Finite Group Databasebeta

← C53×#12252 #27219764

Subgroups — C53×#12252 #27219764

Order 13568.

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

normal subgroups only

This group has 452 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×#2677C211yesmaximal, characteristic
2C53×#2062C211yesmaximal, characteristic
3C53×#1614C211yesmaximal, characteristic
4C53×#1602C211yesmaximal, characteristic
5C53×#1352C211yesmaximal, characteristic
6C53×#1351C211yesmaximal, characteristic
7C53×#1326C211yesmaximal, characteristic
8C53×#543C2211yescharacteristic
9C2×C4⋊Q8×C53—21no—
10C2×C22⋊Q8×C53—21no—
11C53×#484—21no—
12C53×#451—21no—
13C53×#399C2211yescharacteristic
14C53×#398—21no—
15C53×#395—21no—
16C53×#393C2211yescharacteristic
17C53×#388—21no—
18C53×#387—21no—
19C53×#385—21no—
20C53×#385C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2211yescharacteristic
23C53×#328—21no—
24C53×#328—21no—
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×Q8×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C2×Q16×C53—41no—
31C4⋊Q8×C53—44no—
32C22⋊Q8×C53—21no—
33C22⋊Q8×C53—21no—
34C22⋊Q8×C53—41no—
35C22⋊Q8×C53—41no—
36C22⋊Q8×C53—44no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
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×C53D411yescharacteristic
50C2×C22⋊C4×C53C2311yescharacteristic
51C2×C42×C53—21no—
52#107×C53—41no—
53#103×C53—41no—
54#103×C53—41no—
55C22⋊C8×C53—21no—
56C22⋊C8×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—41no—
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—21no—
66C2×Q8×C53—21no—
67C2×Q8×C53—42no—
68C2×Q8×C53—42no—
69C2×Q8×C53—42no—
70C2×Q8×C53—41no—
71C2×Q8×C53—81no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—42no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85Q16×C53—82no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—82no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53Q1612yescomplemented
109C42×C53—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C23×C53C2×Q1611yescharacteristic
116C23×C53#13711yescharacteristic
117Q8×C53—42no—
118Q8×C53—81no—
119Q8×C53—84no—
120Q8×C53—81no—
121Q8×C53—81no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C8×C53—81no—
154C8×C53—81no—
155#12252C5311yesmaximal, characteristic, direct product factor, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—81no—
165C22×C53#45111yescharacteristic
166C22×C53#45611yescharacteristic
167C22×C53B(4,𝔽2)11yescharacteristic
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—82no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179#2677C2×C5311yescharacteristic
180#2062C2×C5311yescharacteristic
181#1614C2×C5311yescharacteristic
182#1602C2×C5311yescharacteristic
183#1352C2×C5311yescharacteristic
184#1351C2×C5311yescharacteristic
185#1326C2×C5311yescharacteristic
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—41no—
189C2×C53—41no—
190C2×C53#160511yescharacteristic
191C2×C53#219611yescharacteristic
192C2×C53#220111yescharacteristic
193#543C22×C5311yescharacteristic
194C2×C4⋊Q8—21no—
195C2×C22⋊Q8—21no—
196#484—21no—
197#451—21no—
198#399C22×C5311yescharacteristic
199#398—21no—
200#395—21no—
201#393C22×C5311yescharacteristic
202#388—21no—
203#387—21no—
204#385—21no—
205#385C22×C5311yescharacteristic
206#342—21no—
207#342C22×C5311yescharacteristic
208#328—21no—
209#328—21no—
210#323C22×C5311yescharacteristic
211#323C22×C5311yescharacteristic
212C53#1225211yescharacteristic, direct product factor, Sylow
213C22×Q8—21no—
214C22×Q8D4×C5311yescharacteristic
215C23×C4D4×C5311yescharacteristic
216C2×Q16—41no—
217C4⋊Q8—44no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4D4×C5311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4D4×C5311yescharacteristic
236C2×C22⋊C4C23×C5311yescharacteristic
237C2×C42—21no—
238#107—41no—
239#103—41no—
240#103—41no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—41no—
248#95D4×C5311yescharacteristic
249#95D4×C5311yescharacteristic
250C24C2×D4×C5311yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C2×D4×C5311yescharacteristic
270C22×C4C2×D4×C5311yescharacteristic
271Q16—82no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C22⋊C4—21no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4Q16×C5312yescomplemented
295C42—42no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23#120×C5311yescharacteristic
301C23C2×Q16×C5311yescharacteristic
302C23#137×C5311yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—84no—
306Q8—81no—
307Q8—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C8—81no—
340C8—81no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22C53×#45111yescharacteristic
351C22C53×#45611yescharacteristic
352C22C53×B(4,𝔽2)11yescharacteristic
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—81no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2C53×#160511yescharacteristic
369C2C53×#219611yescharacteristic
370C2C53×#220111yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC24×C53C2×D4
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
108. C22⋊C4×C53Q16semidirect
155. #12252C53direct (N × H)
212. C53#12252direct (N × H)
294. C22⋊C4Q16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26772
C53×#20622
C53×#16142
C53×#16022
C53×#13522
C53×#13512
C53×#13262
#1225253