Finite Group Databasebeta

← C2×#20289644 #28307911

Subgroups — C2×#20289644 #28307911

Order 13568.

834 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 120 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×#16811468C212yesmaximal, complemented
2#20289644C214yesmaximal, direct product factor
3C2×#16810525C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811468—22no—
6#16811468C2214yescomplemented
7C23×D212C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C22×#11719921—22no—
10#16810525—22no—
11#16810525C412yes—
12#16810525C2212yes—
13C22×C42×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C24×D53—21no—
18C22×D212—21no—
19C22×D212—22no—
20C22×D212—41no—
21C22×D212—41no—
22C22×D212—42no—
23C22×D212C814yescomplemented
24C22×D212C2×C412yes—
25C23×Dic53D412yes—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719921—24no—
32C2×#11719921—24no—
33C2×#11719921—48no—
34C23×C4×C53—21no—
35C23×C4×C53C2×C411yescharacteristic
36C2×C42×C53—21no—
37C2×C42×C53—42no—
38C2×C42×C53C2×C412yes—
39C24×C53C22⋊C411yescharacteristic
40C23×D53—24no—
41C23×D53—22no—
42C23×D53—44no—
43C23×D53—44no—
44C23×D53—41no—
45C23×D53—41no—
46C23×D53—42no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—24no—
51C22×C4×C53—22no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C2×C812yes—
56C22×C4×C53M1612yes—
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—44no—
61C22×Dic53C22⋊C412yes—
62C22×Dic53C2×D412yes—
63C2×D212—24no—
64C2×D212—22no—
65C2×D212—44no—
66C2×D212—44no—
67C2×D212—42no—
68C2×D212—42no—
69C2×D212—44no—
70C2×D212—44no—
71C2×D212—42no—
72C2×D212—42no—
73C2×D212C2×C814yescomplemented
74C2×#88324—44no—
75C2×#88324—84no—
76C42×C53—42no—
77C42×C53—42no—
78#11719921—48no—
79#11719921—48no—
80C4⋊Dic53—41no—
81C4⋊Dic53—42no—
82C4⋊Dic53—41no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C22⋊C812yes—
86C23×C53#9911yescharacteristic
87C23×C53#9911yescharacteristic
88C23×C53#10012yes—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—24no—
91C22×D53—416no—
92C22×D53—44no—
93C22×D53—44no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—88no—
99C22×D53—82no—
100C22×D53—82no—
101C22×D53—82no—
102C22×D53—82no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—24no—
112C2×C4×C53—48no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53C22⋊C812yes—
116C2×C4×C53C22×C811yescharacteristic
117C2×C4×C53C2×M1611yescharacteristic
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—82no—
124D212—44no—
125D212—48no—
126D212—44no—
127#88324—84no—
128C2×#1318—531nomaximal, Sylow
129C22×C53—24no—
130C22×C53—24no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53#32312yes—
136C22×C53#32312yes—
137C22×C53#35312yes—
138C22×C53#35512yes—
139C22×C53C2×C22⋊C811yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10011yescharacteristic
142C2×D53—48no—
143C2×D53—44no—
144C2×D53—84no—
145C2×D53—88no—
146C2×D53—88no—
147C2×D53—82no—
148C2×D53—82no—
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—44no—
153Dic53—42no—
154Dic53—42no—
155C2×#390—531no—
156C2×#324—532no—
157#1318—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—22no—
161C2×C53#131814yescomplemented
162C2×C53C2×#32311yescharacteristic
163C2×C53C2×#35311yescharacteristic
164C2×C53C2×#35511yescharacteristic
165D53—84no—
166C23×D4—531no—
