Finite Group Databasebeta

← #27275811

Subgroups — #27275811

Order 13568.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16811358C211yesmaximal, characteristic
2#20273384C212yesmaximal
3#20273143C212yesmaximal, complemented
4#20273039C211yesmaximal, characteristic, complemented
5C53×#1801C211yesmaximal, characteristic, complemented
6#16811358—24no—
7#16811358—24no—
8C2×#13133214C2211yescharacteristic
9C2×#13133206—21no—
10C2×#13133205C412yes—
11C2×#13133202—22no—
12C2×C4×Dic106C2211yescharacteristic
13C2×C4×D212C412yes—
14C2×#13133179—21no—
15#16810676C2212yes—
16#16810366—24no—
17#16810353—24no—
18#16810349—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#101×C53—21no—
22C53×#343—21no—
23C22×Dic106C2×C411yescharacteristic
24C22×D212—21no—
25C22×D212—21no—
26C22×D212C2×C412yes—
27C22×C4×D53—22no—
28#13133214—22no—
29#13133214D414yes—
30#13133206—28no—
31#13133205—24no—
32#13133205—24no—
33#13133202—216no—
34C4×Dic106—22no—
35C4×Dic106D414yes—
36C4×D212—24no—
37C4×D212—24no—
38#13133179—28no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—22no—
44C2×C4×Dic53—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—22no—
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—21no—
71C2×D212—22no—
72C2×D212—24no—
73C2×D212—24no—
74C2×D212—42no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—44no—
78C2×D212—82no—
79C2×D212C22⋊C414yes—
80C2×C4×D53—28no—
81C2×C4×D53—84no—
82C2×Dic106—21no—
83C2×Dic106—42no—
84C2×Dic106C22⋊C412yes—
85M16×C53—42no—
86M16×C53—42no—
87C2×C8×C53—42no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—24no—
90C42×C53—22no—
91C42×C53—22no—
92C42×C53C2×D412yes—
93C42×C53C2×D412yes—
94#11719921—28no—
95#11719921—28no—
96#11719921—28no—
97C4⋊Dic53—21no—
98C4⋊Dic53C22⋊C412yes—
99#11719919—24no—
100C4×Dic53—24no—
101C2×#88322—42no—
102C23×C53C2×C22⋊C411yescharacteristic
103C22×D53—24no—
104C22×D53—44no—
105C22×D53—44no—
106C22×D53—42no—
107C22×D53—42no—
108Q8×C53—42no—
109Q8×C53—42no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53C2×C22⋊C411yescharacteristic
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53#12012yes—
120C2×C4×C53#12012yes—
121C2×C4×C53#13512yes—
122C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128D212—44no—
129D212—44no—
130D212—44no—
131D212—44no—
132D212—84no—
133C4×D53—88no—
134Dic106—24no—
135Dic106—42no—
136C8×C53—82no—
137#88322—82no—
138#8386—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53#37911yescharacteristic
142C22×C53#41611yescharacteristic
143C22×C53#41711yescharacteristic
144C2×D53—44no—
145C2×D53—44no—
146C2×D53—44no—
147C2×D53—84no—
148C2×D53—84no—
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53#41312yes—
154C4×C53#44912yes—
155C4×C53#45212yes—
156Dic53—42no—
157Dic53—42no—
158C2×#559—531no—
159#1801D5311yescharacteristic, complemented
160#1545—532no—
161#1487—532no—
162#1461—531no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#178511yescharacteristic
166C2×C53#178811yescharacteristic
167C2×C53#179511yescharacteristic
168D53—84no—
169#559—1064no—
170#559—1064no—
171C2×#127—531no—
172C2×#124—532no—
173C2×#124—1061no—
174C2×#123—1062no—
175C2×C4×Q8—531no—
176C2×C4×Q8C2×D5311yescharacteristic
177C2×C4×D4—532no—
178C2×C4×D4—1061no—
179#431—532no—
180#423C2×D5312yes—
181C2×#101—21no—
182#343—21no—
183#331—1064no—
184Q8⋊C8—1064no—
185#324—1062no—
186C53#838611yescharacteristic, complemented, Sylow
187C22×Q8—531no—
188C22×Q8C22⋊D5311yescharacteristic
189C22×D4—532no—
190C22×D4—1061no—
191C22×D4—1061no—
192C23×C4—1062no—
193C2×M16—22no—
194#127—534no—
195#127—1062no—
196#124—1064no—
197#124—1064no—
198#124—1068no—
199#123—10616no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—534no—
203C4×Q8—1062no—
204C4×D4—1064no—
205C4×D4—1064no—
206C4×D4—1068no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4D21212yes—
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C22⋊C4—1062no—
212C2×C22⋊C4—1062no—
213C2×C22⋊C4—1062no—
214C2×C42—1061no—
215C2×C42C4×D5312yes—
216C2×C42C22×D5311yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—1062no—
220#101—42no—
221C22⋊C8—1062no—
222C4×C8—1062no—
223C24—1062no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—532no—
229C2×Q8—1061no—
230C2×Q8—2122no—
231C2×D4—534no—
232C2×D4—1062no—
233C2×D4—1064no—
234C2×D4—1064no—
235C2×D4—4242no—
236C2×D4—2124no—
237C2×D4—2122no—
238C2×D4—2121no—
239C2×D4—2121no—
240C22×C4—1068no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—4244no—
244C22×C4#1171992112yes—
245C22×C4#1171992112yes—
246C22×C4C2×C4×D5311yescharacteristic
247C22×C4C2×D21211yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—42no—
250M16—42no—
251C2×C8—42no—
252C2×C8—2122no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—532no—
256C4⋊C4—1061no—
257C4⋊C4—1064no—
258C22⋊C4—1068no—
259C22⋊C4—1068no—
260C22⋊C4—1068no—
261C42—22no—
262C42—22no—
263C42—1064no—
264C42D4×D5312yes—
265C42D4×D5312yes—
266C23—1064no—
267C23—2124no—
268C23—2124no—
269C23—2122no—
270C23—2122no—
271C23C2×#1171992111yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—1064no—
275Q8—2122no—
276D4—4244no—
277D4—2124no—
278D4—2124no—
279D4—2124no—
280D4—2124no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—1062no—
289C2×C4—1062no—
290C2×C4—4248no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4C22⋊C4×D5311yescharacteristic
294C2×C4C22⋊C4×D5311yescharacteristic
295C2×C4#1313318512yes—
296C2×C4#1313318712yes—
297C2×C4#1313323712yes—
298C2×C4#1313324311yescharacteristic
299C2×C4#1313324811yescharacteristic
300C8—82no—
301C8—4242no—
302C22—21no—
303C22—21no—
304C22—4244no—
305C22—4244no—
306C22—2124no—
307C22—2124no—
308C22—2124no—
309C22#1681047811yescharacteristic
310C22#1681061711yescharacteristic
311C22#1681062211yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—2122no—
317C4—2122no—
318C4#1681060612yes—
319C4#1681076612yes—
320C4#1681078312yes—
321C2—21no—
322C2—21no—
323C2—4244no—
324C2#2027464211yescharacteristic
325C2#2027465311yescharacteristic
326C2#2027468611yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#379
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8386
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
3. #20273143C2semidirect
4. #20273039C2semidirect
5. C53×#1801C2semidirect
159. #1801D53semidirect
186. C53#8386semidirect

Maximal subgroups

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

subgroupindex
C2×#168113582
#202733842
#202731432
#202730392
C53×#18012
#838653