Finite Group Databasebeta

← #27301214

Subgroups — #27301214

Order 13568.

462 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
1#20283632C211yesmaximal, characteristic, complemented
2C2×#16810430C211yesmaximal, characteristic, complemented
3#20276422C211yesmaximal, characteristic, complemented
4#20275614C211yesmaximal, characteristic, complemented
5#20272830C211yesmaximal, characteristic, complemented
6#20272805C211yesmaximal, characteristic, complemented
7C53×#3414C211yesmaximal, characteristic
8#16811390—21no—
9C22×#11719908C2211yescharacteristic
10#16810745—22no—
11#16810740—22no—
12C2×#13133153C2211yescharacteristic
13#16810661C2211yescharacteristic
14#16810640—21no—
15#16810640—21no—
16#16810597—21no—
17#16810578C2211yescharacteristic
18#16810430—22no—
19#16810427—21no—
20#16810424—21no—
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—42no—
33C2×#11719917D411yescharacteristic
34#13133176—21no—
35#13133176D411yescharacteristic
36C2×#11719908—21no—
37C2×#11719908—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×C53C2×C412yes—
57#135×C53D412yes—
58C2×Q16×C53C2×C411yescharacteristic
59C2×Q16×C53D411yescharacteristic
60C2×SD16×C53—21no—
61C2×SD16×C53—21no—
62C2×D8×C53C2×C411yescharacteristic
63C2×D8×C53D411yescharacteristic
64C22×C8×C53C2311yescharacteristic
65Q32×C53—22no—
66SD32×C53—22no—
67SD32×C53—22no—
68D16×C53—22no—
69M32×C53—21no—
70M32×C53C2×C412yes—
71C2×C16×C53—21no—
72#41×C53—22no—
73#41×C53—22no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89Q16×C53—21no—
90Q16×C53—21no—
91Q16×C53—21no—
92Q16×C53C22⋊C412yes—
93SD16×C53—22no—
94SD16×C53—22no—
95SD16×C53—41no—
96SD16×C53—41no—
97D8×C53—21no—
98D8×C53—21no—
99D8×C53—21no—
100D8×C53C22⋊C412yes—
101C2×C8×C53C22×C411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C22⋊Dic53—21no—
108C22⋊Dic53—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—41no—
111C4×Dic53—21no—
112C4×Dic53—41no—
113#11719917—21no—
114#11719917—22no—
115#11719917—44no—
116#11719917—41no—
117#11719917D812yes—
118C2×#88322—21no—
119C2×#88322—44no—
120C2×#88322—41no—
121C16×C53—22no—
122#11719908—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53AGL(1,ℤ/8)11yescharacteristic
126Q8×C53—22no—
127Q8×C53—41no—
128Q8×C53—41no—
129Q8×C53—41no—
130D4×C53—22no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—81no—
137D4×C53—81no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×Dic53—41no—
151C2×Dic53—81no—
152C8×C53C2×C22⋊C411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#88322—22no—
157#88322—42no—
158#13021—531nomaximal, Sylow
159C22×C53—41no—
160C22×C53—42no—
161C22×C53—81no—
162C22×C53—161no—
163C22×C53—161no—
164C22×C53#44211yescharacteristic
165C22×C53#44911yescharacteristic
166C22×C53#45911yescharacteristic
167C4×C53—42no—
168C4×C53—81no—
169C4×C53#38611yescharacteristic
170C4×C53C4×D811yescharacteristic
171C4×C53#44711yescharacteristic
172C4×C53#46311yescharacteristic
173Dic53—81no—
174Dic53—161no—
175#3414D5311yescharacteristic
176#2957—531no—
177C2×#359—531no—
178#2138—531no—
179#1976—531no—
180#1389—531no—
181#1379—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—82no—
186C2×C53—161no—
187C2×C53#187911yescharacteristic
188C2×#135C2×D5311yescharacteristic
189C2×#135C2×D5311yescharacteristic
