Finite Group Databasebeta

← C4×#16810415 #27299255

Subgroups — C4×#16810415 #27299255

Order 13568.

720 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4×#13133213C211yesmaximal, characteristic
2C2×#16810415C212yesmaximal
3C4×#13133132C211yesmaximal, characteristic, complemented
4#20272839C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133213C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×C4×C22⋊D53—21no—
9C2×C4×C22⋊D53—21no—
10C2×#13133132C2212yes—
11#16810526C2211yescharacteristic
12#16810524C411yescharacteristic, complemented
13#16810523—21no—
14#16810523—21no—
15#16810492C411yescharacteristic, complemented
16C4×C22⋊Dic53—21no—
17C42×Dic53C411yescharacteristic, complemented
18#16810415C418yesdirect product factor
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C4×C22⋊C4×C53—21no—
23C53×#342—21no—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27C22×C4×D53—21no—
28#13133213—22no—
29#13133213C2×C414yes—
30C4×D212—41no—
31C4×D212—41no—
32C4×C22⋊D53—44no—
33C4×C22⋊D53—44no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—41no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53C2×C411yescharacteristic
48#13133132—22no—
49#13133132C2×C414yescomplemented
50#13133124—41no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C4×D4×C53—41no—
55C4×D4×C53—41no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C42×C53—41no—
61C2×C42×C53C2×C411yescharacteristic
62#99×C53—22no—
63#99×C53C2×C414yescomplemented
64#95×C53—41no—
65C24×C53C2×D411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C23×D53—21no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53C22×C412yes—
71C2×D4×C53#4112yes—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C2×C22⋊D53—24no—
85C2×C22⋊D53—24no—
86C2×C22⋊D53—42no—
87C2×C22⋊D53—42no—
88C2×C22⋊D53—42no—
89C2×C22⋊D53—42no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—42no—
94C2×D212—21no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212C4212yescomplemented
98C2×C4×D53—42no—
99C2×C4×D53—42no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—42no—
107C42×C53—21no—
108C42×C53—21no—
109C42×C53—82no—
110C22⋊Dic53—24no—
111C22⋊Dic53—42no—
112C22⋊Dic53—42no—
113C22⋊Dic53—42no—
114#11719921—42no—
115#11719921—42no—
116#11719921—42no—
117C4⋊Dic53—41no—
118#11719919—42no—
119#11719919—42no—
120C4×Dic53—24no—
121C4×Dic53—21no—
122C4×Dic53—42no—
123C4×Dic53—41no—
124C4×Dic53—82no—
125C4×Dic53C4212yescomplemented
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53C4×D412yes—
136C23×C53C4×D412yes—
137C22×D53—22no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144D4×C53—42no—
145D4×C53—42no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—44no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—82no—
161C2×C4×C53—82no—
162C2×C4×C53#9911yescharacteristic
163C2×C4×C53#9911yescharacteristic
164C2×C4×C53C2×C4211yescharacteristic
165C2×C4×C53#11711yescharacteristic
166C22⋊D53—48no—
167C22⋊D53—48no—
168C2×Dic53—22no—
169C2×Dic53—22no—
170C2×Dic53—22no—
171C2×Dic53—22no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—44no—
177C2×Dic53—41no—
178C2×Dic53—82no—
179C2×Dic53—82no—
180D212—42no—
181D212—42no—
182C4×D53—82no—
183C4×D53—82no—
184C4×#353—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53C4×C22⋊C411yescharacteristic
198C22×C53C2×#9911yescharacteristic
199C22×C53#41011yescharacteristic
200C2×D53—42no—
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—84no—
214C4×C53#35312yescomplemented
215Dic53—24no—
216Dic53—42no—
217Dic53—84no—
218Dic53—81no—
219C4×#127—531no—
220C2×#353—532no—
221C4×#99—531no—
222C4×#99D5311yescharacteristic, complemented
223#1393—532no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—42no—
227C2×C53—42no—
228C2×C53C4×#9911yescharacteristic
229C2×C53C2×#35311yescharacteristic
230C2×C53#212611yescharacteristic
231D53—82no—
232C2×#127—531no—
233C2×C4×D4—531no—
234C2×C4×D4—1061no—
235C2×C4×D4—1061no—
236C2×C4×D4C2×D5311yescharacteristic
237C2×#99—532no—
238C2×#99C2×D5312yes—
239#390—531no—
240#390—531no—
241#390—1061no—
242#390—1061no—
243#384—531no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—1061no—
