Finite Group Databasebeta

← C2×#20273742 #27356227

Subgroups — C2×#20273742 #27356227

Order 13568.

1180 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1180 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×#16810959C212yesmaximal, complemented
3C2×#16810559C211yesmaximal, characteristic
4#20273742C218yesmaximal, 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×#11719919—21no—
16C22×C22⋊Dic53—21no—
17C22×C22⋊Dic53—21no—
18#16810959—24no—
19#16810959C2218yescomplemented
20C2×#13133151—22no—
21C2×#13133142—22no—
22#16810559—22no—
23#16810559C2214yes—
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×#11719919—24no—
54C2×#11719919—44no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—24no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—44no—
61#13133151—28no—
62#13133142—28no—
63#13133124—21no—
64#13133124C2312yes—
65C25×C53D411yescharacteristic
66C23×C4×C53D411yescharacteristic
67C23×C4×C53C2311yescharacteristic
68C22×C8×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C22⋊C4×C53D412yes—
72C2×C22⋊C4×C53C2312yes—
73C22⋊C8×C53—22no—
74C22⋊C8×C53C2314yes—
75C24×C53—21no—
76C24×C53—21no—
77C24×C53—22no—
78C24×C53—42no—
79C24×C53—42no—
80C24×C53Q1614yes—
81C24×C53C2×D411yescharacteristic
82C24×C53C2×D412yes—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2411yescharacteristic
92C22×Dic53—24no—
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53—22no—
96C22×Dic53—24no—
97C22×Dic53—24no—
98C22×Dic53—22no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—41no—
104C22×Dic53—41no—
105C22×Dic53—82no—
106C22×Dic53—82no—
107C2×Dic106—24no—
108C2×Dic106—22no—
109C2×Dic106—44no—
110C2×Dic106—82no—
111C2×Dic106C2×D414yes—
112C2×C8×C53—24no—
113C2×C8×C53—44no—
114C22⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊Dic53—28no—
117C22⋊Dic53—24no—
118C22⋊Dic53—24no—
119C22⋊Dic53—48no—
120C4⋊Dic53—24no—
121C4⋊Dic53—22no—
122C4⋊Dic53—24no—
123C4⋊Dic53—42no—
124C4⋊Dic53C2×D414yes—
125#11719919—28no—
126#11719919—24no—
127#11719919—24no—
128#11719919—48no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—24no—
133C23×C53—22no—
134C23×C53—24no—
135C23×C53—48no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—84no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53#12011yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53C2×Q1612yes—
146C23×C53C2×Q1614yes—
147C23×C53#13712yes—
148C23×C53C22×D411yescharacteristic
149C2×C4×C53—24no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—81no—
157C2×C4×C53#12012yes—
158C2×C4×C53C22×D412yes—
159C2×Dic53—24no—
160C2×Dic53—24no—
161C2×Dic53—24no—
162C2×Dic53—48no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—44no—
166C2×Dic53—44no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—84no—
171C2×Dic53—84no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174Dic106—44no—
175Dic106—82no—
176Dic106—84no—
177C8×C53—44no—
178C2×#1614—531nomaximal, Sylow
179C22×C53—24no—
180C22×C53—24no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—24no—
186C22×C53—44no—
187C22×C53—48no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—82no—
191C22×C53—84no—
192C22×C53#45112yes—
193C22×C53#45112yes—
194C22×C53#45312yes—
195C22×C53#45812yes—
196C22×C53C2×#12011yescharacteristic
197C22×C53C22×Q1611yescharacteristic
198C22×C53C2×#13711yescharacteristic
199C4×C53—22no—
200C4×C53—22no—
201C4×C53—42no—
202Dic53—44no—
203Dic53—44no—
204Dic53—44no—
205Dic53—82no—
206C2×#543—531no—
207C2×#484—532no—
208C2×#399—531no—
209#1614—538no—
210C2×#328—532no—
211C2×#323D5311yescharacteristic
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—22no—
215C2×C53—44no—
216C2×C53#161414yescomplemented
217C2×C53C2×#45111yescharacteristic
218C2×C53C2×#45311yescharacteristic
219C2×C53C2×#45811yescharacteristic
220#543—534no—
221#543—1064no—
222#543—1062no—
223C2×C22⋊Q8—532no—
224C2×C22⋊Q8—1062no—
225C2×C22⋊Q8—1062no—
226C2×C22⋊Q8—1061no—
227C22×C4⋊C4—531no—
228C22×C4⋊C4—1061no—
