Finite Group Databasebeta

← C53×#7773 #27215285

Subgroups — C53×#7773 #27215285

Order 13568.

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

normal subgroups only

This group has 422 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×#2043C211yesmaximal, characteristic, complemented
3C53×#2024C211yesmaximal, characteristic, complemented
4C53×#2023C211yesmaximal, characteristic
5C53×#1272C211yesmaximal, characteristic, complemented
6C53×#1271C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×#103×C53—21no—
10C2×#103×C53—21no—
11C2×#103×C53—21no—
12C2×#102×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
30C22×C8×C53—21no—
31C22×C8×C53—21no—
32C22×C8×C53—21no—
33C4⋊Q8×C53—42no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C42×C53—21no—
48C2×C42×C53C2311yescharacteristic
49#103×C53—42no—
50#103×C53—42no—
51#103×C53—42no—
52#102×C53—42no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59C24×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—21no—
62C2×Q8×C53—21no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—81no—
108C22⋊C4×C53—81no—
109C22⋊C4×C53—81no—
110C22⋊C4×C53—81no—
111C42×C53—41no—
112C42×C53—81no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53#12011yescharacteristic
121Q8×C53—42no—
122Q8×C53—42no—
123Q8×C53—42no—
124Q8×C53—81no—
125Q8×C53—81no—
126Q8×C53—81no—
127D4×C53—42no—
128D4×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
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—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—42no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C8×C53—81no—
160C8×C53—81no—
161C8×C53—81no—
162#7773C5311yesmaximal, characteristic, direct product factor, Sylow
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53#44911yescharacteristic
171C22×C53#45211yescharacteristic
172C22×C53#45211yescharacteristic
173C22×C53#45411yescharacteristic
174C22×C53#45511yescharacteristic
175C22×C53#45511yescharacteristic
176C22×C53B(4,𝔽2)11yescharacteristic
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—82no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53—81no—
189#2808C2×C5311yescharacteristic
190#2043C2×C5311yescharacteristic, complemented
191#2024C2×C5311yescharacteristic, complemented
192#2023C2×C5311yescharacteristic
193#1272C2×C5311yescharacteristic, complemented
194#1271C2×C5311yescharacteristic, complemented
195#1270C2×C5311yescharacteristic
196C2×C53—161no—
197C2×C53#159711yescharacteristic
198C2×C53#160611yescharacteristic
199C2×C53#160611yescharacteristic
200C2×C53#163811yescharacteristic
201C2×C53#164011yescharacteristic
202C2×C53#164611yescharacteristic
203C2×C53#166211yescharacteristic
204C2×C4⋊Q8C22×C5311yescharacteristic
205C2×#103—21no—
206C2×#103—21no—
207C2×#103—21no—
208C2×#102—21no—
209#397—21no—
210#394—21no—
211#393—21no—
212#391C22×C5311yescharacteristic
213#390C22×C5311yescharacteristic
214#389C22×C5311yescharacteristic
215#388—21no—
216#388—21no—
217#387—21no—
218#340—21no—
219#340—21no—
220#336C22×C5311yescharacteristic
221#336C22×C5311yescharacteristic
222#336C22×C5311yescharacteristic
223C53#777311yescharacteristic, direct product factor, Sylow
224C22×Q8—21no—
225C22×Q8D4×C5311yescharacteristic
226C22×D4D4×C5311yescharacteristic
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C4⋊Q8—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4D4×C5311yescharacteristic
235C2×C4⋊C4D4×C5311yescharacteristic
236C2×C4⋊C4D4×C5311yescharacteristic
237C2×C4⋊C4D4×C5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42—21no—
245C2×C42C23×C5311yescharacteristic
246#103—42no—
247#103—42no—
248#103—42no—
249#102—42no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256C24—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4C2×D4×C5311yescharacteristic
278C22×C4C2×D4×C5311yescharacteristic
279C22×C4C2×D4×C5311yescharacteristic
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—41no—
309C42—81no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#120×C5311yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324D4—42no—
325D4—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
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—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#135×C5311yescharacteristic
351C2×C4#135×C5311yescharacteristic
352C2×C4#135×C5311yescharacteristic
353C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
354C2×C4#137×C5311yescharacteristic
355C2×C4#137×C5311yescharacteristic
356C8—81no—
357C8—81no—
358C8—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22C53×#44911yescharacteristic
367C22C53×#45211yescharacteristic
368C22C53×#45211yescharacteristic
369C22C53×#45411yescharacteristic
370C22C53×#45511yescharacteristic
371C22C53×#45511yescharacteristic
372C22C53×B(4,𝔽2)11yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C2—161no—
386C2C53×#159711yescharacteristic
387C2C53×#160611yescharacteristic
388C2C53×#160611yescharacteristic
389C2C53×#163811yescharacteristic
390C2C53×#164011yescharacteristic
391C2C53×#164611yescharacteristic
392C2C53×#166211yescharacteristic
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×#2043C2semidirect
3. C53×#2024C2semidirect
5. C53×#1272C2semidirect
6. C53×#1271C2semidirect
162. #7773C53direct (N × H)
190. #2043C2×C53semidirect
191. #2024C2×C53semidirect
193. #1272C2×C53semidirect
194. #1271C2×C53semidirect
223. C53#7773direct (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×#20432
C53×#20242
C53×#20232
C53×#12722
C53×#12712
C53×#12702
#777353