Finite Group Databasebeta

← #27300400

Subgroups — #27300400

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D53C211yesmaximal, characteristic
2#20276325C211yesmaximal, characteristic
3#20274750C211yesmaximal, characteristic
4#20272825C211yesmaximal, characteristic, complemented
5#20272824C211yesmaximal, characteristic, complemented
6#20272823C211yesmaximal, characteristic, complemented
7C53×#1387C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×C4⋊C4×D53—21no—
10C2×#13133157C2211yescharacteristic
11C2×#13133153C2211yescharacteristic
12C2×#13133151C2211yescharacteristic
13C2×#13133150—21no—
14#107×D53—21no—
15#107×D53—21no—
16#107×D53C412yes—
17#107×D53C412yes—
18#16810586—21no—
19#16810402—21no—
20#16810402—21no—
21#16810387—21no—
22C2×M32×C53C2211yescharacteristic
23C2×#108×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C22×C4×D53—41no—
26C22×C4×D53D411yescharacteristic
27C2×#11719917—21no—
28C22×#88322D411yescharacteristic
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53C2×C411yescharacteristic
34C2×C8×D53C2×C411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37C2×C4⋊Dic53—21no—
38C2×#11719919—41no—
39C4⋊C4×D53—22no—
40C4⋊C4×D53—22no—
41C4⋊C4×D53—41no—
42C4⋊C4×D53—41no—
43#13133157—21no—
44#13133157—21no—
45C2×#11719908—21no—
46#13133153—21no—
47#13133153—21no—
48#13133151—21no—
49#13133151C2×C412yes—
50#13133150—22no—
51#13133150—22no—
52#13133129—41no—
53#13133129—41no—
54C2×M16×C53—21no—
55C22×C8×C53C2311yescharacteristic
56C2×C4⋊C4×C53—21no—
57M32×C53—21no—
58M32×C53—21no—
59C2×C16×C53—21no—
60#108×C53—21no—
61#108×C53—21no—
62#107×C53—21no—
63#107×C53C2×C412yes—
64C23×D53SD1611yescharacteristic
65C22×C4×C53—41no—
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—41no—
68C22×Dic53SD1611yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53—84no—
76C2×C4×D53C22⋊C411yescharacteristic
77C2×C4×D53C22⋊C411yescharacteristic
78C2×C4×D53D811yescharacteristic
79C2×C4×D53D811yescharacteristic
80M16×C53—41no—
81M16×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53C22×C411yescharacteristic
84C2×C8×C53C2×D411yescharacteristic
85C2×C8×C53#4111yescharacteristic
86C2×C8×C53#4111yescharacteristic
87C2×C8×C53#4111yescharacteristic
88C2×C8×C53#4111yescharacteristic
89C4⋊C4×C53—22no—
90C4⋊C4×C53—41no—
91C4⋊Dic53—22no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95#11719917—41no—
96#11719917—41no—
97C2×#88322—21no—
98C2×#88322—21no—
99C2×#88322—41no—
100C2×#88322C22⋊C411yescharacteristic
101C2×#88322C22⋊C411yescharacteristic
102C8×D53—21no—
103C8×D53—21no—
104C8×D53—21no—
105C8×D53—21no—
106C8×D53—21no—
107C8×D53—21no—
108C8×D53—21no—
109C8×D53—21no—
110C16×C53—41no—
111#11719908—41no—
112C23×C53C2×SD1611yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—22no—
116C22×D53—22no—
117C22×D53#10211yescharacteristic
118C22×D53#10211yescharacteristic
119C2×C4×C53—42no—
120C2×C4×C53—82no—
121C2×C4×C53C2×C22⋊C411yescharacteristic
122C2×C4×C53#12311yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×C4×C53C2×D811yescharacteristic
125C2×C4×C53#13711yescharacteristic
126C2×C4×C53#13711yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—42no—
130C2×Dic53—82no—
131C2×Dic53#10211yescharacteristic
132C2×Dic53#10211yescharacteristic
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—84no—
142C4×D53—84no—
143C8×C53—81no—
144C8×C53#11711yescharacteristic
145C8×C53#11711yescharacteristic
146C8×C53#12311yescharacteristic
147C8×C53#12311yescharacteristic
148#88322—21no—
149#88322—21no—
150#88322—81no—
151#12871—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53C2×#10211yescharacteristic
155C22×C53#48311yescharacteristic
156C22×C53#48311yescharacteristic
157C2×D53—24no—
158C2×D53—24no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53#36112yes—
164C2×D53#36212yes—
165C4×C53—82no—
166C4×C53#38111yescharacteristic
167C4×C53#41911yescharacteristic
168C4×C53#48011yescharacteristic
169C4×C53#48011yescharacteristic
170Dic53—21no—
171Dic53—21no—
172Dic53—82no—
173C22×#107—531no—
174#2112—531no—
175#1810—531no—
176#1387—531no—
