Finite Group Databasebeta

← C2×#20273318 #27352000

Subgroups — C2×#20273318 #27352000

Order 13568.

1050 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133209C211yesmaximal, characteristic
2C2×C4×#11719917C211yesmaximal, characteristic
3#20273318C218yesmaximal, direct product factor
4C2×#16810364C214yesmaximal, complemented
5C2×C53×#432C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133209—21no—
8C2×#13133209—22no—
9C2×#13133209—21no—
10C2×#13133209—22no—
11C2×#13133209C412yes—
12C2×#13133209C414yes—
13C2×#13133209C2212yes—
14C2×#13133209C2214yes—
15C22×C4⋊Dic53C411yescharacteristic
16C22×C4⋊Dic53C411yescharacteristic
17C22×#11719919—21no—
18C22×#11719919—21no—
19C4×#11719917—22no—
20C4×#11719917C2214yes—
21C2×#13133128C2212yes—
22C2×C4×#88322—21no—
23#16810364C22116yescomplemented
24C22×C42×C53C2211yescharacteristic
25C53×#432—22no—
26C53×#432C2214yes—
27C2×C4⋊C8×C53—21no—
28C2×C22⋊C8×C53—21no—
29C2×#97×C53C2212yes—
30C23×Dic53—21no—
31C23×Dic53—21no—
32C2×#11719917—28no—
33C2×#11719917—24no—
34C22×#88322—22no—
35#13133209—24no—
36#13133209—24no—
37#13133209—24no—
38#13133209—24no—
39#13133209—22no—
40#13133209—22no—
41#13133209—24no—
42#13133209C2×C414yes—
43#13133209C2×C418yes—
44#13133209C2314yes—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53C2×C412yes—
50C2×C4⋊Dic53C2×C412yes—
51C2×C4⋊Dic53C2×C414yes—
52C2×C4⋊Dic53C2×C414yes—
53C2×#11719919—24no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—24no—
57C2×#11719919—22no—
58C2×#11719919—24no—
59C2×#11719919—22no—
60C2×#11719919—24no—
61#13133128—22no—
62#13133128C2314yes—
63C4×#88322—24no—
64C23×C4×C53C2×C411yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53D411yescharacteristic
67C22×C8×C53—22no—
68C2×C42×C53C2×C411yescharacteristic
69C2×C42×C53C2×C412yes—
70C2×C42×C53D411yescharacteristic
71C2×C42×C53D412yes—
72C2×C42×C53D411yescharacteristic
73C2×C42×C53D412yes—
74C2×C42×C53C2311yescharacteristic
75C2×C42×C53C2312yes—
76C4⋊C8×C53—24no—
77C22⋊C8×C53—24no—
78#97×C53—22no—
79#97×C53C2314yes—
80C24×C53C22⋊C411yescharacteristic
81C22×C4×C53—24no—
82C22×C4×C53—24no—
83C22×C4×C53—42no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C412yes—
88C22×C4×C53C22⋊C412yes—
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C412yes—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D412yes—
96C22×Dic53—24no—
97C22×Dic53—24no—
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—44no—
101C22×Dic53—44no—
102C2×C8×C53—24no—
103C2×C8×C53—44no—
104C42×C53C22⋊C412yes—
105C42×C53C22⋊C412yes—
106C42×C53C22⋊C412yes—
107C42×C53C22×C411yescharacteristic
108C42×C53C22×C412yes—
109C42×C53C2×D412yes—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D412yes—
113C42×C53C2411yescharacteristic
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53—24no—
117C4⋊Dic53—24no—
118C4⋊Dic53C22×C414yes—
119C4⋊Dic53C22×C414yes—
120#11719919—28no—
121#11719919—24no—
122#11719919—24no—
123#11719919—28no—
124#11719919—24no—
125#11719919—24no—
126#11719917—216no—
127C2×#88322—28no—
128C2×#88322—24no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53#10414yes—
132C23×C53C2×C22⋊C412yes—
133C23×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53—22no—
135C2×C4×C53—24no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
142C2×C4×C53—24no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53#10114yes—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C2×C22⋊C412yes—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C2×C22⋊C412yes—
151C2×C4×C53C2×C22⋊C412yes—
152C2×C4×C53C23×C411yescharacteristic
153C2×C4×C53C22×D411yescharacteristic
154C2×C4×C53C22×D411yescharacteristic
155C2×Dic53—24no—
156C2×Dic53—24no—
157C2×Dic53—48no—
158C2×Dic53—48no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C8×C53—44no—
162#88322—28no—
163C2×#1528—531nomaximal, Sylow
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—28no—
167C22×C53C2×#10112yes—
168C22×C53C2×#10414yes—
169C22×C53C2×#10412yes—
170C22×C53#42112yes—
171C22×C53C22×C22⋊C411yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—22no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—21no—
180C4×C53C2×#10114yes—
181Dic53—44no—
182Dic53—44no—
183C22×#125—531no—
184C2×#432D5311yescharacteristic
185C2×C4×M16—531no—
186#1528—538no—
187C2×#330—534no—
188C2×C53—24no—
189C2×C53#152814yescomplemented
