Finite Group Databasebeta

← C2×#20273723 #27356321

Subgroups — C2×#20273723 #27356321

Order 13568.

1028 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C212yesmaximal
2C2×#16810553C211yesmaximal, characteristic
3#20273723C218yesmaximal, direct product factor
4C2×#16810360C212yesmaximal
5C2×#16810349C211yesmaximal, characteristic
6C2×C53×#544C211yesmaximal, characteristic
7C22×C4⋊Dic53C2211yescharacteristic
8C22×#11719919—21no—
9#16810951—24no—
10#16810951C2218yes—
11C2×#13133148—22no—
12C2×#13133141—22no—
13C2×C22⋊#88322C2212yes—
14#16810553—22no—
15#16810553C2214yes—
16C2×#13133124—21no—
17C2×#13133124C2211yescharacteristic
18#16810360C2218yes—
19#16810349C2214yes—
20C53×#544—24no—
21C53×#544—22no—
22C53×#544C2214yes—
23C2×C4⋊Q8×C53—22no—
24C2×#125×C53—22no—
25C2×C22⋊Q8×C53—21no—
26C2×C22⋊Q8×C53C2212yes—
27C2×#117×C53—21no—
28C22×C4⋊C4×C53C2211yescharacteristic
29C23×Dic53—21no—
30C23×Dic53—21no—
31C22×#88322—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53C2312yes—
36C2×#11719919—24no—
37C2×#11719919—44no—
38#13133148—28no—
39#13133141—28no—
40C22⋊#88322—22no—
41C22⋊#88322C2314yes—
42#13133124—21no—
43#13133124—24no—
44#13133124C2312yes—
45C22×Q8×C53—21no—
46C22×Q8×C53D412yes—
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C4⋊Q8×C53—28no—
51C4⋊Q8×C53—28no—
52#125×C53—28no—
53#125×C53—28no—
54C22⋊Q8×C53—24no—
55C22⋊Q8×C53—24no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53D414yes—
59C22⋊Q8×C53C2314yes—
60#117×C53—24no—
61#117×C53—24no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D412yes—
71C2×C4⋊C4×C53C2312yes—
72C2×C22⋊C4×C53—22no—
73C2×C42×C53—22no—
74C24×C53C2×D411yescharacteristic
75C2×Q8×C53—24no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×Q8×C53—82no—
79C2×Q8×C53C2×D414yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53SD1614yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D412yes—
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2411yescharacteristic
96C22×Dic53—24no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—28no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53C2×D414yes—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—24no—
117C42×C53—28no—
118C4⋊Dic53—24no—
119C4⋊Dic53—42no—
120#11719919—48no—
121C2×#88322—24no—
122C2×#88322—44no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53#12011yescharacteristic
126C23×C53#12011yescharacteristic
127C23×C53C2×SD1612yes—
128C23×C53#13712yes—
129C23×C53C22×D411yescharacteristic
130Q8×C53—44no—
131Q8×C53—82no—
132Q8×C53—84no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—24no—
138C2×C4×C53—22no—
139C2×C4×C53—24no—
140C2×C4×C53—24no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—44no—
147C2×C4×C53—42no—
148C2×C4×C53—81no—
149C2×C4×C53#12012yes—
150C2×C4×C53C2×SD1614yes—
151C2×C4×C53C22×D412yes—
152C2×Dic53—24no—
153C2×Dic53—48no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161#88322—44no—
162C2×#1611—531nomaximal, Sylow
163C22×C53—24no—
164C22×C53—24no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#44812yes—
170C22×C53#44812yes—
171C22×C53#45612yes—
172C22×C53#45812yes—
173C22×C53C2×#12011yescharacteristic
174C22×C53C22×SD1611yescharacteristic
175C22×C53C2×#13711yescharacteristic
176C4×C53—24no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—44no—
180C4×C53—42no—
181C4×C53—44no—
182Dic53—44no—
183Dic53—82no—
184C2×#544D5311yescharacteristic
185C2×#483—532no—
186C2×#398—531no—
187#1611—538no—
188C2×#328—532no—
189C2×#324—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—22no—
