Finite Group Databasebeta

← C53×#12244 #27219756

Subgroups — C53×#12244 #27219756

Order 13568.

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

normal subgroups only

This group has 398 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×#2722C211yesmaximal, characteristic
2C53×#2062C211yesmaximal, characteristic
3C53×#1608C211yesmaximal, characteristic
4C53×#1608C211yesmaximal, characteristic
5C53×#1344C211yesmaximal, characteristic
6C53×#1344C211yesmaximal, characteristic
7C53×#1326C211yesmaximal, characteristic
8C53×#543C2211yescharacteristic
9C2×#126×C53—21no—
10C53×#483—21no—
11C53×#483—21no—
12C53×#398—21no—
13C53×#398—21no—
14C53×#397C2211yescharacteristic
15C53×#397C2211yescharacteristic
16C53×#389—21no—
17C53×#387—21no—
18C53×#385—21no—
19C53×#385C2211yescharacteristic
20C53×#380—21no—
21C53×#342—21no—
22C53×#342C2211yescharacteristic
23C53×#328—21no—
24C53×#328—21no—
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C23×C4×C53D411yescharacteristic
28#126×C53—42no—
29#126×C53—42no—
30C22⋊Q8×C53—21no—
31C22⋊Q8×C53—21no—
32C22⋊Q8×C53—41no—
33C22⋊Q8×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53C2311yescharacteristic
49C2×C42×C53—21no—
50#106×C53—41no—
51#106×C53—41no—
52#103×C53—41no—
53#103×C53—41no—
54C22⋊C8×C53—21no—
55C22⋊C8×C53—21no—
56#95×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—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—81no—
94C4⋊C4×C53—82no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53SD1612yes—
106C42×C53—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53#12011yescharacteristic
112C23×C53C2×SD1611yescharacteristic
113C23×C53AGL(1,ℤ/8)11yescharacteristic
114Q8×C53—81no—
115Q8×C53—81no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
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—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C8×C53—81no—
148C8×C53—81no—
149#12244C5311yesmaximal, characteristic, direct product factor, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53#45011yescharacteristic
160C22×C53#45611yescharacteristic
161C22×C53B(4,𝔽2)11yescharacteristic
162C4×C53—42no—
163C4×C53—41no—
164C4×C53—42no—
165C4×C53—41no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53—82no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—81no—
173#2722C2×C5311yescharacteristic
174#2062C2×C5311yescharacteristic
175#1608C2×C5311yescharacteristic
176#1608C2×C5311yescharacteristic
177#1344C2×C5311yescharacteristic
178#1344C2×C5311yescharacteristic
179#1326C2×C5311yescharacteristic
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—41no—
183C2×C53—41no—
184C2×C53#160411yescharacteristic
185C2×C53#219711yescharacteristic
186C2×C53#220111yescharacteristic
187#543C22×C5311yescharacteristic
188C2×#126—21no—
189#483—21no—
190#483—21no—
191#398—21no—
192#398—21no—
193#397C22×C5311yescharacteristic
194#397C22×C5311yescharacteristic
195#389—21no—
196#387—21no—
197#385—21no—
198#385C22×C5311yescharacteristic
199#380—21no—
200#342—21no—
201#342C22×C5311yescharacteristic
202#328—21no—
203#328—21no—
204#323C22×C5311yescharacteristic
205#323C22×C5311yescharacteristic
206C53#1224411yescharacteristic, direct product factor, Sylow
207C23×C4D4×C5311yescharacteristic
208#126—42no—
209#126—42no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4D4×C5311yescharacteristic
222C2×C4⋊C4D4×C5311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4D4×C5311yescharacteristic
228C2×C22⋊C4C23×C5311yescharacteristic
229C2×C42—21no—
230#106—41no—
231#106—41no—
232#103—41no—
233#103—41no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—41no—
242#95D4×C5311yescharacteristic
243#95D4×C5311yescharacteristic
244C24C2×D4×C5311yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×D4×C5311yescharacteristic
259C22×C4C2×D4×C5311yescharacteristic
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C22⋊C4—21no—
276C22⋊C4—22no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4SD16×C5312yes—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#120×C5311yescharacteristic
292C23C2×SD16×C5311yescharacteristic
293C23AGL(1,ℤ/8)×C5311yescharacteristic
294Q8—81no—
295Q8—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
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—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C8—81no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C53×#45011yescharacteristic
339C22C53×#45611yescharacteristic
340C22C53×B(4,𝔽2)11yescharacteristic
341C4—42no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2C53×#160411yescharacteristic
357C2C53×#219711yescharacteristic
358C2C53×#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
149. #12244C53direct (N × H)
206. C53#12244direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27222
C53×#20622
C53×#16082
C53×#13442
C53×#13262
#1224453