Finite Group Databasebeta

← C53×#10018 #27217530

Subgroups — C53×#10018 #27217530

Order 13568.

662 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C53×#3153C211yesmaximal, characteristic, complemented
2C53×#3109C211yesmaximal, characteristic, complemented
3C53×#2631C211yesmaximal, characteristic, complemented
4C53×#1533C211yesmaximal, characteristic, complemented
5C53×#1514C211yesmaximal, characteristic, complemented
6C53×#1490C211yesmaximal, characteristic, complemented
7C53×#1288C211yesmaximal, characteristic, complemented
8C2×#137×C53—21no—
9C2×AGL(1,ℤ/8)×C53—21no—
10C53×#532C2211yescharacteristic
11C2×#124×C53C2211yescharacteristic
12C2×C4×D4×C53C2211yescharacteristic
13C53×#497—21no—
14C53×#496—21no—
15C53×#495—22no—
16C53×#481—21no—
17C53×#476—21no—
18C53×#459—21no—
19C53×#447—21no—
20C53×#442—22no—
21C53×#432C2211yescharacteristic
22C53×#432C2211yescharacteristic
23C4×M16×C53C2211yescharacteristic
24C53×#397—21no—
25C53×#393—21no—
26C53×#392—21no—
27C53×#392—21no—
28C53×#386—22no—
29C53×#383C2211yescharacteristic
30C53×#329—22no—
31C53×#327—22no—
32C53×D4⋊C8—22no—
33C2×#41×C53—21no—
34C22×Q8×C53D411yescharacteristic
35C22×D4×C53—22no—
36C22×D4×C53D411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53—21no—
39#137×C53—44no—
40AGL(1,ℤ/8)×C53—44no—
41C2×Q16×C53—22no—
42C2×SD16×C53—22no—
43C2×SD16×C53—22no—
44C2×D8×C53—22no—
45C2×D8×C53—22no—
46C2×M16×C53—21no—
47C4⋊Q8×C53D411yescharacteristic, complemented
48#127×C53D411yescharacteristic, complemented
49#124×C53—21no—
50#124×C53—21no—
51#124×C53—21no—
52#124×C53—21no—
53#124×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#106×C53—22no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72#102×C53—22no—
73#102×C53—22no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76#97×C53—21no—
77#97×C53—21no—
78#97×C53D412yescomplemented
79C4×C8×C53D412yescomplemented
80#95×C53—22no—
81#95×C53—21no—
82#95×C53—21no—
83#95×C53D412yes—
84C24×C53—21no—
85C24×C53—21no—
86#41×C53—42no—
87#41×C53—42no—
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53—21no—
91C2×Q8×C53—42no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—42no—
100C2×D4×C53—42no—
101C2×D4×C53—44no—
102C2×D4×C53—41no—
103C2×D4×C53—82no—
104C2×D4×C53—82no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53—82no—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115Q16×C53—44no—
116SD16×C53—44no—
117SD16×C53—44no—
118D8×C53—44no—
119D8×C53—44no—
120M16×C53—24no—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53—41no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—82no—
137C22⋊C4×C53—82no—
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53#4111yescharacteristic
142C23×C53—22no—
143C23×C53—42no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—42no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—42no—
153C23×C53#12011yescharacteristic
154Q8×C53—42no—
155Q8×C53—41no—
156Q8×C53—41no—
157Q8×C53—81no—
158D4×C53—22no—
159D4×C53—22no—
160D4×C53—41no—
161D4×C53—41no—
162D4×C53—42no—
163D4×C53—41no—
164D4×C53—41no—
165D4×C53—42no—
166D4×C53—81no—
167D4×C53—81no—
168D4×C53—88no—
169D4×C53—82no—
170C2×C4×C53—22no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—82no—
182C2×C4×C53—82no—
183C2×C4×C53—82no—
184C2×C4×C53#12011yescharacteristic
185C2×C4×C53#12011yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12711yescharacteristic
190C8×C53—24no—
191C8×C53—42no—
192C8×C53—81no—
193#10018C5311yesmaximal, characteristic, direct product factor, Sylow
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—44no—
197C22×C53—42no—
198C22×C53—82no—
199C22×C53—82no—
200C22×C53—82no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#39211yescharacteristic
210C22×C53#45312yes—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—82no—
218C4×C53—81no—
219C4×C53—81no—
220C4×C53—81no—
221#3153C2×C5311yescharacteristic, complemented
222#3109C2×C5311yescharacteristic, complemented
223#2631C2×C5311yescharacteristic, complemented
224#1533C2×C5311yescharacteristic, complemented
225#1514C2×C5311yescharacteristic, complemented
226#1490C2×C5311yescharacteristic, complemented
