Finite Group Databasebeta

← #27294459

Subgroups — #27294459

Order 13568.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277998C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic, complemented
3#20272741C212yesmaximal, complemented
4#20272717C212yesmaximal
5C4×#101×C53C211yesmaximal, characteristic, complemented
6C2×C4×D212C411yescharacteristic
7#16810676—21no—
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810549—21no—
11#16810540C411yescharacteristic
12#16810540C2211yescharacteristic
13#16810534—21no—
14C4×#11719921—21no—
15C4×#11719921—21no—
16#16810469C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C2×#101×C53—21no—
21C4×M16×C53—21no—
22C53×#343C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C4×D53—21no—
25C4×D212—41no—
26C4×D212—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719921—21no—
29C2×#11719921—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×Q8×C53D411yescharacteristic
45C2×M16×C53—22no—
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#101×C53—24no—
55#97×C53—41no—
56C4×C8×C53—41no—
57C23×D53—21no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×D212—41no—
72C2×D212—42no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77M16×C53—24no—
78M16×C53—42no—
79C2×C8×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89#11719921—42no—
90#11719921—42no—
91#11719921—42no—
92#11719921—42no—
93#11719921—42no—
94C4⋊Dic53—41no—
95C4×Dic53—42no—
96C2×#88322—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98C22×D53—22no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105Q8×C53—44no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#10412yes—
123C2×C4×C53#13512yes—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133D212—42no—
134D212—42no—
135C4×D53—82no—
136C4×D53—82no—
137C8×C53—44no—
138#88322—82no—
139#11734—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53C2×#9911yescharacteristic
143C22×C53#41611yescharacteristic
144C22×C53C2×#10411yescharacteristic
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53—81no—
149C2×D53—81no—
150C2×D53—81no—
151C2×D53—81no—
152C4×C53—21no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53#35512yes—
160Dic53—81no—
161Dic53—81no—
162Dic53—81no—
163#2398—531no—
164C4×#101D5311yescharacteristic, complemented
165#1461—531no—
166#1353—532no—
167#1345—532no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53#150111yescharacteristic
171C2×C53#212011yescharacteristic
172C2×C53C2×#35511yescharacteristic
173D53—82no—
174C2×C4×Q8C2×D5311yescharacteristic
175C2×C4×D4—531no—
176#431—1061no—
177C2×#101—21no—
178C4×M16—21no—
179#400—531no—
180#400—531no—
181#397—1061no—
182#394—531no—
183#394—531no—
184#393—1061no—
185C4×C22⋊C4—1061no—
186C4×C22⋊C4—1061no—
187#376—531no—
188#343C2×D5312yes—
189#324—532no—
190#324—1061no—
191C53#1173411yescharacteristic, complemented, Sylow
192C22×Q8C22⋊D5311yescharacteristic
193C22×D4—531no—
194C23×C4—1061no—
195C2×M16—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—2121no—
199C4×D4—2121no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4C4×D5311yescharacteristic
202C2×C4⋊C4D21211yescharacteristic
203C2×C4⋊C4C22×D5311yescharacteristic
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C42—1061no—
210C2×C42C4×D5311yescharacteristic
211C2×C42D21211yescharacteristic
212C2×C42C22⋊D5311yescharacteristic
213C4⋊C8—2121no—
214#101—24no—
215C22⋊C8—1062no—
216#97—41no—
217C4×C8—41no—
218C4×C8—2121no—
219#95—531no—
220#95—531no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C24—1061no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—2121no—
231C2×D4—2122no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4C2×C4×D5311yescharacteristic
244C22×C4C2×D21211yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—24no—
247M16—42no—
248C2×C8—42no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—2121no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C22⋊C4—2122no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—2122no—
266C23—1062no—
267C23—2121no—
268C23—2121no—
269C23—2121no—
270C23—2121no—
271C23—2121no—
272C23—2121no—
273C23C2×#1171992111yescharacteristic
274Q8—44no—
275D4—2122no—
276D4—2122no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—4242no—
292C2×C4—4242no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4#1313313111yescharacteristic
303C2×C4#1313313111yescharacteristic
304C2×C4#1313314512yes—
305C2×C4#1313323712yes—
306C8—44no—
307C8—4242no—
308C22—21no—
309C22—21no—
310C22—4241no—
311C22—4241no—
312C22—4241no—
313C22—4241no—
314C22—2122no—
315C22—2122no—
316C22—2122no—
317C22C2×#1313313111yescharacteristic
318C22#1681061711yescharacteristic
319C22C2×#1313314511yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—4241no—
328C4—4241no—
329C4—4241no—
330C4#1681041912yes—
331C2—21no—
332C2—21no—
333C2—4242no—
334C2#2027320411yescharacteristic
335C2#2027635711yescharacteristic
336C2C2×#1681041911yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#101D53
hypocenterC53#11734
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #20273039C2semidirect
3. #20272741C2semidirect
5. C4×#101×C53C2semidirect
164. C4×#101D53semidirect
191. C53#11734semidirect

Maximal subgroups

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

subgroupindex
#202779982
#202730392
#202727412
#202727172
C4×#101×C532
#1173453