Finite Group Databasebeta

← C53×#17419 #27224931

Subgroups — C53×#17419 #27224931

Order 13568.

1342 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#550C211yesmaximal, characteristic
2C53×#2880C212yesmaximal, complemented
3C53×#1471C216yesmaximal, complemented
4C53×#1465C213yesmaximal, complemented
5C53×#1459C213yesmaximal, complemented
6C22×#41×C53C411yescharacteristic
7C22×#41×C53C2211yescharacteristic
8C2×#131×C53—22no—
9C53×#550—22no—
10C53×#550—26no—
11C2×#127×C53C413yes—
12C2×#121×C53C413yes—
13C2×#121×C53C2213yes—
14C2×C4×Q8×C53C411yescharacteristic
15C2×C4×D4×C53C2213yes—
16C53×#408—26no—
17C53×#407C2216yes—
18C2×C22⋊C8×C53C2216yes—
19C53×#324C2211yescharacteristic
20C53×#324C2213yes—
21C53×#323C2216yes—
22C53×#323C2216yes—
23C2×#41×C53—23no—
24C2×#41×C53—21no—
25C2×#41×C53—44no—
26C2×#41×C53—44no—
27C2×#41×C53D412yes—
28C2×#41×C53D416yes—
29C22×Q8×C53C2×C411yescharacteristic
30C22×D4×C53C2×C413yes—
31C22×D4×C53C2×C413yes—
32C22×D4×C53C2×C416yes—
33C22×D4×C53C2×C413yes—
34C23×C4×C53C2×C413yes—
35C23×C4×C53C2313yes—
36#131×C53—48no—
37C2×M16×C53—26no—
38C22×C8×C53—26no—
39#127×C53—26no—
40#127×C53—26no—
41#121×C53—26no—
42#121×C53—26no—
43#121×C53—26no—
44#121×C53—26no—
45C4×Q8×C53—23no—
46C4×Q8×C53—21no—
47C4×D4×C53—26no—
48C4×D4×C53—26no—
49C2×C4⋊C4×C53C2×C413yes—
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53C2×C413yes—
52C2×C22⋊C4×C53C2313yes—
53C2×C42×C53C2×C413yes—
54C22⋊C8×C53—26no—
55C22⋊C8×C53—26no—
56C22⋊C8×C53C2312yes—
57C22⋊C8×C53C2316yes—
58C24×C53C22×C413yes—
59C24×C53C22×C413yes—
60#41×C53—28no—
61#41×C53—412no—
62#41×C53—412no—
63#41×C53—44no—
64C2×Q8×C53—24no—
65C2×Q8×C53C22⋊C413yes—
66C2×Q8×C53C22⋊C411yescharacteristic
67C2×D4×C53—26no—
68C2×D4×C53—212no—
69C2×D4×C53—23no—
70C2×D4×C53—23no—
71C2×D4×C53—26no—
72C2×D4×C53—46no—
73C2×D4×C53—43no—
74C2×D4×C53—46no—
75C2×D4×C53—86no—
76C2×D4×C53C22⋊C416yes—
77C2×D4×C53C22⋊C416yes—
78C22×C4×C53—23no—
79C22×C4×C53—23no—
80C22×C4×C53—46no—
81C22×C4×C53—46no—
82C22×C4×C53C22×C413yes—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C22×C413yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D416yes—
87C22×C4×C53C2×D416yes—
88C22×C4×C53C2411yescharacteristic
89M16×C53—412no—
90C2×C8×C53—22no—
91C2×C8×C53—26no—
92C2×C8×C53—412no—
93C4⋊C4×C53—23no—
94C4⋊C4×C53—23no—
95C4⋊C4×C53—22no—
96C22⋊C4×C53—26no—
97C22⋊C4×C53—26no—
98C42×C53—23no—
99C42×C53—23no—
100C23×C53—23no—
101C23×C53—23no—
102C23×C53—23no—
103C23×C53—23no—
104C23×C53—23no—
105C23×C53—43no—
106C23×C53—43no—
107C23×C53—43no—
108C23×C53—43no—
109C23×C53C2×C22⋊C416yes—
