Finite Group Databasebeta

← C2×#20273671 #27356173

Subgroups — C2×#20273671 #27356173

Order 13568.

1308 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1308 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×#16810779C212yesmaximal, complemented
3#20273671C218yesmaximal, direct product factor
4C2×#16810359C212yesmaximal, complemented
5C2×#16810347C211yesmaximal, characteristic
6C2×C53×#393C211yesmaximal, characteristic
7C22×#11719925—22no—
8#16811249—24no—
9#16811249—22no—
10#16811249C2214yes—
11C2×#13133196—21no—
12C2×#13133196C2212yes—
13C2×#13133194—22no—
14C2×#13133194—22no—
15C22×C22⋊Dic53—21no—
16#16810779—24no—
17#16810779C2218yescomplemented
18C2×#13133141—22no—
19C2×C22⋊#88322C2212yes—
20#16810359C2218yescomplemented
21#16810347C2214yes—
22C23×Q8×C53C2211yescharacteristic
23C22×C22⋊C4×C53—21no—
24C22×C22⋊C4×C53C2211yescharacteristic
25C53×#393—22no—
26C53×#393C2214yes—
27C2×#95×C53—21no—
28C2×#95×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106D412yes—
31C23×Dic53—21no—
32C23×Dic53—21no—
33C2×#11719925—28no—
34C2×#11719925—48no—
35C22×#88322—22no—
36#13133196—24no—
37#13133196—24no—
38#13133196—22no—
39#13133196—22no—
40#13133196D414yes—
41#13133196C2314yes—
42#13133194—28no—
43#13133194—28no—
44#13133194—28no—
45#13133194—28no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D412yes—
48C2×#11719919—22no—
49C2×#11719919—22no—
50C2×C22⋊Dic53—22no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—24no—
54#13133141—28no—
55C22⋊#88322—22no—
56C22⋊#88322C2314yes—
57C25×C53D411yescharacteristic
58C22×Q8×C53—21no—
59C22×Q8×C53—22no—
60C22×Q8×C53—22no—
61C22×Q8×C53—44no—
62C22×Q8×C53C2312yes—
63C23×C4×C53—21no—
64C23×C4×C53—21no—
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—44no—
71C2×C22⋊C4×C53D412yes—
72C2×C22⋊C4×C53C2312yes—
73#95×C53—24no—
74#95×C53—21no—
75#95×C53C2312yes—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—22no—
79C24×C53—42no—
80C24×C53—42no—
81C24×C53Q1614yescomplemented
82C24×C53C2×D411yescharacteristic
83C24×C53C2×D412yes—
84C2×Q8×C53—28no—
85C2×Q8×C53—24no—
86C2×Q8×C53—44no—
87C2×Q8×C53—44no—
88C2×Q8×C53—44no—
89C2×Q8×C53—48no—
90C2×Q8×C53—42no—
91C22×C4×C53—24no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—42no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2411yescharacteristic
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—24no—
112C22×Dic53—24no—
113C22×Dic53—82no—
114C22×Dic53—82no—
115C2×Dic106—24no—
116C2×Dic106—22no—
117C2×Dic106—44no—
118C2×Dic106—82no—
119C2×Dic106C2×D414yes—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—48no—
123C22⋊Dic53—28no—
124C22⋊Dic53—24no—
125C22⋊Dic53—24no—
126#11719925—416no—
127C4⋊Dic53—24no—
128C4⋊Dic53—22no—
129C4⋊Dic53C2×D414yes—
130#11719919—28no—
131#11719919—24no—
132#11719919—24no—
133C2×#88322—24no—
134C2×#88322—44no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—24no—
139C23×C53—22no—
140C23×C53—24no—
141C23×C53—48no—
142C23×C53—42no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53—84no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53#12011yescharacteristic
150C23×C53#12011yescharacteristic
151C23×C53C2×Q1612yes—
152C23×C53C2×Q1614yescomplemented
153C23×C53#13712yes—
154C23×C53C22×D411yescharacteristic
155Q8×C53—28no—
156Q8×C53—44no—
157Q8×C53—48no—
158C2×C4×C53—24no—
159C2×C4×C53—24no—
160C2×C4×C53—22no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—44no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—44no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—81no—
174C2×C4×C53—84no—
175C2×C4×C53#12012yes—
176C2×C4×C53C22×D412yes—
177C2×Dic53—24no—
178C2×Dic53—24no—
179C2×Dic53—44no—
180C2×Dic53—44no—
181C2×Dic53—84no—
182C2×Dic53—84no—
183Dic106—44no—
184Dic106—82no—
185Dic106—84no—
186#88322—44no—
187C2×#1602—531nomaximal, Sylow
188C22×C53—24no—
189C22×C53—24no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—24no—
195C22×C53—44no—
196C22×C53—48no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—84no—
201C22×C53#45112yes—
202C22×C53#45112yes—
203C22×C53#45512yes—
204C22×C53B(4,𝔽2)12yes—
205C22×C53C2×#12011yescharacteristic
206C22×C53C22×Q1611yescharacteristic
207C22×C53C2×#13711yescharacteristic
208C4×C53—22no—
209C4×C53—22no—
210C4×C53—42no—
211C4×C53—44no—
212C4×C53—82no—
213Dic53—44no—
214Dic53—44no—
215C2×#543—531no—
216C2×#451—532no—
217C2×#393D5311yescharacteristic
218#1602—538no—
219C2×#328—532no—
