Finite Group Databasebeta

← #27294009

Subgroups — #27294009

Order 13568.

422 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C8×D4×D53C211yesmaximal, characteristic
2#20276517C211yesmaximal, characteristic, complemented
3#20276512C211yesmaximal, characteristic, complemented
4#20272671C211yesmaximal, characteristic, complemented
5D4⋊#11719908C211yesmaximal, characteristic, complemented
6#20272669C211yesmaximal, characteristic, complemented
7C53×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×C8×D53—21no—
9C4×D4×D53C411yescharacteristic
10C8×D212C2211yescharacteristic
11D4×#88322C411yescharacteristic
12#16810688C411yescharacteristic
13C8×C22⋊D53—21no—
14#16810686—21no—
15C4⋊C8×D53C411yescharacteristic
16C22⋊C8×D53—21no—
17C4×C8×D53C2211yescharacteristic
18C4⋊#11719908C2211yescharacteristic, complemented
19#16810441—21no—
20#16810402—21no—
21#16810402—21no—
22#16810398—21no—
23C4×#11719908C2211yescharacteristic, complemented
24C8×D4×C53C2211yescharacteristic
25C53×C4⋊C16C2211yescharacteristic, complemented
26C4×C16×C53C2211yescharacteristic, complemented
27C2×D4×D53C811yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—42no—
34C2×C8×D53—42no—
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37C4×D212C2×C411yescharacteristic
38D4×Dic53C811yescharacteristic
39#13133179C811yescharacteristic
40C4×C22⋊D53—21no—
41#13133177—21no—
42C4⋊C4×D53C811yescharacteristic
43C22⋊C4×D53—21no—
44C42×D53C2×C411yescharacteristic
45C2×#11719908—21no—
46C2×#11719908—21no—
47C4⋊#88322C2×C411yescharacteristic
48#13133146—21no—
49C22⋊#88322—21no—
50#13133129—21no—
51#13133129—21no—
52C4×#88322C2×C411yescharacteristic
53C8×Dic53D411yescharacteristic
54C8×Dic53D411yescharacteristic
55C22×C8×C53—21no—
56C4×D4×C53C2×C411yescharacteristic
57C2×C16×C53—21no—
58C2×C16×C53—21no—
59C4⋊C8×C53C2×C411yescharacteristic
60C22⋊C8×C53—21no—
61C4×C8×C53C2311yescharacteristic
62C23×D53—21no—
63C2×D4×C53C2×C811yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67D4×D53—22no—
68D4×D53—22no—
69C2×D212C2×C811yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53C22⋊C411yescharacteristic
76C2×C4×D53C2×C811yescharacteristic
77C2×C4×D53M1611yescharacteristic
78C2×C8×C53—21no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53C2×D411yescharacteristic
82C2×C8×C53C2×D411yescharacteristic
83C4⋊C4×C53C2×C811yescharacteristic
84C22⋊C4×C53—21no—
85C42×C53C22×C411yescharacteristic
86C22⋊Dic53—21no—
87#11719921—21no—
88#11719921—21no—
89C4⋊Dic53C2×C811yescharacteristic
90#11719919—21no—
91C4×Dic53C22⋊C411yescharacteristic
92C4×Dic53C2×C811yescharacteristic
93C4×Dic53M1611yescharacteristic
94C2×#88322—21no—
95C2×#88322—42no—
96C2×#88322C22⋊C411yescharacteristic
97C2×#88322C22⋊C411yescharacteristic
98C8×D53—21no—
99C8×D53—21no—
100C8×D53—41no—
101C8×D53—42no—
102C8×D53—42no—
103C8×D53—41no—
104C8×D53—41no—
105C16×C53—41no—
106C16×C53—41no—
107#11719908—41no—
108#11719908—41no—
109C23×C53—21no—
110C22×D53—24no—
111C22×D53—21no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53C22⋊C811yescharacteristic
115D4×C53—21no—
116D4×C53M3212yes—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53C2×C22⋊C411yescharacteristic
121C2×C4×C53C22×C811yescharacteristic
122C2×C4×C53C2×M1611yescharacteristic
123C22⋊D53—24no—
124C2×Dic53—21no—
125C2×Dic53—42no—
126C2×Dic53C22⋊C811yescharacteristic
127C2×Dic53C22⋊C811yescharacteristic
128C2×Dic53C22⋊C811yescharacteristic
129D212—21no—
130D212M3212yes—
131C4×D53—22no—
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—42no—
137C4×D53—42no—
138C4×D53—41no—
139C4×D53—41no—
140C8×C53—21no—
141C8×C53—41no—
142C8×C53AGL(1,ℤ/8)11yescharacteristic
143C8×C53AGL(1,ℤ/8)11yescharacteristic
144#88322—21no—
145#88322—21no—
146#88322—41no—
147#11631—531nomaximal, Sylow
148C22×C53—22no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C2×C22⋊C811yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—22no—
155C2×D53—48no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53#35012yes—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53#41811yescharacteristic
162C4×C53#42111yescharacteristic
163C4×C53C2×M3211yescharacteristic
