Finite Group Databasebeta

← #27301522

Subgroups — #27301522

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2#20276451C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4#20274769C211yesmaximal, characteristic, complemented
5#20272828C211yesmaximal, characteristic, complemented
6#20272812C211yesmaximal, characteristic, complemented
7C53×#3067C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—21no—
11C2×#13133157C2211yescharacteristic
12C22×#11719908—21no—
13C22×#11719908C2211yescharacteristic
14C2×#13133153C2211yescharacteristic
15#16810661C2211yescharacteristic
16#16810640—21no—
17#16810433—22no—
18#16810430—21no—
19#16810430—21no—
20#16810395—21no—
21C53×#575—21no—
22C2×#135×C53C2211yescharacteristic
23C2×#131×C53C2211yescharacteristic
24C53×#473C412yescomplemented
25C53×#472C412yescomplemented
26C53×#471—22no—
27C2×#108×C53C2211yescharacteristic
28C53×#413—21no—
29C2×#11719917—21no—
30#13133176—21no—
31#13133157—24no—
32#13133157—21no—
33#13133157D412yes—
34C2×#11719908—24no—
35C2×#11719908—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908D411yescharacteristic
39C2×#11719908D411yescharacteristic
40#13133153—21no—
41#13133153—21no—
42#13133151—21no—
43#13133149—21no—
44#13133144—42no—
45C2×#41×C53—21no—
46C2×#41×C53D411yescharacteristic
47#137×C53—22no—
48#137×C53—41no—
49AGL(1,ℤ/8)×C53—22no—
50AGL(1,ℤ/8)×C53—41no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53—42no—
56C2×Q16×C53C2×C411yescharacteristic
57C2×SD16×C53—21no—
58C2×D8×C53C2×C411yescharacteristic
59#131×C53—21no—
60#131×C53—21no—
61#131×C53C2×C412yes—
62#131×C53D412yes—
63C2×M16×C53—21no—
64C2×M16×C53—21no—
65C2×M16×C53D411yescharacteristic
66C22×C8×C53—21no—
67C22×C8×C53C2311yescharacteristic
68#108×C53—21no—
69#108×C53C2×C412yes—
70#101×C53—22no—
71#100×C53—22no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53C22⋊C412yes—
79#41×C53D812yes—
80C2×Q8×C53—21no—
81C2×Q8×C53SD1611yescharacteristic
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53SD1611yescharacteristic
86C22×C4×C53—21no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89Q16×C53—21no—
90Q16×C53—21no—
91Q16×C53—42no—
92SD16×C53—42no—
93SD16×C53—42no—
94SD16×C53—41no—
95SD16×C53—41no—
96D8×C53—21no—
97D8×C53—21no—
98D8×C53—42no—
99M16×C53—21no—
100M16×C53—22no—
101M16×C53—22no—
102M16×C53—22no—
103M16×C53—41no—
104M16×C53C22⋊C412yes—
105C2×C8×C53—21no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—41no—
109C2×C8×C53C22×C411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C2×C8×C53#4111yescharacteristic
115C22⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4×Dic53—41no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—41no—
121#11719908—21no—
122#11719908—22no—
123#11719908—21no—
124C23×C53—21no—
125C23×C53—41no—
126C23×C53#13511yescharacteristic
127Q8×C53—21no—
128Q8×C53—41no—
129Q8×C53—41no—
130Q8×C53#10212yes—
131D4×C53—22no—
132D4×C53—22no—
133D4×C53—21no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—81no—
137D4×C53—81no—
138D4×C53#10212yes—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×SD1611yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×Dic53—81no—
152C8×C53—22no—
153C8×C53—42no—
154C8×C53C4×D411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12011yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—81no—
159#13071—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—81no—
164C22×C53—161no—
165C22×C53#41611yescharacteristic
166C22×C53#46511yescharacteristic
167C22×C53#46611yescharacteristic
168C4×C53—22no—
169C4×C53—81no—
170C4×C53#38611yescharacteristic
171C4×C53C2×#10211yescharacteristic
172C4×C53#46511yescharacteristic
173C4×C53#46611yescharacteristic
174Dic53—161no—
175C2×#504—531no—
176#3067D5311yescharacteristic
177#2145—531no—
178C2×#359—531no—
179#1814—531no—
180#1388—531no—
