Finite Group Databasebeta

← C53×#12302 #27219814

Subgroups — C53×#12302 #27219814

Order 13568.

310 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3202C211yesmaximal, characteristic
2C53×#2239C212yesmaximal, complemented
3C53×#2098C211yesmaximal, characteristic
4C53×#1340C212yesmaximal, complemented
5C53×#1321C211yesmaximal, characteristic
6C53×#536—21no—
7C53×#485—22no—
8C53×#470C2212yes—
9C53×#467—21no—
10C53×#451—22no—
11C53×#430C2211yescharacteristic
12C53×#424—21no—
13C53×#421—21no—
14C53×#408C2211yescharacteristic
15C53×#365—22no—
16C53×#362—22no—
17C53×#349C2212yes—
18C53×H(3,ℤ/4)C2211yescharacteristic
19C2×#41×C53D411yescharacteristic
20C22×Q8×C53—21no—
21C2×Q16×C53—21no—
22C2×Q16×C53D412yes—
23C2×M16×C53—21no—
24C2×M16×C53C2311yescharacteristic
25C22×C8×C53D411yescharacteristic
26C4⋊Q8×C53—41no—
27#124×C53—41no—
28#123×C53—41no—
29#123×C53—41no—
30C22⋊Q8×C53—22no—
31C22⋊Q8×C53—22no—
32#117×C53—21no—
33#117×C53—21no—
34M32×C53—22no—
35#108×C53—22no—
36#107×C53—21no—
37#106×C53D412yes—
38C4⋊C8×C53—42no—
39#104×C53—42no—
40#103×C53—22no—
41#103×C53—22no—
42C22⋊C8×C53—22no—
43#41×C53—41no—
44#41×C53—41no—
45C2×Q8×C53—22no—
46C2×Q8×C53—41no—
47C2×Q8×C53—81no—
48C2×Q8×C53—81no—
49C2×Q8×C53SD1611yescharacteristic
50C2×D4×C53—21no—
51C2×D4×C53SD1611yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—41no—
54C22×C4×C53C2×D411yescharacteristic
55Q16×C53—42no—
56Q16×C53—81no—
57M16×C53—41no—
58M16×C53—41no—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53—41no—
62C2×C8×C53—41no—
63C2×C8×C53C2×D412yes—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—41no—
71C22⋊C4×C53—42no—
72C42×C53—22no—
73C42×C53—41no—
74C16×C53—42no—
75C23×C53—21no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77Q8×C53—41no—
78Q8×C53—42no—
79Q8×C53—42no—
80Q8×C53—82no—
81D4×C53—41no—
82D4×C53—81no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—82no—
92C2×C4×C53#12011yescharacteristic
93C2×C4×C53C2×SD1611yescharacteristic
94C8×C53—22no—
95C8×C53—41no—
96C8×C53—81no—
97#12302C5311yesmaximal, characteristic, direct product factor, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53#45011yescharacteristic
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—82no—
106C4×C53—82no—
107C4×C53—82no—
108C4×C53#45311yescharacteristic
109C4×C53#45811yescharacteristic
110#3202C2×C5311yescharacteristic
111#2239C2×C5312yescomplemented
112#2098C2×C5311yescharacteristic
113#1340C2×C5312yescomplemented
114#1321C2×C5311yescharacteristic
115C2×C53—21no—
116C2×C53—41no—
117C2×C53—81no—
118C2×C53#161311yescharacteristic
119#536—21no—
120#485—22no—
121#470C22×C5312yes—
122#467—21no—
123#451—22no—
124#430C22×C5311yescharacteristic
125#424—21no—
126#421—21no—
127#408C22×C5311yescharacteristic
128#365—22no—
129#362—22no—
130#349C22×C5312yes—
131H(3,ℤ/4)C22×C5311yescharacteristic
132C53#1230211yescharacteristic, direct product factor, Sylow
133C2×#41D4×C5311yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16D4×C5312yes—
137C2×M16—21no—
138C2×M16C23×C5311yescharacteristic
139C22×C8D4×C5311yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—21no—
148M32—22no—
149#108—22no—
150#107—21no—
151#106D4×C5312yes—
152C4⋊C8—42no—
153#104—42no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8SD16×C5311yescharacteristic
164C2×D4—21no—
165C2×D4SD16×C5311yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×D4×C5311yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—41no—
173M16—41no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8C2×D4×C5312yes—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C42—22no—
187C42—41no—
188C16—42no—
189C23—21no—
190C23AGL(1,ℤ/8)×C5311yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—82no—
206C2×C4#120×C5311yescharacteristic
207C2×C4C2×SD16×C5311yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22C53×#45011yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4C53×#45311yescharacteristic
222C4C53×#45811yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2C53×#161311yescharacteristic
subgroupG/H
center, socleC2×C53#1613
commutator, Frattini, cosocleC2×M16C23×C53
prime radicalC2×#41×C53D4
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×M16×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2239C2semidirect
4. C53×#1340C2semidirect
97. #12302C53direct (N × H)
111. #2239C2×C53semidirect
113. #1340C2×C53semidirect
132. C53#12302direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32022
C53×#22392
C53×#20982
C53×#13402
C53×#13212
#1230253