Finite Group Databasebeta

← #27301003

Subgroups — #27301003

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
1#20287818C211yesmaximal, characteristic, complemented
2#20276343C211yesmaximal, characteristic, complemented
3#20276343C211yesmaximal, characteristic, complemented
4#20276336C211yesmaximal, characteristic, complemented
5#20272799C211yesmaximal, characteristic, complemented
6#20272795C211yesmaximal, characteristic, complemented
7C2×C53×#444C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—22no—
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15#16810407—22no—
16#16810407—22no—
17#16810405—22no—
18C2×#135×C53C411yescharacteristic
19C2×#131×C53C2211yescharacteristic
20C2×#131×C53C2211yescharacteristic
21C53×#444—22no—
22C53×#444—22no—
23C2×#108×C53C411yescharacteristic
24C2×#104×C53C411yescharacteristic
25C2×#104×C53C411yescharacteristic
26C2×#97×C53C2211yescharacteristic
27#13133157—24no—
28#13133157—21no—
29#13133157—21no—
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×#11719908D412yes—
41C2×#41×C53C2×C411yescharacteristic
42C2×#41×C53C2×C411yescharacteristic
43#135×C53—22no—
44#135×C53—22no—
45C2×Q16×C53C811yescharacteristic
46C2×SD16×C53C811yescharacteristic
47C2×SD16×C53C811yescharacteristic
48C2×D8×C53C811yescharacteristic
49#131×C53—21no—
50#131×C53—21no—
51#131×C53—21no—
52#131×C53—21no—
53#131×C53—21no—
54#131×C53—21no—
55#131×C53D412yes—
56#131×C53D412yes—
57C2×M16×C53—21no—
58C2×M16×C53—21no—
59C2×M16×C53C2×C411yescharacteristic
60C2×M16×C53C2×C411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C22×C8×C53C2×C411yescharacteristic
64C22×C8×C53C2311yescharacteristic
65C2×C42×C53C2×C411yescharacteristic
66#108×C53—22no—
67#104×C53—22no—
68#104×C53—22no—
69#97×C53—21no—
70#97×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53C22⋊C412yes—
78#41×C53M1612yes—
79C2×Q8×C53C2×C811yescharacteristic
80C2×Q8×C53C2×C811yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53C2×C811yescharacteristic
84C2×D4×C53C2×C811yescharacteristic
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C22×C411yescharacteristic
89Q16×C53—22no—
90SD16×C53—22no—
91SD16×C53—22no—
92D8×C53—22no—
93M16×C53—21no—
94M16×C53—22no—
95M16×C53—22no—
96M16×C53—21no—
97M16×C53—21no—
98M16×C53—41no—
99M16×C53—41no—
100M16×C53C22⋊C412yes—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—21no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53C2×C811yescharacteristic
110C2×C8×C53C2×C811yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53#4111yescharacteristic
116C2×C8×C53#4111yescharacteristic
117C42×C53—21no—
118C42×C53—21no—
119#11719908—22no—
120#11719908—22no—
121#11719908—41no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53#13111yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53C22⋊C812yes—
130D4×C53—22no—
131D4×C53—22no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—41no—
136D4×C53—41no—
137D4×C53C22⋊C812yes—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—21no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—81no—
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
152C8×C53—22no—
153C8×C53—21no—
154C8×C53—21no—
155C8×C53—41no—
156C8×C53#12011yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53#12311yescharacteristic
160#12986—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—81no—
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
176#3407—531no—
177C2×#444D5311yescharacteristic
178#2117—531no—
179#2117—531no—
180#2115—531no—
181#1377—531no—
182#1375—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#186911yescharacteristic
189C2×#135Dic5311yescharacteristic
190C2×#131C2×D5311yescharacteristic
191C2×#131C2×D5311yescharacteristic
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—531no—
198C2×M32—1062no—
199#444—22no—
200#444—22no—
201C2×#108Dic5311yescharacteristic
202C2×#104Dic5311yescharacteristic
203C2×#104Dic5311yescharacteristic
204C2×#97C2×D5311yescharacteristic
205#350—1062no—
206#350—1062no—
207#349—1062no—
208C53#1298611yescharacteristic, complemented, Sylow
209C2×#41C2×Dic5311yescharacteristic
210C2×#41C2×Dic5311yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16#8832211yescharacteristic
214C2×SD16#8832211yescharacteristic
215C2×SD16#8832211yescharacteristic
216C2×D8#8832211yescharacteristic
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131C22⋊D5312yes—
224#131C22⋊D5312yes—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16C2×Dic5311yescharacteristic
228C2×M16C2×Dic5311yescharacteristic
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C2×Dic5311yescharacteristic
232C22×C8C22×D5311yescharacteristic
233C2×C42C2×Dic5311yescharacteristic
234M32—532no—
235M32—1064no—
236M32—1061no—
237M32—1061no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242M32—1061no—
243C2×C16—532no—
244C2×C16—1064no—
245C2×C16—1061no—
246C2×C16—1061no—
247C2×C16—1061no—
248#108—22no—
249#104—22no—
250#104—22no—
251#97—21no—
252#97—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41#1171991712yes—
260#41C22⋊Dic5312yes—
261C2×Q8C2×#8832211yescharacteristic
262C2×Q8C2×#8832211yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×#8832211yescharacteristic
266C2×D4C2×#8832211yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C22×Dic5311yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—21no—
276M16—22no—
277M16—22no—
278M16—21no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16C22⋊Dic5312yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8C2×#8832211yescharacteristic
292C2×C8C2×#8832211yescharacteristic
293C2×C8D4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C42—21no—
300C42—21no—
301C16—1062no—
302C16—1062no—
303C16—2121no—
304C16—2121no—
305C23—21no—
306C23—21no—
307C23#1313322411yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C22⋊#8832212yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C22⋊#8832212yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4C2×C22⋊Dic5311yescharacteristic
329C2×C4D4×Dic5311yescharacteristic
330C2×C4D4×Dic5311yescharacteristic
331C2×C4C22×#8832211yescharacteristic
332C2×C4C2×#1171991711yescharacteristic
333C2×C4#1313322411yescharacteristic
334C8—22no—
335C8—21no—
336C8—21no—
337C8—41no—
338C8#1313318511yescharacteristic
339C8#1313318911yescharacteristic
340C8#1313318911yescharacteristic
341C8#1313319811yescharacteristic
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
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
centerC4#16810503
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#444C2
socleC2×C53#1869
prime radicalC2×#135×C53C4
cosocleC22×C8×C53C23
hypercenterC2×#444D53
hypocenterC53#12986
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
1. #20287818C2semidirect
2. #20276343C2semidirect
3. #20276343C2semidirect
4. #20276336C2semidirect
5. #20272799C2semidirect
6. #20272795C2semidirect
208. C53#12986semidirect

Maximal subgroups

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

subgroupindex
#202878182
#202763432
#202763362
#202727992
#202727952
C2×C53×#4442
#1298653