Finite Group Databasebeta

← #27275825

Subgroups — #27275825

Order 13568.

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

normal subgroups only

This group has 738 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#20274707C211yesmaximal, characteristic
2#20273387C212yesmaximal, complemented
3#20273144C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic
5C2×C53×#556C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810369—24no—
11Q8⋊#88322—24no—
12#16810347—22no—
13C53×#556—24no—
14C53×#556—24no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C2×#126×C53—22no—
17C2×#124×C53—21no—
18C2×C22⋊Q8×C53C412yes—
19C2×C4×Q8×C53—21no—
20C2×C4×Q8×C53C2211yescharacteristic
21C2×C4×D4×C53C2211yescharacteristic
22C2×#117×C53C412yes—
23C2×#11719917—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133134—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53—21no—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53D411yescharacteristic
33C23×C4×C53C2×C412yes—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53D414yes—
36#126×C53—216no—
37#124×C53—28no—
38C22⋊Q8×C53—24no—
39C22⋊Q8×C53—24no—
40C4×Q8×C53—28no—
41C4×Q8×C53—22no—
42C4×Q8×C53D414yes—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45#117×C53—24no—
46#117×C53—24no—
47C2×C4⋊C4×C53—22no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C412yes—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53D412yes—
56C2×C42×C53—22no—
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C24×C53C22⋊C412yes—
60C2×Q8×C53—21no—
61C2×Q8×C53—24no—
62C2×Q8×C53—42no—
63C2×Q8×C53—42no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53C22⋊C412yes—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—44no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C4⋊C4×C53—28no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53C22⋊C412yes—
88C4⋊C4×C53C22⋊C414yes—
89C22⋊C4×C53—28no—
90C22⋊C4×C53—28no—
91C22⋊C4×C53—24no—
92C42×C53—28no—
93C42×C53—24no—
94C42×C53C2×D412yes—
95C42×C53C2×D412yes—
96#11719917—42no—
97#11719917—42no—
98C2×#88322—42no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—24no—
107Q8×C53—42no—
108Q8×C53—44no—
109Q8×C53—44no—
110D4×C53—42no—
111D4×C53—42no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—24no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—44no—
124C2×C4×C53—42no—
125C2×C4×C53C2×C22⋊C411yescharacteristic
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53#12012yes—
128C2×C4×C53#12012yes—
129C2×C4×C53#13512yes—
130C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C53#13711yescharacteristic
132#88322—82no—
133#88322—82no—
134#8389—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—82no—
141C22×C53—82no—
142C22×C53#37911yescharacteristic
143C22×C53#41611yescharacteristic
144C22×C53#41711yescharacteristic
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—42no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53—44no—
151C4×C53#41312yes—
152C4×C53#44912yes—
153C4×C53#45212yes—
154C2×#556D5311yescharacteristic
155#1800—531no—
156#1546—532no—
157#1487—532no—
158#1460—531no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—82no—
162C2×C53#178511yescharacteristic
163C2×C53#178811yescharacteristic
164C2×C53#179611yescharacteristic
165#556—24no—
166#556—24no—
167C2×C4⋊Q8C2×D5311yescharacteristic
168C2×#126—22no—
169C2×#124—21no—
170C2×C22⋊Q8Dic5312yes—
171C2×C4×Q8—21no—
172C2×C4×Q8C2×D5311yescharacteristic
173C2×C4×D4C2×D5311yescharacteristic
174C2×#117Dic5312yes—
175#431—532no—
176#423—532no—
177C2×#100—1061no—
178#343—1061no—
179#332—1064no—
180Q8⋊C8—1064no—
181#323—1062no—
182C53#838911yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C22×Q8C2×Dic5311yescharacteristic
185C22×D4C22⋊D5311yescharacteristic
186C23×C4C2×Dic5312yes—
187C2×M16—1062no—
188C4⋊Q8—22no—
189C4⋊Q8C22⋊D5314yes—
190#126—216no—
191#124—28no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—28no—
195C4×Q8—22no—
196C4×Q8C22⋊D5314yes—
197C4×D4—22no—
198C4×D4—22no—
199#117—24no—
200#117—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C2×Dic5311yescharacteristic
205C2×C4⋊C4C2×Dic5312yes—
206C2×C4⋊C4C22⋊D5311yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊D5312yes—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42C22×D5311yescharacteristic
213C4⋊C8—1062no—
214C4⋊C8—1062no—
215C4⋊C8—1062no—
216#100—2122no—
217C22⋊C8—1062no—
218C4×C8—1062no—
219C24C22⋊Dic5312yes—
220C2×Q8—21no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8C22⋊Dic5312yes—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4C22⋊Dic5312yes—
238C22×C4C22×Dic5311yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—2122no—
242M16—2122no—
243C2×C8—2122no—
244C2×C8—2122no—
245C4⋊C4—28no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4C22⋊Dic5312yes—
252C4⋊C4C22⋊Dic5314yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—24no—
258C42C2×C22⋊D5312yes—
259C42C2×C22⋊D5312yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×C22⋊Dic5311yescharacteristic
266Q8—24no—
267Q8—42no—
268Q8—44no—
269Q8—44no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4C2×C22⋊Dic5311yescharacteristic
286C2×C4C2×C22⋊Dic5311yescharacteristic
287C2×C4#1313318612yes—
288C2×C4#1313318612yes—
289C2×C4#1313323712yes—
290C2×C4#1313324211yescharacteristic
291C2×C4#1313324711yescharacteristic
292C8—4242no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1681047911yescharacteristic
302C22#1681061911yescharacteristic
303C22#1681062411yescharacteristic
304C4—22no—
305C4—21no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—44no—
310C4#1681060712yes—
311C4#1681076812yes—
312C4#1681078512yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2027464311yescharacteristic
317C2#2027465511yescharacteristic
318C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#556C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterC2×#556D53
hypocenterC53#8389
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. #20273387C2semidirect
3. #20273144C2semidirect
182. C53#8389semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202733872
#202731442
#202730362
C2×C53×#5562
#838953