Finite Group Databasebeta

← #27301010

Subgroups — #27301010

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×#16811040C211yesmaximal, characteristic, complemented
2#20276343C211yesmaximal, characteristic, complemented
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276336C211yesmaximal, characteristic, complemented
5#20272804C211yesmaximal, characteristic, complemented
6#20272801C211yesmaximal, characteristic, complemented
7C53×#2955C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—21no—
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14C22×#11719908—21no—
15C22×#11719908C2211yescharacteristic
16#16810407—21no—
17#16810407—21no—
18#16810407—22no—
19#16810405—22no—
20C53×#575C411yescharacteristic
21C2×#131×C53C2211yescharacteristic
22C2×#131×C53C2211yescharacteristic
23C53×#444—22no—
24C53×#443—22no—
25C53×#430C411yescharacteristic
26C53×#421C411yescharacteristic
27C53×#421C411yescharacteristic
28C53×#405C2211yescharacteristic
29#13133157—24no—
30#13133157—21no—
31#13133157—21no—
32#13133157—21no—
33#13133157—21no—
34#13133157—21no—
35#13133157D412yes—
36C2×#11719908—24no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908D411yescharacteristic
43C2×#11719908D411yescharacteristic
44C2×#41×C53C2×C411yescharacteristic
45C2×#41×C53C2×C411yescharacteristic
46#137×C53—21no—
47#137×C53C812yes—
48AGL(1,ℤ/8)×C53—21no—
49AGL(1,ℤ/8)×C53C812yes—
50#135×C53—22no—
51#131×C53—21no—
52#131×C53—21no—
53#131×C53—21no—
54#131×C53—21no—
55#131×C53—21no—
56#131×C53—21no—
57#131×C53D412yes—
58#131×C53D412yes—
59C2×M16×C53—21no—
60C2×M16×C53—21no—
61C2×M16×C53C2×C411yescharacteristic
62C2×M16×C53C2×C411yescharacteristic
63C2×M16×C53C2×C411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53C2311yescharacteristic
67#117×C53C2×C411yescharacteristic
68#108×C53—22no—
69#104×C53—22no—
70#104×C53—22no—
71#97×C53—21no—
72C4×C8×C53—21no—
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×C53—21no—
96M16×C53—22no—
97M16×C53—22no—
98M16×C53—21no—
99M16×C53—21no—
100M16×C53—41no—
101M16×C53—41no—
102M16×C53C22⋊C412yes—
103M16×C53C2×C812yes—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—21no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53#4111yescharacteristic
116C2×C8×C53#4111yescharacteristic
117C4⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C42×C53—21no—
120#11719908—21no—
121#11719908—22no—
122#11719908—21no—
123#11719908—41no—
124#11719908—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×C53—41no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C22×C811yescharacteristic
151C2×C4×C53C2×M1611yescharacteristic
152C2×C4×C53#13111yescharacteristic
153C8×C53—22no—
154C8×C53—22no—
155C8×C53—41no—
156C8×C53#12011yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53#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—81no—
172C4×C53#38611yescharacteristic
173C4×C53C2×C22⋊C811yescharacteristic
174C4×C53C8×D411yescharacteristic
175C4×C53#43511yescharacteristic
176C2×#504—531no—
177#2955D5311yescharacteristic
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#575Dic5311yescharacteristic
190C2×#131C2×D5311yescharacteristic
191C2×#131C2×D5311yescharacteristic
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—22no—
201#443—22no—
202#430Dic5311yescharacteristic
203#421Dic5311yescharacteristic
204#421Dic5311yescharacteristic
205#405C2×D5311yescharacteristic
206#350—1061no—
207#350—1061no—
208#350—1062no—
209#349—1062no—
210C53#1298711yescharacteristic, complemented, Sylow
211C2×#41C2×Dic5311yescharacteristic
212C2×#41C2×Dic5311yescharacteristic
213#137—21no—
214#137#8832212yes—
215AGL(1,ℤ/8)—21no—
216AGL(1,ℤ/8)#8832212yes—
217#135—22no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131—21no—
224#131C22⋊D5312yes—
225#131C22⋊D5312yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16C2×Dic5311yescharacteristic
229C2×M16C2×Dic5311yescharacteristic
230C2×M16C2×Dic5311yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234#117C2×Dic5311yescharacteristic
235M32—532no—
236M32—1064no—
237M32—1061no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242C2×C16—531no—
243C2×C16—531no—
244C2×C16—1064no—
245C2×C16—1061no—
246C2×C16—1061no—
247C2×C16—1061no—
248C2×C16—1061no—
249C2×C16—1061no—
250#108—22no—
251#104—22no—
252#104—22no—
253#97—21no—
254C4×C8—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260#41—21no—
261#41C2×#8832212yes—
262#41C22⋊Dic5312yes—
263C2×Q8C2×#8832211yescharacteristic
264C2×Q8#1171991711yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×#8832211yescharacteristic
268C2×D4#1171991711yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×Dic5311yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16C2×#8832212yes—
285M16C22⋊Dic5312yes—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8D4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—1061no—
303C16—1061no—
304C16—1062no—
305C16—2121no—
306C16—2121no—
307C23—21no—
308C23—21no—
309C23#1313322411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C22⋊#8832212yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C22⋊#8832212yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊Dic5311yescharacteristic
330C2×C4D4×Dic5311yescharacteristic
331C2×C4D4×Dic5311yescharacteristic
332C2×C4C22×#8832211yescharacteristic
333C2×C4C2×#1171991711yescharacteristic
334C2×C4#1313322411yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8#1313318511yescharacteristic
339C8#1313318911yescharacteristic
340C8#1313318911yescharacteristic
341C8#1313319811yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22#1681059011yescharacteristic
348C22D4×#8832211yescharacteristic
349C22#1681069511yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4#1681050311yescharacteristic
354C4C2×C22⋊#8832211yescharacteristic
355C4D4×#8832211yescharacteristic
356C4#1681069511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2027500811yescharacteristic
subgroupG/H
centerC8#13133189
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2955C2
socleC2×C53#1869
prime radicalC53×#575C4
cosocleC22×C8×C53C23
hypercenter#2955D53
hypocenterC53#12987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
2. #20276343C2semidirect
3. C2×#16810407C2semidirect
4. #20276336C2semidirect
5. #20272804C2semidirect
6. #20272801C2semidirect
210. C53#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
#202763432
C2×#168104072
#202763362
#202728042
#202728012
C53×#29552
#1298753