Finite Group Databasebeta

← #27281942

Subgroups — #27281942

Order 13568.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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#20283739C211yesmaximal, characteristic, complemented
2#20274707C211yesmaximal, characteristic, complemented
3#20273140C212yesmaximal, complemented
4#20272573C212yesmaximal, complemented
5C53×#2535C211yesmaximal, characteristic
6D4×#88322—24no—
7#16810676C2212yes—
8#16810647C2212yes—
9C2×#13133134—21no—
10C2×C22⋊#88322—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—22no—
22C22×#88322—24no—
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—42no—
36#121×C53—42no—
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53C2×C412yes—
48C2×C42×C53C2311yescharacteristic
49#95×C53—22no—
50#95×C53—21no—
51#95×C53C2×C412yes—
52C24×C53—21no—
53C24×C53C22⋊C412yes—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—42no—
58C2×D4×C53—44no—
59C2×D4×C53—44no—
60C2×D4×C53—82no—
61C2×D4×C53D812yes—
62C2×D4×C53SD1612yes—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—42no—
71C22×C4×C53—82no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—82no—
85C42×C53C2×D412yes—
86C42×C53#4112yes—
87#11719917—42no—
88#11719917—42no—
89C2×#88322—24no—
90C2×#88322—42no—
91C2×#88322—84no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10214yes—
104C23×C53C2×C22⋊C411yescharacteristic
105D4×C53—24no—
106D4×C53—24no—
107D4×C53—24no—
108D4×C53—42no—
109D4×C53—88no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—82no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124C2×C4×C53C4×D412yes—
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12112yes—
127C2×C4×C53#12311yescharacteristic
128C2×C4×C53C2×D811yescharacteristic
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53#13512yes—
131#88322—44no—
132#88322—82no—
133#9485—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—24no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—82no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53#38611yescharacteristic
149C22×C53C2×#10211yescharacteristic
150C22×C53#41611yescharacteristic
151C4×C53—22no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—42no—
155C4×C53—82no—
156C4×C53—81no—
157C4×C53#46512yes—
158C4×C53#46612yes—
159#2972—531no—
160#2535D5311yescharacteristic
161#1800—531no—
162#1486—532no—
163#1286—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—44no—
167C2×C53—82no—
168C2×C53#189111yescharacteristic
169C2×C53#189311yescharacteristic
170C2×C53#191111yescharacteristic
171C2×#121—21no—
172C2×C4×D4C2×D5311yescharacteristic
173C2×C4×D4C2×D5311yescharacteristic
174C8×D4—1064no—
175#431—532no—
176#423—532no—
177C2×#100—1061no—
178C2×C22⋊C8—1062no—
179#397Dic5312yes—
180#392Dic5312yes—
181#386—22no—
182#382C2×D5311yescharacteristic
183#380—21no—
184#343—1061no—
185D4⋊C8—1064no—
186C53#948511yescharacteristic, complemented, Sylow
187C22×D4—22no—
188C22×D4C2×Dic5311yescharacteristic
189C22×D4C22⋊D5311yescharacteristic
190C23×C4—21no—
191C23×C4C22⋊D5312yes—
192C2×M16—1062no—
193C22×C8—1064no—
194#121—42no—
195#121—42no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic5311yescharacteristic
202C2×C4⋊C4C22⋊D5311yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×Dic5312yes—
207C2×C42C22×D5311yescharacteristic
208C4⋊C8—1062no—
209C4⋊C8—1062no—
210C4⋊C8—1062no—
211#100—2122no—
212C22⋊C8—1064no—
213C22⋊C8—1062no—
214C4×C8—534no—
215#95—21no—
216#95—22no—
217#95C2×Dic5312yes—
218C24—21no—
219C24C22⋊Dic5312yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4#1171992212yes—
228C2×D4#1171992312yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4C22⋊Dic5312yes—
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—2122no—
243M16—2122no—
244C2×C8—1064no—
245C2×C8—4244no—
246C2×C8—2122no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—82no—
256C42D4×D5312yes—
257C42#1171994712yes—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1313313814yes—
270C23C2×C22⋊Dic5311yescharacteristic
271D4—24no—
272D4—24no—
273D4—24no—
274D4—42no—
275D4—88no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4D4×Dic5312yes—
291C2×C4#1313318511yescharacteristic
292C2×C4#1313318912yes—
293C2×C4#1313319811yescharacteristic
294C2×C4C2×#1171992211yescharacteristic
295C2×C4C2×#1171992311yescharacteristic
296C2×C4#1313323712yes—
297C8—4242no—
298C8—2124no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1681050311yescharacteristic
314C22C2×#1313313811yescharacteristic
315C22#1681061911yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4#1681085012yes—
323C4#1681085712yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#2027513411yescharacteristic
329C2#2027514411yescharacteristic
330C2#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#9485
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. #20283739C2semidirect
2. #20274707C2semidirect
3. #20273140C2semidirect
4. #20272573C2semidirect
29. C4×#88322D4semidirect
186. C53#9485semidirect

Maximal subgroups

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

subgroupindex
#202837392
#202747072
#202731402
#202725732
C53×#25352
#948553