Finite Group Databasebeta

← #27302350

Subgroups — #27302350

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810998C211yesmaximal, characteristic
2C2×#16810437C211yesmaximal, characteristic, complemented
3C2×#16810436C211yesmaximal, characteristic, complemented
4C2×#16810435C211yesmaximal, characteristic, complemented
5M32×Dic53C211yesmaximal, characteristic, complemented
6#20274885C211yesmaximal, characteristic, complemented
7C2×C53×#361C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×#11719922—21no—
10C2×#13133213—21no—
11C2×#13133157C2211yescharacteristic
12#16810998—21no—
13#16810998—21no—
14#16810998C412yes—
15#16810998C412yes—
16C2×#13133153C2211yescharacteristic
17C2×#13133152—21no—
18#16810646—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810437—22no—
21#16810436—22no—
22#16810435—22no—
23#16810398—21no—
24C16×Dic53—21no—
25C22×D8×C53C2211yescharacteristic
26C2×M32×C53C2211yescharacteristic
27C2×#108×C53C2211yescharacteristic
28C53×#361—22no—
29C22×D212—21no—
30C22×C22⋊D53—42no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424C2×C412yes—
37C2×#11719922—22no—
38C2×#11719922—22no—
39C2×#11719922—42no—
40C2×#11719922—42no—
41C2×#11719917—21no—
42C22×#88322D411yescharacteristic
43#13133213—22no—
44#13133213—22no—
45#13133213—42no—
46C2×C4×Dic53D411yescharacteristic
47#13133157—21no—
48#13133157D412yescomplemented
49C2×#11719908—21no—
50#13133153—21no—
51#13133153—21no—
52#13133152—42no—
53#13133146—42no—
54C8×Dic53C2×C411yescharacteristic
55C8×Dic53C2×C411yescharacteristic
56C8×Dic53D411yescharacteristic
57C8×Dic53D411yescharacteristic
58C22×D4×C53—21no—
59C2×D8×C53—21no—
60C2×D8×C53—21no—
61C2×D8×C53—21no—
62C2×D8×C53—21no—
63C2×D8×C53—21no—
64C2×D8×C53C2×C412yes—
65C2×M16×C53—21no—
66C22×C8×C53C2311yescharacteristic
67M32×C53—21no—
68M32×C53D412yescomplemented
69C2×C16×C53—21no—
70#108×C53—21no—
71#108×C53—21no—
72C24×C53—41no—
73C23×D53—41no—
74C2×D4×C53—22no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—44no—
82C2×C22⋊D53—44no—
83C2×C22⋊D53—88no—
84C22×Dic53SD1612yes—
85C2×D212—22no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91D8×C53—22no—
92D8×C53—22no—
93D8×C53—41no—
94D8×C53—41no—
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53C22×C411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C2×C8×C53#4111yescharacteristic
104#11719922—44no—
105#11719922—44no—
106C4×Dic53C22⋊C411yescharacteristic
107C4×Dic53C22⋊C411yescharacteristic
108C4×Dic53D812yes—
109#11719917—41no—
110#11719917—41no—
111C2×#88322—41no—
112C2×#88322C22⋊C411yescharacteristic
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322D812yes—
115C2×#88322SD1612yes—
116D424—22no—
117D424—22no—
118D424—41no—
119D424—41no—
120C16×C53—22no—
121#11719908—22no—
122C23×C53—42no—
123C23×C53—84no—
124C23×C53—81no—
125C23×C53—81no—
126C23×C53C2×SD1611yescharacteristic
127C22×D53—42no—
128C22×D53—84no—
129C22×D53—81no—
130C22×D53—81no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—81no—
134D4×C53—81no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53#12411yescharacteristic
137C2×C4×C53#12711yescharacteristic
138C2×C4×C53C2×D811yescharacteristic
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C22⋊D53—88no—
142C22⋊D53—88no—
143C2×Dic53—24no—
144C2×Dic53#10212yes—
145C2×Dic53#10212yes—
146D212—42no—
147D212—42no—
148D212—81no—
149D212—81no—
150C8×C53—81no—
151C8×C53C4×D411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12111yescharacteristic
155#88322—81no—
156#88322#10212yes—
157#88322#10212yes—
158#13213—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—84no—
161C22×C53—82no—
162C22×C53—162no—
163C22×C53—162no—
164C22×C53C2×#10211yescharacteristic
165C22×C53#48611yescharacteristic
166C22×C53#49211yescharacteristic
167C2×D53—84no—
168C2×D53—82no—
169C2×D53—162no—
170C2×D53—162no—
171C4×C53#39011yescharacteristic
172C4×C53C2×#10211yescharacteristic
173C4×C53#48611yescharacteristic
174C4×C53#49311yescharacteristic
175Dic53—22no—
176Dic53#36114yescomplemented
177C2×#493—531no—
178C2×#361—531no—