177#1387—531no—
178#1387—531no—
179#1387D5311yescharacteristic, complemented
180C2×C53—21no—
181C2×C53—21no—
182C2×C53#182311yescharacteristic
183C2×C53C2×#36111yescharacteristic
184C2×C53C2×#36211yescharacteristic
185D53—22no—
186D53—22no—
187C23×C8—531no—
188C22×C4⋊C4—1061no—
189C2×M32—531no—
190C2×M32C2×D5311yescharacteristic
191C2×#108—531no—
192C2×#108C2×D5311yescharacteristic
193C2×#107—531no—
194C2×#107—532no—
195C2×#107—532no—
196C2×#107—1061no—
197C2×#107—1061no—
198C2×#107—1061no—
199C2×#107C2×D5311yescharacteristic
200#407—1061no—
201C22⋊C16—1061no—
202C22⋊C16—1061no—
203#341—1061no—
204C53#1287111yescharacteristic, complemented, Sylow
205C23×C4—531no—
206C23×C4—2121no—
207C2×M16—21no—
208C2×M16—1061no—
209C22×C8—531no—
210C22×C8—531no—
211C22×C8—531no—
212C22×C8—531no—
213C22×C8—531no—
214C22×C8—1061no—
215C22×C8—1061no—
216C22×C8—1061no—
217C22×C8—1061no—
218C22×C8C22×D5311yescharacteristic
219C2×C4⋊C4—21no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—2121no—
224C2×C4⋊C4—2121no—
225C2×C4⋊C4—2121no—
226M32—21no—
227M32—21no—
228M32—1061no—
229M32—1061no—
230C2×C16—21no—
231C2×C16—1061no—
232#108—21no—
233#108—21no—
234#108—1061no—
235#108—1061no—
236#107—21no—
237#107—532no—
238#107—1062no—
239#107—1062no—
240#107—1061no—
241#107C4×D5312yes—
242C22⋊C8—2121no—
243C22⋊C8—2121no—
244C24—531no—
245C22×C4—41no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—4244no—
256C22×C4—2122no—
257C22×C4—2122no—
258C22×C4—2121no—
259C22×C4D4×D5311yescharacteristic
260M16—41no—
261M16—41no—
262M16—2121no—
263M16—2121no—
264C2×C8—41no—
265C2×C8—531no—
266C2×C8—531no—
267C2×C8—1061no—
268C2×C8—1061no—
269C2×C8—1061no—
270C2×C8—1061no—
271C2×C8—1061no—
272C2×C8—1061no—
273C2×C8—1061no—
274C2×C8—1061no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—2121no—
278C2×C8C2×C4×D5311yescharacteristic
279C2×C8#1171994511yescharacteristic
280C2×C8#1171994511yescharacteristic
281C2×C8D4×D5311yescharacteristic
282C2×C8#1171994711yescharacteristic
283C2×C8#1171994911yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—1062no—
287C4⋊C4—2122no—
288C4⋊C4—2122no—
289C4⋊C4—2121no—
290C16—41no—
291C16—2121no—
292C23—531no—
293C23—531no—
294C23—1062no—
295C23—1062no—
296C23—1061no—
297C23—1061no—
298C23SD16×D5311yescharacteristic
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—531no—
302C2×C4—531no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—4244no—
314C2×C4—4244no—
315C2×C4—4242no—
316C2×C4—2122no—
317C2×C4C22⋊C4×D5311yescharacteristic
318C2×C4#1313319911yescharacteristic
319C2×C4#1313320211yescharacteristic
320C2×C4D8×D5311yescharacteristic
321C2×C4#1313324611yescharacteristic
322C2×C4#1313324911yescharacteristic
323C8—81no—
324C8—1061no—
325C8—1061no—
326C8—4241no—
327C8#1313317411yescharacteristic
328C8#1313317511yescharacteristic
329C8#1313319911yescharacteristic
330C8#1313320211yescharacteristic
331C22—21no—
332C22—21no—
333C22—532no—
334C22—532no—
335C22—1064no—
336C22—1064no—
337C22—1061no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22#102×D5311yescharacteristic
342C22#1681095211yescharacteristic
343C22#1681095511yescharacteristic
344C4—82no—
345C4—1061no—
346C4—1061no—
347C4—4242no—
348C4#1681048511yescharacteristic
349C4#1681063011yescharacteristic
350C4#1681093511yescharacteristic
351C4#1681093811yescharacteristic
352C2—21no—
353C2—21no—
354C2—1062no—
355C2—1062no—
356C2#2027480011yescharacteristic
357C2#361×D5311yescharacteristic
358C2#362×D5311yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1387C2
socleC22×C53C2×#102
prime radicalC23×D53SD16
cosocleC22×C8×C53C23
hypercenter#1387D53
hypocenterC53#12871
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1387

Complements

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

NH ≅ G/Ntype
4. #20272825C2semidirect
5. #20272824C2semidirect
6. #20272823C2semidirect
7. C53×#1387C2semidirect
179. #1387D53semidirect
204. C53#12871semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D532
#202763252
#202747502
#202728252
#202728242
#202728232
C53×#13872
#1287153