Finite Group Databasebeta

← C53×#12779 #27220291

Subgroups — C53×#12779 #27220291

Order 13568.

732 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3550C211yesmaximal, characteristic
2C53×#2129C212yesmaximal, complemented
3C53×#2126C211yesmaximal, characteristic
4C53×#1909C212yesmaximal, complemented
5C53×#1884C211yesmaximal, characteristic
6C53×#568C2211yescharacteristic
7C53×#563—21no—
8C53×#563C412yescomplemented
9C53×#559—21no—
10Q82×C53C412yescomplemented
11C53×#556—22no—
12D4×Q8×C53—22no—
13C53×#542—24no—
14C53×#536—21no—
15C53×#536C2212yes—
16C53×#533—21no—
17C53×#532C2211yescharacteristic
18C53×#421—21no—
19C53×#419—22no—
20C53×#410C2212yes—
21C53×#410C2211yescharacteristic
22C53×#354—21no—
23C53×#354—24no—
24C53×#353—21no—
25C53×#339—21no—
26C53×H(3,ℤ/4)—22no—
27C2×#41×C53D411yescharacteristic
28C2×#41×C53C2311yescharacteristic
29C22×Q8×C53—21no—
30C22×Q8×C53D412yes—
31#131×C53—42no—
32C2×M16×C53—22no—
33C2×M16×C53—21no—
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53—21no—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53—24no—
38C4⋊Q8×C53C2×C411yescharacteristic
39C4⋊Q8×C53C2×C414yes—
40#127×C53D411yescharacteristic
41#126×C53—24no—
42#126×C53—24no—
43#125×C53—22no—
44#124×C53—22no—
45#124×C53—24no—
46#124×C53C2×C411yescharacteristic
47#124×C53D411yescharacteristic
48#124×C53D414yes—
49#123×C53—22no—
50#123×C53—22no—
51#123×C53—22no—
52#123×C53—22no—
53C22⋊Q8×C53—22no—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—24no—
56C22⋊Q8×C53—24no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—22no—
59#121×C53—22no—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C4×Q8×C53—24no—
63C4×D4×C53—24no—
64C4×D4×C53—22no—
65#117×C53—22no—
66#117×C53—21no—
67#117×C53—22no—
68#117×C53—21no—
69#117×C53D412yes—
70C2×C42×C53D411yescharacteristic
71#104×C53—42no—
72#103×C53—44no—
73#99×C53—22no—
74#99×C53—22no—
75#99×C53—22no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—22no—
81C2×Q8×C53—22no—
82C2×Q8×C53—22no—
83C2×Q8×C53—42no—
84C2×Q8×C53—42no—
85C2×Q8×C53—81no—
86C2×Q8×C53—81no—
87C2×Q8×C53C22⋊C414yes—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×Q8×C53C22×C411yescharacteristic
90C2×D4×C53—22no—
91C2×D4×C53—21no—
92C2×D4×C53C22⋊C411yescharacteristic
93C2×D4×C53C2×D412yes—
94C2×D4×C53C2×D411yescharacteristic
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D412yes—
101M16×C53—22no—
102M16×C53—42no—
103M16×C53—41no—
104C2×C8×C53—24no—
105C2×C8×C53—41no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—42no—
116C4⋊C4×C53—41no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—22no—
124C22⋊C4×C53—42no—
125C22⋊C4×C53—41no—
126C42×C53—24no—
127C42×C53—22no—
128C42×C53—21no—
129C42×C53—41no—
130C42×C53—42no—
131C42×C53C22⋊C411yescharacteristic
132C42×C53C22⋊C411yescharacteristic
133C42×C53C22⋊C414yes—
134C23×C53—21no—
135C23×C53#12012yes—
136C23×C53#12011yescharacteristic
137Q8×C53—41no—
138Q8×C53—44no—
139Q8×C53—44no—
140Q8×C53—41no—
141Q8×C53—42no—
142Q8×C53—42no—
143Q8×C53—82no—
144D4×C53—41no—
145D4×C53—42no—
146D4×C53—41no—
147D4×C53—44no—
148D4×C53—81no—
149C2×C4×C53—21no—
150C2×C4×C53—22no—
151C2×C4×C53—21no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—24no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—24no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53—82no—
164C2×C4×C53—81no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53C2×C22⋊C412yes—
167C2×C4×C53#12011yescharacteristic
168C8×C53—42no—
169#12779C5311yesmaximal, characteristic, direct product factor, Sylow
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—41no—
176C22×C53#37911yescharacteristic
177C4×C53—22no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—41no—
