Finite Group Databasebeta

← C53×#8532 #27216044

Subgroups — C53×#8532 #27216044

Order 13568.

800 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3548C211yesmaximal, characteristic
2C53×#1994C211yesmaximal, characteristic, complemented
3C53×#1962C212yesmaximal, complemented
4C53×#1561C211yesmaximal, characteristic, complemented
5C53×#1560C212yesmaximal, complemented
6C53×#562C414yescomplemented
7C53×#554—22no—
8C53×#550—22no—
9C53×#547—22no—
10C53×#540—22no—
11C53×#537—24no—
12C53×#533C2211yescharacteristic
13C53×#532C2212yes—
14C53×#423C2212yes—
15C53×#423C2211yescharacteristic
16C2×#104×C53—24no—
17C43×C53C2211yescharacteristic
18C53×#339—22no—
19C53×#333—22no—
20C53×#332—22no—
21C53×#331—22no—
22C2×#41×C53—24no—
23C2×#41×C53D414yes—
24C2×M16×C53—21no—
25C2×M16×C53—22no—
26C4⋊Q8×C53D412yes—
27C4⋊Q8×C53D412yes—
28#127×C53—22no—
29#127×C53D412yes—
30#125×C53—22no—
31#125×C53C2×C412yes—
32#124×C53C2×C414yes—
33#123×C53—28no—
34C22⋊Q8×C53—24no—
35C22⋊Q8×C53—28no—
36C22⋊Q8×C53—22no—
37#121×C53—28no—
38#121×C53—24no—
39#121×C53—28no—
40#121×C53—22no—
41#121×C53—22no—
42C4×Q8×C53—22no—
43C4×D4×C53—24no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46#117×C53D412yes—
47C2×C4⋊C4×C53—24no—
48C2×C42×C53—21no—
49C2×C42×C53—22no—
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53—22no—
52C4⋊C8×C53—22no—
53C4⋊C8×C53—22no—
54#104×C53—48no—
55#41×C53—44no—
56#41×C53—44no—
57#41×C53—84no—
58#41×C53—84no—
59C2×Q8×C53—22no—
60C2×Q8×C53C22⋊C414yes—
61C2×D4×C53—28no—
62C2×D4×C53—24no—
63C2×D4×C53—22no—
64C2×D4×C53—22no—
65C2×D4×C53C22⋊C414yes—
66C22×C4×C53—28no—
67C22×C4×C53—24no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D412yes—
71M16×C53—44no—
72M16×C53—42no—
73C2×C8×C53—42no—
74C2×C8×C53—41no—
75C4⋊C4×C53—28no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53C22⋊C414yes—
81C22⋊C4×C53—28no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—24no—
85C42×C53—22no—
86C42×C53—24no—
87C42×C53—22no—
88C42×C53—41no—
89C42×C53—42no—
90C42×C53C22×C411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D412yes—
93C23×C53—24no—
94C23×C53#12011yescharacteristic
95Q8×C53—44no—
96Q8×C53—42no—
97Q8×C53—42no—
98D4×C53—44no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—42no—
102D4×C53—42no—
103D4×C53—84no—
104D4×C53—88no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—24no—
110C2×C4×C53—24no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53—44no—
114C2×C4×C53—44no—
115C2×C4×C53—88no—
116C2×C4×C53C2×C22⋊C412yes—
117C2×C4×C53C2×C22⋊C411yescharacteristic
118C2×C4×C53#12012yes—
119C2×C4×C53#12011yescharacteristic
120C8×C53—82no—
121C8×C53—81no—
122#8532C5311yesmaximal, characteristic, direct product factor, Sylow
123C22×C53—22no—
124C22×C53—21no—
125C22×C53—44no—
126C22×C53—44no—
127C22×C53—44no—
128C22×C53#37911yescharacteristic
129C4×C53—22no—
130C4×C53—21no—
131C4×C53—44no—
132C4×C53—44no—
133C4×C53—44no—
134C4×C53#45312yes—
135C4×C53#45411yescharacteristic
136C4×C53#45512yes—
137C4×C53#45611yescharacteristic
138#3548C2×C5311yescharacteristic
139#1994C2×C5311yescharacteristic, complemented
