Finite Group Databasebeta

← C53×#12288 #27219800

Subgroups — C53×#12288 #27219800

Order 13568.

354 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3245C211yesmaximal, characteristic
2C53×#2232C212yesmaximal, complemented
3C53×#2086C211yesmaximal, characteristic, complemented
4C53×#1347C212yesmaximal, complemented
5C53×#1322C211yesmaximal, characteristic, complemented
6C53×#532—21no—
7C53×#474—22no—
8C53×#466—21no—
9C53×#466C2212yes—
10C53×#459—22no—
11C2×#107×C53C2211yescharacteristic
12C2×C4⋊C8×C53C2211yescharacteristic
13C53×#389—21no—
14C53×#366—22no—
15C53×#357—22no—
16C53×C22⋊C16C2212yes—
17C53×#340—21no—
18C53×#340C2211yescharacteristic
19C2×#41×C53—41no—
20C2×D8×C53—21no—
21C2×D8×C53D412yes—
22C22×C8×C53D411yescharacteristic
23C22×C8×C53C2311yescharacteristic
24C4⋊Q8×C53—21no—
25#127×C53—21no—
26#124×C53—41no—
27#121×C53—22no—
28#121×C53—41no—
29C4×D4×C53—22no—
30C4×D4×C53—41no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—41no—
34C2×C42×C53D411yescharacteristic
35C2×C16×C53—22no—
36#107×C53—21no—
37#107×C53—22no—
38#107×C53D412yes—
39C4⋊C8×C53—22no—
40#102×C53—22no—
41#102×C53—22no—
42#95×C53—41no—
43#41×C53—82no—
44#41×C53—82no—
45C2×Q8×C53—41no—
46C2×D4×C53—22no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53—42no—
50C22×C4×C53—42no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53D811yescharacteristic
54C22×C4×C53D811yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56D8×C53—42no—
57D8×C53—81no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C2×C8×C53C2×D412yes—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—82no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C42×C53—21no—
72C42×C53—21no—
73C16×C53—42no—
74C23×C53—42no—
75C23×C53C2×D811yescharacteristic
76Q8×C53—82no—
77D4×C53—42no—
78D4×C53—42no—
79D4×C53—82no—
80D4×C53—82no—
81D4×C53—82no—
82D4×C53—82no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—84no—
92C2×C4×C53—81no—
93C2×C4×C53—81no—
94C2×C4×C53—81no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53D1612yes—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C8×C53—22no—
101C8×C53—41no—
102#12288C5311yesmaximal, characteristic, direct product factor, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—82no—
106C22×C53—82no—
107C22×C53—82no—
108C22×C53#44711yescharacteristic
109C22×C53C2×D1611yescharacteristic
110C22×C53#50911yescharacteristic
111C4×C53—21no—
112C4×C53—22no—
113C4×C53—41no—
114C4×C53—82no—
115C4×C53—82no—
116C4×C53—161no—
117C4×C53#45611yescharacteristic
118C4×C53#45811yescharacteristic
119#3245C2×C5311yescharacteristic
120#2232C2×C5312yescomplemented
121#2086C2×C5311yescharacteristic, complemented
122#1347C2×C5312yescomplemented
123#1322C2×C5311yescharacteristic, complemented
124C2×C53—21no—
125C2×C53—21no—
126C2×C53—162no—
127C2×C53#161011yescharacteristic
128C2×C53#218411yescharacteristic
129C2×C53#219111yescharacteristic
130#532—21no—
131#474—22no—
132#466—21no—
133#466C22×C5312yes—
134#459—22no—
135C2×#107C22×C5311yescharacteristic
136C2×C4⋊C8C22×C5311yescharacteristic
137#389—21no—
138#366—22no—
139#357—22no—
140C22⋊C16C22×C5312yes—
141#340—21no—
142#340C22×C5311yescharacteristic
143C53#1228811yescharacteristic, direct product factor, Sylow
144C2×#41—41no—
145C2×D8—21no—
146C2×D8D4×C5312yes—
147C22×C8D4×C5311yescharacteristic
148C22×C8C23×C5311yescharacteristic
149C4⋊Q8—21no—
150#127—21no—
151#124—41no—
152#121—22no—
153#121—41no—
154C4×D4—22no—
155C4×D4—41no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42D4×C5311yescharacteristic
160C2×C16—22no—
161#107—21no—
162#107—22no—
163#107D4×C5312yes—
164C4⋊C8—22no—
165#102—22no—
166#102—22no—
167#95—41no—
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4D8×C5311yescharacteristic
179C22×C4D8×C5311yescharacteristic
180C22×C4C2×D4×C5311yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×D4×C5312yes—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—82no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—42no—
199C23—42no—
200C23C2×D8×C5311yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4D16×C5312yes—
223C2×C4#120×C5311yescharacteristic
224C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C53×#44711yescharacteristic
233C22C2×D16×C5311yescharacteristic
234C22C53×#50911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—82no—
240C4—161no—
241C4C53×#45611yescharacteristic
242C4C53×#45811yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2C53×#161011yescharacteristic
247C2C53×#218411yescharacteristic
248C2C53×#219111yescharacteristic
subgroupG/H
center, socleC22×C53#447
commutator, Frattini, cosocleC22×C8C23×C53
prime radicalC53×#3245C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4⋊C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2232C2semidirect
3. C53×#2086C2semidirect
4. C53×#1347C2semidirect
5. C53×#1322C2semidirect
102. #12288C53direct (N × H)
120. #2232C2×C53semidirect
121. #2086C2×C53semidirect
122. #1347C2×C53semidirect
123. #1322C2×C53semidirect
143. C53#12288direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32452
C53×#22322
C53×#20862
C53×#13472
C53×#13222
#1228853