Finite Group Databasebeta

← C53×#12975 #27220487

Subgroups — C53×#12975 #27220487

Order 13568.

416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2979C211yesmaximal, characteristic
2C53×#2132C211yesmaximal, characteristic
3C53×#1884C211yesmaximal, characteristic
4C53×#1858C212yesmaximal, complemented
5C53×#1395C212yesmaximal, complemented
6C53×#568C2211yescharacteristic
7C53×#532C2211yescharacteristic
8C53×#436—22no—
9C8×D4×C53—22no—
10C53×#431—21no—
11C53×#430—22no—
12C53×#421—21no—
13C53×#413C2212yes—
14C53×#410C2211yescharacteristic
15C53×#408—21no—
16C53×#408C2212yes—
17C53×#356C412yescomplemented
18C53×#355C412yescomplemented
19C53×#341—22no—
20C53×#339—21no—
21C2×#41×C53D411yescharacteristic
22C2×#41×C53C2311yescharacteristic
23#131×C53—42no—
24C2×M16×C53—21no—
25C2×M16×C53—21no—
26C2×M16×C53—22no—
27C2×M16×C53—22no—
28C2×M16×C53D412yes—
29C22×C8×C53—21no—
30C22×C8×C53D412yes—
31C4⋊Q8×C53C2×C411yescharacteristic
32#127×C53Q811yescharacteristic
33#124×C53C2×C411yescharacteristic
34#124×C53Q811yescharacteristic
35#121×C53—22no—
36C4×D4×C53—22no—
37#117×C53—21no—
38C2×C42×C53D411yescharacteristic
39#108×C53—44no—
40C4⋊C8×C53—21no—
41C4⋊C8×C53—21no—
42#104×C53—42no—
43#101×C53C2×C414yes—
44#100×C53—22no—
45#99×C53—22no—
46C22⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—22no—
49C4×C8×C53—22no—
50#41×C53—41no—
51#41×C53—41no—
52#41×C53—41no—
53#41×C53—41no—
54C2×Q8×C53C4⋊C411yescharacteristic
55C2×Q8×C53C22×C411yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53C4⋊C411yescharacteristic
59C2×D4×C53C2×Q811yescharacteristic
60C2×D4×C53#4112yes—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65M16×C53—22no—
66M16×C53—24no—
67M16×C53—24no—
68M16×C53—42no—
69M16×C53—41no—
70M16×C53—42no—
71M16×C53—81no—
72C2×C8×C53—22no—
73C2×C8×C53—22no—
74C2×C8×C53—42no—
75C2×C8×C53—41no—
76C2×C8×C53—42no—
77C2×C8×C53—42no—
78C2×C8×C53—81no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—41no—
84C42×C53—21no—
85C42×C53—41no—
86C42×C53C4⋊C411yescharacteristic
87C42×C53C4⋊C411yescharacteristic
88C23×C53—21no—
89C23×C53C22⋊Q812yes—
90C23×C53#12311yescharacteristic
91Q8×C53—41no—
92Q8×C53—41no—
93D4×C53—41no—
94D4×C53—42no—
95D4×C53—41no—
96D4×C53—44no—
97D4×C53—81no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—81no—
109C2×C4×C53C2×C4⋊C411yescharacteristic
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12011yescharacteristic
112C8×C53—44no—
113C8×C53—42no—
114C8×C53—42no—
115C8×C53—42no—
116#12975C5311yesmaximal, characteristic, direct product factor, Sylow
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53#38511yescharacteristic
124C4×C53—22no—
125C4×C53—22no—
126C4×C53—21no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—161no—
130C4×C53B(4,𝔽2)11yescharacteristic
131C4×C53#45811yescharacteristic
132#2979C2×C5311yescharacteristic
133#2132C2×C5311yescharacteristic
134#1884C2×C5311yescharacteristic
135#1858C2×C5312yescomplemented
136#1395C2×C5312yescomplemented
137C2×C53—21no—
138C2×C53—41no—
139C2×C53—42no—
140C2×C53—81no—
141C2×C53#186711yescharacteristic
142#568C22×C5311yescharacteristic
143#532C22×C5311yescharacteristic
144#436—22no—
145C8×D4—22no—
146#431—21no—
147#430—22no—
148#421—21no—
149#413C22×C5312yes—
150#410C22×C5311yescharacteristic
151#408—21no—
152#408C22×C5312yes—
153#356C4×C5312yescomplemented
154#355C4×C5312yescomplemented
155#341—22no—
156#339—21no—
157C53#1297511yescharacteristic, direct product factor, Sylow
158C2×#41D4×C5311yescharacteristic
159C2×#41C23×C5311yescharacteristic
160#131—42no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16D4×C5312yes—
166C22×C8—21no—
167C22×C8D4×C5312yes—
168C4⋊Q8C2×C4×C5311yescharacteristic
169#127Q8×C5311yescharacteristic
170#124C2×C4×C5311yescharacteristic
171#124Q8×C5311yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—21no—
175C2×C42D4×C5311yescharacteristic
176#108—44no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179#104—42no—
180#101C2×C4×C5314yes—
181#100—22no—
182#99—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C4×C8—22no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊C4×C5311yescharacteristic
192C2×Q8C22×C4×C5311yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊C4×C5311yescharacteristic
196C2×D4C2×Q8×C5311yescharacteristic
197C2×D4#41×C5312yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×D4×C5311yescharacteristic
201C22×C4C2×D4×C5312yes—
202M16—22no—
203M16—24no—
204M16—24no—
205M16—42no—
206M16—41no—
207M16—42no—
208M16—81no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—81no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—41no—
221C42—21no—
222C42—41no—
223C42C4⋊C4×C5311yescharacteristic
224C42C4⋊C4×C5311yescharacteristic
225C23—21no—
226C23C22⋊Q8×C5312yes—
227C23#123×C5311yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—44no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4C2×C4⋊C4×C5311yescharacteristic
247C2×C4C4×D4×C5312yes—
248C2×C4#120×C5311yescharacteristic
249C8—44no—
250C8—42no—
251C8—42no—
252C8—42no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22C53×#38511yescharacteristic
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—161no—
266C4C53×B(4,𝔽2)11yescharacteristic
267C4C53×#45811yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2C53×#186711yescharacteristic
subgroupG/H
centerC4×C53B(4,𝔽2)
commutatorC2×Q8C22×C4×C53
Frattini, cosocleC2×#41C23×C53
socleC2×C53#1867
prime radicalC53×#532C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1858C2semidirect
5. C53×#1395C2semidirect
17. C53×#356C4semidirect
18. C53×#355C4semidirect
116. #12975C53direct (N × H)
135. #1858C2×C53semidirect
136. #1395C2×C53semidirect
153. #356C4×C53semidirect
154. #355C4×C53semidirect
157. C53#12975direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#29792
C53×#21322
C53×#18842
C53×#18582
C53×#13952
#1297553