Finite Group Databasebeta

← C2×#20273045 #27349933

Subgroups — C2×#20273045 #27349933

Order 13568.

1718 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810587C212yesmaximal
2#20273045C218yesmaximal, direct product factor
3C2×#16810349C214yesmaximal, complemented
4C23×C4⋊C4×C53C211yesmaximal, characteristic
5C22×#11719917—22no—
6#16810587—24no—
7#16810587C2218yes—
8C2×C22⋊#88322—22no—
9C2×C22⋊#88322C2214yes—
10#16810349C22116yescomplemented
11C24×C4×C53C411yescharacteristic
12C24×C4×C53C411yescharacteristic
13C24×C4×C53C2211yescharacteristic
14C22×C4⋊C4×C53—22no—
15C22×C4⋊C4×C53—22no—
16C22×C4⋊C4×C53—24no—
17C22×C4⋊C4×C53C414yes—
18C22×C4⋊C4×C53C412yes—
19C22×C4⋊C4×C53C2212yes—
20C22×C4⋊C4×C53C2214yes—
21C2×#11719917—28no—
22C2×#11719917—48no—
23C22×#88322—24no—
24C22⋊#88322—28no—
25C22⋊#88322—24no—
26C22⋊#88322C2318yes—
27C25×C53C2×C411yescharacteristic
28C23×C4×C53—22no—
29C23×C4×C53—21no—
30C23×C4×C53—21no—
31C23×C4×C53—28no—
32C23×C4×C53—22no—
33C23×C4×C53—21no—
34C23×C4×C53—21no—
35C23×C4×C53—44no—
36C23×C4×C53—44no—
37C23×C4×C53C2×C414yes—
38C23×C4×C53C2×C412yes—
39C23×C4×C53C2×C414yes—
40C23×C4×C53C2×C412yes—
41C23×C4×C53C2×C412yes—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53D412yes—
44C23×C4×C53D412yes—
45C23×C4×C53D414yes—
46C23×C4×C53C2311yescharacteristic
47C23×C4×C53C2312yes—
48C2×C4⋊C4×C53—28no—
49C2×C4⋊C4×C53—24no—
50C2×C4⋊C4×C53—28no—
51C2×C4⋊C4×C53—24no—
52C2×C4⋊C4×C53—24no—
53C2×C4⋊C4×C53—28no—
54C2×C4⋊C4×C53—24no—
55C2×C4⋊C4×C53—28no—
56C2×C4⋊C4×C53C2×C414yes—
57C2×C4⋊C4×C53C2×C418yes—
58C2×C4⋊C4×C53C2314yes—
59C24×C53—24no—
60C24×C53—24no—
61C24×C53—24no—
62C24×C53C22⋊C412yes—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22×C412yes—
66C24×C53C22×C411yescharacteristic
67C22×C4×C53—216no—
68C22×C4×C53—28no—
69C22×C4×C53—28no—
70C22×C4×C53—28no—
71C22×C4×C53—24no—
72C22×C4×C53—24no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—24no—
76C22×C4×C53—216no—
77C22×C4×C53—22no—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—22no—
81C22×C4×C53—44no—
82C22×C4×C53—42no—
83C22×C4×C53—44no—
84C22×C4×C53—48no—
85C22×C4×C53—42no—
86C22×C4×C53—44no—
87C22×C4×C53—48no—
88C22×C4×C53—44no—
89C22×C4×C53—44no—
90C22×C4×C53—48no—
91C22×C4×C53—44no—
92C22×C4×C53C22⋊C414yes—
93C22×C4×C53C22⋊C412yes—
94C22×C4×C53C22⋊C412yes—
95C22×C4×C53C22⋊C414yes—
96C22×C4×C53M1618yescomplemented
97C22×C4×C53C22×C414yes—
98C22×C4×C53C22×C414yes—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C22×C412yes—
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D414yes—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D414yes—
105C22×C4×C53C2411yescharacteristic
106C4⋊C4×C53—28no—
107C4⋊C4×C53—216no—
108C4⋊C4×C53—28no—
109#11719917—416no—
110C2×#88322—28no—
111C2×#88322—48no—
112C23×C53—232no—
113C23×C53—24no—
114C23×C53—24no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—24no—
118C23×C53—22no—
119C23×C53—24no—
120C23×C53—22no—
121C23×C53—24no—
122C23×C53—22no—
123C23×C53—24no—
124C23×C53—22no—
125C23×C53#9914yes—
126C23×C53#10114yes—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53C2×C22⋊C412yes—
129C23×C53C2×C22⋊C412yes—
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53C2×M1614yes—
132C23×C53C23×C411yescharacteristic
133C2×C4×C53—216no—
134C2×C4×C53—216no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—28no—
138C2×C4×C53—28no—
139C2×C4×C53—28no—
140C2×C4×C53—24no—
141C2×C4×C53—24no—
142C2×C4×C53—28no—
143C2×C4×C53—24no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—44no—
150C2×C4×C53—41no—
151C2×C4×C53—44no—
152C2×C4×C53—48no—
153C2×C4×C53—48no—
154C2×C4×C53—44no—
155C2×C4×C53—44no—
156C2×C4×C53C2×C22⋊C412yes—
157C2×C4×C53C2×C22⋊C414yes—
158C2×C4×C53C2×M1618yescomplemented
159C2×C4×C53C22×D412yes—
160#88322—48no—
161C2×#1463—531nomaximal, Sylow
162C22×C53—216no—
163C22×C53—216no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—22no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—28no—
173C22×C53—28no—
174C22×C53—28no—
175C22×C53#40712yes—
176C22×C53#40712yes—
177C22×C53C2×#9914yes—
178C22×C53C2×#9912yes—
179C22×C53C2×#10114yes—
180C22×C53C2×#10112yes—
