Finite Group Databasebeta

← #27299626

Subgroups — #27299626

Order 13568.

1216 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#20276384C214yesmaximal, complemented
2#20274630C212yesmaximal, complemented
3C53×#3484C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yescomplemented
6#16810479—22no—
7#16810418—24no—
8#16810416—24no—
9C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
10C23×D4×C53C2211yescharacteristic
11C53×#546—24no—
12D42×C53—24no—
13C53×#535C2212yes—
14C53×#534—22no—
15C53×#525C412yescomplemented
16C2×#120×C53—22no—
17C53×#518C411yescharacteristic, complemented
18C2×C22⋊Dic53—24no—
19C2×C22⋊Dic53—44no—
20#13133132—28no—
21#13133132—44no—
22C25×C53D412yes—
23H(5,𝔽2)×C53—44no—
24C2×#41×C53—22no—
25C2×#41×C53C2×C412yes—
26C22×D4×C53—22no—
27C22×D4×C53—24no—
28C22×D4×C53—22no—
29C22×D4×C53—24no—
30C22×D4×C53—44no—
31C22×D4×C53C2×C411yescharacteristic
32C22×D4×C53D412yes—
33C22×D4×C53C2311yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35#127×C53D414yes—
36#124×C53D414yes—
37#123×C53—24no—
38#123×C53—22no—
39C22⋊Q8×C53—22no—
40#121×C53—28no—
41#121×C53—22no—
42#121×C53—24no—
43#120×C53—28no—
44#120×C53—24no—
45#120×C53—22no—
46#120×C53—24no—
47C4×D4×C53—24no—
48C4×D4×C53—24no—
49#117×C53C2×C412yes—
50C2×C22⋊C4×C53—24no—
51C2×C22⋊C4×C53—22no—
52C24×C53—22no—
53C24×C53—24no—
54C24×C53—24no—
55C24×C53—42no—
56C24×C53—84no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22⋊C412yes—
59C24×C53C2×D412yes—
60#41×C53—44no—
61#41×C53—48no—
62C2×Q8×C53C22⋊C412yescomplemented
63C2×D4×C53—24no—
64C2×D4×C53—24no—
65C2×D4×C53—24no—
66C2×D4×C53—24no—
67C2×D4×C53—22no—
68C2×D4×C53—24no—
69C2×D4×C53—28no—
70C2×D4×C53—22no—
71C2×D4×C53—42no—
72C2×D4×C53—42no—
73C2×D4×C53—416no—
74C2×D4×C53—48no—
75C2×D4×C53—44no—
76C2×D4×C53—44no—
77C2×D4×C53—88no—
78C2×D4×C53C22⋊C412yescomplemented
79C2×D4×C53C2×D414yes—
80C22×C4×C53—22no—
81C22×C4×C53—24no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—42no—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×Dic53—44no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—22no—
90C22⋊C4×C53—24no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—28no—
95C42×C53C22⋊C414yescomplemented
96C22⋊Dic53—44no—
97C22⋊Dic53—44no—
98C22⋊Dic53—88no—
99C23×C53—22no—
100C23×C53—21no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—48no—
108C23×C53—48no—
109C23×C53—48no—
110C23×C53—44no—
111C23×C53—84no—
112C23×C53—82no—
113C23×C53—82no—
114C23×C53—84no—
115C23×C53—84no—
116C23×C53—84no—
117C23×C53—84no—
118C23×C53#9914yes—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53#12014yes—
121Q8×C53—42no—
122D4×C53—28no—
123D4×C53—44no—
124D4×C53—42no—
125D4×C53—48no—
126D4×C53—48no—
127D4×C53—48no—
128D4×C53—44no—
129D4×C53—88no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—21no—
133C2×C4×C53—24no—
134C2×C4×C53—22no—
135C2×C4×C53—44no—
136C2×C4×C53—44no—
137C2×C4×C53—44no—
138C2×C4×C53—84no—
139C2×C4×C53C2×C22⋊C412yes—
140C2×Dic53—84no—
141C2×Dic53—88no—
142#12707—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53—24no—
147C22×C53—41no—
148C22×C53—44no—
149C22×C53—44no—
150C22×C53—44no—
151C22×C53—44no—
152C22×C53—41no—
153C22×C53—42no—
154C22×C53—44no—
155C22×C53—41no—
156C22×C53—42no—
157C22×C53—82no—
158C22×C53—82no—
159C22×C53—816no—
160C22×C53—164no—
161C22×C53#37911yescharacteristic
162C22×C53C2×#9912yes—
163C4×C53—22no—
164C4×C53—44no—
165C4×C53—41no—
166C4×C53—42no—
167Dic53—164no—
168#3484D5311yescharacteristic
169#2127—534no—
170#1781—532no—
171C2×C53—21no—
172C2×C53—22no—
173C2×C53—44no—
174C2×C53—42no—
175C2×C53—44no—