110C23×C53C23×C411yescharacteristic
111Q8×C53—26no—
112Q8×C53#9914yes—
113D4×C53—212no—
114D4×C53—46no—
115D4×C53—412no—
116D4×C53—46no—
117D4×C53—46no—
118D4×C53—812no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—26no—
122C2×C4×C53—212no—
123C2×C4×C53—21no—
124C2×C4×C53—23no—
125C2×C4×C53—41no—
126C2×C4×C53—46no—
127C2×C4×C53—46no—
128C2×C4×C53—46no—
129C2×C4×C53—41no—
130C2×C4×C53—43no—
131C2×C4×C53C2×C22⋊C413yes—
132C2×C4×C53C2×C22⋊C413yes—
133C2×C4×C53#13112yescomplemented
134C2×C4×C53C22×D412yes—
135C8×C53—42no—
136C8×C53—46no—
137#17419C5311yesmaximal, characteristic, direct product factor, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—26no—
141C22×C53—26no—
142C22×C53—26no—
143C22×C53—43no—
144C22×C53—43no—
145C22×C53—43no—
146C22×C53—83no—
147C22×C53—83no—
148C22×C53—83no—
149C22×C53—83no—
150C22×C53C22×C22⋊C411yescharacteristic
151C22×C53C2×#13111yescharacteristic
152C22×C53#56811yescharacteristic
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—23no—
156C4×C53—41no—
157C4×C53—43no—
158C4×C53C2×#9916yes—
159C2×#550C2×C5311yescharacteristic
160#2880C2×C5312yescomplemented
161#1471C2×C5316yescomplemented
162#1465C2×C5313yescomplemented
163#1459C2×C5313yescomplemented
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—46no—
167C2×C53—83no—
168C2×C53#288011yescharacteristic
169C2×C53C22×#9911yescharacteristic
170C2×C53#288811yescharacteristic
171C22×#41C4×C5311yescharacteristic
172C22×#41C22×C5311yescharacteristic
173C2×#131—22no—
174#550—22no—
175#550—26no—
176C2×#127C4×C5313yes—
177C2×#121C4×C5313yes—
178C2×#121C22×C5313yes—
179C2×C4×Q8C4×C5311yescharacteristic
180C2×C4×D4C22×C5313yes—
181#408—26no—
182#407C22×C5316yes—
183C2×C22⋊C8C22×C5316yes—
184#324C22×C5311yescharacteristic
185#324C22×C5313yes—
186#323C22×C5316yes—
187#323C22×C5316yes—
188C53#1741911yescharacteristic, direct product factor, Sylow
189C2×#41—23no—
190C2×#41—21no—
191C2×#41—44no—
192C2×#41—44no—
193C2×#41D4×C5312yes—
194C2×#41D4×C5316yes—
195C22×Q8C2×C4×C5311yescharacteristic
196C22×D4C2×C4×C5313yes—
197C22×D4C2×C4×C5313yes—
198C22×D4C2×C4×C5316yes—
199C22×D4C2×C4×C5313yes—
200C23×C4C2×C4×C5313yes—
201C23×C4C23×C5313yes—
202#131—48no—
203C2×M16—26no—
204C22×C8—26no—
205#127—26no—
206#127—26no—
207#121—26no—
208#121—26no—
209#121—26no—
210#121—26no—
211C4×Q8—23no—
212C4×Q8—21no—
213C4×D4—26no—
214C4×D4—26no—
215C2×C4⋊C4C2×C4×C5313yes—
216C2×C4⋊C4C23×C5311yescharacteristic
217C2×C22⋊C4C2×C4×C5313yes—
218C2×C22⋊C4C23×C5313yes—
219C2×C42C2×C4×C5313yes—
220C22⋊C8—26no—
221C22⋊C8—26no—
222C22⋊C8C23×C5312yes—