193C2×C53#161114yescomplemented
194C2×C53C2×#44811yescharacteristic
195C2×C53C2×#45611yescharacteristic
196C2×C53C2×#45811yescharacteristic
197#544—24no—
198#544—22no—
199#544C2×D5314yes—
200C2×C4⋊Q8—22no—
201C2×#125—22no—
202C2×C22⋊Q8—21no—
203C2×C22⋊Q8C2×D5312yes—
204C2×#117—21no—
205C22×C4⋊C4—531no—
206C22×C4⋊C4—1061no—
207C22×C4⋊C4C2×D5311yescharacteristic
208#483—538no—
209#483—1064no—
210C2×#106—1062no—
211C2×#103—1062no—
212C2×C22⋊C8—532no—
213#398—534no—
214#398—1062no—
215C2×#95—531no—
216C2×#95—1061no—
217#328—538no—
218#324—534no—
219C53C2×#161111yescharacteristic, complemented, Sylow
220C22×Q8—21no—
221C22×Q8C22⋊D5312yes—
222C23×C4—1061no—
223C23×C4—1061no—
224C23×C4C22⋊D5311yescharacteristic
225C23×C4C22⋊D5311yescharacteristic
226C23×C4C22×D5311yescharacteristic
227C22×C8—1062no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#125—28no—
231#125—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C22⋊D5314yes—
237C22⋊Q8C22×D5314yes—
238#117—24no—
239#117—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—532no—
248C2×C4⋊C4—1062no—
249C2×C4⋊C4—1062no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1064no—
252C2×C4⋊C4—2124no—
253C2×C4⋊C4C22⋊D5312yes—
254C2×C4⋊C4C22⋊D5312yes—
255C2×C4⋊C4C22×D5312yes—
256C2×C22⋊C4—22no—
257C2×C42—22no—
258#106—1068no—
259#103—1068no—
260C22⋊C8—534no—
261C22⋊C8—1062no—
262#95—532no—
263#95—1061no—
264#95—1064no—
265C24C2×C22⋊D5311yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8C2×C22⋊D5314yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—1064no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2122no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4—2122no—
289C22×C4#1171992414yes—
290C22×C4C2×C22⋊D5312yes—
291C22×C4C2×C22⋊D5312yes—
292C22×C4C2×C22⋊D5312yes—
293C22×C4C2×C22⋊D5311yescharacteristic
294C22×C4C2×C22⋊D5312yes—
295C22×C4C23×D5311yescharacteristic
296C2×C8—1064no—
297C2×C8—2124no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—1064no—
308C4⋊C4—2122no—
309C4⋊C4—2128no—
310C4⋊C4C2×C22⋊D5314yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C42—28no—
314C23—22no—
315C23—22no—
316C23#1313318611yescharacteristic
317C23#1313318611yescharacteristic
318C23C2×#1171992412yes—
319C23#1313324812yes—
320C23C22×C22⋊D5311yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—84no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—1064no—
341C2×C4—4242no—
342C2×C4—4242no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2128no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4#1313318612yes—
350C2×C4C2×#1171992414yes—
351C2×C4C22×C22⋊D5312yes—
352C8—2124no—
353C22—24no—
354C22—24no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1681076212yes—
360C22#1681076212yes—
361C22#1681080612yes—
362C22#1681081312yes—
363C22C2×#1313318611yescharacteristic
364C22C22×#1171992411yescharacteristic
365C22C2×#1313324811yescharacteristic
366C4—22no—
367C4—22no—
368C4—24no—
369C4—44no—
370C4—42no—
371C4—44no—
372C4—4242no—
373C4—2124no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2#2027372314yesdirect product factor
378C2C2×#1681076211yescharacteristic
379C2C2×#1681080611yescharacteristic
380C2C2×#1681081311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenterC2×#544D53
hypocenterC53C2×#1611
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20273723C2direct (N × H)
193. C2×C53#1611semidirect
219. C53C2×#1611semidirect
377. C2#20273723direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168105532
#202737232
C2×#168103602
C2×#168103492
C2×C53×#5442
C2×#161153