Finite Group Databasebeta

← #27275856

Subgroups — #27275856

Order 13568.

782 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16811330C211yesmaximal, characteristic
2#20273401C212yesmaximal, complemented
3#20273143C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic
5C53×#1800C211yesmaximal, characteristic
6#16811330—24no—
7#16811330—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133208—21no—
10C2×#13133196C412yes—
11C2×#13133194—22no—
12C2×C4×Dic106C2211yescharacteristic
13C2×D4×Dic53—21no—
14C22×C4⋊Dic53C412yes—
15#16810676C2212yes—
16#16810368—24no—
17#16810353—24no—
18#16810347—22no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#100×C53—21no—
22C53×#343—21no—
23C22×Dic106—21no—
24C22×Dic106C2×C411yescharacteristic
25C23×Dic53—22no—
26#13133217—22no—
27#13133217D414yes—
28#13133208—28no—
29#13133196—24no—
30#13133196—24no—
31#13133194—216no—
32C4×Dic106—22no—
33C4×Dic106D414yes—
34D4×Dic53—28no—
35C2×C4⋊Dic53—24no—
36C2×C4⋊Dic53—24no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C412yes—
41C2×#11719919—22no—
42C2×#11719919—21no—
43C2×C22⋊Dic53—22no—
44C2×C4×Dic53—21no—
45C4⋊#88322—22no—
46C22⋊#88322—22no—
47C4×#88322—22no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53C2×C412yes—
50C2×M16×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58#100×C53—42no—
59C24×C53C22⋊C412yes—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53SD1612yes—
63C2×D4×C53Q1612yes—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—44no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—28no—
75C22×Dic53—84no—
76C2×Dic106—21no—
77C2×Dic106—24no—
78C2×Dic106—42no—
79C2×Dic106—82no—
80C2×Dic106C22⋊C412yes—
81M16×C53—42no—
82M16×C53—42no—
83C2×C8×C53—42no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—24no—
86C42×C53C2×D412yes—
87C42×C53C2×D412yes—
88C22⋊Dic53—28no—
89C4⋊Dic53—28no—
90C4⋊Dic53—21no—
91C4⋊Dic53—22no—
92C4⋊Dic53—24no—
93C4⋊Dic53C22⋊C412yes—
94C4⋊Dic53C22⋊C414yes—
95#11719919—28no—
96#11719919—24no—
97C4×Dic53—24no—
98C2×#88322—42no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53#10314yes—
104C23×C53C2×C22⋊C411yescharacteristic
105D4×C53—24no—
106D4×C53—24no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—44no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53C2×C22⋊C411yescharacteristic
114C2×C4×C53C2×C22⋊C411yescharacteristic
115C2×C4×C53#12012yes—
116C2×C4×C53#12012yes—
117C2×C4×C53C2×SD1611yescharacteristic
118C2×C4×C53C2×Q1611yescharacteristic
119C2×C4×C53#13712yes—
120C2×Dic53—24no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—42no—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—88no—
127Dic106—24no—
128Dic106—42no—
129Dic106—84no—
130C8×C53—82no—
131#88322—82no—
132#8395—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—24no—
136C22×C53—44no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53#37911yescharacteristic
141C22×C53C2×#10311yescharacteristic
142C22×C53#41911yescharacteristic
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—42no—
146C4×C53#41312yes—
147C4×C53#44812yes—
148C4×C53#45112yes—
149Dic53—44no—
150Dic53—42no—
151Dic53—42no—
152C2×#554—531no—
153#1800D5311yescharacteristic
154#1549—532no—
155#1487—532no—
156#1460—531no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—44no—
160C2×C53#178511yescharacteristic
161C2×C53#178711yescharacteristic
162C2×C53#179511yescharacteristic
163#554—1064no—
164#554—1064no—
165C2×C4⋊Q8—531no—
166C2×#125—1061no—
167C2×C22⋊Q8—532no—
168C2×C22⋊Q8—1062no—
169C2×C4×Q8—531no—
170C2×C4×D4—1061no—
171C2×C4×D4C2×D5311yescharacteristic
172C22×C4⋊C4—532no—
173#431—532no—
174#423C2×D5312yes—
175C2×#100—21no—
176#343—21no—
177#332—1064no—
178Q8⋊C8—1064no—
179#323—1062no—
180C53#839511yescharacteristic, complemented, Sylow
181C22×Q8—531no—
182C22×Q8—1061no—
183C22×D4C22⋊D5311yescharacteristic
184C23×C4—1062no—
185C23×C4C4×D5312yes—
186C2×M16—22no—
187C4⋊Q8—534no—
188C4⋊Q8—1062no—
189#125—1068no—
190C22⋊Q8—10616no—
191C22⋊Q8—1064no—
192C22⋊Q8—1064no—
193C4×Q8—534no—
194C4×Q8—1062no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—1068no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—532no—
200C2×C4⋊C4—1064no—
201C2×C4⋊C4—1064no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4—1062no—
206C2×C4⋊C4C22⋊D5311yescharacteristic
207C2×C22⋊C4—1062no—
208C2×C22⋊C4D21212yes—
209C2×C42—1061no—
210C2×C42C22×D5311yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—1062no—
214#100—42no—
215C22⋊C8—1062no—
216C4×C8—1062no—
217C24#1171992112yes—
218C2×Q8—532no—
219C2×Q8—1061no—
220C2×Q8—1064no—
221C2×Q8—4242no—
222C2×Q8—2122no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4#1171992312yes—
226C2×D4#1171992512yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—1068no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4—1062no—
234C22×C4—4244no—
235C22×C4#1171992112yes—
236C22×C4C2×C4×D5311yescharacteristic
237C22×C4C2×D21211yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239M16—42no—
240M16—42no—
241C2×C8—42no—
242C2×C8—2122no—
243C4⋊C4—22no—
244C4⋊C4—532no—
245C4⋊C4—534no—
246C4⋊C4—1068no—
247C4⋊C4—1068no—
248C4⋊C4—1061no—
249C4⋊C4—1062no—
250C4⋊C4—1064no—
251C4⋊C4—1064no—
252C22⋊C4—24no—
253C22⋊C4—1068no—
254C42—1064no—
255C42D4×D5312yes—
256C42D4×D5312yes—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23#1313314014yes—
262C23C2×#1171992111yescharacteristic
263Q8—1064no—
264Q8—4244no—
265Q8—2122no—
266D4—24no—
267D4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—1064no—
277C2×C4—4248no—
278C2×C4—2122no—
279C2×C4—2124no—
280C2×C4—2122no—
281C2×C4C22⋊C4×D5311yescharacteristic
282C2×C4C22⋊C4×D5311yescharacteristic
283C2×C4#1313318512yes—
284C2×C4#1313318712yes—
285C2×C4C2×#1171992311yescharacteristic
286C2×C4C2×#1171992511yescharacteristic
287C2×C4#1313324712yes—
288C8—82no—
289C8—4242no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22#1681047811yescharacteristic
298C22C2×#1313314011yescharacteristic
299C22#1681063111yescharacteristic
300C4—21no—
301C4—22no—
302C4—42no—
303C4—2122no—
304C4—2122no—
305C4—2124no—
306C4#1681060412yes—
307C4#1681076112yes—
308C4#1681077812yes—
309C2—21no—
310C2—21no—
311C2—44no—
312C2#2027464011yescharacteristic
313C2#2027464911yescharacteristic
314C2#2027468611yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#8395
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20273401C2semidirect
3. #20273143C2semidirect
180. C53#8395semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
#202734012
#202731432
#202730362
C53×#18002
#839553