Finite Group Databasebeta

← C2×#20273708 #27356221

Subgroups — C2×#20273708 #27356221

Order 13568.

1136 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16811249C211yesmaximal, characteristic
2C2×#16810954C212yesmaximal, complemented
3C2×#16810548C211yesmaximal, characteristic
4#20273708C218yesmaximal, direct product factor
5C2×#16810361C212yesmaximal, complemented
6C2×C53×#323C211yesmaximal, characteristic
7#16811249—24no—
8#16811249—22no—
9#16811249C2214yes—
10C2×#13133196—21no—
11C2×#13133196C2212yes—
12C2×#13133194—22no—
13C2×#13133194—22no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53—21no—
16C22×C22⋊Dic53—21no—
17#16810954—24no—
18#16810954C2218yescomplemented
19C2×#13133149—22no—
20C2×#13133142—22no—
21#16810548—22no—
22#16810548C2214yes—
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#16810361C2218yescomplemented
26C22×C22⋊C4×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2212yes—
28C53×#323C2214yes—
29C22×Dic106—21no—
30C22×Dic106D412yes—
31C23×Dic53—21no—
32C23×Dic53—21no—
33C23×Dic53—21no—
34C23×Dic53—21no—
35#13133196—24no—
36#13133196—24no—
37#13133196—22no—
38#13133196—22no—
39#13133196D414yes—
40#13133196C2314yes—
41#13133194—28no—
42#13133194—28no—
43#13133194—28no—
44#13133194—28no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53D412yes—
50C2×C4⋊Dic53C2312yes—
51C2×#11719919—22no—
52C2×#11719919—22no—
53C2×C22⋊Dic53—22no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—24no—
57C2×C22⋊Dic53—24no—
58C2×C22⋊Dic53—44no—
59#13133149—28no—
60#13133142—28no—
61#13133124—21no—
62#13133124—24no—
63#13133124C2312yes—
64C25×C53D411yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C22×C8×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53D412yes—
71C2×C22⋊C4×C53C2312yes—
72C22⋊C8×C53—22no—
73C22⋊C8×C53C2314yes—
74C24×C53—21no—
75C24×C53—21no—
76C24×C53—22no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53SD1614yes—
80C24×C53C2×D411yescharacteristic
81C24×C53C2×D412yes—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2411yescharacteristic
91C22×Dic53—22no—
92C22×Dic53—22no—
93C22×Dic53—24no—
94C22×Dic53—24no—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—41no—
100C22×Dic53—41no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C22×Dic53—82no—
105C22×Dic53—82no—
106C2×Dic106—24no—
107C2×Dic106—22no—
108C2×Dic106—44no—
109C2×Dic106—82no—
110C2×Dic106C2×D414yes—
111C2×C8×C53—24no—
112C2×C8×C53—44no—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—24no—
115C22⋊Dic53—28no—
116C22⋊Dic53—24no—
117C22⋊Dic53—24no—
118C22⋊Dic53—48no—
119C4⋊Dic53—24no—
120C4⋊Dic53—22no—
121C4⋊Dic53—24no—
122C4⋊Dic53—42no—
123C4⋊Dic53C2×D414yes—
124#11719919—28no—
125#11719919—24no—
126#11719919—24no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—22no—
130C23×C53—24no—
131C23×C53—22no—
132C23×C53—24no—
133C23×C53—48no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—84no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53#12011yescharacteristic
142C23×C53#12011yescharacteristic
143C23×C53C2×SD1612yes—
144C23×C53C2×SD1614yes—
145C23×C53#13712yes—
146C23×C53C22×D411yescharacteristic
147C2×C4×C53—24no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—81no—
155C2×C4×C53#12012yes—
156C2×C4×C53C22×D412yes—
157C2×Dic53—24no—
158C2×Dic53—24no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—84no—
168C2×Dic53—84no—
169C2×Dic53—84no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172Dic106—44no—
173Dic106—82no—
174Dic106—84no—
175C8×C53—44no—
176C2×#1608—531nomaximal, Sylow
177C22×C53—24no—
178C22×C53—24no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—24no—
184C22×C53—44no—
185C22×C53—48no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—82no—
189C22×C53—84no—
190C22×C53#44812yes—
191C22×C53#44812yes—
192C22×C53#45512yes—
193C22×C53#45812yes—
194C22×C53C2×#12011yescharacteristic
195C22×C53C22×SD1611yescharacteristic
196C22×C53C2×#13711yescharacteristic
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—42no—
200Dic53—44no—
201Dic53—44no—
202Dic53—82no—
203Dic53—82no—
204C2×#543—531no—
205C2×#483—532no—
206C2×#397—531no—
207#1608—538no—
208C2×#328—532no—
209C2×#323D5311yescharacteristic
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—44no—
