Finite Group Databasebeta

← #27301011

Subgroups — #27301011

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811039C211yesmaximal, characteristic, complemented
2#20283616C211yesmaximal, characteristic
3#20276344C211yesmaximal, characteristic, complemented
4#20276336C211yesmaximal, characteristic, complemented
5#20272803C211yesmaximal, characteristic, complemented
6#20272800C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8#16811421C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10#16811039—22no—
11#16811039—22no—
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14C2×#13133156—21no—
15C2×C16×D53—21no—
16#16810743—22no—
17#16810738—22no—
18#16810671C411yescharacteristic
19#16810640C411yescharacteristic
20#16810639C411yescharacteristic
21#16810578C2211yescharacteristic
22#16810408—22no—
23#16810405—22no—
24C2×#131×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C22×C16×C53C2211yescharacteristic
27C53×#350—21no—
28C53×#350—21no—
29C2×#11719945C2×C411yescharacteristic
30#13133248C812yes—
31#13133247—21no—
32#13133243C812yes—
33#13133242—21no—
34#13133237—22no—
35#13133223—21no—
36#13133223—21no—
37#13133223D412yes—
38#13133223D412yes—
39C2×#11719917C2×C411yescharacteristic
40C2×#11719917C2×C411yescharacteristic
41C2×#11719912—21no—
42C2×C8×D53—21no—
43#13133176C2×C411yescharacteristic
44#13133157—21no—
45#13133157—21no—
46#13133157—21no—
47#13133157D412yes—
48#13133156—24no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51C16×D53—24no—
52#13133152—22no—
53#13133144—22no—
54#13133143—22no—
55#13133127—21no—
56C8×Dic53—21no—
57C2×#41×C53C2×C411yescharacteristic
58#131×C53—21no—
59#131×C53—21no—
60#131×C53—21no—
61#131×C53—21no—
62C2×M16×C53—21no—
63C2×M16×C53C2×C411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66M32×C53—21no—
67M32×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53D411yescharacteristic
72C2×C16×C53D411yescharacteristic
73#41×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77C2×Q8×C53C2×C811yescharacteristic
78C2×D4×C53—21no—
79C2×D4×C53C2×C811yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53C22×C411yescharacteristic
82C2×C22⋊D53—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945C22⋊C412yes—
86#11719945C2×C812yes—
87C2×D212M1611yescharacteristic
88C2×C4×D53—21no—
89C2×Dic106M1611yescharacteristic
90M16×C53—21no—
91M16×C53—21no—
92M16×C53—41no—
93M16×C53—41no—
94C2×C8×C53—21no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C22⋊Dic53—21no—
104#11719925—22no—
105#11719924—22no—
106#11719923—22no—
107#11719922—22no—
108C4⋊Dic53—21no—
109C4×Dic53—21no—
110#11719917—21no—
111#11719917—21no—
112#11719917C22⋊C412yes—
113#11719917C2×C812yes—
114C2×#88322—21no—
115C2×#88322—21no—
116#11719912—22no—
117#11719912—22no—
118C8×D53—22no—
119C8×D53—22no—
120C16×C53—21no—
121C16×C53—21no—
122C16×C53—41no—
123#11719908—22no—
124#11719908—41no—
125C23×C53—21no—
126C23×C53#13111yescharacteristic
127C22×D53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—41no—
133D4×C53—41no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C22×C811yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53#13111yescharacteristic
143C22⋊D53—22no—
144C22⋊D53—22no—
145C2×Dic53—21no—
146C2×Dic53—41no—
147D212—21no—
148D212C22⋊C812yes—
149C4×D53—22no—
150C4×D53—22no—
151Dic106—21no—
152Dic106C22⋊C812yes—
153C8×C53—41no—
154C8×C53#12011yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—22no—
159#88322—22no—
160#12987—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53#40811yescharacteristic
165C22×C53C8×D411yescharacteristic
166C22×C53#43511yescharacteristic
167C2×D53—22no—
168C2×D53—42no—
169C4×C53—41no—
170C4×C53#38611yescharacteristic
171C4×C53C2×C22⋊C811yescharacteristic
172C4×C53C8×D411yescharacteristic
173C4×C53#43511yescharacteristic
174Dic53—22no—
175Dic53—81no—
176C2×#504—531no—
177#2955—531no—
178#2117—531no—