246C43—531no—
247#353—538no—
248#342—21no—
249#342—1061no—
250C53C4×#35311yescharacteristic, complemented, Sylow
251C22×D4—531no—
252C22×D4—1061no—
253C22×D4—1061no—
254C22×D4C22×D5311yescharacteristic
255C23×C4—1061no—
256C23×C4D21211yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258#127—534no—
259#127—1062no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—2121no—
263C4×D4—2121no—
264C4×D4—2124no—
265C4×D4—2124no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4C4×D5311yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—1062no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4—1061no—
275C2×C22⋊C4—1061no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4D21211yescharacteristic
278C2×C22⋊C4C22⋊D5311yescharacteristic
279C2×C42—41no—
280C2×C42—531no—
281C2×C42—531no—
282C2×C42—1061no—
283C2×C42—1061no—
284C2×C42—2121no—
285C2×C42C4×D5311yescharacteristic
286#99—22no—
287#99—534no—
288#99—1062no—
289#99C4×D5314yescomplemented
290#95—41no—
291#95—2121no—
292C24—1061no—
293C24C2×D21211yescharacteristic
294C24C2×C22⋊D5311yescharacteristic
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—532no—
298C2×D4—1064no—
299C2×D4—1064no—
300C2×D4—1061no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2122no—
304C2×D4—2122no—
305C2×D4—2122no—
306C2×D4—2122no—
307C2×D4C2×C4×D5312yes—
308C2×D4#1171994512yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—2122no—
320C22×C4—2122no—
321C22×C4—2122no—
322C22×C4—2121no—
323C22×C4—2121no—
324C22×C4#1171992111yescharacteristic
325C22×C4#1171992111yescharacteristic
326C22×C4#1171992111yescharacteristic
327C22×C4#1171992111yescharacteristic
328C22×C4C2×C4×D5311yescharacteristic
329C4⋊C4—41no—
330C4⋊C4—2121no—
331C4⋊C4—2122no—
332C4⋊C4—2122no—
333C22⋊C4—24no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—1064no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C22⋊C4—2122no—
341C22⋊C4—2122no—
342C22⋊C4—2122no—
343C22⋊C4—2122no—
344C42—21no—
345C42—21no—
346C42—82no—
347C42—532no—
348C42—1064no—
349C42—1061no—
350C42—4242no—
351C42—2121no—
352C42—2122no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—1062no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23—2121no—
366C23—2121no—
367C23—2121no—
368C23C2×#1171992111yescharacteristic
369C23C4×C22⋊D5312yes—
370C23C4×D21212yes—
371D4—42no—
372D4—42no—
373D4—2122no—
374D4—2122no—
375D4—2128no—
376D4—2128no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—1062no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—1062no—
397C2×C4—4242no—
398C2×C4—4242no—
399C2×C4—4242no—
400C2×C4—4242no—
401C2×C4—2124no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4#1313313111yescharacteristic
408C2×C4#1313313111yescharacteristic
409C2×C4C42×D5311yescharacteristic
410C2×C4#1313317411yescharacteristic
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—42no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—81no—
422C22—81no—
423C22—4241no—
424C22—4241no—
425C22—4241no—
426C22—4241no—
427C22—2122no—
428C22—2122no—
429C22—2122no—
430C22C4×#1171992111yescharacteristic
431C22C2×#1313313111yescharacteristic
432C22#1681059611yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—84no—
440C4—1064no—
441C4—4244no—
442C4—4241no—
443C4—2122no—
444C4#1681041512yesdirect product factor
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—4242no—
450C2C4×#1313313111yescharacteristic
451C2C2×#1681041511yescharacteristic
452C2#2027637811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53C4×#353
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. C4×#13133132C2semidirect
4. #20272839C2semidirect
5. C4×#99×C53C2semidirect
7. C2×C4×D212C4semidirect
12. #16810524C4semidirect
15. #16810492C4semidirect
17. C42×Dic53C4semidirect
18. #16810415C4direct (N × H)
49. #13133132C2×C4semidirect
63. #99×C53C2×C4semidirect
97. C2×D212C42semidirect
125. C4×Dic53C42semidirect
214. C4×C53#353semidirect
222. C4×#99D53semidirect
250. C53C4×#353semidirect
289. #99C4×D53semidirect
444. C4#16810415direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#131332132
C2×#168104152
C4×#131331322
#202728392
C4×#99×C532
C4×#35353