Finite Group Databasebeta

← #27281962

Subgroups — #27281962

Order 13568.

602 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#20283756C211yesmaximal, characteristic, complemented
2#20274707C211yesmaximal, characteristic, complemented
3#20273140C212yesmaximal, complemented
4#20272573C212yesmaximal, complemented
5C53×#2535C211yesmaximal, characteristic
6#16810701—24no—
7#16810676C2212yes—
8#16810647C2212yes—
9C2×#13133134—21no—
10#16810587—22no—
11#16810392—21no—
12D4⋊#88322—24no—
13C2×#121×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#397C412yes—
17C53×#392C412yes—
18C53×#386—22no—
19C53×#382C2211yescharacteristic
20C53×#380—21no—
21C2×#11719917—24no—
22C2×#11719917—22no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133134—42no—
27C22⋊#88322—24no—
28C22⋊#88322—22no—
29C4×#88322D414yescomplemented
30C22×D4×C53—22no—
31C22×D4×C53C2×C411yescharacteristic
32C22×D4×C53D411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53D412yes—
35#121×C53—44no—
36C4×D4×C53—22no—
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C42×C53C2311yescharacteristic
48#95×C53—22no—
49#95×C53—21no—
50#95×C53C2×C412yes—
51C24×C53—21no—
52C24×C53C22⋊C412yes—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—21no—
57C2×D4×C53—21no—
58C2×D4×C53—42no—
59C2×D4×C53—44no—
60C2×D4×C53—44no—
61C2×D4×C53—82no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—82no—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—42no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—82no—
83C42×C53C2×D412yes—
84C42×C53#4112yes—
85#11719917—42no—
86#11719917—42no—
87#11719917—84no—
88C2×#88322—24no—
89C2×#88322—42no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103D4×C53—24no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53—42no—
107D4×C53—88no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—82no—
120C2×C4×C53—82no—
121C2×C4×C53—82no—
122C2×C4×C53C4×D412yes—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12112yes—
125C2×C4×C53#12311yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53AGL(1,ℤ/8)12yes—
128C2×C4×C53#13711yescharacteristic
129#88322—44no—
130#88322—82no—
131#9489—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—42no—
135C22×C53—44no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53#38611yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#41811yescharacteristic
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53—82no—
154C4×C53—81no—
155C4×C53#46812yes—
156C4×C53#46912yes—
157#2975—531no—
158#2535D5311yescharacteristic
159#1800—531no—
160#1486—532no—
161#1286—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—82no—
165C2×C53—82no—
166C2×C53#189111yescharacteristic
167C2×C53#189511yescharacteristic
168C2×C53#191111yescharacteristic
169C2×#121—21no—
170C2×C4×D4C2×D5311yescharacteristic
171C2×C4×D4C2×D5311yescharacteristic
172#436—1064no—
173#431—532no—
174#423—532no—
175C2×#100—1061no—
176#407—1062no—
177#397Dic5312yes—
178#392Dic5312yes—
179#386—22no—
180#382C2×D5311yescharacteristic
181#380—21no—
182#343—1061no—
183D4⋊C8—1064no—
184C53#948911yescharacteristic, complemented, Sylow
185C22×D4—22no—
186C22×D4C2×Dic5311yescharacteristic
187C22×D4C22⋊D5311yescharacteristic
188C23×C4—21no—
189C23×C4C22⋊D5312yes—
190C2×M16—1064no—
191C2×M16—1062no—
192#121—44no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C4⋊C4C22⋊D5311yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×Dic5312yes—
204C2×C42C22×D5311yescharacteristic
205C4⋊C8—1062no—
206C4⋊C8—1062no—
207C4⋊C8—1062no—
208#100—2122no—
209C22⋊C8—1064no—
210C22⋊C8—1062no—
211C4×C8—534no—
212#95—21no—
213#95—22no—
214#95C2×Dic5312yes—
215C24—21no—
216C24C22⋊Dic5312yes—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4C22⋊Dic5312yes—
236C22×C4C22×Dic5311yescharacteristic
237C22×C4C2×C22⋊D5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239M16—4244no—
240M16—2122no—
241M16—2122no—
242C2×C8—1064no—
243C2×C8—2122no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C22⋊C4—24no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—82no—
252C42D4×D5312yes—
253C42#1171994712yes—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic5311yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4D4×Dic5312yes—
287C2×C4#1313318511yescharacteristic
288C2×C4#1313318912yes—
289C2×C4#1313319811yescharacteristic
290C2×C4#1313324211yescharacteristic
291C2×C4#1313324312yes—
292C2×C4#1313324711yescharacteristic
293C8—4242no—
294C8—2124no—
295C22—21no—
296C22—21no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#1681050311yescharacteristic
310C22#1681062411yescharacteristic
311C22#1681062811yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—42no—
316C4—82no—
317C4—81no—
318C4#1681086712yes—
319C4#1681087412yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#2027513411yescharacteristic
325C2#2027515411yescharacteristic
326C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2535C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#2535D53
hypocenterC53#9489
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #20283756C2semidirect
2. #20274707C2semidirect
3. #20273140C2semidirect
4. #20272573C2semidirect
29. C4×#88322D4semidirect
184. C53#9489semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202747072
#202731402
#202725732
C53×#25352
#948953