179C2×#350D5311yescharacteristic, complemented
180#2115—531no—
181#1379—531no—
182#1378—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—81no—
187C2×C53#186911yescharacteristic
188D53—42no—
189#575—531no—
190C2×#131—531no—
191C2×#131C2×D5311yescharacteristic
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—1061no—
197C2×M32C2×D5311yescharacteristic
198C22×C16—1061no—
199C22×C16C2×D5311yescharacteristic
200#444—1062no—
201#443—1062no—
202#430—531no—
203#421—531no—
204#421—531no—
205#405—531no—
206#350—21no—
207#350—21no—
208#350—1062no—
209#349—1062no—
210C53#1298711yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41C4×D5311yescharacteristic
213#137—532no—
214#137—1061no—
215AGL(1,ℤ/8)—532no—
216AGL(1,ℤ/8)—1061no—
217#135—1062no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—532no—
223#131—532no—
224#131—1061no—
225#131—1061no—
226C2×M16—21no—
227C2×M16—531no—
228C2×M16—531no—
229C2×M16—1061no—
230C2×M16C4×D5311yescharacteristic
231C22×C8—21no—
232C22×C8—1061no—
233C22×C8C22×D5311yescharacteristic
234#117—531no—
235M32—21no—
236M32—21no—
237M32—532no—
238M32—1064no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1064no—
246C2×C16—1061no—
247C2×C16—1061no—
248C2×C16C22⋊D5311yescharacteristic
249C2×C16C22⋊D5311yescharacteristic
250#108—1062no—
251#104—1062no—
252#104—1062no—
253#97—1061no—
254C4×C8—1061no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—532no—
260#41—532no—
261#41—1061no—
262#41—1061no—
263C2×Q8—531no—
264C2×Q8C8×D5311yescharacteristic
265C2×D4—21no—
266C2×D4—531no—
267C2×D4—1061no—
268C2×D4C8×D5311yescharacteristic
269C22×C4—21no—
270C22×C4—1061no—
271C22×C4C2×C4×D5311yescharacteristic
272Q16—1062no—
273SD16—1062no—
274SD16—1062no—
275D8—1062no—
276M16—21no—
277M16—21no—
278M16—41no—
279M16—41no—
280M16—532no—
281M16—532no—
282M16—1061no—
283M16—1061no—
284M16—1062no—
285M16—1062no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8#1171994511yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—1061no—
300C22⋊C4—1061no—
301C42—1061no—
302C16—21no—
303C16—21no—
304C16—41no—
305C16—1062no—
306C16—2121no—
307C23—21no—
308C23—1061no—
309C23#1313322511yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—532no—
313Q8—1061no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—41no—
318D4—532no—
319D4—1062no—
320D4—1062no—
321D4—1061no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—1061no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—2121no—
329C2×C4C22⋊C4×D5311yescharacteristic
330C2×C4#1313317711yescharacteristic
331C2×C4C4×C22⋊D5311yescharacteristic
332C2×C4C2×C8×D5311yescharacteristic
333C2×C4M16×D5311yescharacteristic
334C2×C4#1313322311yescharacteristic
335C8—41no—
336C8—1062no—
337C8—1062no—
338C8#1313318511yescharacteristic
339C8#1313318811yescharacteristic
340C8#1313319011yescharacteristic
341C8#1313319911yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—1062no—
346C22—2122no—
347C22#1681058811yescharacteristic
348C22#1681068611yescharacteristic
349C22#1681069211yescharacteristic
350C4—41no—
351C4—1062no—
352C4—4241no—
353C4#1681049911yescharacteristic
354C4C22⋊C8×D5311yescharacteristic
355C4C8×C22⋊D5311yescharacteristic
356C4#1681069711yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—2122no—
362C2#2027500411yescharacteristic
subgroupG/H
centerC8#13133188
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC2×C53#1869
prime radical#16811421C4
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#12987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. C2×#16811039C2semidirect
3. #20276344C2semidirect
4. #20276336C2semidirect
5. #20272803C2semidirect
6. #20272800C2semidirect
7. C2×C53×#350C2semidirect
179. C2×#350D53semidirect
210. C53#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168110392
#202836162
#202763442
#202763362
#202728032
#202728002
C2×C53×#3502
#1298753