Finite Group Databasebeta

← C53×#12225 #27219737

Subgroups — C53×#12225 #27219737

Order 13568.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2809C211yesmaximal, characteristic
2C53×#2028C211yesmaximal, characteristic
3C53×#1617C212yesmaximal
4C53×#1353C212yesmaximal
5C53×#1319C211yesmaximal, characteristic
6C53×#544C2211yescharacteristic
7C53×#484—22no—
8C53×#419—21no—
9C53×#400C2212yes—
10C53×#398—22no—
11C53×#391—21no—
12C53×#391C2211yescharacteristic
13C53×#389—21no—
14C53×#387—22no—
15C53×#343C2211yescharacteristic
16C53×#339—21no—
17C53×#328—22no—
18C53×#324C2212yes—
19C22×Q8×C53D411yescharacteristic
20C2×M16×C53—21no—
21C4⋊Q8×C53—41no—
22#125×C53—41no—
23C22⋊Q8×C53—22no—
24#117×C53—21no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53D412yes—
30C2×C4⋊C4×C53C2311yescharacteristic
31C2×C42×C53—21no—
32C2×C42×C53D411yescharacteristic
33#107×C53—42no—
34#103×C53—42no—
35#103×C53—41no—
36#103×C53—41no—
37C22⋊C8×C53—22no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C2×Q8×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53Q1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54M16×C53—41no—
55M16×C53—41no—
56C2×C8×C53—42no—
57C2×C8×C53—41no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—21no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—44no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—81no—
67C4⋊C4×C53—81no—
68C22⋊C4×C53—41no—
69C42×C53—41no—
70C42×C53—41no—
71C42×C53—81no—
72C23×C53#12011yescharacteristic
73Q8×C53—42no—
74Q8×C53—81no—
75Q8×C53—82no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53C2×Q1611yescharacteristic
95C2×C4×C53#13711yescharacteristic
96C8×C53—81no—
97C8×C53—82no—
98#12225C5311yesmaximal, characteristic, direct product factor, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53#45111yescharacteristic
102C22×C53#45511yescharacteristic
103C22×C53B(4,𝔽2)11yescharacteristic
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—42no—
107C4×C53—42no—
108C4×C53—42no—
109C4×C53—82no—
110C4×C53—81no—
111C4×C53—81no—
112C4×C53—82no—
113#2809C2×C5311yescharacteristic
114#2028C2×C5311yescharacteristic
115#1617C2×C5312yes—
116#1353C2×C5312yes—
117#1319C2×C5311yescharacteristic
118C2×C53—21no—
119C2×C53—21no—
120C2×C53#160211yescharacteristic
121C2×C53#220111yescharacteristic
122C2×C53#220411yescharacteristic
123#544C22×C5311yescharacteristic
124#484—22no—
125#419—21no—
126#400C22×C5312yes—
127#398—22no—
128#391—21no—
129#391C22×C5311yescharacteristic
130#389—21no—
131#387—22no—
132#343C22×C5311yescharacteristic
133#339—21no—
134#328—22no—
135#324C22×C5312yes—
136C53#1222511yescharacteristic, direct product factor, Sylow
137C22×Q8D4×C5311yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D4×C5312yes—
148C2×C4⋊C4C23×C5311yescharacteristic
149C2×C42—21no—
150C2×C42D4×C5311yescharacteristic
151#107—42no—
152#103—41no—
153#103—41no—
154#103—42no—
155C22⋊C8—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95D4×C5312yes—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8Q16×C5312yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×D4×C5311yescharacteristic
171C22×C4C2×D4×C5312yes—
172M16—41no—
173M16—41no—
174C2×C8—42no—
175C2×C8—41no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—81no—
190C23#120×C5311yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×Q16×C5311yescharacteristic
213C2×C4#137×C5311yescharacteristic
214C8—81no—
215C8—82no—
216C22—21no—
217C22—21no—
218C22C53×#45111yescharacteristic
219C22C53×#45511yescharacteristic
220C22C53×B(4,𝔽2)11yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2C53×#160211yescharacteristic
233C2C53×#220111yescharacteristic
234C2C53×#220411yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC23×C53#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
98. #12225C53direct (N × H)
136. C53#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28092
C53×#20282
C53×#16172
C53×#13532
C53×#13192
#1222553