229C22×C22⋊C4—1061no—
230C22×C22⋊C4—1061no—
231C22×C22⋊C4C2×D5311yescharacteristic
232#484—538no—
233#484—1064no—
234C2×#107—1062no—
235C2×#103—1062no—
236C2×C22⋊C8C2×D5312yes—
237#399—534no—
238#399—1062no—
239C2×#95—531no—
240#328—538no—
241#323C2×D5314yes—
242C53C2×#161411yescharacteristic, complemented, Sylow
243C25D21211yescharacteristic
244C22×Q8—532no—
245C22×Q8—1061no—
246C23×C4—1061no—
247C23×C4—1061no—
248C23×C4—1061no—
249C23×C4—1061no—
250C23×C4D21211yescharacteristic
251C23×C4C22×D5311yescharacteristic
252C22×C8—22no—
253C22⋊Q8—534no—
254C22⋊Q8—534no—
255C22⋊Q8—1068no—
256C22⋊Q8—1068no—
257C22⋊Q8—1068no—
258C22⋊Q8—1068no—
259C22⋊Q8—1064no—
260C22⋊Q8—1062no—
261C22⋊Q8—1064no—
262C22⋊Q8—1062no—
263C2×C4⋊C4—532no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—1062no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1064no—
272C2×C4⋊C4—2124no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—1062no—
276C2×C22⋊C4—1064no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1064no—
279C2×C22⋊C4—1064no—
280C2×C22⋊C4—2124no—
281C2×C22⋊C4D21212yes—
282C2×C22⋊C4C22×D5312yes—
283#107—1068no—
284#103—1068no—
285C22⋊C8—22no—
286C22⋊C8C22×D5314yes—
287#95—532no—
288#95—1061no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24Dic21214yes—
295C24C2×D21211yescharacteristic
296C24C2×D21212yes—
297C2×Q8—534no—
298C2×Q8—1064no—
299C2×Q8—1062no—
300C2×Q8—4242no—
301C2×Q8—2124no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—1064no—
307C22×C4—1064no—
308C22×C4—1064no—
309C22×C4—1062no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—4242no—
314C22×C4—4242no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4—2122no—
320C22×C4—2122no—
321C22×C4C2×D21212yes—
322C22×C4C2×D21211yescharacteristic
323C22×C4D4×D5312yes—
324C22×C4D4×D5312yes—
325C22×C4C23×D5311yescharacteristic
326C2×C8—24no—
327C2×C8—44no—
328C4⋊C4—534no—
329C4⋊C4—1068no—
330C4⋊C4—1064no—
331C4⋊C4—1062no—
332C4⋊C4—1064no—
333C4⋊C4—1064no—
334C4⋊C4—1064no—
335C4⋊C4—2128no—
336C4⋊C4—2122no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—1068no—
340C22⋊C4—1064no—
341C22⋊C4—1064no—
342C22⋊C4—2128no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—42no—
351C23—48no—
352C23—42no—
353C23—42no—
354C23—84no—
355C23—81no—
356C23—81no—
357C23#1313318711yescharacteristic
358C23#1313318711yescharacteristic
359C23C2×Dic21212yes—
360C23C2×Dic21214yes—
361C23#1313325012yes—
362C23C22×D21211yescharacteristic
363Q8—4242no—
364Q8—4244no—
365Q8—2124no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—1064no—
375C2×C4—1064no—
376C2×C4—1064no—
377C2×C4—4242no—
378C2×C4—4242no—
379C2×C4—4244no—
380C2×C4—4244no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4—2128no—
385C2×C4—2122no—
386C2×C4—2124no—
387C2×C4—2124no—
388C2×C4—2122no—
389C2×C4#1313318712yes—
390C2×C4C2×D4×D5312yes—
391C8—44no—
392C22—24no—
393C22—24no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1681078112yes—
406C22#1681078112yes—
407C22#1681079312yes—
408C22#1681081412yes—
409C22C2×#1313318711yescharacteristic
410C22C22×Dic21211yescharacteristic
411C22C2×#1313325011yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—4242no—
416C4—2124no—
417C4—2124no—
418C4—2124no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—44no—
423C2#2027374214yesdirect product factor
424C2C2×#1681078111yescharacteristic
425C2C2×#1681079311yescharacteristic
426C2C2×#1681081411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#323C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#323D53
hypocenterC53C2×#1614
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×#16810959C2semidirect
4. #20273742C2direct (N × H)
5. C2×#16810361C2semidirect
19. #16810959C22semidirect
25. #16810361C22semidirect
216. C2×C53#1614semidirect
242. C53C2×#1614semidirect
423. C2#20273742direct (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×#168109592
C2×#168105592
#202737422
C2×#168103612
C2×C53×#3232
C2×#161453