190C2×C53C22×#10111yescharacteristic
191C2×C53C22×#10411yescharacteristic
192C2×C53C2×#42111yescharacteristic
193C22×M16—1061no—
194C2×#125—534no—
195C2×#125—532no—
196C2×#125—532no—
197C2×#125—534no—
198C2×#125—1061no—
199C2×#125—1062no—
200C2×#125—1061no—
201C2×#125—1062no—
202C22×C4⋊C4—531no—
203C22×C4⋊C4—531no—
204C22×C4⋊C4—1061no—
205C22×C4⋊C4—1061no—
206C22×C42C2×D5311yescharacteristic
207#432—22no—
208#432C2×D5314yes—
209C2×C4⋊C8—21no—
210C2×C22⋊C8—21no—
211C4×M16—534no—
212C4×M16—1062no—
213C2×#97—532no—
214C2×#97C2×D5312yes—
215C2×C4×C8—1061no—
216#330—5316no—
217C53C2×#152811yescharacteristic, complemented, Sylow
218C23×C4—1061no—
219C23×C4—1061no—
220C23×C4C4×D5311yescharacteristic
221C23×C4D21211yescharacteristic
222C23×C4C22⋊D5311yescharacteristic
223C2×M16—1068no—
224C2×M16—1064no—
225C22×C8—22no—
226C22×C8—1062no—
227#125—534no—
228#125—538no—
229#125—534no—
230#125—1064no—
231#125—1062no—
232#125—1062no—
233#125—1064no—
234#125—1064no—
235#125—1064no—
236#125—1064no—
237C2×C4⋊C4—534no—
238C2×C4⋊C4—534no—
239C2×C4⋊C4—532no—
240C2×C4⋊C4—532no—
241C2×C4⋊C4—1064no—
242C2×C4⋊C4—1062no—
243C2×C4⋊C4—1064no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1062no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1064no—
250C2×C4⋊C4—1064no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4—1062no—
253C2×C42C4×D5312yes—
254C2×C42C4×D5311yescharacteristic
255C2×C42D21211yescharacteristic
256C2×C42D21212yes—
257C2×C42C22⋊D5311yescharacteristic
258C2×C42C22⋊D5312yes—
259C2×C42C22×D5311yescharacteristic
260C2×C42C22×D5312yes—
261C4⋊C8—24no—
262C22⋊C8—24no—
263#97—22no—
264#97—534no—
265#97—1062no—
266#97C22×D5314yes—
267C4×C8—1064no—
268C24#1171992111yescharacteristic
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—1064no—
273C22×C4—1064no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—2124no—
277C22×C4—2124no—
278C22×C4#1171992112yes—
279C22×C4#1171992112yes—
280C22×C4#1171992111yescharacteristic
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992112yes—
283C22×C4#1171992111yescharacteristic
284C22×C4C2×C4×D5312yes—
285C22×C4C2×C4×D5311yescharacteristic
286C22×C4C2×D21212yes—
287C22×C4C2×D21211yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289C22×C4C2×C22⋊D5312yes—
290M16—10616no—
291C2×C8—24no—
292C2×C8—44no—
293C2×C8—1068no—
294C2×C8—1064no—
295C4⋊C4—534no—
296C4⋊C4—534no—
297C4⋊C4—1062no—
298C4⋊C4—1062no—
299C4⋊C4—1064no—
300C4⋊C4—1064no—
301C4⋊C4—1068no—
302C4⋊C4—1064no—
303C4⋊C4—1064no—
304C4⋊C4—1068no—
305C4⋊C4—1064no—
306C4⋊C4—1064no—
307C42#1171992112yes—
308C42#1171992112yes—
309C42#1171992112yes—
310C42C2×C4×D5311yescharacteristic
311C42C2×C4×D5312yes—
312C42C2×D21212yes—
313C42C2×D21211yescharacteristic
314C42C2×C22⋊D5311yescharacteristic
315C42C2×C22⋊D5312yes—
316C42C23×D5311yescharacteristic
317C23—22no—
318C23—22no—
319C23#1313314314yes—
320C23C2×#1171992112yes—
321C23C2×#1171992111yescharacteristic
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—1064no—
334C2×C4—1064no—
335C2×C4—2128no—
336C2×C4—2128no—
337C2×C4—2124no—
338C2×C4—2124no—
339C2×C4#1313313514yes—
340C2×C4C2×#1171992112yes—
341C2×C4C2×#1171992111yescharacteristic
342C2×C4C2×#1171992111yescharacteristic
343C2×C4C2×#1171992112yes—
344C2×C4C2×#1171992111yescharacteristic
345C2×C4C2×#1171992112yes—
346C2×C4C22×C4×D5311yescharacteristic
347C2×C4C22×C22⋊D5311yescharacteristic
348C2×C4C22×D21211yescharacteristic
349C8—44no—
350C8—1068no—
351C22—22no—
352C22—22no—
353C22—28no—
354C22C2×#1313313512yes—
355C22C2×#1313314314yes—
356C22C2×#1313314312yes—
357C22#1681064112yes—
358C22C22×#1171992111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—2124no—
368C4—2124no—
369C4C2×#1313313514yes—
370C2—24no—
371C2#2027331814yesdirect product factor
372C2C22×#1313313511yescharacteristic
373C2C22×#1313314311yescharacteristic
374C2C2×#1681064111yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×#432C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC42×C53C24
hypercenterC2×#432D53
hypocenterC53C2×#1528
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. #20273318C2direct (N × H)
4. C2×#16810364C2semidirect
23. #16810364C22semidirect
189. C2×C53#1528semidirect
217. C53C2×#1528semidirect
371. C2#20273318direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332092
C2×C4×#117199172
#202733182
C2×#168103642
C2×C53×#4322
C2×#152853