Finite Group Databasebeta

← C53×#7494 #27215006

Subgroups — C53×#7494 #27215006

Order 13568.

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2126C211yesmaximal, characteristic
2C53×#1884C211yesmaximal, characteristic
3C53×#1536C211yesmaximal, characteristic
4C53×#568C411yescharacteristic, complemented
5C53×#532C2211yescharacteristic
6C53×#432—21no—
7C53×#421—21no—
8C53×#410—21no—
9C53×#410C411yescharacteristic, complemented
10C53×#354—21no—
11C53×#353C412yescomplemented
12C53×#339—21no—
13C53×#329—21no—
14C53×#329C412yescomplemented
15C2×#41×C53C2×C411yescharacteristic
16C2×#41×C53D411yescharacteristic
17#131×C53—42no—
18C2×M16×C53—21no—
19C2×M16×C53—41no—
20C4⋊Q8×C53Q811yescharacteristic
21#127×C53C2×C411yescharacteristic
22#124×C53C2×C411yescharacteristic
23#124×C53D411yescharacteristic
24#121×C53—41no—
25C4×D4×C53—41no—
26#117×C53—21no—
27#117×C53—41no—
28C2×C42×C53D411yescharacteristic
29C4⋊C8×C53—41no—
30#104×C53—42no—
31#99×C53—22no—
32#99×C53—41no—
33#99×C53—41no—
34C22⋊C8×C53—41no—
35#97×C53—22no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C4⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4211yescharacteristic
46C2×D4×C53C22⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—22no—
52M16×C53—41no—
53M16×C53—42no—
54C2×C8×C53—41no—
55C2×C8×C53—42no—
56C2×C8×C53—42no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—81no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—41no—
64C42×C53—21no—
65C42×C53—41no—
66C42×C53—81no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C4⋊C411yescharacteristic
69C23×C53—21no—
70C23×C53—21no—
71C23×C53#9911yescharacteristic
72Q8×C53—41no—
73Q8×C53—41no—
74D4×C53—41no—
75D4×C53—41no—
76D4×C53—42no—
77D4×C53—81no—
78D4×C53—82no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—81no—
90C2×C4×C53—81no—
91C2×C4×C53—82no—
92C2×C4×C53#9511yescharacteristic
93C2×C4×C53#9911yescharacteristic
94C8×C53—42no—
95C8×C53—82no—
96#7494C5311yesmaximal, characteristic, direct product factor, Sylow
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53—81no—
103C22×C53#34211yescharacteristic
104C4×C53—21no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—161no—
110C4×C53—162no—
111C4×C53AΣL(1,𝔽16)11yescharacteristic
112C4×C53#35211yescharacteristic
113#2126C2×C5311yescharacteristic
114#1884C2×C5311yescharacteristic
115#1536C2×C5311yescharacteristic
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—81no—
119C2×C53—81no—
120C2×C53#139011yescharacteristic
121#568C4×C5311yescharacteristic, complemented
122#532C22×C5311yescharacteristic
123#432—21no—
124#421—21no—
125#410—21no—
126#410C4×C5311yescharacteristic, complemented
127#354—21no—
128#353C4×C5312yescomplemented
129#339—21no—
130#329—21no—
131#329C4×C5312yescomplemented
132C53#749411yescharacteristic, direct product factor, Sylow
133C2×#41C2×C4×C5311yescharacteristic
134C2×#41D4×C5311yescharacteristic
135#131—42no—
136C2×M16—21no—
137C2×M16—41no—
138C4⋊Q8Q8×C5311yescharacteristic
139#127C2×C4×C5311yescharacteristic
140#124C2×C4×C5311yescharacteristic
141#124D4×C5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145#117—41no—
146C2×C42D4×C5311yescharacteristic
147C4⋊C8—41no—
148#104—42no—
149#99—22no—
150#99—41no—
151#99—41no—
152C22⋊C8—41no—
153#97—22no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8C4⋊C4×C5311yescharacteristic
159C2×Q8C4⋊C4×C5311yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C42×C5311yescharacteristic
164C2×D4C22⋊C4×C5311yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C22⋊C4×C5311yescharacteristic
169M16—22no—
170M16—41no—
171M16—42no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42—81no—
185C42C22⋊C4×C5311yescharacteristic
186C42C4⋊C4×C5311yescharacteristic
187C23—21no—
188C23—21no—
189C23#99×C5311yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4#95×C5311yescharacteristic
211C2×C4#99×C5311yescharacteristic
212C8—42no—
213C8—82no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22C53×#34211yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—161no—
227C4—162no—
228C4C53×AΣL(1,𝔽16)11yescharacteristic
229C4C53×#35211yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2C53×#139011yescharacteristic
subgroupG/H
center, socleC2×C53#1390
commutatorC2×D4C42×C53
Frattini, cosocle#532C22×C53
prime radicalC53×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#532 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#568C4semidirect
9. C53×#410C4semidirect
11. C53×#353C4semidirect
14. C53×#329C4semidirect
96. #7494C53direct (N × H)
121. #568C4×C53semidirect
126. #410C4×C53semidirect
128. #353C4×C53semidirect
131. #329C4×C53semidirect
132. C53#7494direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#21262
C53×#18842
C53×#15362
#749453