Finite Group Databasebeta

← C53×#7779 #27215291

Subgroups — C53×#7779 #27215291

Order 13568.

454 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2808C211yesmaximal, characteristic
2C53×#2057C211yesmaximal, characteristic, complemented
3C53×#2024C211yesmaximal, characteristic, complemented
4C53×#2013C211yesmaximal, characteristic
5C53×#1272C211yesmaximal, characteristic, complemented
6C53×#1271C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C22×SD16×C53—21no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×#107×C53—21no—
11C2×#103×C53—21no—
12C2×#103×C53—21no—
13C53×#397—21no—
14C53×#394—21no—
15C53×#393—21no—
16C53×#391C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#389C2211yescharacteristic
19C53×#388—21no—
20C53×#388—21no—
21C53×#387—21no—
22C53×#340—21no—
23C53×#340—21no—
24C53×#336C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Q8×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C22×D4×C53D411yescharacteristic
30C2×SD16×C53—42no—
31C2×SD16×C53—42no—
32C2×SD16×C53—42no—
33C22×C8×C53—21no—
34C22×C8×C53—21no—
35C22×C8×C53—21no—
36C4⋊Q8×C53—42no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
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×C42×C53—21no—
51C2×C42×C53C2311yescharacteristic
52#107×C53—42no—
53#103×C53—42no—
54#103×C53—42no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—21no—
64C2×Q8×C53—21no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85SD16×C53—48no—
86C2×C8×C53—22no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—81no—
111C22⋊C4×C53—81no—
112C22⋊C4×C53—81no—
113C22⋊C4×C53—81no—
114C42×C53—41no—
115C42×C53—81no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53#12011yescharacteristic
124Q8×C53—42no—
125Q8×C53—42no—
126Q8×C53—42no—
127Q8×C53—81no—
128Q8×C53—81no—
129Q8×C53—81no—
130D4×C53—42no—
131D4×C53—81no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C8×C53—42no—
163C8×C53—42no—
164C8×C53—81no—
165#7779C5311yesmaximal, characteristic, direct product factor, Sylow
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#44911yescharacteristic
174C22×C53#45211yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45311yescharacteristic
177C22×C53#45511yescharacteristic
178C22×C53#45611yescharacteristic
179C22×C53B(4,𝔽2)11yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—81no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—81no—
192#2808C2×C5311yescharacteristic
193#2057C2×C5311yescharacteristic, complemented
194#2024C2×C5311yescharacteristic, complemented
195#2013C2×C5311yescharacteristic
196#1272C2×C5311yescharacteristic, complemented
197#1271C2×C5311yescharacteristic, complemented
198#1270C2×C5311yescharacteristic
199C2×C53—161no—
200C2×C53#160011yescharacteristic
201C2×C53#160311yescharacteristic
202C2×C53#160611yescharacteristic
203C2×C53#163011yescharacteristic
204C2×C53#164611yescharacteristic
205C2×C53#165011yescharacteristic
206C2×C53#166111yescharacteristic
207C22×SD16—21no—
208C2×C4⋊Q8C22×C5311yescharacteristic
209C2×#107—21no—
210C2×#103—21no—
211C2×#103—21no—
212#397—21no—
213#394—21no—
214#393—21no—
215#391C22×C5311yescharacteristic
216#390C22×C5311yescharacteristic
217#389C22×C5311yescharacteristic
218#388—21no—
219#388—21no—
220#387—21no—
221#340—21no—
222#340—21no—
223#336C22×C5311yescharacteristic
224#336C22×C5311yescharacteristic
225#336C22×C5311yescharacteristic
226C53#777911yescharacteristic, direct product factor, Sylow
227C22×Q8—21no—
228C22×Q8D4×C5311yescharacteristic
229C22×D4D4×C5311yescharacteristic
230C2×SD16—42no—
231C2×SD16—42no—
232C2×SD16—42no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C4⋊Q8—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C5311yescharacteristic
241C2×C4⋊C4D4×C5311yescharacteristic
242C2×C4⋊C4D4×C5311yescharacteristic
243C2×C4⋊C4D4×C5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42C23×C5311yescharacteristic
252#107—42no—
253#103—42no—
254#103—42no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4C2×D4×C5311yescharacteristic
283C22×C4C2×D4×C5311yescharacteristic
284C22×C4C2×D4×C5311yescharacteristic
285SD16—48no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—81no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#120×C5311yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—81no—
330D4—42no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#135×C5311yescharacteristic
357C2×C4#135×C5311yescharacteristic
358C2×C4#135×C5311yescharacteristic
359C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
360C2×C4#137×C5311yescharacteristic
361C2×C4#137×C5311yescharacteristic
362C8—42no—
363C8—42no—
364C8—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C53×#44911yescharacteristic
373C22C53×#45211yescharacteristic
374C22C53×#45211yescharacteristic
375C22C53×#45311yescharacteristic
376C22C53×#45511yescharacteristic
377C22C53×#45611yescharacteristic
378C22C53×B(4,𝔽2)11yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—81no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—161no—
392C2C53×#160011yescharacteristic
393C2C53×#160311yescharacteristic
394C2C53×#160611yescharacteristic
395C2C53×#163011yescharacteristic
396C2C53×#164611yescharacteristic
397C2C53×#165011yescharacteristic
398C2C53×#166111yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2057C2semidirect
3. C53×#2024C2semidirect
5. C53×#1272C2semidirect
6. C53×#1271C2semidirect
165. #7779C53direct (N × H)
193. #2057C2×C53semidirect
194. #2024C2×C53semidirect
196. #1272C2×C53semidirect
197. #1271C2×C53semidirect
226. C53#7779direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28082
C53×#20572
C53×#20242
C53×#20132
C53×#12722
C53×#12712
C53×#12702
#777953