167C22×C4⋊C4—531no—
168C22×C22⋊C4—1062no—
169C22×C42#615711yescharacteristic
170C2×C22⋊C8—1062no—
171#390—532no—
172#390—532no—
173#390—1062no—
174#324—534no—
175#324—1062no—
176C53C2×#131811yescharacteristic, complemented, Sylow
177C25—1061no—
178C22×D4—532no—
179C22×D4—534no—
180C22×D4—1062no—
181C22×D4—1061no—
182C22×D4—2121no—
183C22×D4—2121no—
184C22×D4—2122no—
185C23×C4—21no—
186C23×C4—532no—
187C23×C4C2×#615711yescharacteristic
188C22×C8—2122no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—2121no—
193C2×C4⋊C4—2121no—
194C2×C22⋊C4—1064no—
195C2×C22⋊C4—1064no—
196C2×C22⋊C4—2128no—
197C2×C42—21no—
198C2×C42—42no—
199C2×C42C2×#615712yes—
200C22⋊C8—1064no—
201C22⋊C8—2122no—
202C24—1064no—
203C24—1062no—
204C24—2124no—
205C24—2124no—
206C24—2121no—
207C24—2121no—
208C24—2122no—
209C24C22⋊#615711yescharacteristic
210C2×D4—534no—
211C2×D4—1064no—
212C2×D4—1062no—
213C2×D4—2122no—
214C2×D4—2122no—
215C2×D4—2124no—
216C2×D4—2124no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2124no—
220C2×D4—2124no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—2124no—
233C22×C4D53⋊C812yes—
234C22×C4#1171993612yes—
235C22×C4C22⋊#615711yescharacteristic
236C22×C4C22×#615711yescharacteristic
237C2×C8—4244no—
238C2×C8—2124no—
239C4⋊C4—2122no—
240C4⋊C4—2121no—
241C4⋊C4—2121no—
242C22⋊C4—2128no—
243C22⋊C4—2128no—
244C42—42no—
245C42—42no—
246C23—22no—
247C23—22no—
248C23—1064no—
249C23—4248no—
250C23—4242no—
251C23—4242no—
252C23—4242no—
253C23—4242no—
254C23—2124no—
255C23—21216no—
256C23—2124no—
257C23—2122no—
258C23—2122no—
259C23—2124no—
260C23—2124no—
261C23#1313327412yes—
262C23#1313327611yescharacteristic
263C23#1313327611yescharacteristic
264C23#1313327712yes—
265C23C2×C22⋊#615711yescharacteristic
266D4—2124no—
267D4—2128no—
268D4—2124no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—48no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—1062no—
282C2×C4—1062no—
283C2×C4—4242no—
284C2×C4—2124no—
285C2×C4—2122no—
286C2×C4—2122no—
287C2×C4#1313327412yes—
288C2×C4C2×D53⋊C811yescharacteristic
289C2×C4C2×#1171993611yescharacteristic
290C8—4244no—
291C22—24no—
292C22—24no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—4244no—
298C22—4248no—
299C22—4248no—
300C22—4242no—
301C22—4242no—
302C22—2128no—
303C22—2124no—
304C22#1681146712yes—
305C22#1681146712yes—
306C22#1681152812yes—
307C22#1681153212yes—
308C22C2×#1313327411yescharacteristic
309C22C2×#1313327611yescharacteristic
310C22C2×#1313327711yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—44no—
315C4—2122no—
316C4—2122no—
317C2—21no—
318C2—22no—
319C2—21no—
320C2—4244no—
321C2#2028964414yesdirect product factor
322C2C2×#1681146711yescharacteristic
323C2C2×#1681152811yescharacteristic
324C2C2×#1681153211yescharacteristic
subgroupG/H
centerC23#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC23×C53#99
prime radicalC23×D212C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53C2×#1318
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C2×#16811468C2semidirect
2. #20289644C2direct (N × H)
6. #16811468C22semidirect
23. C22×D212C8semidirect
73. C2×D212C2×C8semidirect
161. C2×C53#1318semidirect
176. C53C2×#1318semidirect
321. C2#20289644direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896442
C2×#168105252
C2×#131853