176C2×C53—82no—
177C2×C53#178111yescharacteristic
178C2×H(5,𝔽2)Dic5311yescharacteristic, complemented
179C23×D4C2×D5311yescharacteristic
180#546—24no—
181D42—24no—
182#535C2×D5312yes—
183#534—22no—
184#525Dic5312yescomplemented
185C2×#120—22no—
186#518Dic5311yescharacteristic, complemented
187C2×#99—534no—
188C2×#99—1062no—
189#379—1062no—
190#354—1064no—
191#353—1064no—
192C53#1270711yescharacteristic, complemented, Sylow
193C25C22⋊D5312yes—
194H(5,𝔽2)—44no—
195C2×#41—22no—
196C2×#41C2×Dic5312yes—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—44no—
202C22×D4C2×Dic5311yescharacteristic
203C22×D4C22⋊D5312yes—
204C22×D4C22×D5311yescharacteristic
205C23×C4C2×Dic5311yescharacteristic
206#127C22⋊D5314yes—
207#124C22⋊D5314yes—
208#123—24no—
209#123—22no—
210C22⋊Q8—22no—
211#121—28no—
212#121—24no—
213#121—22no—
214#120—28no—
215#120—24no—
216#120—24no—
217#120—22no—
218C4×D4—24no—
219C4×D4—24no—
220#117C2×Dic5312yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—1064no—
224C2×C22⋊C4—2124no—
225#99—1068no—
226#99—2124no—
227C24—22no—
228C24—24no—
229C24—24no—
230C24—42no—
231C24—84no—
232C24C22⋊Dic5311yescharacteristic
233C24C22⋊Dic5312yes—
234C24C2×C22⋊D5312yes—
235#41—44no—
236#41—48no—
237C2×Q8C22⋊Dic5312yescomplemented
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—28no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—416no—
249C2×D4—48no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—88no—
253C2×D4C22⋊Dic5312yescomplemented
254C2×D4C2×C22⋊D5314yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—2124no—
261C22×C4C22⋊Dic5311yescharacteristic
262C22×C4C22×Dic5311yescharacteristic
263C4⋊C4—24no—
264C4⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—22no—
269C22⋊C4—28no—
270C22⋊C4—4248no—
271C22⋊C4—2124no—
272C22⋊C4—2124no—
273C42C22⋊Dic5314yescomplemented
274C23—22no—
275C23—22no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—48no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—84no—
287C23—82no—
288C23—82no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23#1313313214yes—
294C23C2×C22⋊Dic5311yescharacteristic
295C23#1313318614yes—
296Q8—42no—
297D4—28no—
298D4—44no—
299D4—42no—
300D4—48no—
301D4—48no—
302D4—48no—
303D4—44no—
304D4—88no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—4244no—
315C2×C4—4248no—
316C2×C4C2×C22⋊Dic5312yes—
317C22—22no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—41no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—41no—
327C22—42no—
328C22—44no—
329C22—41no—
330C22—42no—
331C22—82no—
332C22—82no—
333C22—816no—
334C22—164no—
335C22#1681047911yescharacteristic
336C22C2×#1313313212yes—
337C4—22no—
338C4—44no—
339C4—41no—
340C4—42no—
341C4—8484no—
342C2—21no—
343C2—22no—
344C2—44no—
345C2—42no—
346C2—44no—
347C2—82no—
348C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC53×#3484C2
socleC2×C53#1781
cosocleC22×D4×C53C23
hypercenter#3484D53
hypocenterC53#12707
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #3484

Complements

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

NH ≅ G/Ntype
1. #20276384C2semidirect
2. #20274630C2semidirect
5. C2×#13133132C22semidirect
9. C2×H(5,𝔽2)×C53C4semidirect
15. C53×#525C4semidirect
17. C53×#518C4semidirect
62. C2×Q8×C53C22⋊C4semidirect
78. C2×D4×C53C22⋊C4semidirect
95. C42×C53C22⋊C4semidirect
178. C2×H(5,𝔽2)Dic53semidirect
184. #525Dic53semidirect
186. #518Dic53semidirect
192. C53#12707semidirect
237. C2×Q8C22⋊Dic53semidirect
253. C2×D4C22⋊Dic53semidirect
273. C42C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202763842
#202746302
C53×#34842
#1270753