Finite Group Databasebeta

← #27301179

Subgroups — #27301179

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283632C211yesmaximal, characteristic, complemented
2#20276451C211yesmaximal, characteristic, complemented
3#20276443C211yesmaximal, characteristic, complemented
4#20275614C211yesmaximal, characteristic, complemented
5#20272811C211yesmaximal, characteristic, complemented
6#20272803C211yesmaximal, characteristic, complemented
7C2×C53×#508C211yesmaximal, characteristic
8#16811390—21no—
9C2×#13133157C2211yescharacteristic
10#16810745—22no—
11#16810740—22no—
12C2×#13133153C2211yescharacteristic
13#16810661C2211yescharacteristic
14#16810640—21no—
15#16810640—21no—
16#16810597—21no—
17#16810578C2211yescharacteristic
18#16810433—22no—
19#16810430—22no—
20C2×#135×C53C2211yescharacteristic
21C2×#135×C53C2211yescharacteristic
22C53×#508—22no—
23C53×#508—22no—
24C2×Q32×C53C411yescharacteristic, complemented
25C2×SD32×C53C411yescharacteristic, complemented
26C2×SD32×C53C411yescharacteristic, complemented
27C2×D16×C53C411yescharacteristic, complemented
28C22×C16×C53C2211yescharacteristic
29#13133224—24no—
30#13133224—42no—
31C2×#11719917—21no—
32C2×#11719917—42no—
33C2×#11719917D411yescharacteristic
34#13133176—21no—
35#13133176D411yescharacteristic
36#13133157—21no—
37#13133157—21no—
38C2×#11719908—21no—
39#13133153—21no—
40#13133153D412yes—
41#13133151—21no—
42#13133149—21no—
43#13133144—24no—
44#13133144—42no—
45#13133132—42no—
46#13133127—21no—
47C8×Dic53—21no—
48C2×#41×C53—21no—
49C2×#41×C53—21no—
50#135×C53—21no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53D412yes—
57#135×C53D412yes—
58C2×Q16×C53C2×C411yescharacteristic
59C2×Q16×C53C2×C411yescharacteristic
60C2×SD16×C53—21no—
61C2×SD16×C53—21no—
62C2×D8×C53C2×C411yescharacteristic
63C2×D8×C53C2×C411yescharacteristic
64C22×C8×C53C2311yescharacteristic
65Q32×C53—22no—
66SD32×C53—22no—
67SD32×C53—22no—
68D16×C53—22no—
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53C2×C411yescharacteristic
72C2×C16×C53C2×C411yescharacteristic
73#41×C53—22no—
74#41×C53—22no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90Q16×C53—21no—
91Q16×C53—21no—
92Q16×C53—21no—
93Q16×C53C22⋊C412yes—
94SD16×C53—22no—
95SD16×C53—22no—
96SD16×C53—41no—
97SD16×C53—41no—
98D8×C53—21no—
99D8×C53—21no—
100D8×C53—21no—
101D8×C53C22⋊C412yes—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C22⋊Dic53—21no—
109C22⋊Dic53—41no—
110C4⋊Dic53—21no—
111C4⋊Dic53—41no—
112C4×Dic53—21no—
113C4×Dic53—41no—
114#11719917—21no—
115#11719917—22no—
116#11719917—44no—
117#11719917—41no—
118#11719917D812yes—
119C2×#88322—21no—
120C2×#88322—44no—
121C2×#88322—41no—
122C16×C53—21no—
123C16×C53—21no—
124#11719908—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128Q8×C53—22no—
129Q8×C53—41no—
130Q8×C53—41no—
131Q8×C53—41no—
132D4×C53—22no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—41no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—81no—
139D4×C53—81no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×Dic53—41no—
153C2×Dic53—81no—
154C8×C53C2×C22⋊C411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—22no—
159#88322—42no—
160#13016—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—81no—
164C22×C53—161no—
165C22×C53—161no—
166C22×C53#44211yescharacteristic
167C22×C53#44911yescharacteristic
168C22×C53#45911yescharacteristic
169C4×C53—42no—
170C4×C53—81no—
171C4×C53#38611yescharacteristic
172C4×C53C4×D811yescharacteristic
173C4×C53#44711yescharacteristic
174C4×C53#46311yescharacteristic
175Dic53—81no—
176Dic53—161no—
177C2×#508D5311yescharacteristic
178#2957—531no—
179#2145—531no—
180#2143—531no—
181#1976—531no—
182#1382—531no—
183#1379—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—82no—
188C2×C53—161no—
189C2×C53#187911yescharacteristic
