Finite Group Databasebeta

← #27301235

Subgroups — #27301235

Order 13568.

466 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×#16810745C211yesmaximal, characteristic, complemented
2#20276464C211yesmaximal, characteristic, complemented
3#20276443C211yesmaximal, characteristic, complemented
4#20275609C211yesmaximal, characteristic, complemented
5#20272827C211yesmaximal, characteristic, complemented
6#20272798C211yesmaximal, characteristic, complemented
7C53×#3414C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133157C2211yescharacteristic
10#16810745—22no—
11#16810745—22no—
12C2×#13133153C2211yescharacteristic
13C2×#13133153C2211yescharacteristic
14C2×#13133144—21no—
15C2×#13133144—21no—
16#16810590—21no—
17C2×#13133127C2211yescharacteristic
18#16810439—21no—
19#16810436—21no—
20#16810430—22no—
21C2×#135×C53C2211yescharacteristic
22C2×#135×C53C2211yescharacteristic
23C53×#510—21no—
24C53×#510C412yescomplemented
25C53×#509—21no—
26C53×#509C412yescomplemented
27C53×#508—22no—
28C2×M32×C53C2211yescharacteristic
29#13133224—24no—
30#13133224—42no—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—41no—
34C22×#88322—41no—
35C22×#88322D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37#13133157—21no—
38#13133157—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133153—21no—
43#13133153D412yes—
44#13133144—24no—
45#13133144—42no—
46C22⋊#88322—42no—
47#13133127—21no—
48#13133127—21no—
49C2×#41×C53—21no—
50C2×#41×C53—21no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53—21no—
57#135×C53C2×C412yes—
58#135×C53D412yes—
59C2×Q16×C53C2×C411yescharacteristic
60C2×Q16×C53D411yescharacteristic
61C2×SD16×C53—21no—
62C2×SD16×C53—21no—
63C2×D8×C53C2×C411yescharacteristic
64C2×D8×C53D411yescharacteristic
65C22×C8×C53C2311yescharacteristic
66Q32×C53—22no—
67SD32×C53—22no—
68SD32×C53—22no—
69D16×C53—22no—
70M32×C53—21no—
71M32×C53C2×C412yes—
72C2×C16×C53—21no—
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
90C22×Dic53—21no—
91Q16×C53—21no—
92Q16×C53—21no—
93Q16×C53—21no—
94Q16×C53C22⋊C412yes—
95SD16×C53—22no—
96SD16×C53—22no—
97SD16×C53—41no—
98SD16×C53—41no—
99D8×C53—21no—
100D8×C53—21no—
101D8×C53—21no—
102D8×C53C22⋊C412yes—
103C2×C8×C53C22×C411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C4×Dic53—21no—
110C4×Dic53—21no—
111#11719917—22no—
112#11719917—44no—
113#11719917—41no—
114#11719917—41no—
115#11719917—41no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—44no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322D812yes—
122C16×C53—22no—
123#11719908—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#13511yescharacteristic
127Q8×C53—22no—
128Q8×C53—41no—
129Q8×C53—41no—
130Q8×C53—41no—
131D4×C53—22no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—41no—
137D4×C53—81no—
138D4×C53—81no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×Dic53—41no—
152C2×Dic53—81no—
153C8×C53C2×C22⋊C411yescharacteristic
154C8×C53C4×D411yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12311yescharacteristic
157#88322—22no—
158#88322—42no—
159#88322—81no—
160#13024—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—81no—
166C22×C53C4×D811yescharacteristic
167C22×C53#44911yescharacteristic
168C22×C53#46311yescharacteristic
169C4×C53—42no—
170C4×C53—81no—
171C4×C53#38611yescharacteristic
172C4×C53#44211yescharacteristic
173C4×C53#44711yescharacteristic
174C4×C53#45911yescharacteristic
175Dic53—81no—
176#3414D5311yescharacteristic
177C2×#444—531no—
178#2148—531no—
179#2143—531no—
180#1975—531no—
181#1388—531no—
182#1377—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—82no—
