Finite Group Databasebeta

← #28311600

Subgroups — #28311600

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×#16811681C211yesmaximal, characteristic, complemented
2#20290957C211yesmaximal, characteristic, complemented
3C2×#16811521C211yesmaximal, characteristic, complemented
4#20290951C211yesmaximal, characteristic, complemented
5#20289770C211yesmaximal, characteristic, complemented
6#20289766C211yesmaximal, characteristic, complemented
7#20283618C211yesmaximal, characteristic
8#16811681—22no—
9#16811681—22no—
10C2×#13133291—21no—
11C2×#13133291C2211yescharacteristic
12C2×#13133291C2211yescharacteristic
13C2×#13133291C2211yescharacteristic
14C22×#11719910—21no—
15C22×#11719910C2211yescharacteristic
16#16811521—21no—
17#16811521—21no—
18#16811521—22no—
19#16811519—22no—
20C2×#13133224C2211yescharacteristic
21C2×#13133224C2211yescharacteristic
22#16810745—22no—
23#16810740—22no—
24#16810672C411yescharacteristic
25#16810640C411yescharacteristic
26#16810640C411yescharacteristic
27#16810581C2211yescharacteristic
28C53×#575C411yescharacteristic
29#13133291—24no—
30#13133291—21no—
31#13133291—21no—
32#13133291—21no—
33#13133291—21no—
34#13133291—21no—
35#13133291D412yes—
36C2×#11719910—24no—
37C2×#11719910—21no—
38C2×#11719910—21no—
39C2×#11719910—21no—
40C2×#11719910—21no—
41C2×#11719910—21no—
42C2×#11719910D411yescharacteristic
43C2×#11719910D411yescharacteristic
44#13133224—21no—
45#13133224—21no—
46#13133224—21no—
47#13133224—21no—
48#13133224—21no—
49#13133224—21no—
50#13133224D412yes—
51#13133224D412yes—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917C2×C411yescharacteristic
55C2×#11719917C2×C411yescharacteristic
56C22×#88322—21no—
57C22×#88322—21no—
58C22×#88322C2311yescharacteristic
59#13133176C2×C411yescharacteristic
60#13133153—22no—
61#13133144—22no—
62#13133144—22no—
63#13133127—21no—
64C8×Dic53—21no—
65C2×#41×C53C2×C411yescharacteristic
66C2×#41×C53C2×C411yescharacteristic
67#137×C53—21no—
68#137×C53C812yes—
69AGL(1,ℤ/8)×C53—21no—
70AGL(1,ℤ/8)×C53C812yes—
71#135×C53—22no—
72C2×M16×C53C2×C411yescharacteristic
73#41×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—21no—
79#41×C53C22⋊C412yes—
80#41×C53C2×C812yes—
81C2×Q8×C53C2×C811yescharacteristic
82C2×Q8×C53M1611yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53C2×C811yescharacteristic
86C2×D4×C53M1611yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53C22×C411yescharacteristic
90Q16×C53—22no—
91SD16×C53—22no—
92SD16×C53—22no—
93D8×C53—22no—
94M16×C53—21no—
95M16×C53C2×C812yes—
96C2×C8×C53—21no—
97C22⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4×Dic53—21no—
100#11719917—21no—
101#11719917—22no—
102#11719917—22no—
103#11719917—21no—
104#11719917—21no—
105#11719917—41no—
106#11719917—41no—
107#11719917C22⋊C412yes—
108C2×#88322—21no—
109C2×#88322—22no—
110C2×#88322—22no—
111C2×#88322—21no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322C2×D411yescharacteristic
115C2×#88322C2×D411yescharacteristic
116C2×#88322C2×D411yescharacteristic
117C2×#88322C2×D411yescharacteristic
118C2×#88322#4111yescharacteristic
119C2×#88322#4111yescharacteristic
120#11719910—21no—
121#11719910—22no—
122#11719910—21no—
123#11719910—41no—
124#11719910—41no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53#13111yescharacteristic
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53C22⋊C812yes—
132D4×C53—22no—
133D4×C53—22no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53C22⋊C812yes—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C22×C811yescharacteristic
150C2×C4×C53C2×M1611yescharacteristic
151C2×C4×C53#13111yescharacteristic
152C2×Dic53—41no—
153C8×C53—22no—
154#88322—22no—
155#88322—41no—
156#88322#12011yescharacteristic
157#88322#12111yescharacteristic
158#88322#12111yescharacteristic
159#88322#12311yescharacteristic
160#12987—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—41no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—81no—
