Finite Group Databasebeta

← #27301270

Subgroups — #27301270

Order 13568.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810740C211yesmaximal, characteristic, complemented
2C2×#16810436C211yesmaximal, characteristic, complemented
3#20276427C211yesmaximal, characteristic, complemented
4#20275582C211yesmaximal, characteristic, complemented
5#20272824C211yesmaximal, characteristic, complemented
6#20272796C211yesmaximal, characteristic, complemented
7C2×C53×#509C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133157C2211yescharacteristic
10#16810740—22no—
11#16810740—22no—
12C2×#13133153C2211yescharacteristic
13C2×#13133151C2211yescharacteristic
14C2×#13133144—21no—
15C2×#13133138—21no—
16C2×C8×Dic53C2211yescharacteristic
17#16810526—21no—
18#16810436—22no—
19#16810427—21no—
20#16810424—21no—
21C2×#135×C53C2211yescharacteristic
22C22×D8×C53C2211yescharacteristic
23C53×#509—22no—
24C53×#509C414yescomplemented
25C2×SD32×C53—21no—
26C2×D16×C53—21no—
27C2×M32×C53C2211yescharacteristic
28#13133224—24no—
29#13133224—42no—
30C2×#11719917—21no—
31C2×#11719917—41no—
32C22×#88322—41no—
33C22×#88322D411yescharacteristic
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—42no—
36C2×C4×Dic53D411yescharacteristic
37#13133157—21no—
38#13133157—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153D412yes—
42#13133151—21no—
43#13133151—21no—
44#13133144—24no—
45#13133138—41no—
46#13133138—41no—
47C8×Dic53—21no—
48C8×Dic53—21no—
49C2×#41×C53—21no—
50C22×D4×C53—21no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53C2×C412yes—
54#135×C53D412yes—
55C2×Q16×C53D411yescharacteristic
56C2×SD16×C53—21no—
57C2×D8×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53—21no—
60C2×D8×C53—21no—
61C2×D8×C53—21no—
62C2×D8×C53C2×C412yes—
63C2×D8×C53D411yescharacteristic
64C22×C8×C53C2311yescharacteristic
65SD32×C53—24no—
66D16×C53—24no—
67M32×C53—21no—
68M32×C53C2×C412yes—
69C2×C16×C53—21no—
70C24×C53—41no—
71#41×C53—22no—
72#41×C53—22no—
73#41×C53—41no—
74#41×C53—41no—
75C2×Q8×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—21no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—21no—
87C22×Dic53—41no—
88Q16×C53—21no—
89Q16×C53C22⋊C412yescomplemented
90SD16×C53—22no—
91SD16×C53—22no—
92D8×C53—22no—
93D8×C53—22no—
94D8×C53—21no—
95D8×C53—41no—
96D8×C53—41no—
97D8×C53C22⋊C412yescomplemented
98C2×C8×C53C22×C411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C22⋊Dic53—84no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109#11719917—22no—
110#11719917—44no—
111#11719917—41no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—44no—
115C2×#88322—41no—
116C2×#88322D811yescharacteristic
117C2×#88322D811yescharacteristic
118C16×C53—22no—
119#11719908—41no—
120C23×C53—41no—
121C23×C53—42no—
122C23×C53—82no—
123C23×C53—82no—
124C23×C53—81no—
125C23×C53—81no—
126C23×C53AGL(1,ℤ/8)11yescharacteristic
127Q8×C53—22no—
128Q8×C53—41no—
129D4×C53—22no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—81no—
136D4×C53—81no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×Dic53—22no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—82no—
150C2×Dic53—81no—
151C8×C53C2×C22⋊C411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—21no—
156#88322—21no—
157#88322—42no—
158#13029—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—42no—
162C22×C53—84no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—82no—
166C22×C53—162no—
167C22×C53—161no—
168C22×C53—161no—
169C22×C53#44211yescharacteristic
170C22×C53#44911yescharacteristic
171C22×C53#45911yescharacteristic
172C4×C53—42no—
173C4×C53#38611yescharacteristic
174C4×C53C4×D811yescharacteristic
175C4×C53#44711yescharacteristic
176C4×C53#46311yescharacteristic
177Dic53—42no—
178Dic53—161no—
179C2×#509D5311yescharacteristic
180C2×#443—531no—
181C2×#361—531no—
182#2139—531no—
183#1970—531no—
184#1387—531no—
