Finite Group Databasebeta

← #27287726

Subgroups — #27287726

Order 13568.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16810905C211yesmaximal, characteristic
2#20274708C211yesmaximal, characteristic
3#20273487C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic
5#20273111C211yesmaximal, characteristic
6#20272611C211yesmaximal, characteristic
7C53×#1548C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic
9#16810905—22no—
10#16810905—22no—
11#16810676C2211yescharacteristic
12C2×#13133148—21no—
13#16810647C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133140—21no—
16C2×#13133135—21no—
17#16810391—21no—
18#16810373—22no—
19#16810369—22no—
20#16810355—22no—
21C2×C4⋊Q8×C53C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C53×#423C2211yescharacteristic
24C53×#333—22no—
25C22×Dic106D411yescharacteristic
26C2×#11719917—21no—
27C22×#88322—21no—
28C4×Dic106—21no—
29C4×Dic106—21no—
30C4×Dic106—21no—
31C4×Dic106—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×C4×Dic53—21no—
35#13133148—24no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322D412yes—
40C4⋊#88322D412yes—
41#13133140—24no—
42#13133135—42no—
43C22⋊#88322—21no—
44C4×#88322—21no—
45C22×Q8×C53—21no—
46C2×M16×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53D412yes—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C42×C53C2311yescharacteristic
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C22⋊C8×C53—21no—
58C4×C8×C53—21no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C2×Q8×C53—82no—
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C2×Dic106—21no—
69C2×Dic106—21no—
70C2×Dic106—21no—
71C2×Dic106—21no—
72C2×Dic106—21no—
73C2×Dic106SD1612yes—
74M16×C53—41no—
75M16×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C42×C53C2×D411yescharacteristic
83C42×C53C2×D411yescharacteristic
84C42×C53C2×D411yescharacteristic
85C42×C53#4111yescharacteristic
86C4⋊Dic53—21no—
87C4⋊Dic53—21no—
88#11719919—41no—
89#11719919—41no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92#11719917—41no—
93#11719917—41no—
94C2×#88322—22no—
95C2×#88322—22no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53#12311yescharacteristic
100Q8×C53—82no—
101Q8×C53—82no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—82no—
106C2×C4×C53#12011yescharacteristic
107C2×C4×C53#12111yescharacteristic
108C2×C4×C53#12111yescharacteristic
109C2×C4×C53#12111yescharacteristic
110C2×C4×C53#12311yescharacteristic
111C2×C4×C53#12411yescharacteristic
112C2×C4×C53C2×SD1611yescharacteristic
113C2×C4×C53#13511yescharacteristic
114C2×C4×C53#13711yescharacteristic
115C2×C4×C53#13711yescharacteristic
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—81no—
121Dic106—22no—
122Dic106—22no—
123Dic106—41no—
124Dic106—41no—
125C8×C53—81no—
126C8×C53—81no—
127#88322—42no—
128#88322—42no—
129#88322—81no—
130#10450—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#39411yescharacteristic
134C22×C53#48311yescharacteristic
135C22×C53#48511yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—82no—
140C4×C53#44811yescharacteristic
141C4×C53#45211yescharacteristic
142C4×C53#46511yescharacteristic
143C4×C53#47011yescharacteristic
144C4×C53#47112yes—
145Dic53—42no—
146Dic53—81no—
147C2×#475—531no—
148#1801—531no—
149#1570—531no—
150#1549—531no—
151#1548D5311yescharacteristic
152#1479—531no—
153#1301—531no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53#203511yescharacteristic
157C2×C53#204611yescharacteristic
158C2×C53#204911yescharacteristic
159C2×C4⋊Q8C2×D5311yescharacteristic
160C2×C4×Q8—531no—
161#475—1062no—
162#475—1062no—
163#431—531no—
164#431C2×D5311yescharacteristic
165C2×#106—1061no—
166#423—531no—
167#423C2×D5311yescharacteristic
168C2×C4⋊C8—531no—
169C2×#103—1061no—
170C2×#101—1061no—
171#343—1061no—
172#334—1062no—
173#333—22no—
174#332—1062no—
175Q8⋊C8—1062no—
176C53#1045011yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C22×Q8—531no—
179C2×M16—21no—
180C2×M16—1061no—
181C22×C8—1061no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8C22⋊D5312yes—
186C4×Q8—1061no—
187C4×Q8—1061no—
188C4×Q8—1061no—
189C4×Q8—1061no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C42—1061no—
195C2×C42C22×D5311yescharacteristic
196#106—1064no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—532no—
201C4⋊C8—532no—
202C4⋊C8—1061no—
203C4⋊C8—1061no—
204C4⋊C8—1061no—
205#103—1064no—
206#101—2122no—
207C22⋊C8—21no—
208C22⋊C8—1061no—
209C4×C8—21no—
210C4×C8—1061no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—532no—
215C2×Q8—1061no—
216C2×Q8—1061no—
217C2×Q8—1061no—
218C2×Q8—1061no—
219C2×Q8—1061no—
220C22×C4—41no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4#1171994511yescharacteristic
224C22×C4D4×D5311yescharacteristic
225C22×C4#1171994711yescharacteristic
226M16—41no—
227M16—41no—
228M16—2121no—
229M16—2121no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—1062no—
233C2×C8—1062no—
234C2×C8—2121no—
235C2×C8—2121no—
236C2×C8—2121no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—1061no—
242C4⋊C4—1061no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C42—2121no—
246C42—2121no—
247C42#1171994511yescharacteristic
248C42D4×D5311yescharacteristic
249C42D4×D5311yescharacteristic
250C42C2×C22⋊D5311yescharacteristic
251C23#1313319911yescharacteristic
252Q8—82no—
253Q8—82no—
254Q8—1062no—
255Q8—1062no—
256Q8—2121no—
257Q8—2121no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—1062no—
263C2×C4—4241no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313318511yescharacteristic
268C2×C4#1313318911yescharacteristic
269C2×C4#1313319011yescharacteristic
270C2×C4#1313319011yescharacteristic
271C2×C4#1313320011yescharacteristic
272C2×C4#1313320311yescharacteristic
273C2×C4SD16×D5311yescharacteristic
274C2×C4#1313324011yescharacteristic
275C2×C4#1313324611yescharacteristic
276C2×C4#1313324911yescharacteristic
277C8—81no—
278C8—81no—
279C8—4241no—
280C8—2122no—
281C8—2122no—
282C22—21no—
283C22—21no—
284C22#1681053811yescharacteristic
285C22#1681095211yescharacteristic
286C22#1681096311yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—82no—
291C4—4241no—
292C4—2122no—
293C4#1681076011yescharacteristic
294C4#1681078411yescharacteristic
295C4#1681085511yescharacteristic
296C4#1681088011yescharacteristic
297C4#1681088512yes—
298C2—21no—
299C2—21no—
300C2#2027590911yescharacteristic
301C2#2027597611yescharacteristic
302C2#2027599311yescharacteristic
subgroupG/H
centerC22#16810538
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1548C2
socleC22×C53#394
prime radicalC23×C53#123
hypercenter#1548D53
hypocenterC53#10450
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
176. C53#10450semidirect

Maximal subgroups

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

subgroupindex
C2×#168109052
#202747082
#202734872
#202734002
#202731112
#202726112
C53×#15482
#1045053