190#568—1061no—
191#510—21no—
192#510Dic5312yescomplemented
193#509—21no—
194#509Dic5312yescomplemented
195#508—22no—
196C2×M32C2×D5311yescharacteristic
197C22×C16—531no—
198#444—1062no—
199#443—1062no—
200C2×#108—531no—
201#427—531no—
202#421—1061no—
203#421—1061no—
204#410—1061no—
205#405—531no—
206#359—1062no—
207#358—1061no—
208#357—1061no—
209C53#1302111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135C2×Dic5312yes—
219#135C22⋊D5312yes—
220C2×Q16C2×Dic5311yescharacteristic
221C2×Q16C22⋊D5311yescharacteristic
222C2×SD16—21no—
223C2×SD16—21no—
224C2×D8C2×Dic5311yescharacteristic
225C2×D8C22⋊D5311yescharacteristic
226#131—1064no—
227#131—2122no—
228C2×M16—531no—
229C2×M16—1061no—
230C2×M16—2122no—
231C22×C8C22×D5311yescharacteristic
232#117—531no—
233#117—1061no—
234Q32—22no—
235SD32—22no—
236SD32—22no—
237D16—22no—
238M32—21no—
239M32C2×Dic5312yes—
240C2×C16—21no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244#108—532no—
245#108—1061no—
246#107—1061no—
247#106—1061no—
248#104—1064no—
249#104—2122no—
250#99—2122no—
251#97—1061no—
252C4×C8—1061no—
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4D4×D5311yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—21no—
273Q16C22⋊Dic5312yes—
274SD16—22no—
275SD16—22no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8C22⋊Dic5312yes—
282M16—532no—
283M16—1061no—
284M16—1062no—
285M16—2124no—
286M16—2121no—
287C2×C8—1061no—
288C2×C8—2124no—
289C2×C8—2121no—
290C2×C8D4×D5311yescharacteristic
291C2×C8#1171994711yescharacteristic
292C2×C8#1171994711yescharacteristic
293C2×C8C22×Dic5311yescharacteristic
294C2×C8C2×C22⋊D5311yescharacteristic
295C2×C8C2×C22⋊D5311yescharacteristic
296C4⋊C4—1061no—
297C4⋊C4—2121no—
298C22⋊C4—1061no—
299C22⋊C4—2121no—
300C42—1061no—
301C42—2121no—
302C16—22no—
303C16—2121no—
304C23—41no—
305C23—41no—
306C23#1313324111yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4D4×Dic5311yescharacteristic
328C2×C4#1313318511yescharacteristic
329C2×C4#1313318911yescharacteristic
330C2×C4D8×D5311yescharacteristic
331C2×C4#1313323611yescharacteristic
332C2×C4#1313324111yescharacteristic
333C8—1062no—
334C8—2122no—
335C8C2×C22⋊Dic5311yescharacteristic
336C8D4×Dic5311yescharacteristic
337C8#1313318911yescharacteristic
338C8#1313319811yescharacteristic
339C22—41no—
340C22—42no—
341C22—81no—
342C22—161no—
343C22—161no—
344C22#1681073511yescharacteristic
345C22#1681076511yescharacteristic
346C22#1681081611yescharacteristic
347C4—42no—
348C4—81no—
349C4—8481no—
350C4—4241no—
351C4#1681050311yescharacteristic
352C4D8×Dic5311yescharacteristic
353C4#1681075511yescharacteristic
354C4#1681083611yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—161no—
360C2#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#13021
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. #20283632C2semidirect
2. C2×#16810430C2semidirect
3. #20276422C2semidirect
4. #20275614C2semidirect
5. #20272830C2semidirect
6. #20272805C2semidirect
24. C53×#510C4semidirect
26. C53×#509C4semidirect
192. #510Dic53semidirect
194. #509Dic53semidirect
209. C53#13021semidirect

Maximal subgroups

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

subgroupindex
#202836322
C2×#168104302
#202764222
#202756142
#202728302
#202728052
C53×#34142
#1302153