Finite Group Databasebeta

← C53×#12094 #27219606

Subgroups — C53×#12094 #27219606

Order 13568.

570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2815C211yesmaximal, characteristic, complemented
2C53×#2022C211yesmaximal, characteristic, complemented
3C53×#1595C212yesmaximal, complemented
4C53×#1343C212yesmaximal, complemented
5C53×#1317C211yesmaximal, characteristic, complemented
6C53×#534C2211yescharacteristic
7C2×#123×C53—21no—
8C2×#121×C53—21no—
9C53×#447—22no—
10C2×#99×C53—21no—
11C53×#397—22no—
12C53×#392—22no—
13C53×#392—21no—
14C53×#392C2212yes—
15C53×#381C2211yescharacteristic
16C53×#342C2211yescharacteristic
17C53×#327—22no—
18C53×#323C2212yes—
19C22×D4×C53—21no—
20C22×D4×C53—21no—
21C22×D4×C53—22no—
22C22×D4×C53—21no—
23C22×D4×C53D412yes—
24C23×C4×C53D411yescharacteristic
25C2×D8×C53—42no—
26#123×C53—41no—
27#123×C53—42no—
28#123×C53—42no—
29#121×C53—22no—
30#121×C53—42no—
31#121×C53—42no—
32#120×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—41no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53C2311yescharacteristic
42#102×C53—42no—
43#99×C53—42no—
44C22⋊C8×C53—22no—
45#95×C53—21no—
46#95×C53—22no—
47#95×C53D412yes—
48C24×C53—21no—
49C24×C53—22no—
50C24×C53C2×D411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—42no—
59C2×D4×C53—41no—
60C2×D4×C53—42no—
61C2×D4×C53—41no—
62C2×D4×C53—82no—
63C2×D4×C53—81no—
64C2×D4×C53—81no—
65C2×D4×C53—82no—
66C2×D4×C53—82no—
67C2×D4×C53—82no—
68C2×D4×C53—81no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53D812yescomplemented
78C22×C4×C53C2×D412yes—
79D8×C53—84no—
80C2×C8×C53—42no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—44no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—44no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—82no—
92C22⋊C4×C53—82no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—42no—
96C23×C53—41no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—41no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—41no—
105C23×C53—42no—
106C23×C53—82no—
107C23×C53#12011yescharacteristic
108C23×C53C2×D811yescharacteristic
109C23×C53AGL(1,ℤ/8)11yescharacteristic
110D4×C53—44no—
111D4×C53—82no—
112D4×C53—82no—
113D4×C53—84no—
114D4×C53—81no—
115D4×C53—81no—
116D4×C53—84no—
117D4×C53—84no—
118D4×C53—84no—
119D4×C53—81no—
120D4×C53—81no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—81no—
137C2×C4×C53—82no—
138C8×C53—82no—
139#12094C5311yesmaximal, characteristic, direct product factor, Sylow
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—42no—
146C22×C53—81no—
147C22×C53—82no—
148C22×C53—82no—
149C22×C53—82no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53—81no—
154C22×C53—82no—
155C22×C53—82no—
156C22×C53—82no—
157C22×C53—161no—
158C22×C53#44711yescharacteristic
159C22×C53#45311yescharacteristic
160C22×C53B(4,𝔽2)11yescharacteristic
161C4×C53—42no—
162C4×C53—42no—
163C4×C53—82no—
164C4×C53—81no—
165C4×C53—82no—
166C4×C53—81no—
167#2815C2×C5311yescharacteristic, complemented
168#2022C2×C5311yescharacteristic, complemented
169#1595C2×C5312yescomplemented
170#1343C2×C5312yescomplemented
171#1317C2×C5311yescharacteristic, complemented
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—41no—
175C2×C53—41no—
176C2×C53—162no—
177C2×C53—161no—
178C2×C53#159511yescharacteristic
179C2×C53#220012yes—
180#534C22×C5311yescharacteristic
181C2×#123—21no—
182C2×#121—21no—
183#447—22no—
184C2×#99—21no—
185#397—22no—
186#392—22no—
187#392—21no—
188#392C22×C5312yes—
189#381C22×C5311yescharacteristic
190#342C22×C5311yescharacteristic
191#327—22no—
192#323C22×C5312yes—
193C53#1209411yescharacteristic, direct product factor, Sylow
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—21no—
198C22×D4D4×C5312yes—
199C23×C4D4×C5311yescharacteristic
200C2×D8—42no—
201#123—42no—
202#123—41no—
203#123—42no—
204#121—22no—
205#121—42no—
206#121—42no—
207#120—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4D4×C5311yescharacteristic
216C2×C22⋊C4C23×C5311yescharacteristic
217#102—42no—
218#99—42no—
219C22⋊C8—22no—
220#95—21no—
221#95—22no—
222#95D4×C5312yes—
223C24—21no—
224C24—22no—
225C24C2×D4×C5311yescharacteristic
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4D8×C5312yescomplemented
253C22×C4C2×D4×C5312yes—
254D8—84no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23—82no—
282C23#120×C5311yescharacteristic
283C23C2×D8×C5311yescharacteristic
284C23AGL(1,ℤ/8)×C5311yescharacteristic
285D4—44no—
286D4—82no—
287D4—82no—
288D4—84no—
289D4—81no—
290D4—81no—
291D4—84no—
292D4—84no—
293D4—84no—
294D4—81no—
295D4—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C8—82no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—161no—
332C22C53×#44711yescharacteristic
333C22C53×#45311yescharacteristic
334C22C53×B(4,𝔽2)11yescharacteristic
335C4—42no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—82no—
340C4—81no—
341C2—21no—
342C2—21no—
343C2—41no—
344C2—41no—
345C2—162no—
346C2—161no—
347C2C53×#159511yescharacteristic
348C2C53×#220012yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2815C2semidirect
2. C53×#2022C2semidirect
3. C53×#1595C2semidirect
4. C53×#1343C2semidirect
5. C53×#1317C2semidirect
77. C22×C4×C53D8semidirect
139. #12094C53direct (N × H)
167. #2815C2×C53semidirect
168. #2022C2×C53semidirect
169. #1595C2×C53semidirect
170. #1343C2×C53semidirect
171. #1317C2×C53semidirect
193. C53#12094direct (N × H)
252. C22×C4D8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28152
C53×#20222
C53×#15952
C53×#13432
C53×#13172
#1209453