227#1288C2×C5311yescharacteristic, complemented
228C2×C53—22no—
229C2×C53—82no—
230C2×C53—161no—
231C2×C53—161no—
232C2×C53#200211yescharacteristic
233C2×C53#200811yescharacteristic
234C2×C53#201011yescharacteristic
235C2×#137—21no—
236C2×AGL(1,ℤ/8)—21no—
237#532C22×C5311yescharacteristic
238C2×#124C22×C5311yescharacteristic
239C2×C4×D4C22×C5311yescharacteristic
240#497—21no—
241#496—21no—
242#495—22no—
243#481—21no—
244#476—21no—
245#459—21no—
246#447—21no—
247#442—22no—
248#432C22×C5311yescharacteristic
249#432C22×C5311yescharacteristic
250C4×M16C22×C5311yescharacteristic
251#397—21no—
252#393—21no—
253#392—21no—
254#392—21no—
255#386—22no—
256#383C22×C5311yescharacteristic
257#329—22no—
258#327—22no—
259D4⋊C8—22no—
260C53#1001811yescharacteristic, direct product factor, Sylow
261C2×#41—21no—
262C22×Q8D4×C5311yescharacteristic
263C22×D4—22no—
264C22×D4D4×C5311yescharacteristic
265C22×D4D4×C5311yescharacteristic
266C23×C4—21no—
267#137—44no—
268AGL(1,ℤ/8)—44no—
269C2×Q16—22no—
270C2×SD16—22no—
271C2×SD16—22no—
272C2×D8—22no—
273C2×D8—22no—
274C2×M16—21no—
275C4⋊Q8D4×C5311yescharacteristic, complemented
276#127D4×C5311yescharacteristic, complemented
277#124—21no—
278#124—21no—
279#124—21no—
280#124—21no—
281#124—21no—
282#121—21no—
283#121—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C2×C4⋊C4D4×C5311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C42C23×C5311yescharacteristic
297#106—22no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300#102—22no—
301#102—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304#97—21no—
305#97—21no—
306#97D4×C5312yescomplemented
307C4×C8D4×C5312yescomplemented
308#95—22no—
309#95—21no—
310#95—21no—
311#95D4×C5312yes—
312C24—21no—
313C24—21no—
314#41—42no—
315#41—42no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—42no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—41no—
331C2×D4—82no—
332C2×D4—82no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—82no—
340C22×C4C2×D4×C5311yescharacteristic
341C22×C4C2×D4×C5311yescharacteristic
342C22×C4C2×D4×C5311yescharacteristic
343Q16—44no—
344SD16—44no—
345SD16—44no—
346D8—44no—
347D8—44no—
348M16—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—41no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42C2×D4×C5311yescharacteristic
367C42C2×D4×C5311yescharacteristic
368C42C2×D4×C5311yescharacteristic
369C42#41×C5311yescharacteristic
370C23—22no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23#120×C5311yescharacteristic
382Q8—41no—
383Q8—42no—
384Q8—41no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—42no—
392D4—42no—
393D4—41no—
394D4—81no—
395D4—81no—
396D4—88no—
397D4—82no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#120×C5311yescharacteristic
413C2×C4#120×C5311yescharacteristic
414C2×C4#121×C5311yescharacteristic
415C2×C4#121×C5311yescharacteristic
416C2×C4#121×C5311yescharacteristic
417C2×C4#127×C5311yescharacteristic
418C8—24no—
419C8—42no—
420C8—81no—
421C22—21no—
422C22—21no—
423C22—44no—
424C22—42no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22C53×#39211yescharacteristic
437C22C53×#45312yes—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—81no—
448C2—22no—
449C2—82no—
450C2—161no—
451C2—161no—
452C2C53×#200211yescharacteristic
453C2C53×#200811yescharacteristic
454C2C53×#201011yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3153C2semidirect
2. C53×#3109C2semidirect
3. C53×#2631C2semidirect
4. C53×#1533C2semidirect
5. C53×#1514C2semidirect
6. C53×#1490C2semidirect
7. C53×#1288C2semidirect
47. C4⋊Q8×C53D4semidirect
48. #127×C53D4semidirect
78. #97×C53D4semidirect
79. C4×C8×C53D4semidirect
193. #10018C53direct (N × H)
221. #3153C2×C53semidirect
222. #3109C2×C53semidirect
223. #2631C2×C53semidirect
224. #1533C2×C53semidirect
225. #1514C2×C53semidirect
226. #1490C2×C53semidirect
227. #1288C2×C53semidirect
260. C53#10018direct (N × H)
275. C4⋊Q8D4×C53semidirect
276. #127D4×C53semidirect
306. #97D4×C53semidirect
307. C4×C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31532
C53×#31092
C53×#26312
C53×#15332
C53×#15142
C53×#14902
C53×#12882
#1001853