Finite Group Databasebeta

← C53×#7602 #27215114

Subgroups — C53×#7602 #27215114

Order 13568.

408 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C4×#125×C53C211yesmaximal, characteristic
2C53×#1831C212yesmaximal
3C53×#1271C214yesmaximal
4C2×#125×C53C411yescharacteristic
5C2×C4⋊C8×C53—22no—
6C53×#389C2212yes—
7C53×#387—22no—
8C53×#382C411yescharacteristic
9C4×C4⋊C4×C53—22no—
10C4×C4⋊C4×C53C412yes—
11C43×C53C2211yescharacteristic
12C53×#336—22no—
13C53×#336C2214yes—
14C22×C8×C53—24no—
15#125×C53—22no—
16#125×C53C814yescomplemented
17C2×C4⋊C4×C53—22no—
18C2×C4⋊C4×C53—22no—
19C2×C4⋊C4×C53C2×C412yes—
20C2×C4⋊C4×C53C2×C412yes—
21C2×C42×C53—22no—
22C2×C42×C53C2×C411yescharacteristic
23C2×C42×C53C2×C411yescharacteristic
24C2×C42×C53D412yes—
25C2×C42×C53D412yes—
26C2×C42×C53C2311yescharacteristic
27C4⋊C8×C53—44no—
28#95×C53—22no—
29#95×C53—22no—
30#95×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53—22no—
33C22×C4×C53C22⋊C411yescharacteristic
34C22×C4×C53C22⋊C412yes—
35C22×C4×C53C22⋊C411yescharacteristic
36C22×C4×C53C22×C411yescharacteristic
37C22×C4×C53C2×D412yes—
38C2×C8×C53—44no—
39C2×C8×C53—44no—
40C2×C8×C53—44no—
41C4⋊C4×C53—28no—
42C4⋊C4×C53—22no—
43C4⋊C4×C53—42no—
44C4⋊C4×C53—42no—
45C4⋊C4×C53—42no—
46C4⋊C4×C53C2×C814yes—
47C42×C53—22no—
48C42×C53—22no—
49C42×C53—22no—
50C42×C53—42no—
51C42×C53—41no—
52C42×C53—44no—
53C42×C53C2×C812yes—
54C42×C53M1612yes—
55C23×C53C2×C22⋊C411yescharacteristic
56C2×C4×C53—21no—
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—24no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53C22⋊C814yes—
72C2×C4×C53#10112yes—
73C2×C4×C53C22×C811yescharacteristic
74C2×C4×C53C2×M1611yescharacteristic
75C2×C4×C53AGL(1,ℤ/8)12yes—
76C2×C4×C53#13712yes—
77C8×C53—84no—
78#7602C5311yesmaximal, characteristic, direct product factor, Sylow
79C22×C53C2×C22⋊C811yescharacteristic
80C22×C53#41011yescharacteristic
81C22×C53C2×#10111yescharacteristic
82C22×C53#41712yes—
83C22×C53#42112yes—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—24no—
87C4×C53—22no—
88C4×C53—22no—
89C4×C53—22no—
90C4×C53—41no—
91C4×C53—44no—
92C4×C53—82no—
93C4×#125C2×C5311yescharacteristic
94#1831C2×C5312yes—
95#1271C2×C5314yes—
96C2×C53#146111yescharacteristic
97C2×C53#147712yes—
98C2×C53#151212yes—
99C2×C53#153411yescharacteristic
100C2×C53#156111yescharacteristic
101C2×#125C4×C5311yescharacteristic
102C2×C4⋊C8—22no—
103#389C22×C5312yes—
104#387—22no—
105#382C4×C5311yescharacteristic
106C4×C4⋊C4—22no—
107C4×C4⋊C4C4×C5312yes—
108C43C22×C5311yescharacteristic
109#336—22no—
110#336C22×C5314yes—
111C53#760211yescharacteristic, direct product factor, Sylow
112C22×C8—24no—
113#125—22no—
114#125C8×C5314yescomplemented
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4C2×C4×C5312yes—
118C2×C4⋊C4C2×C4×C5312yes—
119C2×C42—22no—
120C2×C42C2×C4×C5311yescharacteristic
121C2×C42C2×C4×C5311yescharacteristic
122C2×C42D4×C5312yes—
123C2×C42D4×C5312yes—
124C2×C42C23×C5311yescharacteristic
125C4⋊C8—44no—
126#95—22no—
127#95—22no—
128#95—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4C22⋊C4×C5311yescharacteristic
132C22×C4C22⋊C4×C5312yes—
133C22×C4C22⋊C4×C5311yescharacteristic
134C22×C4C22×C4×C5311yescharacteristic
135C22×C4C2×D4×C5312yes—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—44no—
139C4⋊C4—28no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4C2×C8×C5314yes—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42—41no—
150C42—44no—
151C42C2×C8×C5312yes—
152C42M16×C5312yes—
153C23C2×C22⋊C4×C5311yescharacteristic
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4C22⋊C8×C5314yes—
170C2×C4#101×C5312yes—
171C2×C4C22×C8×C5311yescharacteristic
172C2×C4C2×M16×C5311yescharacteristic
173C2×C4AGL(1,ℤ/8)×C5312yes—
174C2×C4#137×C5312yes—
175C8—84no—
176C22C2×C22⋊C8×C5311yescharacteristic
177C22C53×#41011yescharacteristic
178C22C2×#101×C5311yescharacteristic
179C22C53×#41712yes—
180C22C53×#42112yes—
181C4—21no—
182C4—21no—
183C4—24no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—44no—
188C4—41no—
189C4—82no—
190C2C53×#146111yescharacteristic
191C2C53×#147712yes—
192C2C53×#151212yes—
193C2C53×#153411yescharacteristic
194C2C53×#156111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C53C2×C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C43×C53 ◁ G

Complements

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

NH ≅ G/Ntype
16. #125×C53C8semidirect
78. #7602C53direct (N × H)
111. C53#7602direct (N × H)
114. #125C8×C53semidirect

Maximal subgroups

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

subgroupindex
C4×#125×C532
C53×#18312
C53×#12712
#760253