214C2×C53#160814yescomplemented
215C2×C53C2×#44811yescharacteristic
216C2×C53C2×#45511yescharacteristic
217C2×C53C2×#45811yescharacteristic
218#543—534no—
219#543—1064no—
220#543—1062no—
221C2×C22⋊Q8—532no—
222C2×C22⋊Q8—1062no—
223C2×C22⋊Q8—1062no—
224C2×C22⋊Q8—1061no—
225C22×C4⋊C4—531no—
226C22×C22⋊C4—1061no—
227C22×C22⋊C4—1061no—
228C22×C22⋊C4C2×D5311yescharacteristic
229#483—538no—
230#483—1064no—
231C2×#106—1062no—
232C2×#103—1062no—
233C2×C22⋊C8C2×D5312yes—
234#397—534no—
235#397—1062no—
236C2×#95—531no—
237C2×#95—1061no—
238#328—538no—
239#323C2×D5314yes—
240C53C2×#160811yescharacteristic, complemented, Sylow
241C25D21211yescharacteristic
242C22×Q8—532no—
243C22×Q8—1061no—
244C23×C4—1061no—
245C23×C4—1061no—
246C23×C4—1061no—
247C23×C4—1061no—
248C23×C4D21211yescharacteristic
249C23×C4C22×D5311yescharacteristic
250C22×C8—22no—
251C22⋊Q8—534no—
252C22⋊Q8—534no—
253C22⋊Q8—1068no—
254C22⋊Q8—1068no—
255C22⋊Q8—1068no—
256C22⋊Q8—1068no—
257C22⋊Q8—1064no—
258C22⋊Q8—1062no—
259C22⋊Q8—1064no—
260C22⋊Q8—1062no—
261C2×C4⋊C4—532no—
262C2×C4⋊C4—532no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1062no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4—1061no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—1062no—
272C2×C22⋊C4—1064no—
273C2×C22⋊C4—1062no—
274C2×C22⋊C4—1064no—
275C2×C22⋊C4—1064no—
276C2×C22⋊C4—2124no—
277C2×C22⋊C4D21212yes—
278C2×C22⋊C4C22×D5312yes—
279#106—1068no—
280#103—1068no—
281C22⋊C8—22no—
282C22⋊C8C22×D5314yes—
283#95—532no—
284#95—1061no—
285#95—1064no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24#1171991314yes—
292C24C2×D21211yescharacteristic
293C24C2×D21212yes—
294C2×Q8—534no—
295C2×Q8—1064no—
296C2×Q8—1062no—
297C2×Q8—4242no—
298C2×Q8—2124no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—1064no—
304C22×C4—1064no—
305C22×C4—1062no—
306C22×C4—1062no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—4242no—
310C22×C4—4242no—
311C22×C4—2122no—
312C22×C4—2122no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4C2×D21212yes—
319C22×C4C2×D21211yescharacteristic
320C22×C4D4×D5312yes—
321C22×C4D4×D5312yes—
322C22×C4C23×D5311yescharacteristic
323C2×C8—24no—
324C2×C8—44no—
325C4⋊C4—534no—
326C4⋊C4—1068no—
327C4⋊C4—1064no—
328C4⋊C4—1062no—
329C4⋊C4—1064no—
330C4⋊C4—1064no—
331C4⋊C4—1064no—
332C4⋊C4—2122no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—1068no—
336C22⋊C4—1064no—
337C22⋊C4—1064no—
338C22⋊C4—2128no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—48no—
348C23—42no—
349C23—42no—
350C23—84no—
351C23—81no—
352C23—81no—
353C23#1313318711yescharacteristic
354C23#1313318711yescharacteristic
355C23C2×#1171991312yes—
356C23C2×#1171991314yes—
357C23#1313325012yes—
358C23C22×D21211yescharacteristic
359Q8—4242no—
360Q8—4244no—
361Q8—2124no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—1064no—
371C2×C4—1064no—
372C2×C4—4244no—
373C2×C4—4242no—
374C2×C4—4242no—
375C2×C4—4244no—
376C2×C4—4244no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2124no—
384C2×C4—2124no—
385C2×C4#1313318712yes—
386C2×C4C2×D4×D5312yes—
387C8—44no—
388C22—24no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—42no—
399C22—84no—
400C22—82no—
401C22#1681076412yes—
402C22#1681076412yes—
403C22#1681080312yes—
404C22#1681081412yes—
405C22C2×#1313318711yescharacteristic
406C22C22×#1171991311yescharacteristic
407C22C2×#1313325011yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—4242no—
412C4—4242no—
413C4—2124no—
414C4—2124no—
415C2—21no—
416C2—22no—
417C2—21no—
418C2—44no—
419C2#2027370814yesdirect product factor
420C2C2×#1681076411yescharacteristic
421C2C2×#1681080311yescharacteristic
422C2C2×#1681081411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#323C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#323D53
hypocenterC53C2×#1608
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#16810954C2semidirect
4. #20273708C2direct (N × H)
5. C2×#16810361C2semidirect
18. #16810954C22semidirect
25. #16810361C22semidirect
214. C2×C53#1608semidirect
240. C53C2×#1608semidirect
419. C2#20273708direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112492
C2×#168109542
C2×#168105482
#202737082
C2×#168103612
C2×C53×#3232
C2×#160853