179C2×#361—531no—
180C2×#361—531no—
181C2×#361D5311yescharacteristic, complemented
182C4×M32—531no—
183#1848—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—162no—
187C2×C53#195711yescharacteristic
188C2×C53C2×#36112yes—
189D53—162no—
190C22×D8—531no—
191C22×D8—1061no—
192C22×D8C2×D5311yescharacteristic
193C2×#127—1061no—
194C2×M32—531no—
195C2×M32C2×D5311yescharacteristic
196#493—532no—
197#493—532no—
198#493—1061no—
199#493—1061no—
200C2×#108—531no—
201C2×#108—1061no—
202C2×#108C2×D5311yescharacteristic
203#423—1061no—
204C2×C4×C8—531no—
205#361—22no—
206#361—1062no—
207#361—1062no—
208#361—1062no—
209#346—1061no—
210C4×C16—1061no—
211C53#1321311yescharacteristic, complemented, Sylow
212C22×D4—21no—
213C22×D4—1061no—
214C22×D4—2122no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—532no—
221C2×D8—1062no—
222C2×D8—1062no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C2×D8—2122no—
229C2×D8—2122no—
230C2×D8C4×D5312yes—
231C2×M16—21no—
232C2×M16—1061no—
233C22×C8—531no—
234C22×C8C22×D5311yescharacteristic
235#127—1062no—
236#127—1062no—
237#127—2122no—
238C2×C42—531no—
239M32—21no—
240M32—532no—
241M32—1061no—
242M32C22⋊D5312yescomplemented
243C2×C16—21no—
244C2×C16—1061no—
245#108—21no—
246#108—21no—
247#108—1061no—
248#108—1061no—
249#108—2122no—
250C4⋊C8—2122no—
251C4×C8—531no—
252C4×C8—531no—
253C4×C8—531no—
254C4×C8—531no—
255C24—41no—
256C24—2121no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—1062no—
264C2×D4—4248no—
265C2×D4—2124no—
266C2×D4—2124no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C22×C4—532no—
273C22×C4D4×D5311yescharacteristic
274D8—22no—
275D8—22no—
276D8—41no—
277D8—41no—
278D8—1062no—
279D8—1062no—
280D8—2124no—
281D8—2124no—
282D8—2121no—
283D8—2121no—
284M16—41no—
285M16—41no—
286M16—2121no—
287M16—2121no—
288C2×C8—41no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—532no—
292C2×C8—532no—
293C2×C8—2121no—
294C2×C8C2×C4×D5311yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8D4×D5311yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C42—531no—
301C42—531no—
302C42—532no—
303C16—22no—
304C16—1062no—
305C23—42no—
306C23—84no—
307C23—81no—
308C23—81no—
309C23—4244no—
310C23—4241no—
311C23—4241no—
312C23—2122no—
313C23SD16×D5311yescharacteristic
314D4—42no—
315D4—42no—
316D4—81no—
317D4—81no—
318D4—4248no—
319D4—4248no—
320D4—4241no—
321D4—4241no—
322D4—2122no—
323D4—2122no—
324C2×C4—532no—
325C2×C4—532no—
326C2×C4—1064no—
327C2×C4C22⋊C4×D5311yescharacteristic
328C2×C4#1313320311yescharacteristic
329C2×C4#1313321311yescharacteristic
330C2×C4D8×D5311yescharacteristic
331C2×C4D8×D5311yescharacteristic
332C2×C4SD16×D5311yescharacteristic
333C8—81no—
334C8—532no—
335C8—532no—
336C8—4241no—
337C8#1313317711yescharacteristic
338C8C4×C22⋊D5311yescharacteristic
339C8#1313318911yescharacteristic
340C8#1313319011yescharacteristic
341C22—22no—
342C22—82no—
343C22—84no—
344C22—162no—
345C22—162no—
346C22—8482no—
347C22—8482no—
348C22—4242no—
349C22—4244no—
350C22#102×D5311yescharacteristic
351C22#1681096811yescharacteristic
352C22#1681099411yescharacteristic
353C4—534no—
354C4—1062no—
355C4#1681052311yescharacteristic
356C4#102×D5311yescharacteristic
357C4#1681096811yescharacteristic
358C4#1681099811yescharacteristic
359C2—21no—
360C2—21no—
361C2—162no—
362C2—8482no—
363C2#2027551911yescharacteristic
364C2#361×D5312yes—
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#361C2
socleC22×C53C2×#102
prime radicalC2×#16810998C2
cosocleC22×C8×C53C23
hypercenterC2×#361D53
hypocenterC53#13213
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
2. C2×#16810437C2semidirect
3. C2×#16810436C2semidirect
4. C2×#16810435C2semidirect
5. M32×Dic53C2semidirect
6. #20274885C2semidirect
7. C2×C53×#361C2semidirect
48. #13133157D4semidirect
68. M32×C53D4semidirect
176. Dic53#361semidirect
181. C2×#361D53semidirect
211. C53#13213semidirect
242. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109982
C2×#168104372
C2×#168104362
C2×#168104352
M32×Dic532
#202748852
C2×C53×#3612
#1321353