190C2×#135C2×D5311yescharacteristic
191C2×#135C2×D5311yescharacteristic
192#568—1061no—
193#508—22no—
194#508—22no—
195C2×Q32Dic5311yescharacteristic, complemented
196C2×SD32Dic5311yescharacteristic, complemented
197C2×SD32Dic5311yescharacteristic, complemented
198C2×D16Dic5311yescharacteristic, complemented
199C2×M32—531no—
200C22×C16C2×D5311yescharacteristic
201#444—1062no—
202#443—1062no—
203C2×#108—531no—
204#427—531no—
205#421—1061no—
206#421—1061no—
207#410—1061no—
208#405—531no—
209#360—1062no—
210#359—1062no—
211C53#1301611yescharacteristic, complemented, Sylow
212C2×#41—21no—
213C2×#41—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135C22⋊D5312yes—
221#135C22⋊D5312yes—
222C2×Q16C2×Dic5311yescharacteristic
223C2×Q16C2×Dic5311yescharacteristic
224C2×SD16—21no—
225C2×SD16—21no—
226C2×D8C2×Dic5311yescharacteristic
227C2×D8C2×Dic5311yescharacteristic
228#131—1064no—
229#131—2122no—
230C2×M16—531no—
231C2×M16—1061no—
232C2×M16—2122no—
233C22×C8C22×D5311yescharacteristic
234#117—531no—
235#117—1061no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—1061no—
241M32—1061no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—1061no—
245C2×C16C2×Dic5311yescharacteristic
246C2×C16C2×Dic5311yescharacteristic
247#108—532no—
248#108—1061no—
249#107—1061no—
250#106—1061no—
251#104—1064no—
252#104—2122no—
253#99—2122no—
254#97—1061no—
255C4×C8—1061no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4D4×D5311yescharacteristic
273Q16—21no—
274Q16—21no—
275Q16—21no—
276Q16C22⋊Dic5312yes—
277SD16—22no—
278SD16—22no—
279SD16—41no—
280SD16—41no—
281D8—21no—
282D8—21no—
283D8—21no—
284D8C22⋊Dic5312yes—
285M16—532no—
286M16—1061no—
287M16—1062no—
288M16—2124no—
289M16—2121no—
290C2×C8—1061no—
291C2×C8—2124no—
292C2×C8—2121no—
293C2×C8D4×D5311yescharacteristic
294C2×C8#1171994711yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8C22×Dic5311yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—1061no—
300C4⋊C4—2121no—
301C22⋊C4—1061no—
302C22⋊C4—2121no—
303C42—1061no—
304C42—2121no—
305C16—21no—
306C16—21no—
307C16—2121no—
308C23—41no—
309C23—41no—
310C23#1313324111yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—4241no—
330C2×C4—2121no—
331C2×C4D4×Dic5311yescharacteristic
332C2×C4#1313318511yescharacteristic
333C2×C4#1313318911yescharacteristic
334C2×C4D8×D5311yescharacteristic
335C2×C4#1313323611yescharacteristic
336C2×C4#1313324111yescharacteristic
337C8—1062no—
338C8—2122no—
339C8C2×C22⋊Dic5311yescharacteristic
340C8D4×Dic5311yescharacteristic
341C8#1313318911yescharacteristic
342C8#1313319811yescharacteristic
343C22—41no—
344C22—42no—
345C22—81no—
346C22—161no—
347C22—161no—
348C22#1681073511yescharacteristic
349C22#1681076511yescharacteristic
350C22#1681081611yescharacteristic
351C4—42no—
352C4—81no—
353C4—8481no—
354C4—4241no—
355C4#1681050311yescharacteristic
356C4D8×Dic5311yescharacteristic
357C4#1681075511yescharacteristic
358C4#1681083611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—161no—
364C2#2027506511yescharacteristic
subgroupG/H
centerC4#16810755
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#508C2
socleC2×C53#1879
cosocleC22×C8×C53C23
hypercenterC2×#508D53
hypocenterC53#13016
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. #20283632C2semidirect
2. #20276451C2semidirect
3. #20276443C2semidirect
4. #20275614C2semidirect
5. #20272811C2semidirect
6. #20272803C2semidirect
24. C2×Q32×C53C4semidirect
25. C2×SD32×C53C4semidirect
26. C2×SD32×C53C4semidirect
27. C2×D16×C53C4semidirect
195. C2×Q32Dic53semidirect
196. C2×SD32Dic53semidirect
197. C2×SD32Dic53semidirect
198. C2×D16Dic53semidirect
211. C53#13016semidirect

Maximal subgroups

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

subgroupindex
#202836322
#202764512
#202764432
#202756142
#202728112
#202728032
C2×C53×#5082
#1301653