185#1376—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—82no—
189C2×C53—162no—
190C2×C53#187911yescharacteristic
191C2×C53#221211yescharacteristic
192C2×C53#221311yescharacteristic
193C2×#135C2×D5311yescharacteristic
194C22×D8C2×D5311yescharacteristic
195C2×#131—1061no—
196#509—22no—
197#509Dic5314yescomplemented
198C2×SD32—21no—
199C2×D16—21no—
200C2×M32—531no—
201C2×M32C2×D5311yescharacteristic
202#443—1062no—
203#443—1062no—
204C2×#108—531no—
205C2×#107—531no—
206C2×#104—1061no—
207C2×#102—1061no—
208C2×C4×C8—531no—
209#390—1061no—
210#361—1062no—
211#358—1061no—
212#357—1061no—
213C53#1302911yescharacteristic, complemented, Sylow
214C2×#41—21no—
215C22×D4—21no—
216#135—21no—
217#135—21no—
218#135C2×Dic5312yes—
219#135C22⋊D5312yes—
220C2×Q16C22⋊D5311yescharacteristic
221C2×SD16—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8C2×Dic5312yes—
228C2×D8C22⋊D5311yescharacteristic
229#131—1064no—
230#131—2122no—
231C2×M16—1061no—
232C2×M16—2121no—
233C22×C8—531no—
234C22×C8—2121no—
235C22×C8C22×D5311yescharacteristic
236C2×C4⋊C4—1061no—
237C2×C22⋊C4—2122no—
238C2×C42—531no—
239SD32—24no—
240D16—24no—
241M32—21no—
242M32—1061no—
243M32—1061no—
244M32C2×Dic5312yes—
245C2×C16—21no—
246C2×C16—1061no—
247#108—532no—
248#108—1061no—
249#107—1061no—
250#107—1061no—
251#104—1064no—
252#102—2121no—
253#102—2121no—
254C4×C8—1061no—
255C4×C8—1061no—
256C24—41no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4D4×D5311yescharacteristic
274Q16—21no—
275Q16C22⋊Dic5312yescomplemented
276SD16—22no—
277SD16—22no—
278D8—22no—
279D8—22no—
280D8—21no—
281D8—41no—
282D8—41no—
283D8C22⋊Dic5312yescomplemented
284M16—1062no—
285M16—2124no—
286M16—2121no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—1061no—
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—2121no—
300C4⋊C4—2121no—
301C22⋊C4—4244no—
302C42—1061no—
303C42—1061no—
304C16—22no—
305C16—2121no—
306C23—41no—
307C23—42no—
308C23—82no—
309C23—82no—
310C23—81no—
311C23—81no—
312C23#1313324111yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—41no—
320D4—42no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—1062no—
327C2×C4—4242no—
328C2×C4—4241no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4D4×Dic5311yescharacteristic
332C2×C4#1313318511yescharacteristic
333C2×C4#1313318911yescharacteristic
334C2×C4D8×D5311yescharacteristic
335C2×C4#1313323611yescharacteristic
336C2×C4#1313324111yescharacteristic
337C8—1061no—
338C8—1061no—
339C8—2122no—
340C8C2×C22⋊Dic5311yescharacteristic
341C8D4×Dic5311yescharacteristic
342C8#1313318911yescharacteristic
343C8#1313319811yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—84no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—162no—
352C22—161no—
353C22—161no—
354C22#1681073511yescharacteristic
355C22#1681076511yescharacteristic
356C22#1681081611yescharacteristic
357C4—42no—
358C4—8481no—
359C4—2122no—
360C4#1681050311yescharacteristic
361C4D8×Dic5311yescharacteristic
362C4#1681075511yescharacteristic
363C4#1681083611yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—162no—
368C2#2027506511yescharacteristic
369C2#2027678211yescharacteristic
370C2#2027678911yescharacteristic
subgroupG/H
centerC22#16810816
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#509C2
socleC22×C53#459
cosocleC22×C8×C53C23
hypercenterC2×#509D53
hypocenterC53#13029
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16810740C2semidirect
2. C2×#16810436C2semidirect
3. #20276427C2semidirect
4. #20275582C2semidirect
5. #20272824C2semidirect
6. #20272796C2semidirect
24. C53×#509C4semidirect
89. Q16×C53C22⋊C4semidirect
97. D8×C53C22⋊C4semidirect
197. #509Dic53semidirect
213. C53#13029semidirect
275. Q16C22⋊Dic53semidirect
283. D8C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107402
C2×#168104362
#202764272
#202755822
#202728242
#202727962
C2×C53×#5092
#1302953