Finite Group Databasebeta

← C53×#12127 #27219639

Subgroups — C53×#12127 #27219639

Order 13568.

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

normal subgroups only

This group has 454 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×#2804C211yesmaximal, characteristic, complemented
2C53×#2006C211yesmaximal, characteristic
3C53×#1616C212yesmaximal, complemented
4C53×#1353C212yesmaximal, complemented
5C53×#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C53—21no—
7C53×#537C2211yescharacteristic
8C53×#481—22no—
9C2×#104×C53—21no—
10C53×#400C2212yes—
11C53×#393—22no—
12C53×#391—21no—
13C53×#390C2211yescharacteristic
14C53×#386—21no—
15C53×#385—22no—
16C53×#343C2211yescharacteristic
17C53×#327—22no—
18C53×#324C2212yes—
19C2×#41×C53—21no—
20C22×Q8×C53—21no—
21C22×D4×C53D411yescharacteristic
22C23×C4×C53—21no—
23AGL(1,ℤ/8)×C53—42no—
24AGL(1,ℤ/8)×C53—42no—
25C2×SD16×C53—41no—
26C2×D8×C53—41no—
27C2×M16×C53—21no—
28C22⋊Q8×C53—41no—
29#121×C53—22no—
30#121×C53—41no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53C2311yescharacteristic
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—22no—
37C2×C42×C53D411yescharacteristic
38#106×C53—42no—
39#104×C53—42no—
40#102×C53—42no—
41C22⋊C8×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yes—
46C24×C53—21no—
47#41×C53—41no—
48#41×C53—41no—
49#41×C53—41no—
50#41×C53—41no—
51C2×Q8×C53—21no—
52C2×Q8×C53—42no—
53C2×Q8×C53—81no—
54C2×Q8×C53—81no—
55C2×D4×C53—21no—
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53SD1612yes—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—41no—
66C22×C4×C53—82no—
67C22×C4×C53—81no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D412yes—
70SD16×C53—44no—
71D8×C53—44no—
72M16×C53—22no—
73M16×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53—42no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—81no—
80C22⋊C4×C53—44no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—82no—
83C22⋊C4×C53—81no—
84C42×C53—41no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53#12011yescharacteristic
92Q8×C53—41no—
93Q8×C53—41no—
94Q8×C53—84no—
95D4×C53—42no—
96D4×C53—41no—
97D4×C53—41no—
98D4×C53—81no—
99D4×C53—82no—
100D4×C53—81no—
101D4×C53—81no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—81no—
117C2×C4×C53—81no—
118C2×C4×C53—84no—
119C2×C4×C53C2×SD1611yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C8×C53—42no—
122C8×C53—82no—
123#12127C5311yesmaximal, characteristic, direct product factor, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53—81no—
134C22×C53#44811yescharacteristic
135C22×C53#45311yescharacteristic
136C22×C53B(4,𝔽2)11yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53—42no—
141C4×C53—82no—
142C4×C53—81no—
143C4×C53—82no—
144#2804C2×C5311yescharacteristic, complemented
145#2006C2×C5311yescharacteristic
146#1616C2×C5312yescomplemented
147#1353C2×C5312yescomplemented
148#1318C2×C5311yescharacteristic, complemented
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—82no—
152C2×C53—161no—
153C2×C53#159611yescharacteristic
154C2×C53#219811yescharacteristic
155C2×C53#220211yescharacteristic
156C2×AGL(1,ℤ/8)—21no—
157#537C22×C5311yescharacteristic
158#481—22no—
159C2×#104—21no—
160#400C22×C5312yes—
161#393—22no—
162#391—21no—
163#390C22×C5311yescharacteristic
164#386—21no—
165#385—22no—
166#343C22×C5311yescharacteristic
167#327—22no—
168#324C22×C5312yes—
169C53#1212711yescharacteristic, direct product factor, Sylow
170C2×#41—21no—
171C22×Q8—21no—
172C22×D4D4×C5311yescharacteristic
173C23×C4—21no—
174AGL(1,ℤ/8)—42no—
175AGL(1,ℤ/8)—42no—
176C2×SD16—41no—
177C2×D8—41no—
178C2×M16—21no—
179C22⋊Q8—41no—
180#121—22no—
181#121—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4D4×C5312yes—
184C2×C4⋊C4C23×C5311yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42D4×C5311yescharacteristic
189#106—42no—
190#104—42no—
191#102—42no—
192C22⋊C8—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95D4×C5312yes—
197C24—21no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4SD16×C5312yes—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4C2×D4×C5311yescharacteristic
220C22×C4C2×D4×C5312yes—
221SD16—44no—
222D8—44no—
223M16—22no—
224M16—41no—
225C2×C8—41no—
226C2×C8—42no—
227C4⋊C4—22no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C22⋊C4—44no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C22⋊C4—81no—
235C42—41no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#120×C5311yescharacteristic
243Q8—41no—
244Q8—41no—
245Q8—84no—
246D4—42no—
247D4—41no—
248D4—41no—
249D4—81no—
250D4—82no—
251D4—81no—
252D4—81no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4C2×SD16×C5311yescharacteristic
271C2×C4#137×C5311yescharacteristic
272C8—42no—
273C8—82no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—42no—
278C22—82no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22C53×#44811yescharacteristic
285C22C53×#45311yescharacteristic
286C22C53×B(4,𝔽2)11yescharacteristic
287C4—21no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—82no—
292C4—82no—
293C4—81no—
294C2—21no—
295C2—21no—
296C2—82no—
297C2—161no—
298C2C53×#159611yescharacteristic
299C2C53×#219811yescharacteristic
300C2C53×#220211yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC53×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2804C2semidirect
3. C53×#1616C2semidirect
4. C53×#1353C2semidirect
5. C53×#1318C2semidirect
123. #12127C53direct (N × H)
144. #2804C2×C53semidirect
146. #1616C2×C53semidirect
147. #1353C2×C53semidirect
148. #1318C2×C53semidirect
169. C53#12127direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28042
C53×#20062
C53×#16162
C53×#13532
C53×#13182
#1212753