223C22⋊C8C23×C5316yes—
224C24C22×C4×C5313yes—
225C24C22×C4×C5313yes—
226#41—28no—
227#41—412no—
228#41—412no—
229#41—44no—
230C2×Q8—24no—
231C2×Q8C22⋊C4×C5313yes—
232C2×Q8C22⋊C4×C5311yescharacteristic
233C2×D4—26no—
234C2×D4—212no—
235C2×D4—23no—
236C2×D4—23no—
237C2×D4—26no—
238C2×D4—46no—
239C2×D4—43no—
240C2×D4—46no—
241C2×D4—86no—
242C2×D4C22⋊C4×C5316yes—
243C2×D4C22⋊C4×C5316yes—
244C22×C4—23no—
245C22×C4—23no—
246C22×C4—46no—
247C22×C4—46no—
248C22×C4C22×C4×C5313yes—
249C22×C4C22×C4×C5311yescharacteristic
250C22×C4C22×C4×C5313yes—
251C22×C4C22×C4×C5311yescharacteristic
252C22×C4C2×D4×C5316yes—
253C22×C4C2×D4×C5316yes—
254C22×C4C24×C5311yescharacteristic
255M16—412no—
256C2×C8—22no—
257C2×C8—26no—
258C2×C8—412no—
259C4⋊C4—23no—
260C4⋊C4—23no—
261C4⋊C4—22no—
262C22⋊C4—26no—
263C22⋊C4—26no—
264C42—23no—
265C42—23no—
266C23—23no—
267C23—23no—
268C23—23no—
269C23—23no—
270C23—23no—
271C23—43no—
272C23—43no—
273C23—43no—
274C23—43no—
275C23C2×C22⋊C4×C5316yes—
276C23C23×C4×C5311yescharacteristic
277Q8—26no—
278Q8#99×C5314yes—
279D4—212no—
280D4—46no—
281D4—412no—
282D4—46no—
283D4—46no—
284D4—812no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—212no—
288C2×C4—26no—
289C2×C4—23no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—46no—
293C2×C4—46no—
294C2×C4—46no—
295C2×C4—41no—
296C2×C4—43no—
297C2×C4C2×C22⋊C4×C5313yes—
298C2×C4C2×C22⋊C4×C5313yes—
299C2×C4#131×C5312yescomplemented
300C2×C4C22×D4×C5312yes—
301C8—42no—
302C8—46no—
303C22—21no—
304C22—21no—
305C22—26no—
306C22—26no—
307C22—26no—
308C22—43no—
309C22—43no—
310C22—43no—
311C22—83no—
312C22—83no—
313C22—83no—
314C22—83no—
315C22C22×C22⋊C4×C5311yescharacteristic
316C22C2×#131×C5311yescharacteristic
317C22C53×#56811yescharacteristic
318C4—22no—
319C4—22no—
320C4—23no—
321C4—41no—
322C4—43no—
323C4C2×#99×C5316yes—
324C2—21no—
325C2—21no—
326C2—46no—
327C2—83no—
328C2C53×#288011yescharacteristic
329C2C22×#99×C5311yescharacteristic
330C2C53×#288811yescharacteristic
subgroupG/H
center, socleC22×C53C22×C22⋊C4
commutatorC23C23×C4×C53
Frattini, cosocleC22×C4C24×C53
prime radicalC2×C53×#550C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2880C2semidirect
3. C53×#1471C2semidirect
4. C53×#1465C2semidirect
5. C53×#1459C2semidirect
133. C2×C4×C53#131semidirect
137. #17419C53direct (N × H)
160. #2880C2×C53semidirect
161. #1471C2×C53semidirect
162. #1465C2×C53semidirect
163. #1459C2×C53semidirect
188. C53#17419direct (N × H)
299. C2×C4#131×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5502
C53×#28802
C53×#14712
C53×#14652
C53×#14592
#1741953