Finite Group Databasebeta

← C53×#12970 #27220482

Subgroups — C53×#12970 #27220482

Order 13568.

472 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2379C211yesmaximal, characteristic
2C2×C53×#355C211yesmaximal, characteristic
3C53×#1888C211yesmaximal, characteristic
4C53×#1857C212yesmaximal, complemented
5C53×#1394C212yesmaximal, complemented
6C2×#124×C53C2211yescharacteristic
7C53×#519C2211yescharacteristic
8C53×#428—22no—
9C53×#418—21no—
10C2×#101×C53C2211yescharacteristic
11C2×#101×C53C2212yes—
12C53×#391C2212yes—
13C53×#388—22no—
14C53×#386—21no—
15C53×#385—22no—
16C53×#382—21no—
17C53×#355C414yescomplemented
18C53×#339—22no—
19C53×#339—21no—
20C22×Q8×C53C2311yescharacteristic
21C22×D4×C53D411yescharacteristic
22C23×C4×C53—21no—
23C2×M16×C53—22no—
24C2×M16×C53—21no—
25#124×C53—22no—
26#124×C53C2×C412yes—
27#124×C53Q812yes—
28C4×Q8×C53—41no—
29C4×Q8×C53—41no—
30#117×C53—21no—
31#117×C53—22no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53D412yes—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53D411yescharacteristic
39C2×C42×C53D412yes—
40#107×C53—42no—
41#106×C53—42no—
42#102×C53—41no—
43#102×C53—41no—
44#101×C53—22no—
45#101×C53—22no—
46#101×C53C2×C414yescomplemented
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C22×C411yescharacteristic
54C2×Q8×C53C2×Q811yescharacteristic
55C2×Q8×C53#4112yes—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53C4⋊C412yes—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—82no—
64C22×C4×C53—81no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D412yes—
67M16×C53—24no—
68M16×C53—42no—
69M16×C53—41no—
70M16×C53—41no—
71C2×C8×C53—42no—
72C2×C8×C53—41no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—81no—
79C4⋊C4×C53—81no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—24no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—82no—
84C22⋊C4×C53—81no—
85C42×C53—21no—
86C42×C53—41no—
87C42×C53—42no—
88C42×C53—42no—
89C42×C53C4⋊C412yes—
90C23×C53—22no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12011yescharacteristic
96Q8×C53—41no—
97Q8×C53—41no—
98Q8×C53—42no—
99D4×C53—42no—
100D4×C53—81no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
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—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—84no—
117C2×C4×C53C2×C4⋊C411yescharacteristic
118C2×C4×C53C4×D412yes—
119C2×C4×C53C22⋊Q812yes—
120C2×C4×C53#12311yescharacteristic
121C8×C53—44no—
122C8×C53—81no—
123#12970C5311yesmaximal, 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#38511yescharacteristic
132C22×C53B(4,𝔽2)11yescharacteristic
133C22×C53#45811yescharacteristic
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—42no—
139C4×C53—82no—
140C4×C53—81no—
141C4×C53—82no—
142C4×C53—81no—
143#2379C2×C5311yescharacteristic
144C2×#355C2×C5311yescharacteristic
145#1888C2×C5311yescharacteristic
146#1857C2×C5312yescomplemented
147#1394C2×C5312yescomplemented
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—82no—
151C2×C53#186711yescharacteristic
152C2×C53#220211yescharacteristic
153C2×C53#220311yescharacteristic
154C2×#124C22×C5311yescharacteristic
155#519C22×C5311yescharacteristic
156#428—22no—
157#418—21no—
158C2×#101C22×C5311yescharacteristic
159C2×#101C22×C5312yes—
160#391C22×C5312yes—
161#388—22no—
162#386—21no—
163#385—22no—
164#382—21no—
165#355C4×C5314yescomplemented
166#339—22no—
167#339—21no—
168C53#1297011yescharacteristic, direct product factor, Sylow
169C22×Q8C23×C5311yescharacteristic
170C22×D4D4×C5311yescharacteristic
171C23×C4—21no—
172C2×M16—22no—
173C2×M16—21no—
174#124—22no—
175#124C2×C4×C5312yes—
176#124Q8×C5312yes—
177C4×Q8—41no—
178C4×Q8—41no—
179#117—21no—
180#117—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4D4×C5312yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42D4×C5311yescharacteristic
188C2×C42D4×C5312yes—
189#107—42no—
190#106—42no—
191#102—41no—
192#102—41no—
193#101—22no—
194#101—22no—
195#101C2×C4×C5314yescomplemented
196#95—22no—
197#95—22no—
198#95—22no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C22×C4×C5311yescharacteristic
203C2×Q8C2×Q8×C5311yescharacteristic
204C2×Q8#41×C5312yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊C4×C5312yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—82no—
213C22×C4—81no—
214C22×C4C2×D4×C5311yescharacteristic
215C22×C4C2×D4×C5312yes—
216M16—24no—
217M16—42no—
218M16—41no—
219M16—41no—
220C2×C8—42no—
221C2×C8—41no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—82no—
233C22⋊C4—81no—
234C42—21no—
235C42—42no—
236C42—42no—
237C42—41no—
238C42C4⋊C4×C5312yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#120×C5311yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4C2×C4⋊C4×C5311yescharacteristic
267C2×C4C4×D4×C5312yes—
268C2×C4C22⋊Q8×C5312yes—
269C2×C4#123×C5311yescharacteristic
270C8—44no—
271C8—81no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22C53×#38511yescharacteristic
280C22C53×B(4,𝔽2)11yescharacteristic
281C22C53×#45811yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—82no—
289C4—81no—
290C4—81no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2C53×#186711yescharacteristic
295C2C53×#220211yescharacteristic
296C2C53×#220311yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×Q8C22×C4×C53
Frattini, cosocleC22×Q8C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1857C2semidirect
5. C53×#1394C2semidirect
17. C53×#355C4semidirect
46. #101×C53C2×C4semidirect
123. #12970C53direct (N × H)
146. #1857C2×C53semidirect
147. #1394C2×C53semidirect
165. #355C4×C53semidirect
168. C53#12970direct (N × H)
195. #101C2×C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#23792
C2×C53×#3552
C53×#18882
C53×#18572
C53×#13942
#1297053