Finite Group Databasebeta

← C53×#12710 #27220222

Subgroups — C53×#12710 #27220222

Order 13568.

952 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3486C211yesmaximal, characteristic
2C53×#2133C214yesmaximal, complemented
3C53×#1784C212yesmaximal, complemented
4C2×2−1+4×C53C411yescharacteristic
5C23×Q8×C53C2211yescharacteristic
6D4×Q8×C53—24no—
7C53×#548—24no—
8C53×#536—22no—
9C53×#536C2212yes—
10C53×#525—22no—
11C53×#525C412yes—
12C53×#519C411yescharacteristic
13C2×#101×C53—22no—
14C2×#101×C53C2214yes—
15C53×#407—22no—
16C53×#356—24no—
17C53×#355—24no—
182−1+4×C53—44no—
19C2×#41×C53—22no—
20C2×#41×C53—22no—
21C2×#41×C53C2×C412yes—
22C22×Q8×C53—22no—
23C22×Q8×C53—24no—
24C22×Q8×C53—44no—
25C22×Q8×C53C2×C411yescharacteristic
26C22×Q8×C53D412yes—
27C22×Q8×C53C2311yescharacteristic
28C23×C4×C53C2×C411yescharacteristic
29C23×C4×C53D412yes—
30C2×M16×C53—24no—
31C4⋊Q8×C53—24no—
32C4⋊Q8×C53D414yes—
33#124×C53—24no—
34#124×C53D414yes—
35#123×C53—22no—
36#123×C53—24no—
37C22⋊Q8×C53—24no—
38C22⋊Q8×C53—28no—
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—24no—
41#121×C53—24no—
42#121×C53—22no—
43#120×C53—22no—
44C4×Q8×C53—24no—
45C4×D4×C53—28no—
46C4×D4×C53—24no—
47#117×C53—22no—
48#117×C53C2×C412yes—
49#101×C53—28no—
50#101×C53—44no—
51C22⋊C8×C53—44no—
52C24×C53C22⋊C411yescharacteristic
53#41×C53—44no—
54#41×C53—44no—
55#41×C53—48no—
56#41×C53—44no—
57C2×Q8×C53—22no—
58C2×Q8×C53—24no—
59C2×Q8×C53—24no—
60C2×Q8×C53—22no—
61C2×Q8×C53—24no—
62C2×Q8×C53—42no—
63C2×Q8×C53—42no—
64C2×Q8×C53—416no—
65C2×Q8×C53—48no—
66C2×Q8×C53C22⋊C412yes—
67C2×Q8×C53C2×D414yes—
68C2×D4×C53—24no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53C22⋊C412yes—
72C22×C4×C53—24no—
73C22×C4×C53—24no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—24no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D412yes—
83M16×C53—44no—
84M16×C53—44no—
85C2×C8×C53—44no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—28no—
90C22⋊C4×C53—24no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—28no—
94C42×C53—24no—
95C42×C53C22⋊C414yes—
96C23×C53—21no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101Q8×C53—28no—
102Q8×C53—44no—
103Q8×C53—48no—
104Q8×C53—44no—
105Q8×C53—42no—
106D4×C53—42no—
107D4×C53—44no—
108D4×C53—48no—
109D4×C53—44no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—24no—
118C2×C4×C53—22no—
119C2×C4×C53—48no—
120C2×C4×C53—44no—
121C2×C4×C53—48no—
122C2×C4×C53—44no—
123C2×C4×C53#9914yes—
124C2×C4×C53C2×C22⋊C412yes—
125C2×C4×C53#12014yes—
126C8×C53—84no—
127#12710C5311yesmaximal, characteristic, direct product factor, Sylow
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—41no—
135C22×C53—42no—
136C22×C53#37911yescharacteristic
137C22×C53C2×#9912yes—
138C4×C53—24no—
139C4×C53—24no—
140C4×C53—22no—
141C4×C53—44no—
142C4×C53—41no—
143C4×C53—42no—
144#3486C2×C5311yescharacteristic
145#2133C2×C5314yescomplemented
146#1784C2×C5312yescomplemented
147C2×C53—21no—
148C2×C53—22no—
149C2×C53—42no—
150C2×C53—82no—
151C2×C53#178111yescharacteristic
152C2×2−1+4C4×C5311yescharacteristic
153C23×Q8C22×C5311yescharacteristic
154D4×Q8—24no—
155#548—24no—
156#536—22no—
157#536C22×C5312yes—
158#525—22no—
159#525C4×C5312yes—
160#519C4×C5311yescharacteristic
161C2×#101—22no—
162C2×#101C22×C5314yes—
163#407—22no—
164#356—24no—
165#355—24no—
166C53#1271011yescharacteristic, direct product factor, Sylow
1672−1+4—44no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41C2×C4×C5312yes—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—44no—
174C22×Q8C2×C4×C5311yescharacteristic
175C22×Q8D4×C5312yes—
176C22×Q8C23×C5311yescharacteristic
177C23×C4C2×C4×C5311yescharacteristic
178C23×C4D4×C5312yes—
179C2×M16—24no—
180C4⋊Q8—24no—
181C4⋊Q8D4×C5314yes—
182#124—24no—
183#124D4×C5314yes—
184#123—24no—
185#123—22no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—24no—
189C22⋊Q8—22no—
190#121—24no—
191#121—22no—
192#120—22no—
193C4×Q8—24no—
194C4×D4—28no—
195C4×D4—24no—
196#117—22no—
197#117C2×C4×C5312yes—
198#101—28no—
199#101—44no—
200C22⋊C8—44no—
201C24C22⋊C4×C5311yescharacteristic
202#41—44no—
203#41—44no—
204#41—48no—
205#41—44no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—416no—
214C2×Q8—48no—
215C2×Q8C22⋊C4×C5312yes—
216C2×Q8C2×D4×C5314yes—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22⋊C4×C5312yes—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊C4×C5311yescharacteristic
229C22×C4C22⋊C4×C5312yes—
230C22×C4C22×C4×C5311yescharacteristic
231C22×C4C2×D4×C5312yes—
232M16—44no—
233M16—44no—
234C2×C8—44no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—28no—
243C42—24no—
244C42C22⋊C4×C5314yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23C2×C22⋊C4×C5311yescharacteristic
250Q8—28no—
251Q8—44no—
252Q8—48no—
253Q8—42no—
254Q8—44no—
255D4—42no—
256D4—44no—
257D4—48no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4#99×C5314yes—
273C2×C4C2×C22⋊C4×C5312yes—
274C2×C4#120×C5314yes—
275C8—84no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22—41no—
283C22—42no—
284C22C53×#37911yescharacteristic
285C22C2×#99×C5312yes—
286C4—24no—
287C4—24no—
288C4—22no—
289C4—44no—
290C4—41no—
291C4—42no—
292C2—21no—
293C2—22no—
294C2—42no—
295C2—82no—
296C2C53×#178111yescharacteristic
subgroupG/H
center, socleC2×C53#1781
commutatorC22×C4C22×C4×C53
Frattini, cosocleC22×Q8C23×C53
prime radicalC53×#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C23×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2133C2semidirect
3. C53×#1784C2semidirect
127. #12710C53direct (N × H)
145. #2133C2×C53semidirect
146. #1784C2×C53semidirect
166. C53#12710direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34862
C53×#21332
C53×#17842
#1271053