Finite Group Databasebeta

← C53×#12367 #27219879

Subgroups — C53×#12367 #27219879

Order 13568.

336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3293C211yesmaximal, characteristic
2C53×#2230C212yesmaximal, complemented
3C53×#1716C211yesmaximal, characteristic, complemented
4C53×#1358C211yesmaximal, characteristic, complemented
5C53×#1336C212yesmaximal, complemented
6C53×#578—22no—
7C53×#498—21no—
8C53×#498C2211yescharacteristic
9C53×#494C2211yescharacteristic
10C53×#487—21no—
11C53×#473—22no—
12C53×#456—21no—
13C53×#456—21no—
14C53×#443C2212yes—
15C53×#365—22no—
16C53×#364C2212yes—
17C53×#360—22no—
18C53×#322C2211yescharacteristic
192−1+4×C53—21no—
202−1+4×C53—21no—
21#137×C53—22no—
22#137×C53—22no—
23#137×C53—42no—
24#137×C53—42no—
25#135×C53—42no—
26#135×C53D412yes—
27C2×Q16×C53—22no—
28C2×Q16×C53—42no—
29#131×C53—22no—
30C4⋊Q8×C53—21no—
31C4⋊Q8×C53D411yescharacteristic
32C4⋊Q8×C53D411yescharacteristic
33M32×C53—22no—
34#108×C53D412yes—
35#106×C53—22no—
36#106×C53—42no—
37#104×C53—22no—
38#103×C53—42no—
39#101×C53—21no—
40#101×C53—21no—
41C4×C8×C53—21no—
42C4×C8×C53C2311yescharacteristic
43#41×C53—22no—
44#41×C53—42no—
45#41×C53—42no—
46#41×C53—81no—
47#41×C53—81no—
48C2×Q8×C53—21no—
49C2×Q8×C53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53Q16×C53—42no—
54Q16×C53—42no—
55Q16×C53—42no—
56Q16×C53—42no—
57Q16×C53SD1612yes—
58SD16×C53—22no—
59SD16×C53—42no—
60SD16×C53—42no—
61D8×C53SD1612yes—
62M16×C53—22no—
63M16×C53—42no—
64C2×C8×C53—22no—
65C2×C8×C53—42no—
66C2×C8×C53C2×D412yes—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C42×C53C2×D411yescharacteristic
72C16×C53—42no—
73Q8×C53—22no—
74Q8×C53—42no—
75Q8×C53—42no—
76Q8×C53—82no—
77Q8×C53—81no—
78Q8×C53—81no—
79D4×C53—22no—
80D4×C53—42no—
81D4×C53—81no—
82D4×C53—81no—
83C2×C4×C53—21no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—82no—
88C2×C4×C53—82no—
89C2×C4×C53—81no—
90C2×C4×C53#12011yescharacteristic
91C8×C53—42no—
92C8×C53—42no—
93C8×C53C2×SD1612yes—
94C8×C53AGL(1,ℤ/8)12yes—
95#12367C5311yesmaximal, characteristic, direct product factor, Sylow
96C22×C53—42no—
97C22×C53#45311yescharacteristic
98C4×C53—22no—
99C4×C53—42no—
100C4×C53—81no—
101C4×C53—81no—
102C4×C53—161no—
103C4×C53#45012yes—
104#3293C2×C5311yescharacteristic
105#2230C2×C5312yescomplemented
106#1716C2×C5311yescharacteristic, complemented
107#1358C2×C5311yescharacteristic, complemented
108#1336C2×C5312yescomplemented
109C2×C53—21no—
110C2×C53—82no—
111C2×C53#162911yescharacteristic
112#578—22no—
113#498—21no—
114#498C22×C5311yescharacteristic
115#494C22×C5311yescharacteristic
116#487—21no—
117#473—22no—
118#456—21no—
119#456—21no—
120#443C22×C5312yes—
121#365—22no—
122#364C22×C5312yes—
123#360—22no—
124#322C22×C5311yescharacteristic
125C53#1236711yescharacteristic, direct product factor, Sylow
1262−1+4—21no—
1272−1+4—21no—
128#137—22no—
129#137—22no—
130#137—42no—
131#137—42no—
132#135—42no—
133#135D4×C5312yes—
134C2×Q16—22no—
135C2×Q16—42no—
136#131—22no—
137C4⋊Q8—21no—
138C4⋊Q8D4×C5311yescharacteristic
139C4⋊Q8D4×C5311yescharacteristic
140M32—22no—
141#108D4×C5312yes—
142#106—22no—
143#106—42no—
144#104—22no—
145#103—42no—
146#101—21no—
147#101—21no—
148C4×C8—21no—
149C4×C8C23×C5311yescharacteristic
150#41—22no—
151#41—42no—
152#41—42no—
153#41—81no—
154#41—81no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—42no—
159C2×Q8—42no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163Q16—42no—
164Q16SD16×C5312yes—
165SD16—22no—
166SD16—42no—
167SD16—42no—
168D8SD16×C5312yes—
169M16—22no—
170M16—42no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8C2×D4×C5312yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C42C2×D4×C5311yescharacteristic
179C16—42no—
180Q8—22no—
181Q8—42no—
182Q8—42no—
183Q8—82no—
184Q8—81no—
185Q8—81no—
186D4—22no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4#120×C5311yescharacteristic
198C8—42no—
199C8—42no—
200C8C2×SD16×C5312yes—
201C8AGL(1,ℤ/8)×C5312yes—
202C22—42no—
203C22C53×#45311yescharacteristic
204C4—22no—
205C4—42no—
206C4—81no—
207C4—81no—
208C4—161no—
209C4C53×#45012yes—
210C2—21no—
211C2—82no—
212C2C53×#162911yescharacteristic
subgroupG/H
center, socleC2×C53#1629
commutator, Frattini, cosocleC4×C8C23×C53
prime radicalC53×#3293C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2230C2semidirect
3. C53×#1716C2semidirect
4. C53×#1358C2semidirect
5. C53×#1336C2semidirect
95. #12367C53direct (N × H)
105. #2230C2×C53semidirect
106. #1716C2×C53semidirect
107. #1358C2×C53semidirect
108. #1336C2×C53semidirect
125. C53#12367direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32932
C53×#22302
C53×#17162
C53×#13582
C53×#13362
#1236753