181#1382—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—42no—
186C2×C53—161no—
187C2×C53#189311yescharacteristic
188#575—21no—
189C2×#135C2×D5311yescharacteristic
190C2×#131C2×D5311yescharacteristic
191#504—1062no—
192#504—1062no—
193C2×M32—531no—
194C2×M32—1061no—
195C22×C16—531no—
196C22×C16—1061no—
197#473Dic5312yescomplemented
198#472Dic5312yescomplemented
199#471—22no—
200C2×#108—531no—
201C2×#108C2×D5311yescharacteristic
202#427—531no—
203#421—1061no—
204#413—21no—
205#360—1062no—
206#359—1061no—
207#359—1061no—
208#344—1061no—
209C53#1307111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41C22⋊D5311yescharacteristic
212#137—22no—
213#137—41no—
214AGL(1,ℤ/8)—22no—
215AGL(1,ℤ/8)—41no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135—42no—
221C2×Q16C2×Dic5311yescharacteristic
222C2×SD16—21no—
223C2×D8C2×Dic5311yescharacteristic
224#131—21no—
225#131—21no—
226#131C2×Dic5312yes—
227#131C22⋊D5312yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1061no—
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234#117—1061no—
235M32—532no—
236M32—1064no—
237M32—1061no—
238C2×C16—531no—
239C2×C16—531no—
240C2×C16—1064no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244#108—21no—
245#108—1061no—
246#108—1061no—
247#108C2×Dic5312yes—
248#107—1061no—
249#106—1061no—
250#104—2122no—
251#101—22no—
252#100—22no—
253#41—21no—
254#41—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41#1171992212yes—
260#41C22⋊Dic5312yes—
261C2×Q8—21no—
262C2×Q8#1171992311yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#1171992311yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×C22⋊D5311yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—42no—
273SD16—42no—
274SD16—42no—
275SD16—41no—
276SD16—41no—
277D8—21no—
278D8—21no—
279D8—42no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—41no—
285M16—2121no—
286M16—2121no—
287M16C22⋊Dic5312yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—2121no—
293C2×C8D4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8C22×Dic5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—2121no—
300C22⋊C4—2121no—
301C42—2121no—
302C16—1061no—
303C16—1061no—
304C16—1062no—
305C23—21no—
306C23—41no—
307C23#1313323711yescharacteristic
308Q8—21no—
309Q8—41no—
310Q8—41no—
311Q8#1313313812yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4#1313313812yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—4241no—
327C2×C4C2×C22⋊Dic5311yescharacteristic
328C2×C4#1313318911yescharacteristic
329C2×C4#1313318911yescharacteristic
330C2×C4C2×#1171992211yescharacteristic
331C2×C4C2×#1171992311yescharacteristic
332C2×C4#1313323711yescharacteristic
333C8—22no—
334C8—42no—
335C8—4241no—
336C8D4×Dic5311yescharacteristic
337C8D4×Dic5311yescharacteristic
338C8#1313318511yescharacteristic
339C8#1313319811yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22#1681061911yescharacteristic
346C22#1681085011yescharacteristic
347C22#1681085711yescharacteristic
348C4—22no—
349C4—81no—
350C4—8481no—
351C4#1681050311yescharacteristic
352C4C2×#1313313811yescharacteristic
353C4#1681085011yescharacteristic
354C4#1681085711yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2#2027514411yescharacteristic
subgroupG/H
centerC4#16810857
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3067C2
socleC2×C53#1893
cosocleC22×C8×C53C23
hypercenter#3067D53
hypocenterC53#13071
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
2. #20276451C2semidirect
3. C2×#16810430C2semidirect
4. #20274769C2semidirect
5. #20272828C2semidirect
6. #20272812C2semidirect
24. C53×#473C4semidirect
25. C53×#472C4semidirect
197. #473Dic53semidirect
198. #472Dic53semidirect
209. C53#13071semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
#202764512
C2×#168104302
#202747692
#202728282
#202728122
C53×#30672
#1307153