220C2×#323—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53—44no—
225C2×C53#160214yescomplemented
226C2×C53C2×#45111yescharacteristic
227C2×C53C2×#45511yescharacteristic
228C2×C53C2×B(4,𝔽2)11yescharacteristic
229C23×Q8C2×D5311yescharacteristic
230C22×Q16—1062no—
231#543—534no—
232#543—1064no—
233#543—1062no—
234C2×C22⋊Q8—532no—
235C2×C22⋊Q8—1062no—
236C2×C22⋊Q8—1062no—
237C2×C22⋊Q8—1061no—
238C22×C22⋊C4—21no—
239C22×C22⋊C4—1061no—
240C22×C22⋊C4C2×D5311yescharacteristic
241#451—538no—
242#451—1064no—
243C2×#103—1062no—
244C2×C22⋊C8—532no—
245#393—22no—
246#393C2×D5314yes—
247C2×#95—21no—
248C2×#95C2×D5311yescharacteristic
249#328—538no—
250#323—534no—
251C53C2×#160211yescharacteristic, complemented, Sylow
252C25C22⋊D5311yescharacteristic
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—44no—
257C22×Q8—532no—
258C22×Q8—1061no—
259C22×Q8C22×D5312yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—1061no—
263C23×C4—1061no—
264C23×C4C22⋊D5311yescharacteristic
265C23×C4C22×D5311yescharacteristic
266C2×Q16—1068no—
267C2×Q16—2128no—
268C22×C8—1062no—
269C22⋊Q8—534no—
270C22⋊Q8—534no—
271C22⋊Q8—1068no—
272C22⋊Q8—1068no—
273C22⋊Q8—1068no—
274C22⋊Q8—1068no—
275C22⋊Q8—1064no—
276C22⋊Q8—1062no—
277C22⋊Q8—1062no—
278C22⋊Q8—1064no—
279C2×C4⋊C4—532no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1062no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1064no—
289C2×C22⋊C4—1062no—
290C2×C22⋊C4—1064no—
291C2×C22⋊C4C22⋊D5312yes—
292C2×C22⋊C4C22×D5312yes—
293#103—1068no—
294C22⋊C8—534no—
295C22⋊C8—1062no—
296#95—24no—
297#95—21no—
298#95C22×D5312yes—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—42no—
303C24—42no—
304C24#1171992514yescomplemented
305C24C2×C22⋊D5311yescharacteristic
306C24C2×C22⋊D5312yes—
307C2×Q8—28no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—534no—
315C2×Q8—1064no—
316C2×Q8—1062no—
317C2×Q8—4242no—
318C2×Q8—2124no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—1064no—
333C22×C4—1064no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—4242no—
337C22×C4—4242no—
338C22×C4D4×D5312yes—
339C22×C4D4×D5312yes—
340C22×C4C2×C22⋊D5312yes—
341C22×C4C2×C22⋊D5311yescharacteristic
342C22×C4C23×D5311yescharacteristic
343Q16—21216no—
344C2×C8—1064no—
345C2×C8—2124no—
346C4⋊C4—534no—
347C4⋊C4—1068no—
348C4⋊C4—1064no—
349C4⋊C4—1062no—
350C4⋊C4—1064no—
351C4⋊C4—1064no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—48no—
355C22⋊C4—1068no—
356C22⋊C4—1064no—
357C22⋊C4—1064no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—24no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—48no—
367C23—42no—
368C23—42no—
369C23—84no—
370C23—81no—
371C23—81no—
372C23#1313318511yescharacteristic
373C23#1313318511yescharacteristic
374C23C2×#1171992512yes—
375C23C2×#1171992514yescomplemented
376C23#1313324712yes—
377C23C22×C22⋊D5311yescharacteristic
378Q8—28no—
379Q8—44no—
380Q8—48no—
381Q8—4242no—
382Q8—4244no—
383Q8—2124no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—84no—
401C2×C4—1064no—
402C2×C4—1064no—
403C2×C4—4244no—
404C2×C4—4244no—
405C2×C4—2124no—
406C2×C4—2124no—
407C2×C4#1313318512yes—
408C2×C4C2×D4×D5312yes—
409C8—2124no—
410C22—24no—
411C22—24no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—24no—
417C22—44no—
418C22—48no—
419C22—42no—
420C22—42no—
421C22—82no—
422C22—84no—
423C22#1681077812yes—
424C22#1681077812yes—
425C22#1681080012yes—
426C22#1681080912yes—
427C22C2×#1313318511yescharacteristic
428C22C22×#1171992511yescharacteristic
429C22C2×#1313324711yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—44no—
434C4—82no—
435C4—2124no—
436C4—2124no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—44no—
441C2#2027367114yesdirect product factor
442C2C2×#1681077811yescharacteristic
443C2C2×#1681080011yescharacteristic
444C2C2×#1681080911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#393C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#393D53
hypocenterC53C2×#1602
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#16810779C2semidirect
3. #20273671C2direct (N × H)
4. C2×#16810359C2semidirect
17. #16810779C22semidirect
20. #16810359C22semidirect
81. C24×C53Q16semidirect
152. C23×C53C2×Q16semidirect
225. C2×C53#1602semidirect
251. C53C2×#1602semidirect
304. C24#11719925semidirect
375. C23C2×#11719925semidirect
441. C2#20273671direct (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×#168107792
#202736712
C2×#168103592
C2×#168103472
C2×C53×#3932
C2×#160253