Finite Group Databasebeta

← C53×#12315 #27219827

Subgroups — C53×#12315 #27219827

Order 13568.

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

normal subgroups only

This group has 362 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×#3291C211yesmaximal, characteristic
2C53×#2214C212yesmaximal, complemented
3C53×#1722C211yesmaximal, characteristic, complemented
4C53×#1359C211yesmaximal, characteristic, complemented
5C53×#1336C212yesmaximal, complemented
6C53×#577—22no—
7C53×#510—22no—
8C53×#500—21no—
9C53×#494C2211yescharacteristic, complemented
10C53×#492C2211yescharacteristic
11C53×#486—21no—
12C53×#471—22no—
13C53×#456—21no—
14C53×#453—21no—
15C53×#443C2212yes—
16C53×#364C2212yescomplemented
17C53×#362—22no—
18C53×#322C2211yescharacteristic, complemented
192−1+4×C53—21no—
20H(5,𝔽2)×C53—21no—
21#137×C53—22no—
22#137×C53—42no—
23AGL(1,ℤ/8)×C53—22no—
24AGL(1,ℤ/8)×C53—42no—
25#135×C53—42no—
26#135×C53D412yes—
27C2×Q16×C53—22no—
28C2×SD16×C53—22no—
29C2×SD16×C53—42no—
30#131×C53—22no—
31C4⋊Q8×C53—21no—
32C4⋊Q8×C53D411yescharacteristic
33#127×C53D411yescharacteristic
34Q32×C53—42no—
35SD32×C53—42no—
36M32×C53—22no—
37#108×C53D412yes—
38#107×C53—42no—
39#104×C53—22no—
40#102×C53—42no—
41#101×C53—21no—
42#100×C53—21no—
43C4×C8×C53—21no—
44C4×C8×C53C2311yescharacteristic
45#41×C53—22no—
46#41×C53—42no—
47#41×C53—42no—
48#41×C53—81no—
49C2×Q8×C53—21no—
50C2×Q8×C53—42no—
51C2×Q8×C53—41no—
52C2×D4×C53—21no—
53C2×D4×C53—41no—
54C2×D4×C53—42no—
55C2×D4×C53—81no—
56Q16×C53—42no—
57Q16×C53—42no—
58Q16×C53—42no—
59Q16×C53D812yes—
60SD16×C53—22no—
61SD16×C53—42no—
62SD16×C53—42no—
63SD16×C53—42no—
64SD16×C53—42no—
65D8×C53—42no—
66D8×C53D812yes—
67M16×C53—22no—
68M16×C53—42no—
69C2×C8×C53—22no—
70C2×C8×C53—42no—
71C2×C8×C53C2×D412yes—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C42×C53C2×D411yescharacteristic
76C16×C53—42no—
77C23×C53—41no—
78C23×C53—81no—
79Q8×C53—22no—
80Q8×C53—42no—
81Q8×C53—81no—
82Q8×C53—82no—
83D4×C53—22no—
84D4×C53—42no—
85D4×C53—42no—
86D4×C53—82no—
87D4×C53—82no—
88D4×C53—81no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—81no—
93C2×C4×C53—82no—
94C2×C4×C53—81no—
95C2×C4×C53#12011yescharacteristic
96C8×C53—42no—
97C8×C53—42no—
98C8×C53C2×D812yes—
99C8×C53AGL(1,ℤ/8)12yes—
100#12315C5311yesmaximal, characteristic, direct product factor, Sylow
101C22×C53—42no—
102C22×C53—81no—
103C22×C53—161no—
104C22×C53—321no—
105C22×C53#45311yescharacteristic
106C4×C53—22no—
107C4×C53—42no—
108C4×C53—81no—
109C4×C53—161no—
110C4×C53#44712yes—
111#3291C2×C5311yescharacteristic
112#2214C2×C5312yescomplemented
113#1722C2×C5311yescharacteristic, complemented
114#1359C2×C5311yescharacteristic, complemented
115#1336C2×C5312yescomplemented
116C2×C53—21no—
117C2×C53—82no—
118C2×C53—161no—
119C2×C53#161911yescharacteristic
120#577—22no—
121#510—22no—
122#500—21no—
123#494C22×C5311yescharacteristic, complemented
124#492C22×C5311yescharacteristic
125#486—21no—
126#471—22no—
127#456—21no—
128#453—21no—
129#443C22×C5312yes—
130#364C22×C5312yescomplemented
131#362—22no—
132#322C22×C5311yescharacteristic, complemented
133C53#1231511yescharacteristic, direct product factor, Sylow
1342−1+4—21no—
135H(5,𝔽2)—21no—
136#137—22no—
137#137—42no—
138AGL(1,ℤ/8)—22no—
139AGL(1,ℤ/8)—42no—
140#135—42no—
141#135D4×C5312yes—
142C2×Q16—22no—
143C2×SD16—22no—
144C2×SD16—42no—
145#131—22no—
146C4⋊Q8—21no—
147C4⋊Q8D4×C5311yescharacteristic
148#127D4×C5311yescharacteristic
149Q32—42no—
150SD32—42no—
151M32—22no—
152#108D4×C5312yes—
153#107—42no—
154#104—22no—
155#102—42no—
156#101—21no—
157#100—21no—
158C4×C8—21no—
159C4×C8C23×C5311yescharacteristic
160#41—22no—
161#41—42no—
162#41—42no—
163#41—81no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—81no—
171Q16—42no—
172Q16—42no—
173Q16—42no—
174Q16D8×C5312yes—
175SD16—22no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179SD16—42no—
180D8—42no—
181D8D8×C5312yes—
182M16—22no—
183M16—42no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8C2×D4×C5312yes—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C42C2×D4×C5311yescharacteristic
191C16—42no—
192C23—41no—
193C23—81no—
194Q8—22no—
195Q8—42no—
196Q8—81no—
197Q8—82no—
198D4—22no—
199D4—42no—
200D4—42no—
201D4—82no—
202D4—82no—
203D4—81no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4#120×C5311yescharacteristic
211C8—42no—
212C8—42no—
213C8C2×D8×C5312yes—
214C8AGL(1,ℤ/8)×C5312yes—
215C22—42no—
216C22—81no—
217C22—161no—
218C22—321no—
219C22C53×#45311yescharacteristic
220C4—22no—
221C4—42no—
222C4—81no—
223C4—161no—
224C4C53×#44712yes—
225C2—21no—
226C2—82no—
227C2—161no—
228C2C53×#161911yescharacteristic
subgroupG/H
center, socleC2×C53#1619
commutator, Frattini, cosocleC4×C8C23×C53
prime radicalC53×#3291C2
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×#2214C2semidirect
3. C53×#1722C2semidirect
4. C53×#1359C2semidirect
5. C53×#1336C2semidirect
9. C53×#494C22semidirect
16. C53×#364C22semidirect
18. C53×#322C22semidirect
100. #12315C53direct (N × H)
112. #2214C2×C53semidirect
113. #1722C2×C53semidirect
114. #1359C2×C53semidirect
115. #1336C2×C53semidirect
123. #494C22×C53semidirect
130. #364C22×C53semidirect
132. #322C22×C53semidirect
133. C53#12315direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32912
C53×#22142
C53×#17222
C53×#13592
C53×#13362
#1231553