166C22×C53#40811yescharacteristic
167C22×C53C8×D411yescharacteristic
168C22×C53#43511yescharacteristic
169C4×C53—22no—
170C4×C53—41no—
171C4×C53#38611yescharacteristic
172C4×C53C2×C22⋊C811yescharacteristic
173C4×C53C8×D411yescharacteristic
174C4×C53#43511yescharacteristic
175Dic53—81no—
176C2×#504—531no—
177#2955—531no—
178#2117—531no—
179C2×#350—531no—
180#2115—531no—
181#1379—531no—
182#1378—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#186911yescharacteristic
189#575#615711yescharacteristic
190C2×#131—531no—
191C2×#131—531no—
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—1061no—
198C22×C16—531no—
199C22×C16—1061no—
200#444—1062no—
201#443—1062no—
202#430—531no—
203#421—531no—
204#421—531no—
205#405—531no—
206#350—1061no—
207#350—1061no—
208#350—1062no—
209#349—1062no—
210C53#1298711yescharacteristic, complemented, Sylow
211C2×#41C2×#615711yescharacteristic
212C2×#41C2×#615711yescharacteristic
213#137—21no—
214#137#8832412yes—
215AGL(1,ℤ/8)—21no—
216AGL(1,ℤ/8)#8832412yes—
217#135—22no—
218#131—532no—
219#131—532no—
220#131—1061no—
221#131—1061no—
222#131—1061no—
223#131—1061no—
224#131—1061no—
225#131—1061no—
226C2×M16—531no—
227C2×M16—531no—
228C2×M16—1061no—
229C2×M16—1061no—
230C2×M16C2×#615711yescharacteristic
231C22×C8—531no—
232C22×C8—1061no—
233C22×C8—1061no—
234#117—531no—
235M32—532no—
236M32—1061no—
237M32—1064no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242C2×C16—531no—
243C2×C16—531no—
244C2×C16—1061no—
245C2×C16—1061no—
246C2×C16—1061no—
247C2×C16—1064no—
248C2×C16—1061no—
249C2×C16—1061no—
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—21no—
260#41—21no—
261#41C2×#8832412yes—
262#41C22⋊#615712yes—
263C2×Q8C2×#8832411yescharacteristic
264C2×Q8#1171994011yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×#8832411yescharacteristic
268C2×D4#1171994011yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×#615711yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—532no—
278M16—1062no—
279M16—1062no—
280M16—1061no—
281M16—1061no—
282M16—1061no—
283M16—2121no—
284M16—2121no—
285M16C2×#8832412yes—
286C2×C8—21no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—531no—
292C2×C8—531no—
293C2×C8—1062no—
294C2×C8—1062no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—1061no—
300C22⋊C4—1061no—
301C42—1061no—
302C16—1061no—
303C16—1061no—
304C16—1062no—
305C16—2121no—
306C16—2121no—
307C23—21no—
308C23—21no—
309C23#1313330411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C22⋊#8832412yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C22⋊#8832412yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—2121no—
329C2×C4C2×C22⋊#615711yescharacteristic
330C2×C4D4×#615711yescharacteristic
331C2×C4D4×#615711yescharacteristic
332C2×C4C22×#8832411yescharacteristic
333C2×C4C2×#1171994011yescharacteristic
334C2×C4#1313330411yescharacteristic
335C8—22no—
336C8—531no—
337C8—531no—
338C8—531no—
339C8—531no—
340C8—1062no—
341C8—2121no—
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22#1681159611yescharacteristic
348C22D4×#8832411yescharacteristic
349C22#1681165411yescharacteristic
350C4—22no—
351C4—41no—
352C4—4241no—
353C4#1681157111yescharacteristic
354C4C2×C22⋊#8832411yescharacteristic
355C4D4×#8832411yescharacteristic
356C4#1681165411yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2029068811yescharacteristic
subgroupG/H
centerC4#16811571
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
Fitting, prime radicalC53×#575C4
socleC2×C53#1869
cosocleC22×#88322C23
hypercenter#575#6157
hypocenterC53#12987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16811681C2semidirect
2. #20290957C2semidirect
3. C2×#16811521C2semidirect
4. #20290951C2semidirect
5. #20289770C2semidirect
6. #20289766C2semidirect
210. C53#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168116812
#202909572
C2×#168115212
#202909512
#202897702
#202897662
#202836182
#1298753