Finite Group Databasebeta

← #27287713

Subgroups — #27287713

Order 13568.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16810900C211yesmaximal, characteristic
2#20274704C211yesmaximal, characteristic
3#20273486C211yesmaximal, characteristic, complemented
4#20273396C211yesmaximal, characteristic, complemented
5#20273105C211yesmaximal, characteristic
6#20272610C211yesmaximal, characteristic, complemented
7C53×#1549C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810900—22no—
10#16810900—22no—
11#16810676C2211yescharacteristic
12C2×#13133150—21no—
13#16810647C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—21no—
16C2×#13133133—21no—
17#16810391—21no—
18#16810374—22no—
19#16810372—22no—
20#16810350—22no—
21C2×C4⋊Q8×C53C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C53×#423C2211yescharacteristic
24C53×#332—22no—
25C22×D212D411yescharacteristic
26C22×C4×D53—21no—
27C2×#11719917—21no—
28C22×#88322—21no—
29C4×D212—21no—
30C4×D212—21no—
31C4×D212—21no—
32C4×D212—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719921—21no—
35#13133150—24no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D412yes—
41#13133137—24no—
42#13133133—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×C53D412yes—
57C22⋊C8×C53—21no—
58C4×C8×C53—21no—
59C23×D53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53—82no—
63C22×C4×C53—41no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×C4×C53#4111yescharacteristic
67C22×Dic53—21no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212SD1612yes—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—82no—
77M16×C53—41no—
78M16×C53—41no—
79C2×C8×C53—22no—
80C2×C8×C53—41no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C42×C53C2×D411yescharacteristic
86C42×C53C2×D411yescharacteristic
87C42×C53C2×D411yescharacteristic
88C42×C53#4111yescharacteristic
89#11719921—41no—
90#11719921—41no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93#11719917—41no—
94#11719917—41no—
95C2×#88322—22no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C23×C53#12311yescharacteristic
101C22×D53—22no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—81no—
105C22×D53—81no—
106Q8×C53—82no—
107Q8×C53—82no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—82no—
112C2×C4×C53#12011yescharacteristic
113C2×C4×C53#12111yescharacteristic
114C2×C4×C53#12111yescharacteristic
115C2×C4×C53#12111yescharacteristic
116C2×C4×C53#12311yescharacteristic
117C2×C4×C53#12411yescharacteristic
118C2×C4×C53C2×SD1611yescharacteristic
119C2×C4×C53#13511yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C2×C4×C53#13711yescharacteristic
122C2×Dic53—41no—
123C2×Dic53—81no—
124D212—22no—
125D212—22no—
126D212—41no—
127D212—41no—
128C4×D53—82no—
129C4×D53—82no—
130C8×C53—42no—
131C8×C53—81no—
132#88322—42no—
133#88322—81no—
134#88322—81no—
135#10448—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53#39411yescharacteristic
139C22×C53#48311yescharacteristic
140C22×C53#48511yescharacteristic
141C2×D53—42no—
142C2×D53—82no—
143C2×D53—81no—
144C2×D53—81no—
145C2×D53—82no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—82no—
150C4×C53#44811yescharacteristic
151C4×C53#45211yescharacteristic
152C4×C53#46511yescharacteristic
153C4×C53#47011yescharacteristic
154C4×C53#47212yes—
155Dic53—81no—
156C2×#474—531no—
157#1800—531no—
158#1570—531no—
159#1549D5311yescharacteristic, complemented
160#1548—531no—
161#1478—531no—
162#1301—531no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#203511yescharacteristic
166C2×C53#204511yescharacteristic
