Finite Group Databasebeta

← #27275871

Subgroups — #27275871

Order 13568.

782 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16811333C211yesmaximal, characteristic
2#20273401C212yesmaximal, complemented
3#20273139C212yesmaximal
4#20273039C211yesmaximal, characteristic, complemented
5C53×#1801C211yesmaximal, characteristic, complemented
6#16811333—24no—
7#16811333—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133209C412yes—
10C2×#13133195—22no—
11C2×#13133193—21no—
12C2×C4×D212C412yes—
13C2×C4×D212C2211yescharacteristic
14C2×C4⋊C4×D53—21no—
15#16810676C2212yes—
16#16810368—24no—
17#16810350—24no—
18#16810349—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#101×C53—21no—
22C53×#343—21no—
23C22×Dic106—21no—
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26C22×C4×D53—22no—
27#13133217—22no—
28#13133217D414yes—
29#13133209—24no—
30#13133209—24no—
31#13133195—216no—
32#13133193—28no—
33C4×D212—22no—
34C4×D212—24no—
35C4×D212—24no—
36C4×D212D414yes—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C412yes—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—22no—
41C2×#11719919—22no—
42C4⋊C4×D53—28no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C4⋊#88322—22no—
46C22⋊#88322—22no—
47C4×#88322—22no—
48C22×Q8×C53D411yescharacteristic
49C2×M16×C53—22no—
50C4×Q8×C53—22no—
51C4×Q8×C53—22no—
52C2×C4⋊C4×C53D412yes—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C42×C53C2×C412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58#101×C53—42no—
59C23×D53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—22no—
62C2×Q8×C53—21no—
63C2×Q8×C53—21no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—22no—
71C2×D212—21no—
72C2×D212—24no—
73C2×D212C22⋊C412yes—
74C2×C4×D53—24no—
75C2×C4×D53—28no—
76C2×C4×D53—42no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—84no—
80C2×Dic106—24no—
81C2×Dic106—82no—
82M16×C53—42no—
83M16×C53—42no—
84C2×C8×C53—42no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—24no—
87C42×C53—22no—
88C42×C53—22no—
89C42×C53C2×D412yes—
90C42×C53C2×D412yes—
91#11719921—28no—
92#11719921—24no—
93C4⋊Dic53—21no—
94C4⋊Dic53—22no—
95C4⋊Dic53—24no—
96C4⋊Dic53C22⋊C414yes—
97C4⋊Dic53C22⋊C412yes—
98#11719919—28no—
99#11719919—28no—
100C2×#88322—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102C22×D53—24no—
103C22×D53—22no—
104C22×D53—42no—
105C22×D53—41no—
106C22×D53—41no—
107Q8×C53—42no—
108Q8×C53—42no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53C2×C22⋊C411yescharacteristic
117C2×C4×C53C2×C22⋊C411yescharacteristic
118C2×C4×C53#12012yes—
119C2×C4×C53#12012yes—
120C2×C4×C53#13512yes—
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53#13711yescharacteristic
123C2×Dic53—24no—
124C2×Dic53—22no—
125C2×Dic53—44no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128D212—24no—
129D212—22no—
130D212#10114yescomplemented
131C4×D53—44no—
132C4×D53—44no—
133C4×D53—88no—
134Dic106—84no—
135C8×C53—82no—
136#88322—82no—
137#8398—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53#37911yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53#41711yescharacteristic
143C2×D53—24no—
144C2×D53—44no—
145C2×D53—44no—
146C2×D53—42no—
147C2×D53—42no—
148C4×C53—22no—
149C4×C53—21no—
150C4×C53—42no—
151C4×C53—42no—
152C4×C53C2×#10112yes—
153C4×C53#44912yes—
154C4×C53#45212yes—
155Dic53—44no—
156Dic53—42no—
157C2×#554—531no—
158#1801D5311yescharacteristic, complemented
159#1549—532no—
160#1486—532no—
161#1461—531no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53#178411yescharacteristic
