Finite Group Databasebeta

← C53×#12101 #27219613

Subgroups — C53×#12101 #27219613

Order 13568.

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

normal subgroups only

This group has 442 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×#2734C211yesmaximal, characteristic, complemented
2C53×#2007C211yesmaximal, characteristic
3C53×#1610C212yesmaximal
4C53×#1346C212yesmaximal, complemented
5C53×#1319C211yesmaximal, characteristic, complemented
6C2×#137×C53—21no—
7C53×#537C2211yescharacteristic
8C2×C4⋊Q8×C53—21no—
9C53×#480—22no—
10C2×#104×C53—21no—
11C53×#398C2212yes—
12C53×#395—22no—
13C53×#391C2211yescharacteristic
14C53×#389—21no—
15C53×#387—22no—
16C53×#343C2211yescharacteristic
17C53×#327—22no—
18C53×#324C2212yes—
19C2×#41×C53—21no—
20C22×Q8×C53—21no—
21C22×Q8×C53D411yescharacteristic
22#137×C53—42no—
23#137×C53—42no—
24C2×Q16×C53—41no—
25C2×SD16×C53—41no—
26C2×M16×C53—21no—
27C4⋊Q8×C53—42no—
28C4⋊Q8×C53—42no—
29C22⋊Q8×C53—41no—
30#121×C53—22no—
31#121×C53—41no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D412yes—
37C2×C4⋊C4×C53C2311yescharacteristic
38C2×C42×C53—21no—
39C2×C42×C53D411yescharacteristic
40#107×C53—42no—
41#104×C53—42no—
42#102×C53—42no—
43C22⋊C8×C53—22no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53D412yes—
48#41×C53—41no—
49#41×C53—41no—
50#41×C53—41no—
51#41×C53—41no—
52C2×Q8×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—42no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53—81no—
58C2×Q8×C53—81no—
59C2×Q8×C53D812yes—
60C2×D4×C53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D412yes—
70Q16×C53—44no—
71SD16×C53—44no—
72M16×C53—22no—
73M16×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53—42no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—44no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—81no—
84C22⋊C4×C53—42no—
85C42×C53—42no—
86C42×C53—41no—
87C23×C53—41no—
88C23×C53#12011yescharacteristic
89Q8×C53—42no—
90Q8×C53—41no—
91Q8×C53—41no—
92Q8×C53—81no—
93Q8×C53—84no—
94D4×C53—41no—
95D4×C53—41no—
96D4×C53—82no—
97D4×C53—81no—
98D4×C53—81no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—24no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
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×C53C2×D811yescharacteristic
119C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
120C8×C53—42no—
121C8×C53—82no—
122#12101C5311yesmaximal, characteristic, direct product factor, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—81no—
126C22×C53—81no—
127C22×C53—81no—
128C22×C53#44711yescharacteristic
129C22×C53#45311yescharacteristic
130C22×C53B(4,𝔽2)11yescharacteristic
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—44no—
134C4×C53—42no—
135C4×C53—42no—
136C4×C53—42no—
137C4×C53—82no—
138C4×C53—81no—
139#2734C2×C5311yescharacteristic, complemented
140#2007C2×C5311yescharacteristic
141#1610C2×C5312yes—
142#1346C2×C5312yescomplemented
143#1319C2×C5311yescharacteristic, complemented
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—161no—
147C2×C53#159511yescharacteristic
148C2×C53#220111yescharacteristic
149C2×C53#220511yescharacteristic
150C2×#137—21no—
151#537C22×C5311yescharacteristic
152C2×C4⋊Q8—21no—
153#480—22no—
154C2×#104—21no—
155#398C22×C5312yes—
156#395—22no—
157#391C22×C5311yescharacteristic
158#389—21no—
159#387—22no—
160#343C22×C5311yescharacteristic
161#327—22no—
162#324C22×C5312yes—
163C53#1210111yescharacteristic, direct product factor, Sylow
164C2×#41—21no—
165C22×Q8—21no—
166C22×Q8D4×C5311yescharacteristic
167#137—42no—
168#137—42no—
169C2×Q16—41no—
170C2×SD16—41no—
171C2×M16—21no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4D4×C5312yes—
182C2×C4⋊C4C23×C5311yescharacteristic
183C2×C42—21no—
184C2×C42D4×C5311yescharacteristic
185#107—42no—
186#104—42no—
187#102—42no—
188C22⋊C8—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95D4×C5312yes—
193#41—41no—
194#41—41no—
195#41—41no—
196#41—41no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8D8×C5312yes—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4C2×D4×C5311yescharacteristic
214C22×C4C2×D4×C5312yes—
215Q16—44no—
216SD16—44no—
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C2×C8—42no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—81no—
229C22⋊C4—42no—
230C42—41no—
231C42—42no—
232C23—41no—
233C23#120×C5311yescharacteristic
234Q8—42no—
235Q8—41no—
236Q8—41no—
237Q8—81no—
238Q8—84no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×D8×C5311yescharacteristic
264C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
265C8—42no—
266C8—82no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22C53×#44711yescharacteristic
273C22C53×#45311yescharacteristic
274C22C53×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—44no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2C53×#159511yescharacteristic
287C2C53×#220111yescharacteristic
288C2C53×#220511yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC53×#537C22
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×#2734C2semidirect
4. C53×#1346C2semidirect
5. C53×#1319C2semidirect
122. #12101C53direct (N × H)
139. #2734C2×C53semidirect
142. #1346C2×C53semidirect
143. #1319C2×C53semidirect
163. C53#12101direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27342
C53×#20072
C53×#16102
C53×#13462
C53×#13192
#1210153