Finite Group Databasebeta

← #27277698

Subgroups — #27277698

Order 13568.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#13133134C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic
3#20272608C212yesmaximal, complemented
4#20272579C212yesmaximal, complemented
5C53×#2353C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×#13133134—21no—
8C4×#11719917C2212yes—
9#16810392—21no—
10#16810347—21no—
11#16810347—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C53×#388—21no—
14C53×#387—21no—
15C53×#386—21no—
16C53×#383C2211yescharacteristic
17C53×#381C411yescharacteristic
18C53×#380C411yescharacteristic
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53C411yescharacteristic
21C53×#376C2211yescharacteristic
22C2×#95×C53C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—22no—
25#13133134—24no—
26C22⋊#88322—22no—
27#13133128—22no—
28#13133128—22no—
29C4×#88322—22no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C4×D4×C53—22no—
35C4×D4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53C2311yescharacteristic
46#95×C53—21no—
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53C2×C411yescharacteristic
52#95×C53C2×C411yescharacteristic
53#95×C53C2×C411yescharacteristic
54#95×C53C2×C411yescharacteristic
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—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—22no—
79C4⋊C4×C53—81no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—81no—
85C42×C53—42no—
86C42×C53—81no—
87C42×C53#4112yes—
88C42×C53#4112yes—
89#11719917—24no—
90#11719917—42no—
91C2×#88322—42no—
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—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—84no—
130C2×C4×C53#11711yescharacteristic
131C2×C4×C53#11711yescharacteristic
132C2×C4×C53#12312yes—
133C2×C4×C53#12312yes—
134#88322—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—42no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—81no—
160C4×C53—81no—
161C4×C53—81no—
162#2353D5311yescharacteristic
163C4×#100—531no—
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—1061no—
177C4×M16—532no—
178#388—21no—
179#387—21no—
180#386—21no—
181#383C2×D5311yescharacteristic
182#381Dic5311yescharacteristic
183#380Dic5311yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4Dic5311yescharacteristic
186#376C2×D5311yescharacteristic
187C2×#95Dic5311yescharacteristic
188#343—1061no—
189#323—1061no—
190#323—1061no—
191C53#877211yescharacteristic, complemented, Sylow
192C22×D4C22⋊D5311yescharacteristic
193C23×C4—21no—
194C23×C4C2×Dic5311yescharacteristic
195C23×C4C2×Dic5311yescharacteristic
196C2×M16—1062no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C22⋊D5311yescharacteristic
205C2×C22⋊C4C22⋊D5311yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22×D5311yescharacteristic
209C4⋊C8—1062no—
210#100—1064no—
211C22⋊C8—1062no—
212#97—1062no—
213#97—1062no—
214C4×C8—1062no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×Dic5311yescharacteristic
221#95C2×Dic5311yescharacteristic
222#95C2×Dic5311yescharacteristic
223#95C2×Dic5311yescharacteristic
224C24C22⋊Dic5311yescharacteristic
225C24C22⋊Dic5311yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22×Dic5311yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—1064no—
248M16—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C4⋊C4—22no—
252C4⋊C4—81no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—81no—
258C42—42no—
259C42—81no—
260C42#1171994512yes—
261C42#1171994512yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1313314412yes—
272C23C2×C22⋊Dic5311yescharacteristic
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—84no—
299C2×C4#1313317611yescharacteristic
300C2×C4#1313317611yescharacteristic
301C2×C4#1313320012yes—
302C2×C4#1313320012yes—
303C8—4242no—
304C8—2124no—
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#1681048611yescharacteristic
318C22C2×#1313314411yescharacteristic
319C22#1681064011yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#2027481411yescharacteristic
335C2#2027481911yescharacteristic
336C2#2027482611yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2353C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2353D53
hypocenterC53#8772
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #20272608C2semidirect
4. #20272579C2semidirect
191. C53#8772semidirect

Maximal subgroups

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

subgroupindex
C4×#131331342
#202730562
#202726082
#202725792
C53×#23532
#877253