187C2×C53—161no—
188C2×C53#187911yescharacteristic
189C2×#135C2×D5311yescharacteristic
190C2×#135C2×D5311yescharacteristic
191C2×#131—1061no—
192#510—21no—
193#510Dic5312yescomplemented
194#509—21no—
195#509Dic5312yescomplemented
196#508—22no—
197C2×M32—531no—
198C2×M32C2×D5311yescharacteristic
199#444—1062no—
200#444—1062no—
201C2×#108—531no—
202C2×#108—531no—
203C2×#104—1061no—
204C2×#104—1061no—
205#408—1061no—
206C2×#97—531no—
207#362—1061no—
208#361—1061no—
209#359—1062no—
210C53#1302411yescharacteristic, complemented, Sylow
211C2×#41—21no—
212C2×#41—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135C2×Dic5312yes—
220#135C22⋊D5312yes—
221C2×Q16C2×Dic5311yescharacteristic
222C2×Q16C22⋊D5311yescharacteristic
223C2×SD16—21no—
224C2×SD16—21no—
225C2×D8C2×Dic5311yescharacteristic
226C2×D8C22⋊D5311yescharacteristic
227#131—1064no—
228#131—2122no—
229C2×M16—1061no—
230C2×M16—1061no—
231C2×M16—2121no—
232C22×C8—531no—
233C22×C8—2121no—
234C22×C8C22×D5311yescharacteristic
235C2×C42—531no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—21no—
241M32—1061no—
242M32—1061no—
243M32C2×Dic5312yes—
244C2×C16—21no—
245C2×C16—1061no—
246#108—532no—
247#108—1061no—
248#108—1061no—
249#108—1061no—
250#104—1064no—
251#104—2122no—
252C22⋊C8—2122no—
253#97—1061no—
254#97—1061no—
255#41—22no—
256#41—22no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—1061no—
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—1062no—
286M16—2124no—
287M16—2121no—
288M16—2121no—
289M16—2121no—
290C2×C8—532no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—2124no—
294C2×C8—2121no—
295C2×C8—2121no—
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8C22×Dic5311yescharacteristic
300C2×C8C2×C22⋊D5311yescharacteristic
301C2×C8C2×C22⋊D5311yescharacteristic
302C42—1061no—
303C42—1061no—
304C16—22no—
305C16—2121no—
306C23—41no—
307C23—41no—
308C23#1313323611yescharacteristic
309Q8—22no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313D4—22no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—4241no—
328C2×C4—2121no—
329C2×C4D4×Dic5311yescharacteristic
330C2×C4#1313318511yescharacteristic
331C2×C4#1313318911yescharacteristic
332C2×C4D8×D5311yescharacteristic
333C2×C4#1313324111yescharacteristic
334C2×C4#1313324111yescharacteristic
335C8—1062no—
336C8—4241no—
337C8—2122no—
338C8C2×C22⋊Dic5311yescharacteristic
339C8D4×Dic5311yescharacteristic
340C8#1313318911yescharacteristic
341C8#1313319811yescharacteristic
342C22—41no—
343C22—42no—
344C22—82no—
345C22—82no—
346C22—81no—
347C22D8×Dic5311yescharacteristic
348C22#1681076511yescharacteristic
349C22#1681083611yescharacteristic
350C4—42no—
351C4—81no—
352C4—4241no—
353C4#1681050311yescharacteristic
354C4#1681073511yescharacteristic
355C4#1681075511yescharacteristic
356C4#1681081611yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—161no—
362C2#2027506511yescharacteristic
subgroupG/H
centerC4#16810755
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3414C2
socleC2×C53#1879
cosocleC22×C8×C53C23
hypercenter#3414D53
hypocenterC53#13024
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. C2×#16810745C2semidirect
2. #20276464C2semidirect
3. #20276443C2semidirect
4. #20275609C2semidirect
5. #20272827C2semidirect
6. #20272798C2semidirect
24. C53×#510C4semidirect
26. C53×#509C4semidirect
193. #510Dic53semidirect
195. #509Dic53semidirect
210. C53#13024semidirect

Maximal subgroups

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

subgroupindex
C2×#168107452
#202764642
#202764432
#202756092
#202728272
#202727982
C53×#34142
#1302453