Finite Group Databasebeta

← C53×#12092 #27219604

Subgroups — C53×#12092 #27219604

Order 13568.

544 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2600C211yesmaximal, characteristic, complemented
2C53×#2022C211yesmaximal, characteristic, complemented
3C53×#1607C211yesmaximal, characteristic, complemented
4C53×#1595C211yesmaximal, characteristic
5C53×#1344C211yesmaximal, characteristic, complemented
6C53×#1343C211yesmaximal, characteristic, complemented
7C53×#1317C211yesmaximal, characteristic, complemented
8C53×#534C2211yescharacteristic
9C2×#123×C53—21no—
10C2×C4×D4×C53—21no—
11C53×#480—21no—
12C53×#447—21no—
13C2×#99×C53—21no—
14C53×#397C2211yescharacteristic
15C53×#394—21no—
16C53×#392C2211yescharacteristic
17C53×#388—21no—
18C53×#387—21no—
19C53×#386—21no—
20C53×#385—21no—
21C53×#381C2211yescharacteristic
22C53×#342C2211yescharacteristic
23C53×#327—21no—
24C53×#327—21no—
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53D411yescharacteristic
30C23×C4×C53—21no—
31C23×C4×C53D411yescharacteristic
32C2×D8×C53—41no—
33#123×C53—41no—
34#123×C53—42no—
35#123×C53—42no—
36#121×C53—21no—
37#121×C53—21no—
38#120×C53—41no—
39C4×D4×C53—42no—
40C4×D4×C53—42no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—41no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53—21no—
55#107×C53—41no—
56#102×C53—41no—
57#102×C53—41no—
58#99×C53—42no—
59C22⋊C8×C53—21no—
60C22⋊C8×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53D411yescharacteristic
66#95×C53D411yescharacteristic
67C24×C53—21no—
68C24×C53—21no—
69C24×C53C2×D411yescharacteristic
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—42no—
73C2×D4×C53—41no—
74C2×D4×C53—42no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—42no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53D812yescomplemented
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100D8×C53—82no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—42no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—44no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—44no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—81no—
122C42×C53—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—22no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53#12011yescharacteristic
140C23×C53C2×D811yescharacteristic
141C23×C53AGL(1,ℤ/8)11yescharacteristic
142D4×C53—42no—
143D4×C53—44no—
144D4×C53—44no—
145D4×C53—81no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—81no—
177C2×C4×C53—84no—
178C2×C4×C53—81no—
179C2×C4×C53—81no—
180C8×C53—81no—
181C8×C53—81no—
182#12092C5311yesmaximal, characteristic, direct product factor, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—42no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—82no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—161no—
204C22×C53#44711yescharacteristic
205C22×C53#45311yescharacteristic
206C22×C53B(4,𝔽2)11yescharacteristic
207C4×C53—42no—
208C4×C53—42no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—81no—
212C4×C53—81no—
213C4×C53—82no—
214C4×C53—81no—
215C4×C53—81no—
216#2600C2×C5311yescharacteristic, complemented
217#2022C2×C5311yescharacteristic, complemented
218#1607C2×C5311yescharacteristic, complemented
219#1595C2×C5311yescharacteristic
220#1344C2×C5311yescharacteristic, complemented
221#1343C2×C5311yescharacteristic, complemented
222#1317C2×C5311yescharacteristic, complemented
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—41no—
226C2×C53—41no—
227C2×C53—82no—
228C2×C53—161no—
229C2×C53#159511yescharacteristic
230C2×C53#219711yescharacteristic
231C2×C53#219811yescharacteristic
232#534C22×C5311yescharacteristic
233C2×#123—21no—
234C2×C4×D4—21no—
235#480—21no—
236#447—21no—
237C2×#99—21no—
238#397C22×C5311yescharacteristic
239#394—21no—
240#392C22×C5311yescharacteristic
241#388—21no—
242#387—21no—
243#386—21no—
244#385—21no—
245#381C22×C5311yescharacteristic
246#342C22×C5311yescharacteristic
247#327—21no—
248#327—21no—
249#323C22×C5311yescharacteristic
250#323C22×C5311yescharacteristic
251C53#1209211yescharacteristic, direct product factor, Sylow
252C22×D4—21no—
253C22×D4—21no—
254C22×D4D4×C5311yescharacteristic
255C23×C4—21no—
256C23×C4D4×C5311yescharacteristic
257C2×D8—41no—
258#123—42no—
259#123—41no—
260#123—42no—
261#121—21no—
262#121—21no—
263#120—41no—
264C4×D4—42no—
265C4×D4—42no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4D4×C5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4D4×C5311yescharacteristic
278C2×C22⋊C4C23×C5311yescharacteristic
279C2×C42—21no—
280#107—41no—
281#102—41no—
282#102—41no—
283#99—42no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95D4×C5311yescharacteristic
291#95D4×C5311yescharacteristic
292C24—21no—
293C24—21no—
294C24C2×D4×C5311yescharacteristic
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4D8×C5312yescomplemented
323C22×C4C2×D4×C5311yescharacteristic
324C22×C4C2×D4×C5311yescharacteristic
325D8—82no—
326C2×C8—41no—
327C2×C8—41no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—81no—
347C42—42no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—81no—
363C23—81no—
364C23#120×C5311yescharacteristic
365C23C2×D8×C5311yescharacteristic
366C23AGL(1,ℤ/8)×C5311yescharacteristic
367D4—42no—
368D4—44no—
369D4—44no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—84no—
403C2×C4—81no—
404C2×C4—81no—
405C8—81no—
406C8—81no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—161no—
428C22C53×#44711yescharacteristic
429C22C53×#45311yescharacteristic
430C22C53×B(4,𝔽2)11yescharacteristic
431C4—42no—
432C4—41no—
433C4—41no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4—82no—
438C4—81no—
439C4—81no—
440C2—21no—
441C2—21no—
442C2—41no—
443C2—41no—
444C2—82no—
445C2—161no—
446C2C53×#159511yescharacteristic
447C2C53×#219711yescharacteristic
448C2C53×#219811yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#1595C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2600C2semidirect
2. C53×#2022C2semidirect
3. C53×#1607C2semidirect
5. C53×#1344C2semidirect
6. C53×#1343C2semidirect
7. C53×#1317C2semidirect
97. C22×C4×C53D8semidirect
182. #12092C53direct (N × H)
216. #2600C2×C53semidirect
217. #2022C2×C53semidirect
218. #1607C2×C53semidirect
220. #1344C2×C53semidirect
221. #1343C2×C53semidirect
222. #1317C2×C53semidirect
251. C53#12092direct (N × H)
322. C22×C4D8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26002
C53×#20222
C53×#16072
C53×#15952
C53×#13442
C53×#13432
C53×#13172
#1209253