Finite Group Databasebeta

← #27277699

Subgroups — #27277699

Order 13568.

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

normal subgroups only

This group has 446 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
1#20277655C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic
3#20272609C212yesmaximal, complemented
4#20272580C212yesmaximal, complemented
5C4×#100×C53C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810516—21no—
8#16810509—21no—
9#16810503—21no—
10#16810491C2211yescharacteristic
11#16810486C411yescharacteristic
12#16810483C411yescharacteristic
13C22⋊C4×Dic53—21no—
14C4×C22⋊Dic53C411yescharacteristic
15#16810469C2211yescharacteristic
16C2×#13133124C411yescharacteristic
17#16810347—21no—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#100×C53—21no—
21C4×M16×C53C2212yes—
22C53×#343—21no—
23C23×Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—21no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53—21no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133124—21no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124C2×C411yescharacteristic
39#13133124C2×C411yescharacteristic
40#13133124C2×C411yescharacteristic
41#13133124C2×C411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C2×M16×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52#100×C53—24no—
53#97×C53—22no—
54C4×C8×C53—22no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—81no—
65C22×C4×C53—81no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C22×Dic53—81no—
78M16×C53—24no—
79M16×C53—42no—
80C2×C8×C53—42no—
81C4⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C42×C53#4112yes—
85C42×C53#4112yes—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—81no—
89C4⋊Dic53—81no—
90C4×Dic53—42no—
91C4×Dic53—81no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53#10412yes—
103C23×C53C2×C22⋊C411yescharacteristic
104D4×C53—44no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—81no—
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53—81no—
117C2×C4×C53#11711yescharacteristic
118C2×C4×C53#11711yescharacteristic
119C2×C4×C53#12312yes—
120C2×C4×C53#12312yes—
121C2×Dic53—22no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—84no—
134C8×C53—44no—
135#88322—82no—
136#8772—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53#38111yescharacteristic
150C22×C53C2×#10411yescharacteristic
151C22×C53#42111yescharacteristic
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—42no—
159Dic53—81no—
160Dic53—81no—
161Dic53—81no—
162#2353—531no—
163C4×#100D5311yescharacteristic
164#1466—531no—
165#1300—532no—
166#1288—532no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—42no—
170C2×C53—81no—
171C2×C53#182611yescharacteristic
172C2×C53#182711yescharacteristic
173C2×C53#182911yescharacteristic
174C2×C4×D4C2×D5311yescharacteristic
175#432—532no—
176C2×#100—21no—
177C4×M16C2×D5312yes—
178#388—1061no—
179#387—1061no—
180#386—1061no—
181#383—531no—
182#381—531no—
183#380—531no—
184C4×C22⋊C4—531no—
185C4×C22⋊C4—1061no—
186#376—531no—
187C2×#95—531no—
188#343—21no—
189#323—1061no—
190#323—1061no—
191C53#877211yescharacteristic, complemented, Sylow
192C22×D4C22⋊D5311yescharacteristic
193C23×C4—1061no—
194C23×C4C4×D5311yescharacteristic
195C23×C4C4×D5311yescharacteristic
196C2×M16—22no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1061no—
203C2×C22⋊C4—1061no—
204C2×C22⋊C4D21211yescharacteristic
205C2×C22⋊C4D21211yescharacteristic
206C2×C42—1061no—
207C2×C42—1061no—
208C2×C42C22×D5311yescharacteristic
209C4⋊C8—1062no—
210#100—24no—
211C22⋊C8—1062no—
212#97—22no—
213#97—1062no—
214C4×C8—22no—
215#95—531no—
216#95—531no—
217#95—531no—
218#95—531no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224C24#1171992111yescharacteristic
225C24#1171992111yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—4241no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4C2×D21211yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C2×C8—2122no—
251C4⋊C4—22no—
252C4⋊C4—4241no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—4241no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C42—4241no—
259C42—2122no—
260C42#1171994712yes—
261C42#1171994712yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1313314512yes—
272C23C2×#1171992111yescharacteristic
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—1062no—
287C2×C4—4244no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4#1313317311yescharacteristic
300C2×C4#1313317311yescharacteristic
301C2×C4#1313319812yes—
302C2×C4#1313320112yes—
303C8—44no—
304C8—4242no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22#1681048411yescharacteristic
318C22C2×#1313314511yescharacteristic
319C22#1681064111yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—4241no—
327C4—4241no—
328C4—4241no—
329C4—2122no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#2027481511yescharacteristic
335C2#2027482011yescharacteristic
336C2#2027482411yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#100×C53C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterC4×#100D53
hypocenterC53#8772
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #20272609C2semidirect
4. #20272580C2semidirect
191. C53#8772semidirect

Maximal subgroups

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

subgroupindex
#202776552
#202730562
#202726092
#202725802
C4×#100×C532
#877253