Finite Group Databasebeta

← C53×#10551 #27218063

Subgroups — C53×#10551 #27218063

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3141C211yesmaximal, characteristic
2C53×#1950C211yesmaximal, characteristic, complemented
3C53×#1549C211yesmaximal, characteristic, complemented
4C53×#1547C211yesmaximal, characteristic
5C2×C53×#333C211yesmaximal, characteristic, complemented
6C53×#1480C211yesmaximal, characteristic
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C2×C4⋊Q8×C53C2211yescharacteristic
10C53×#489—22no—
11C53×#487—21no—
12C53×#486—21no—
13C53×#431C2211yescharacteristic
14C2×#108×C53—21no—
15C53×#423C2211yescharacteristic
16C2×C4⋊C8×C53C2211yescharacteristic
17C2×C4⋊C8×C53C2211yescharacteristic
18C53×#417—21no—
19C53×#417—21no—
20C4×M16×C53C2211yescharacteristic
21C53×#341—21no—
22C53×#333—22no—
23C53×#332—21no—
24C53×#332—22no—
25C53×#331—21no—
26C53×D4⋊C8—22no—
27C2×#41×C53—21no—
28C22×Q8×C53—21no—
29C2×M16×C53—21no—
30C2×M16×C53D411yescharacteristic
31C2×M16×C53D411yescharacteristic
32C22×C8×C53—21no—
33C22×C8×C53—21no—
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53—21no—
36C4⋊Q8×C53—21no—
37C4⋊Q8×C53D411yescharacteristic
38C4⋊Q8×C53D412yes—
39#127×C53D411yescharacteristic
40#124×C53—21no—
41#121×C53—21no—
42#121×C53—21no—
43C4×D4×C53—21no—
44C4×D4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2311yescharacteristic
48#108×C53—42no—
49C4⋊C8×C53—21no—
50C4⋊C8×C53—21no—
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53D412yes—
56#103×C53—22no—
57#103×C53—22no—
58#102×C53—22no—
59#102×C53—22no—
60C22⋊C8×C53—21no—
61#97×C53—21no—
62C4×C8×C53—21no—
63C4×C8×C53—21no—
64#41×C53—41no—
65#41×C53—41no—
66#41×C53—41no—
67#41×C53—41no—
68C2×Q8×C53—21no—
69C2×Q8×C53—22no—
70C2×Q8×C53—41no—
71C2×Q8×C53—82no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81M16×C53—21no—
82M16×C53—21no—
83M16×C53—21no—
84M16×C53—21no—
85M16×C53—41no—
86M16×C53—41no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—22no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—81no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C23×C53—41no—
109C23×C53#12411yescharacteristic
110Q8×C53—41no—
111Q8×C53—41no—
112Q8×C53—82no—
113Q8×C53—82no—
114D4×C53—41no—
115D4×C53—41no—
116D4×C53—41no—
117D4×C53—41no—
118D4×C53—81no—
119D4×C53—81no—
120D4×C53—81no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—82no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C8×C53—41no—
140C8×C53—41no—
141C8×C53—42no—
142C8×C53—81no—
143C8×C53—81no—
144C8×C53—81no—
145#10551C5311yesmaximal, characteristic, direct product factor, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53#39411yescharacteristic
152C22×C53#48811yescharacteristic
153C22×C53#48911yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—82no—
158C4×C53—81no—
159C4×C53#44911yescharacteristic
160C4×C53#45211yescharacteristic
161C4×C53#46311yescharacteristic
162C4×C53#46411yescharacteristic
163C4×C53#47111yescharacteristic
164C4×C53#47311yescharacteristic
165#3141C2×C5311yescharacteristic
166#1950C2×C5311yescharacteristic, complemented
167#1549C2×C5311yescharacteristic, complemented
168#1547C2×C5311yescharacteristic
169C2×#333C2×C5311yescharacteristic, complemented
170#1480C2×C5311yescharacteristic
171#1283C2×C5311yescharacteristic, complemented
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—161no—
175C2×C53#204211yescharacteristic
176C2×C53#204811yescharacteristic
177C2×C53#205211yescharacteristic
178#532C22×C5311yescharacteristic
179C2×C4⋊Q8C22×C5311yescharacteristic
180#489—22no—
181#487—21no—
182#486—21no—
183#431C22×C5311yescharacteristic
184C2×#108—21no—
185#423C22×C5311yescharacteristic
186C2×C4⋊C8C22×C5311yescharacteristic
187C2×C4⋊C8C22×C5311yescharacteristic
188#417—21no—
189#417—21no—
190C4×M16C22×C5311yescharacteristic
191#341—21no—
192#333—22no—
193#332—21no—
194#332—22no—
195#331—21no—
196D4⋊C8—22no—
197C53#1055111yescharacteristic, direct product factor, Sylow
198C2×#41—21no—
199C22×Q8—21no—
200C2×M16—21no—
201C2×M16D4×C5311yescharacteristic
202C2×M16D4×C5311yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8D4×C5312yes—
209C4⋊Q8D4×C5311yescharacteristic
210#127D4×C5311yescharacteristic
211#124—21no—
212#121—21no—
213#121—21no—
214C4×D4—21no—
215C4×D4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C42C23×C5311yescharacteristic
219#108—42no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8D4×C5312yes—
227#103—22no—
228#103—22no—
229#102—22no—
230#102—22no—
231C22⋊C8—21no—
232#97—21no—
233C4×C8—21no—
234C4×C8—21no—
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×D4×C5311yescharacteristic
250C22×C4#41×C5311yescharacteristic
251C22×C4#41×C5311yescharacteristic
252M16—21no—
253M16—21no—
254M16—21no—
255M16—21no—
256M16—41no—
257M16—41no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—81no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42C2×D4×C5311yescharacteristic
276C42C2×D4×C5311yescharacteristic
277C42C2×D4×C5311yescharacteristic
278C42#41×C5311yescharacteristic
279C23—41no—
280C23#124×C5311yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—82no—
284Q8—82no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—81no—
290D4—81no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4#120×C5311yescharacteristic
301C2×C4#121×C5311yescharacteristic
302C2×C4#121×C5311yescharacteristic
303C2×C4#121×C5311yescharacteristic
304C2×C4#123×C5311yescharacteristic
305C2×C4#123×C5311yescharacteristic
306C2×C4#135×C5311yescharacteristic
307C2×C4#135×C5311yescharacteristic
308C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
309C2×C4#137×C5311yescharacteristic
310C8—41no—
311C8—41no—
312C8—42no—
313C8—81no—
314C8—81no—
315C8—81no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C53×#39411yescharacteristic
322C22C53×#48811yescharacteristic
323C22C53×#48911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—82no—
329C4C53×#44911yescharacteristic
330C4C53×#45211yescharacteristic
331C4C53×#46311yescharacteristic
332C4C53×#46411yescharacteristic
333C4C53×#47111yescharacteristic
334C4C53×#47311yescharacteristic
335C2—21no—
336C2—21no—
337C2—161no—
338C2C53×#204211yescharacteristic
339C2C53×#204811yescharacteristic
340C2C53×#205211yescharacteristic
subgroupG/H
center, socleC22×C53#394
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1950C2semidirect
3. C53×#1549C2semidirect
5. C2×C53×#333C2semidirect
7. C53×#1283C2semidirect
145. #10551C53direct (N × H)
166. #1950C2×C53semidirect
167. #1549C2×C53semidirect
169. C2×#333C2×C53semidirect
171. #1283C2×C53semidirect
197. C53#10551direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31412
C53×#19502
C53×#15492
C53×#15472
C2×C53×#3332
C53×#14802
C53×#12832
#1055153