164C4×C53C2×M3211yescharacteristic
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169C2×C8×D4—531no—
170#2163—531no—
171#2162—531no—
172D4⋊C16—531no—
173D4⋊C16—531no—
174D4⋊C16—531no—
175D4⋊C16D5311yescharacteristic, complemented
176C2×C53—42no—
177C2×C53#147811yescharacteristic
178C2×C53#211211yescharacteristic
179C2×C53C2×#35011yescharacteristic
180D53—22no—
181D53—42no—
182C23×C8—1061no—
183C2×C4×D4—531no—
184C8×D4—531no—
185C8×D4—531no—
186C8×D4—531no—
187C8×D4—1061no—
188C8×D4—1061no—
189C8×D4C2×D5311yescharacteristic
190C2×C4⋊C8—531no—
191C2×C22⋊C8—1061no—
192C2×C4×C8—531no—
193C4⋊C16—531no—
194C4⋊C16—1061no—
195C4⋊C16C2×D5311yescharacteristic, complemented
196C22⋊C16—1061no—
197C22⋊C16—1061no—
198#346—1061no—
199C4×C16—531no—
200C4×C16C2×D5311yescharacteristic, complemented
201C53#1163111yescharacteristic, complemented, Sylow
202C22×D4—531no—
203C23×C4—1061no—
204C22×C8—21no—
205C22×C8—531no—
206C22×C8—531no—
207C22×C8—1061no—
208C22×C8—1061no—
209C22×C8—1061no—
210C22×C8—1061no—
211C22×C8—2122no—
212C22×C8—2122no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—1061no—
217C4×D4—1061no—
218C4×D4C4×D5311yescharacteristic
219C2×C4⋊C4—531no—
220C2×C22⋊C4—1061no—
221C2×C42—531no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—1061no—
225C2×C16—1061no—
226C4⋊C8—531no—
227C4⋊C8—1061no—
228C4⋊C8C4×D5311yescharacteristic
229C22⋊C8—21no—
230C22⋊C8—1061no—
231C22⋊C8—1061no—
232C22⋊C8—1061no—
233C4×C8—531no—
234C4×C8—531no—
235C4×C8—531no—
236C4×C8C22×D5311yescharacteristic
237C24—1061no—
238C2×D4—531no—
239C2×D4—1061no—
240C2×D4—1062no—
241C2×D4—1062no—
242C2×D4C8×D5311yescharacteristic
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—2122no—
252C22×C4—2122no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—531no—
257C2×C8—531no—
258C2×C8—1061no—
259C2×C8—1061no—
260C2×C8—1061no—
261C2×C8—2121no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8D4×D5311yescharacteristic
268C2×C8D4×D5311yescharacteristic
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4C8×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C42—531no—
277C42—531no—
278C42—531no—
279C42C2×C4×D5311yescharacteristic
280C16—41no—
281C16—41no—
282C16—2121no—
283C16—2121no—
284C23—21no—
285C23—531no—
286C23—1064no—
287C23—1061no—
288C23—2122no—
289C23—2122no—
290D4—21no—
291D4—532no—
292D4—1064no—
293D4—1061no—
294D4#1313315612yes—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—531no—
299C2×C4—531no—
300C2×C4—531no—
301C2×C4—1062no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4C22⋊C4×D5311yescharacteristic
313C2×C4C2×C8×D5311yescharacteristic
314C2×C4M16×D5311yescharacteristic
315C8—21no—
316C8—41no—
317C8—1061no—
318C8—1061no—
319C8—2121no—
320C8#1313324111yescharacteristic
321C8#1313324411yescharacteristic
322C22—22no—
323C22—41no—
324C22—41no—
325C22—532no—
326C22—1061no—
327C22—1061no—
328C22—1062no—
329C22—2128no—
330C22—2121no—
331C22—2121no—
332C22C22⋊C8×D5311yescharacteristic
333C4—21no—
334C4—41no—
335C4—1061no—
336C4—1061no—
337C4—1061no—
338C4—2121no—
339C4#1681062611yescharacteristic
340C4#1681063811yescharacteristic
341C4M32×D5311yescharacteristic
342C4C2×#1313315611yescharacteristic
343C2—42no—
344C2—1062no—
345C2—2122no—
346C2#2027310411yescharacteristic
347C2#2027632411yescharacteristic
348C2#350×D5311yescharacteristic
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC4×C8C22×D53
FittingC53×D4⋊C16C2
socleC22×C53C2×C22⋊C8
prime radicalC2×D4×D53C8
cosocleC4×C8×C53C23
hypercenterD4⋊C16D53
hypocenterC53#11631
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
2. #20276517C2semidirect
3. #20276512C2semidirect
4. #20272671C2semidirect
5. D4⋊#11719908C2semidirect
6. #20272669C2semidirect
7. C53×D4⋊C16C2semidirect
18. C4⋊#11719908C22semidirect
23. C4×#11719908C22semidirect
25. C53×C4⋊C16C22semidirect
26. C4×C16×C53C22semidirect
175. D4⋊C16D53semidirect
195. C4⋊C16C2×D53semidirect
200. C4×C16C2×D53semidirect
201. C53#11631semidirect

Maximal subgroups

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

subgroupindex
C8×D4×D532
#202765172
#202765122
#202726712
D4⋊#117199082
#202726692
C53×D4⋊C162
#1163153