181C4×C53—44no—
182C4×C53—41no—
183C4×C53—44no—
184C4×C53—162no—
185C4×C53—161no—
186C4×C53B(4,𝔽2)11yescharacteristic
187C4×C53#45811yescharacteristic
188#3550C2×C5311yescharacteristic
189#2129C2×C5312yescomplemented
190#2126C2×C5311yescharacteristic
191#1909C2×C5312yescomplemented
192#1884C2×C5311yescharacteristic
193C2×C53—21no—
194C2×C53—42no—
195C2×C53—41no—
196C2×C53—81no—
197C2×C53#179011yescharacteristic
198#568C22×C5311yescharacteristic
199#563—21no—
200#563C4×C5312yescomplemented
201#559—21no—
202Q82C4×C5312yescomplemented
203#556—22no—
204D4×Q8—22no—
205#542—24no—
206#536—21no—
207#536C22×C5312yes—
208#533—21no—
209#532C22×C5311yescharacteristic
210#421—21no—
211#419—22no—
212#410C22×C5312yes—
213#410C22×C5311yescharacteristic
214#354—21no—
215#354—24no—
216#353—21no—
217#339—21no—
218H(3,ℤ/4)—22no—
219C53#1277911yescharacteristic, direct product factor, Sylow
220C2×#41D4×C5311yescharacteristic
221C2×#41C23×C5311yescharacteristic
222C22×Q8—21no—
223C22×Q8D4×C5312yes—
224#131—42no—
225C2×M16—22no—
226C2×M16—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—22no—
230C4⋊Q8—24no—
231C4⋊Q8C2×C4×C5311yescharacteristic
232C4⋊Q8C2×C4×C5314yes—
233#127D4×C5311yescharacteristic
234#126—24no—
235#126—24no—
236#125—22no—
237#124—22no—
238#124—24no—
239#124C2×C4×C5311yescharacteristic
240#124D4×C5311yescharacteristic
241#124D4×C5314yes—
242#123—22no—
243#123—22no—
244#123—22no—
245#123—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—24no—
256C4×D4—24no—
257C4×D4—22no—
258#117—22no—
259#117—21no—
260#117—22no—
261#117—21no—
262#117D4×C5312yes—
263C2×C42D4×C5311yescharacteristic
264#104—42no—
265#103—44no—
266#99—22no—
267#99—22no—
268#99—22no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×Q8C22⋊C4×C5314yes—
281C2×Q8C22⋊C4×C5311yescharacteristic
282C2×Q8C22×C4×C5311yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C22⋊C4×C5311yescharacteristic
286C2×D4C2×D4×C5312yes—
287C2×D4C2×D4×C5311yescharacteristic
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4C2×D4×C5311yescharacteristic
293C22×C4C2×D4×C5312yes—
294M16—22no—
295M16—42no—
296M16—41no—
297C2×C8—24no—
298C2×C8—41no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C22⋊C4—22no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C42—24no—
320C42—22no—
321C42—21no—
322C42—41no—
323C42—42no—
324C42C22⋊C4×C5311yescharacteristic
325C42C22⋊C4×C5311yescharacteristic
326C42C22⋊C4×C5314yes—
327C23—21no—
328C23#120×C5312yes—
329C23#120×C5311yescharacteristic
330Q8—41no—
331Q8—44no—
332Q8—41no—
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—82no—
337D4—41no—
338D4—42no—
339D4—41no—
340D4—44no—
341D4—81no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4C2×C22⋊C4×C5312yes—
359C2×C4C2×C22⋊C4×C5311yescharacteristic
360C2×C4#120×C5311yescharacteristic
361C8—42no—
362C22—22no—
363C22—21no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22C53×#37911yescharacteristic
369C4—22no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—44no—
374C4—41no—
375C4—44no—
376C4—162no—
377C4—161no—
378C4C53×B(4,𝔽2)11yescharacteristic
379C4C53×#45811yescharacteristic
380C2—21no—
381C2—42no—
382C2—41no—
383C2—81no—
384C2C53×#179011yescharacteristic
subgroupG/H
center, socleC2×C53#1790
commutatorC2×Q8C22×C4×C53
Frattini, cosocleC2×#41C23×C53
prime radicalC53×#532C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2129C2semidirect
4. C53×#1909C2semidirect
8. C53×#563C4semidirect
10. Q82×C53C4semidirect
169. #12779C53direct (N × H)
189. #2129C2×C53semidirect
191. #1909C2×C53semidirect
200. #563C4×C53semidirect
202. Q82C4×C53semidirect
219. C53#12779direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35502
C53×#21292
C53×#21262
C53×#19092
C53×#18842
#1277953