140#1962C2×C5312yescomplemented
141#1561C2×C5311yescharacteristic, complemented
142#1560C2×C5312yescomplemented
143C2×C53—41no—
144C2×C53—84no—
145C2×C53#179212yes—
146C2×C53#179311yescharacteristic
147#562C4×C5314yescomplemented
148#554—22no—
149#550—22no—
150#547—22no—
151#540—22no—
152#537—24no—
153#533C22×C5311yescharacteristic
154#532C22×C5312yes—
155#423C22×C5311yescharacteristic
156#423C22×C5312yes—
157C2×#104—24no—
158C43C22×C5311yescharacteristic
159#339—22no—
160#333—22no—
161#332—22no—
162#331—22no—
163C53#853211yescharacteristic, direct product factor, Sylow
164C2×#41—24no—
165C2×#41D4×C5314yes—
166C2×M16—21no—
167C2×M16—22no—
168C4⋊Q8D4×C5312yes—
169C4⋊Q8D4×C5312yes—
170#127—22no—
171#127D4×C5312yes—
172#125—22no—
173#125C2×C4×C5312yes—
174#124C2×C4×C5314yes—
175#123—28no—
176C22⋊Q8—24no—
177C22⋊Q8—28no—
178C22⋊Q8—22no—
179#121—28no—
180#121—24no—
181#121—28no—
182#121—22no—
183#121—22no—
184C4×Q8—22no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—22no—
188#117D4×C5312yes—
189C2×C4⋊C4—24no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42C23×C5311yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196#104—48no—
197#41—44no—
198#41—44no—
199#41—84no—
200#41—84no—
201C2×Q8—22no—
202C2×Q8C22⋊C4×C5314yes—
203C2×D4—28no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4C22⋊C4×C5314yes—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—41no—
211C22×C4C2×D4×C5311yescharacteristic
212C22×C4C2×D4×C5312yes—
213M16—44no—
214M16—42no—
215C2×C8—42no—
216C2×C8—41no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4C22⋊C4×C5314yes—
223C22⋊C4—28no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C42—22no—
228C42—24no—
229C42—22no—
230C42—41no—
231C42—42no—
232C42C22×C4×C5311yescharacteristic
233C42C2×D4×C5311yescharacteristic
234C42C2×D4×C5312yes—
235C23—24no—
236C23#120×C5311yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—44no—
241D4—42no—
242D4—42no—
243D4—42no—
244D4—42no—
245D4—84no—
246D4—88no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—88no—
258C2×C4C2×C22⋊C4×C5311yescharacteristic
259C2×C4C2×C22⋊C4×C5312yes—
260C2×C4#120×C5312yes—
261C2×C4#120×C5311yescharacteristic
262C8—82no—
263C8—81no—
264C22—22no—
265C22—21no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22C53×#37911yescharacteristic
270C4—22no—
271C4—21no—
272C4—44no—
273C4—44no—
274C4—44no—
275C4C53×#45312yes—
276C4C53×#45411yescharacteristic
277C4C53×#45512yes—
278C4C53×#45611yescharacteristic
279C2—41no—
280C2—84no—
281C2C53×#179212yes—
282C2C53×#179311yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3548C2
LCSG ▷ 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×#1994C2semidirect
3. C53×#1962C2semidirect
4. C53×#1561C2semidirect
5. C53×#1560C2semidirect
6. C53×#562C4semidirect
122. #8532C53direct (N × H)
139. #1994C2×C53semidirect
140. #1962C2×C53semidirect
141. #1561C2×C53semidirect
142. #1560C2×C53semidirect
147. #562C4×C53semidirect
163. C53#8532direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35482
C53×#19942
C53×#19622
C53×#15612
C53×#15602
#853253