Finite Group Databasebeta

← C53×#12971 #27220483

Subgroups — C53×#12971 #27220483

Order 13568.

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

normal subgroups only

This group has 424 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×#2381C211yesmaximal, characteristic
2C2×C53×#356C211yesmaximal, characteristic
3C53×#1886C211yesmaximal, characteristic
4C53×#1857C212yesmaximal
5C53×#1394C212yesmaximal
6C2×C4⋊Q8×C53C2211yescharacteristic
7C53×#519C2211yescharacteristic
8C53×#428—22no—
9C53×#419—21no—
10C2×#101×C53C2211yescharacteristic
11C2×#101×C53C2212yes—
12C53×#391—21no—
13C53×#391C2212yes—
14C53×#387—22no—
15C53×#387—22no—
16C53×#383—21no—
17C53×#356C414yescomplemented
18C53×#339—22no—
19C53×#339—21no—
20C22×Q8×C53D411yescharacteristic
21C22×Q8×C53C2311yescharacteristic
22C2×M16×C53—22no—
23C2×M16×C53—21no—
24C4⋊Q8×C53—22no—
25C4⋊Q8×C53C2×C412yes—
26C4⋊Q8×C53Q812yes—
27C4×Q8×C53—42no—
28#117×C53—21no—
29#117×C53—22no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53D412yes—
33C2×C42×C53—21no—
34C2×C42×C53D411yescharacteristic
35C2×C42×C53D412yes—
36#107×C53—42no—
37#106×C53—42no—
38#103×C53—42no—
39#101×C53—22no—
40#101×C53—22no—
41#101×C53C2×C414yes—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53—21no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C4⋊C412yes—
54C2×Q8×C53C22×C411yescharacteristic
55C2×Q8×C53C2×Q811yescharacteristic
56C2×Q8×C53#4112yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63M16×C53—24no—
64M16×C53—42no—
65M16×C53—41no—
66M16×C53—41no—
67C2×C8×C53—42no—
68C2×C8×C53—41no—
69C4⋊C4×C53—24no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—81no—
76C22⋊C4×C53—41no—
77C42×C53—21no—
78C42×C53—41no—
79C42×C53—42no—
80C42×C53—42no—
81C42×C53—81no—
82C42×C53C4⋊C412yes—
83C23×C53#12011yescharacteristic
84Q8×C53—41no—
85Q8×C53—41no—
86Q8×C53—42no—
87Q8×C53—42no—
88Q8×C53—81no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—44no—
99C2×C4×C53—44no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53C2×C4⋊C411yescharacteristic
104C2×C4×C53C4×D412yes—
105C2×C4×C53C22⋊Q812yes—
106C2×C4×C53#12311yescharacteristic
107C8×C53—44no—
108C8×C53—81no—
109#12971C5311yesmaximal, characteristic, direct product factor, Sylow
110C22×C53—22no—
111C22×C53#38511yescharacteristic
112C22×C53B(4,𝔽2)11yescharacteristic
113C22×C53#45811yescharacteristic
114C4×C53—22no—
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—22no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53—82no—
121C4×C53—82no—
122C4×C53—82no—
123#2381C2×C5311yescharacteristic
124C2×#356C2×C5311yescharacteristic
125#1886C2×C5311yescharacteristic
126#1857C2×C5312yes—
127#1394C2×C5312yes—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#186711yescharacteristic
131C2×C53#220312yes—
132C2×C4⋊Q8C22×C5311yescharacteristic
133#519C22×C5311yescharacteristic
134#428—22no—
135#419—21no—
136C2×#101C22×C5311yescharacteristic
137C2×#101C22×C5312yes—
138#391—21no—
139#391C22×C5312yes—
140#387—22no—
141#387—22no—
142#383—21no—
143#356C4×C5314yescomplemented
144#339—22no—
145#339—21no—
146C53#1297111yescharacteristic, direct product factor, Sylow
147C22×Q8D4×C5311yescharacteristic
148C22×Q8C23×C5311yescharacteristic
149C2×M16—22no—
150C2×M16—21no—
151C4⋊Q8—22no—
152C4⋊Q8C2×C4×C5312yes—
153C4⋊Q8Q8×C5312yes—
154C4×Q8—42no—
155#117—21no—
156#117—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4D4×C5312yes—
160C2×C42—21no—
161C2×C42D4×C5311yescharacteristic
162C2×C42D4×C5312yes—
163#107—42no—
164#106—42no—
165#103—42no—
166#101—22no—
167#101—22no—
168#101C2×C4×C5314yes—
169#95—21no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95—22no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊C4×C5312yes—
181C2×Q8C22×C4×C5311yescharacteristic
182C2×Q8C2×Q8×C5311yescharacteristic
183C2×Q8#41×C5312yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4C2×D4×C5311yescharacteristic
189C22×C4C2×D4×C5312yes—
190M16—24no—
191M16—42no—
192M16—41no—
193M16—41no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C22⋊C4—41no—
204C42—21no—
205C42—42no—
206C42—42no—
207C42—41no—
208C42—81no—
209C42C4⋊C4×C5312yes—
210C23#120×C5311yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—42no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4C2×C4⋊C4×C5311yescharacteristic
231C2×C4C4×D4×C5312yes—
232C2×C4C22⋊Q8×C5312yes—
233C2×C4#123×C5311yescharacteristic
234C8—44no—
235C8—81no—
236C22—22no—
237C22C53×#38511yescharacteristic
238C22C53×B(4,𝔽2)11yescharacteristic
239C22C53×#45811yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—82no—
249C2—21no—
250C2—21no—
251C2C53×#186711yescharacteristic
252C2C53×#220312yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×Q8C22×C4×C53
Frattini, cosocleC22×Q8C23×C53
prime radicalC23×C53#120
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
17. C53×#356C4semidirect
109. #12971C53direct (N × H)
143. #356C4×C53semidirect
146. C53#12971direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#23812
C2×C53×#3562
C53×#18862
C53×#18572
C53×#13942
#1297153