Finite Group Databasebeta

← C53×#7754 #27215266

Subgroups — C53×#7754 #27215266

Order 13568.

536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2683C211yesmaximal, characteristic
2C53×#2058C212yesmaximal, complemented
3C53×#2013C211yesmaximal, characteristic
4C53×#1271C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C22×SD16×C53—21no—
7C2×C4⋊Q8×C53C2211yescharacteristic
8C2×C22⋊Q8×C53—22no—
9C2×#106×C53—22no—
10C2×#103×C53—21no—
11C53×#396—22no—
12C53×#395—21no—
13C53×#390C2211yescharacteristic
14C53×#389C2212yes—
15C53×#386—21no—
16C53×#340—22no—
17C53×#336C2211yescharacteristic
18C53×#336C2212yes—
19C22×Q8×C53—21no—
20C22×Q8×C53—21no—
21C22×D4×C53D411yescharacteristic
22C23×C4×C53—21no—
23C2×SD16×C53—42no—
24C2×SD16×C53—42no—
25C2×SD16×C53—42no—
26C22×C8×C53—21no—
27C22×C8×C53—22no—
28C4⋊Q8×C53—42no—
29C22⋊Q8×C53—44no—
30C22⋊Q8×C53—44no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53D412yes—
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2311yescharacteristic
41#106×C53—44no—
42#103×C53—42no—
43#95×C53—21no—
44#95×C53—22no—
45C24×C53—21no—
46C2×Q8×C53—22no—
47C2×Q8×C53—44no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—81no—
54C2×D4×C53—21no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53SD1612yes—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—44no—
62C22×C4×C53—41no—
63C22×C4×C53—81no—
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66SD16×C53—48no—
67C2×C8×C53—22no—
68C2×C8×C53—24no—
69C2×C8×C53—42no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—42no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—44no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53SD1614yescomplemented
82C22⋊C4×C53—44no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—81no—
86C42×C53—41no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53#12011yescharacteristic
93Q8×C53—42no—
94Q8×C53—42no—
95Q8×C53—81no—
96Q8×C53—81no—
97Q8×C53—84no—
98D4×C53—42no—
99D4×C53—81no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—24no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—84no—
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53C2×SD1611yescharacteristic
117C2×C4×C53C2×SD1612yes—
118C2×C4×C53AGL(1,ℤ/8)12yes—
119C2×C4×C53#13711yescharacteristic
120C8×C53—44no—
121C8×C53—42no—
122#7754C5311yesmaximal, characteristic, direct product factor, Sylow
123C22×C53—42no—
124C22×C53—44no—
125C22×C53—82no—
126C22×C53—81no—
127C22×C53—81no—
128C22×C53#44811yescharacteristic
129C22×C53#45012yes—
130C22×C53#45311yescharacteristic
131C22×C53#45612yes—
132C22×C53B(4,𝔽2)11yescharacteristic
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—44no—
138C4×C53—82no—
139C4×C53—82no—
140C4×C53—81no—
141#2683C2×C5311yescharacteristic
142#2058C2×C5312yescomplemented
143#2013C2×C5311yescharacteristic
144#1271C2×C5312yescomplemented
145#1270C2×C5311yescharacteristic
146C2×C53—82no—
147C2×C53#159611yescharacteristic
148C2×C53#160412yes—
149C2×C53#162911yescharacteristic
150C2×C53#165112yes—
151C2×C53#166011yescharacteristic
152C22×SD16—21no—
153C2×C4⋊Q8C22×C5311yescharacteristic
154C2×C22⋊Q8—22no—
155C2×#106—22no—
156C2×#103—21no—
157#396—22no—
158#395—21no—
159#390C22×C5311yescharacteristic
160#389C22×C5312yes—
161#386—21no—
162#340—22no—
163#336C22×C5312yes—
164#336C22×C5311yescharacteristic
165C53#775411yescharacteristic, direct product factor, Sylow
166C22×Q8—21no—
167C22×Q8—21no—
168C22×D4D4×C5311yescharacteristic
169C23×C4—21no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×SD16—42no—
173C22×C8—21no—
174C22×C8—22no—
175C4⋊Q8—42no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4D4×C5312yes—
182C2×C4⋊C4D4×C5312yes—
183C2×C4⋊C4D4×C5311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42C23×C5311yescharacteristic
188#106—44no—
189#103—42no—
190#95—21no—
191#95—22no—
192C24—21no—
193C2×Q8—22no—
194C2×Q8—44no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—81no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4SD16×C5312yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—44no—
209C22×C4—41no—
210C22×C4—81no—
211C22×C4C2×D4×C5312yes—
212C22×C4C2×D4×C5311yescharacteristic
213SD16—48no—
214C2×C8—22no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—42no—
220C4⋊C4—22no—
221C4⋊C4—24no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4SD16×C5314yescomplemented
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23#120×C5311yescharacteristic
240Q8—42no—
241Q8—42no—
242Q8—81no—
243Q8—81no—
244Q8—84no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—84no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×SD16×C5311yescharacteristic
264C2×C4C2×SD16×C5312yes—
265C2×C4AGL(1,ℤ/8)×C5312yes—
266C2×C4#137×C5311yescharacteristic
267C8—44no—
268C8—42no—
269C22—42no—
270C22—44no—
271C22—81no—
272C22—82no—
273C22—81no—
274C22C53×#44811yescharacteristic
275C22C53×#45012yes—
276C22C53×#45311yescharacteristic
277C22C53×#45612yes—
278C22C53×B(4,𝔽2)11yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—81no—
287C2—82no—
288C2C53×#159611yescharacteristic
289C2C53×#160412yes—
290C2C53×#162911yescharacteristic
291C2C53×#165112yes—
292C2C53×#166011yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2058C2semidirect
4. C53×#1271C2semidirect
81. C4⋊C4×C53SD16semidirect
122. #7754C53direct (N × H)
142. #2058C2×C53semidirect
144. #1271C2×C53semidirect
165. C53#7754direct (N × H)
228. C4⋊C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26832
C53×#20582
C53×#20132
C53×#12712
C53×#12702
#775453