165C2×C53#178811yescharacteristic
166C2×C53#179511yescharacteristic
167D53—44no—
168#554—1064no—
169#554—1064no—
170C2×C4⋊Q8—531no—
171C2×#125—532no—
172C2×C22⋊Q8—1061no—
173C2×C22⋊Q8—1062no—
174C2×C4×Q8C2×D5311yescharacteristic
175C2×C4×D4—532no—
176C2×C4×D4—531no—
177C22×C4⋊C4—1061no—
178#431—532no—
179#423C2×D5312yes—
180C2×#101—21no—
181#343—21no—
182#332—1064no—
183D4⋊C8—1064no—
184#324—1062no—
185C53#839811yescharacteristic, complemented, Sylow
186C22×Q8—1061no—
187C22×Q8C22⋊D5311yescharacteristic
188C22×D4—531no—
189C23×C4—1062no—
190C23×C4—1061no—
191C2×M16—22no—
192C4⋊Q8—534no—
193C4⋊Q8—1062no—
194#125—1064no—
195#125—1064no—
196C22⋊Q8—10616no—
197C22⋊Q8—1068no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—534no—
201C4×D4—1064no—
202C4×D4—1064no—
203C4×D4—1062no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—532no—
206C2×C4⋊C4—1068no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4D21212yes—
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1062no—
214C2×C42C4×D5312yes—
215C2×C42C22×D5311yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—1062no—
219#101—42no—
220C22⋊C8—1062no—
221C4×C8—1062no—
222C24—1061no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—1064no—
228C2×Q8—4242no—
229C2×D4—532no—
230C2×D4—1064no—
231C2×D4—1061no—
232C22×C4—1068no—
233C22×C4—1061no—
234C22×C4—1062no—
235C22×C4—1064no—
236C22×C4—4244no—
237C22×C4—2122no—
238C22×C4—2121no—
239C22×C4—2121no—
240C22×C4#1171992112yes—
241C22×C4#1171992112yes—
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4C2×D21211yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—2122no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—532no—
252C4⋊C4—534no—
253C4⋊C4—1068no—
254C4⋊C4—1068no—
255C4⋊C4—1061no—
256C4⋊C4—1062no—
257C4⋊C4—1064no—
258C22⋊C4—1068no—
259C22⋊C4—1064no—
260C42—22no—
261C42—22no—
262C42D4×D5312yes—
263C42D4×D5312yes—
264C23—1064no—
265C23—1062no—
266C23—2122no—
267C23—2121no—
268C23—2121no—
269C23C2×#1171992111yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—4244no—
273D4—534no—
274D4—1064no—
275D4—1062no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—1062no—
284C2×C4—1064no—
285C2×C4—4248no—
286C2×C4—2124no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2124no—
290C2×C4—2124no—
291C2×C4C22⋊C4×D5311yescharacteristic
292C2×C4C22⋊C4×D5311yescharacteristic
293C2×C4#1313318512yes—
294C2×C4#1313318712yes—
295C2×C4#1313323712yes—
296C2×C4#1313324311yescharacteristic
297C2×C4#1313324811yescharacteristic
298C8—82no—
299C8—4242no—
300C22—21no—
301C22—21no—
302C22—1064no—
303C22—2124no—
304C22—2124no—
305C22—2122no—
306C22—2122no—
307C22#1681047811yescharacteristic
308C22#1681061711yescharacteristic
309C22#1681062211yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—2122no—
315C4—2124no—
316C4#101×D5312yes—
317C4#1681076612yes—
318C4#1681078312yes—
319C2—21no—
320C2—21no—
321C2—2124no—
322C2#2027463711yescharacteristic
323C2#2027465311yescharacteristic
324C2#2027468611yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#379
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8398
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #20273401C2semidirect
4. #20273039C2semidirect
5. C53×#1801C2semidirect
130. D212#101semidirect
158. #1801D53semidirect
185. C53#8398semidirect

Maximal subgroups

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

subgroupindex
C2×#168113332
#202734012
#202731392
#202730392
C53×#18012
#839853