181C22×C53C22×C22⋊C411yescharacteristic
182C22×C53C22×M1612yes—
183C4×C53—28no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—24no—
187C4×C53—44no—
188C23×C4⋊C4D5311yescharacteristic
189C2×#407—532no—
190#1463—538no—
191C2×#324—534no—
192C2×C53—22no—
193C2×C53—22no—
194C2×C53—24no—
195C2×C53—24no—
196C2×C53#146314yescomplemented
197C2×C53C2×#40711yescharacteristic
198C2×C53C22×#9911yescharacteristic
199C2×C53C22×#10111yescharacteristic
200C24×C4Dic5311yescharacteristic
201C24×C4Dic5311yescharacteristic
202C24×C4C2×D5311yescharacteristic
203C22×M16—1062no—
204C22×C4⋊C4—22no—
205C22×C4⋊C4—22no—
206C22×C4⋊C4—24no—
207C22×C4⋊C4Dic5314yes—
208C22×C4⋊C4Dic5312yes—
209C22×C4⋊C4C2×D5312yes—
210C22×C4⋊C4C2×D5314yes—
211#407—538no—
212#407—1064no—
213C2×C22⋊C8—534no—
214C2×C22⋊C8—1062no—
215#324—5316no—
216C53C2×#146311yescharacteristic, complemented, Sylow
217C25C2×Dic5311yescharacteristic
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—28no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×Dic5314yes—
228C23×C4C2×Dic5312yes—
229C23×C4C2×Dic5314yes—
230C23×C4C2×Dic5312yes—
231C23×C4C2×Dic5312yes—
232C23×C4C2×Dic5311yescharacteristic
233C23×C4C22⋊D5312yes—
234C23×C4C22⋊D5312yes—
235C23×C4C22⋊D5314yes—
236C23×C4C22×D5311yescharacteristic
237C23×C4C22×D5312yes—
238C2×M16—1068no—
239C2×M16—2128no—
240C22×C8—1064no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4C2×Dic5314yes—
250C2×C4⋊C4C2×Dic5318yes—
251C2×C4⋊C4C22×D5314yes—
252C22⋊C8—538no—
253C22⋊C8—1064no—
254C22⋊C8—1068no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24C22⋊Dic5312yes—
259C24C22⋊Dic5311yescharacteristic
260C24C22⋊Dic5311yescharacteristic
261C24C22×Dic5312yes—
262C24C22×Dic5311yescharacteristic
263C22×C4—216no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—216no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4#1171991718yescomplemented
289C22×C4C22⋊Dic5314yes—
290C22×C4C22⋊Dic5312yes—
291C22×C4C22⋊Dic5312yes—
292C22×C4C22⋊Dic5314yes—
293C22×C4C22×Dic5314yes—
294C22×C4C22×Dic5314yes—
295C22×C4C22×Dic5311yescharacteristic
296C22×C4C22×Dic5312yes—
297C22×C4C2×C22⋊D5314yes—
298C22×C4C2×C22⋊D5312yes—
299C22×C4C2×C22⋊D5312yes—
300C22×C4C2×C22⋊D5314yes—
301C22×C4C23×D5311yescharacteristic
302M16—21216no—
303C2×C8—1068no—
304C2×C8—2128no—
305C4⋊C4—216no—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C23—232no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—24no—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—24no—
317C23—22no—
318C23—24no—
319C23—22no—
320C23—24no—
321C23#1313313214yes—
322C23#1313313614yes—
323C23C2×C22⋊Dic5311yescharacteristic
324C23C2×C22⋊Dic5312yes—
325C23C2×C22⋊Dic5312yes—
326C23C2×C22⋊Dic5311yescharacteristic
327C23C2×#1171991714yes—
328C23C23×Dic5311yescharacteristic
329C2×C4—216no—
330C2×C4—216no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4C2×C22⋊Dic5314yes—
353C2×C4C2×C22⋊Dic5312yes—
354C2×C4C2×#1171991718yescomplemented
355C2×C4C22×C22⋊D5312yes—
356C8—2128no—
357C22—216no—
358C22—216no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—28no—
368C22—28no—
369C22—28no—
370C22#1681058712yes—
371C22#1681058712yes—
372C22C2×#1313313214yes—
373C22C2×#1313313212yes—
374C22C2×#1313313614yes—
375C22C2×#1313313612yes—
376C22C22×C22⋊Dic5311yescharacteristic
377C22C22×#1171991712yes—
378C4—28no—
379C4—22no—
380C4—22no—
381C4—24no—
382C4—44no—
383C2—22no—
384C2—24no—
385C2—24no—
386C2—22no—
387C2#2027304514yesdirect product factor
388C2C2×#1681058711yescharacteristic
389C2C22×#1313313211yescharacteristic
390C2C22×#1313313611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC23×C4⋊C4×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenterC23×C4⋊C4D53
hypocenterC53C2×#1463
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20273045C2direct (N × H)
3. C2×#16810349C2semidirect
10. #16810349C22semidirect
96. C22×C4×C53M16semidirect
158. C2×C4×C53C2×M16semidirect
196. C2×C53#1463semidirect
216. C53C2×#1463semidirect
288. C22×C4#11719917semidirect
354. C2×C4C2×#11719917semidirect
387. C2#20273045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105872
#202730452
C2×#168103492
C23×C4⋊C4×C532
C2×#146353