167C2×C53#204911yescharacteristic
168D53—82no—
169C2×C4⋊Q8C2×D5311yescharacteristic
170C2×C4×D4—531no—
171#474—1062no—
172#474—1062no—
173#431—531no—
174#431C2×D5311yescharacteristic
175C2×#107—1061no—
176#423—531no—
177#423C2×D5311yescharacteristic
178C2×C4⋊C8—531no—
179C2×#102—1061no—
180C2×#100—1061no—
181#343—1061no—
182#334—1062no—
183#333—1062no—
184#332—22no—
185D4⋊C8—1062no—
186C53#1044811yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4—531no—
189C23×C4—1061no—
190C2×M16—21no—
191C2×M16—1061no—
192C22×C8—1061no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8D21212yes—
197C4×D4—1061no—
198C4×D4—1061no—
199C4×D4—1061no—
200C4×D4—1061no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—531no—
204C2×C22⋊C4—1061no—
205C2×C42C22×D5311yescharacteristic
206#107—1064no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—532no—
210C4⋊C8—1061no—
211C4⋊C8—1061no—
212C4⋊C8—1061no—
213C4⋊C8—1061no—
214C4⋊C8C22⋊D5312yes—
215#102—1064no—
216#100—2122no—
217C22⋊C8—21no—
218C22⋊C8—1061no—
219C4×C8—21no—
220C4×C8—1061no—
221C24—1061no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—532no—
226C2×D4—1061no—
227C2×D4—1061no—
228C2×D4—1061no—
229C2×D4—1061no—
230C2×D4—1061no—
231C22×C4—41no—
232C22×C4—1061no—
233C22×C4—4242no—
234C22×C4—2121no—
235C22×C4—2121no—
236C22×C4#1171994511yescharacteristic
237C22×C4D4×D5311yescharacteristic
238C22×C4#1171994711yescharacteristic
239M16—41no—
240M16—41no—
241M16—2121no—
242M16—2121no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—1062no—
246C2×C8—2121no—
247C2×C8—2121no—
248C2×C8—2121no—
249C2×C8—2121no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C42C2×D21211yescharacteristic
259C42D4×D5311yescharacteristic
260C42#1171994711yescharacteristic
261C42C2×C22⋊D5311yescharacteristic
262C23—1062no—
263C23—4241no—
264C23—4241no—
265C23—2121no—
266C23—2121no—
267C23#1313319911yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—1062no—
271D4—1062no—
272D4—2121no—
273D4—2121no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—4241no—
281C2×C4—2121no—
282C2×C4#1313318711yescharacteristic
283C2×C4#1313318811yescharacteristic
284C2×C4#1313318911yescharacteristic
285C2×C4#1313319111yescharacteristic
286C2×C4#1313320111yescharacteristic
287C2×C4#1313320311yescharacteristic
288C2×C4SD16×D5311yescharacteristic
289C2×C4#1313323811yescharacteristic
290C2×C4#1313324911yescharacteristic
291C2×C4#1313324911yescharacteristic
292C8—42no—
293C8—81no—
294C8—4241no—
295C8—4241no—
296C8—2122no—
297C22—21no—
298C22—21no—
299C22—4242no—
300C22—4242no—
301C22—4241no—
302C22—4241no—
303C22—2122no—
304C22#1681053911yescharacteristic
305C22#1681095211yescharacteristic
306C22#1681096211yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—4241no—
312C4#1681076311yescharacteristic
313C4#1681078811yescharacteristic
314C4#1681084911yescharacteristic
315C4#1681087811yescharacteristic
316C4#1681089112yes—
317C2—21no—
318C2—21no—
319C2—4242no—
320C2#2027591011yescharacteristic
321C2#2027597011yescharacteristic
322C2#2027599411yescharacteristic
subgroupG/H
centerC22#16810539
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1549C2
socleC22×C53#394
prime radicalC22×D212D4
hypercenter#1549D53
hypocenterC53#10448
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. #20273486C2semidirect
4. #20273396C2semidirect
6. #20272610C2semidirect
7. C53×#1549C2semidirect
159. #1549D53semidirect
186. C53#10448semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202747042
#202734862
#202733962
#202731052
#202726102
C53×#15492
#1044853