Finite Group Databasebeta

← #27275810

Subgroups — #27275810

Order 13568.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#20274711C211yesmaximal, characteristic, complemented
2#20273383C212yesmaximal
3#20273144C212yesmaximal, complemented
4#20273039C211yesmaximal, characteristic, complemented
5C2×C53×#559C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810392—21no—
10#16810367—24no—
11Q8⋊#88322—24no—
12#16810349—22no—
13C53×#559—24no—
14C53×#559—24no—
15C2×#127×C53C2211yescharacteristic
16C2×#124×C53—21no—
17C2×#124×C53C412yes—
18C2×#123×C53—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C2×C4×Q8×C53C2211yescharacteristic
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C412yes—
23C2×#11719917—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133136—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53C2×C412yes—
35C23×C4×C53—22no—
36#127×C53—22no—
37#127×C53D414yes—
38#124×C53—28no—
39#124×C53—24no—
40#124×C53—24no—
41#123×C53—216no—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45C4×Q8×C53D414yes—
46C4×D4×C53—28no—
47C4×D4×C53—24no—
48C4×D4×C53—24no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D412yes—
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—22no—
56C2×C42×C53—21no—
57C2×C42×C53C2×C412yes—
58C2×C42×C53C2311yescharacteristic
59C24×C53—22no—
60C2×Q8×C53—21no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—21no—
64C2×Q8×C53—21no—
65C2×Q8×C53—42no—
66C2×Q8×C53C22⋊C412yes—
67C2×D4×C53—22no—
68C2×D4×C53—24no—
69C2×D4×C53—24no—
70C2×D4×C53—44no—
71C2×D4×C53—42no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—82no—
75C2×D4×C53C22⋊C414yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—28no—
79C22×C4×C53—84no—
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22⋊C412yes—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53C22⋊C412yes—
90C22⋊C4×C53—28no—
91C22⋊C4×C53—28no—
92C22⋊C4×C53—28no—
93C42×C53—22no—
94C42×C53—22no—
95C42×C53—24no—
96C42×C53C2×D412yes—
97C42×C53C2×D412yes—
98#11719917—42no—
99#11719917—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—24no—
103C23×C53—44no—
104C23×C53—44no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53C2×C22⋊C411yescharacteristic
108Q8×C53—24no—
109Q8×C53—42no—
110Q8×C53—42no—
111Q8×C53—42no—
112D4×C53—44no—
113D4×C53—44no—
114D4×C53—44no—
115D4×C53—44no—
116D4×C53—84no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—88no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53#12012yes—
132C2×C4×C53#12012yes—
133C2×C4×C53#13512yes—
134C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C53#13711yescharacteristic
136#88322—82no—
137#88322—82no—
138#8386—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—44no—
142C22×C53—44no—
143C22×C53—44no—
144C22×C53—84no—
145C22×C53—84no—
146C22×C53#37911yescharacteristic
147C22×C53#41611yescharacteristic
148C22×C53#41711yescharacteristic
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—42no—
154C4×C53—42no—
155C4×C53#41312yes—
156C4×C53#44912yes—
157C4×C53#45212yes—
158C2×#559D5311yescharacteristic
159#1801—531no—
160#1545—532no—
161#1487—532no—
162#1461—531no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—84no—
166C2×C53#178511yescharacteristic
167C2×C53#178811yescharacteristic
168C2×C53#179511yescharacteristic
169#559—24no—
170#559—24no—
171C2×#127C2×D5311yescharacteristic
172C2×#124—21no—
173C2×#124Dic5312yes—
174C2×#123—22no—
175C2×C4×Q8C2×D5311yescharacteristic
176C2×C4×Q8C2×D5311yescharacteristic
177C2×C4×D4—21no—
178C2×C4×D4Dic5312yes—
179#431—532no—
180#423—532no—
181C2×#101—1061no—
182#343—1061no—
183#331—1064no—
184Q8⋊C8—1064no—
185#324—1062no—
186C53#838611yescharacteristic, complemented, Sylow
187C22×Q8C2×Dic5311yescharacteristic
188C22×Q8C22⋊D5311yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C2×Dic5312yes—
192C23×C4—22no—
193C2×M16—1062no—
194#127—22no—
195#127C22⋊D5314yes—
196#124—28no—
197#124—24no—
198#124—24no—
199#123—216no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C22⋊D5314yes—
204C4×D4—28no—
205C4×D4—24no—
206C4×D4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic5311yescharacteristic
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C4⋊C4C22⋊D5312yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42C2×Dic5312yes—
216C2×C42C22×D5311yescharacteristic
217C4⋊C8—1062no—
218C4⋊C8—1062no—
219C4⋊C8—1062no—
220#101—2122no—
221C22⋊C8—1062no—
222C4×C8—1062no—
223C24—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8C22⋊Dic5312yes—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4C22⋊Dic5314yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—28no—
243C22×C4—84no—
244C22×C4C22⋊Dic5312yes—
245C22×C4C22⋊Dic5312yes—
246C22×C4C22×Dic5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—2122no—
250M16—2122no—
251C2×C8—2122no—
252C2×C8—2122no—
253C4⋊C4—21no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4C22⋊Dic5312yes—
258C22⋊C4—28no—
259C22⋊C4—28no—
260C22⋊C4—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42C2×C22⋊D5312yes—
265C42C2×C22⋊D5312yes—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23C2×C22⋊Dic5311yescharacteristic
272Q8—24no—
273Q8—42no—
274Q8—42no—
275Q8—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—84no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—88no—
293C2×C4C2×C22⋊Dic5311yescharacteristic
294C2×C4C2×C22⋊Dic5311yescharacteristic
295C2×C4#1313318612yes—
296C2×C4#1313318612yes—
297C2×C4#1313323712yes—
298C2×C4#1313324211yescharacteristic
299C2×C4#1313324711yescharacteristic
300C8—4242no—
301C8—4242no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—84no—
308C22—84no—
309C22#1681047911yescharacteristic
310C22#1681061911yescharacteristic
311C22#1681062411yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4#1681060712yes—
319C4#1681076812yes—
320C4#1681078512yes—
321C2—21no—
322C2—21no—
323C2—84no—
324C2#2027464311yescharacteristic
325C2#2027465511yescharacteristic
326C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#559C2
socleC22×C53#379
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenterC2×#559D53
hypocenterC53#8386
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. #20274711C2semidirect
3. #20273144C2semidirect
4. #20273039C2semidirect
186. C53#8386semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202733832
#202731442
#202730392
C2×C53×#5592
#838653