23C4×C22⋊D5314yes—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—416no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—4244no—
293C2×C4—4244no—
294C2×C4—2128no—
295C2×C4#1313313211yescharacteristic
296C2×C4#1313313211yescharacteristic
297C2×C4C2×C4×Dic5311yescharacteristic
298C2×C4#1313317611yescharacteristic
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22C4×C22⋊Dic5311yescharacteristic
312C22C2×#1313313211yescharacteristic
313C22#1681059711yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—81no—
322C4—4248no—
323C4#1681041612yesdirect product factor
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2C4×#1313313211yescharacteristic
329C2C2×#1681041611yescharacteristic
330C2#2027638011yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#127×C53C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenterC4×#127D53
hypocenterC53C4×#353
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C4×#13133132C2semidirect
3. #20272840C2semidirect
7. #16810416C4direct (N × H)
11. C2×C4×D4×C53C4semidirect
14. C53×#390C4semidirect
16. C53×#384C4semidirect
17. C43×C53C4semidirect
21. #13133132C2×C4semidirect
57. C2×D4×C53C42semidirect
87. C42×C53C42semidirect
157. C4×C53#353semidirect
172. C2×C4×D4Dic53semidirect
176. #390Dic53semidirect
178. #384Dic53semidirect
180. C43Dic53semidirect
183. C53C4×#353semidirect
223. C2×D4C4×Dic53semidirect
256. C42C4×Dic53semidirect
323. C4#16810416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168104162
C4×#131331322
#202728402
C4×#127×C532
C4×#35353