Finite Group Databasebeta

← #27283151

Subgroups — #27283151

Order 13568.

622 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
1#20283073C211yesmaximal, characteristic
2#20274866C211yesmaximal, characteristic, complemented
3C4⋊C8×Dic53C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6C2×#16810366C211yesmaximal, characteristic
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133206—22no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133146—22no—
13C2×C8×Dic53—21no—
14#16810524—21no—
15#16810524—21no—
16#16810524C414yes—
17C42×Dic53C2211yescharacteristic
18#16810377—21no—
19#16810369—22no—
20#16810366—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#332—22no—
26C22×D212—21no—
27C22×D212—21no—
28C22×D212C2×C412yes—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214—21no—
35#13133206—44no—
36#13133206—44no—
37C2×#11719921—24no—
38C2×#11719921—24no—
39C2×C4×Dic53C2×C412yes—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D412yes—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133146—44no—
48C8×Dic53—42no—
49C22×Q8×C53—21no—
50C22×C8×C53—21no—
51C22×C8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C412yes—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53D412yes—
63C23×D53—22no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—42no—
67C2×Q8×C53—42no—
68C22×C4×C53—22no—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53C22⋊C412yes—
73C22×Dic53C22⋊C412yes—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—44no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—42no—
82C2×D212—42no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—42no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53#4111yescharacteristic
99C42×C53#4111yescharacteristic
100#11719921—48no—
101#11719921—48no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—42no—
107C4×Dic53D814yes—
108C4×Dic53SD1614yes—
109C2×#88322—22no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—42no—
114C23×C53C2×C22⋊C411yescharacteristic
115C22×D53—42no—
116C22×D53—44no—
117C22×D53—44no—
118C22×D53—42no—
119C22×D53—42no—
120Q8×C53—82no—
121Q8×C53—81no—
122Q8×C53—81no—
123C2×C4×C53—22no—
124C2×C4×C53—42no—
125C2×C4×C53—44no—
126C2×C4×C53C4×D411yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53#12711yescharacteristic
132C2×C4×C53C2×D812yes—
133C2×C4×C53C2×SD1612yes—
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C2×Dic53—22no—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53#10214yes—
143C2×Dic53#10214yes—
144D212—82no—
145D212—82no—
146D212—81no—
147D212—81no—
148D212—81no—
149D212—81no—
150D212—82no—
151D212—81no—
152D212—81no—
153C8×C53—42no—
154C8×C53—81no—
155#88322—42no—
156#88322—81no—
157#9698—531nomaximal, Sylow
158C22×C53#39011yescharacteristic
159C22×C53#41312yes—
160C22×C53C2×#10212yes—
161C22×C53#41611yescharacteristic
162C22×C53#41711yescharacteristic
163C2×D53—84no—
164C2×D53—84no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—82no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—82no—
171C4×C53#46311yescharacteristic
172C4×C53#46311yescharacteristic
173C4×C53#46411yescharacteristic
174C4×C53#46411yescharacteristic
175C4×C53#48612yes—
176C4×C53#49211yescharacteristic
177C4×C53#49311yescharacteristic
178Dic53—24no—
179Dic53—24no—
180#2858—531no—
181#1842—531no—
182C4×C4⋊C8—531no—
183C2×#332—531no—
184C2×#332D5311yescharacteristic, complemented
185C2×#331—531no—
186C2×#331—531no—
187C2×C53#154012yes—
188C2×C53#154712yes—
189C2×C53#195711yescharacteristic
190C2×C53#196711yescharacteristic
191C2×C53#198311yescharacteristic
192D53—162no—
193C2×C4⋊Q8C2×D5311yescharacteristic
194C2×#127—531no—
195C2×#124—1062no—
196C2×C4⋊C8—531no—
197C2×C4⋊C8—531no—
198C2×C4⋊C8—1062no—
199C2×C4⋊C8C2×D5311yescharacteristic
200C2×C4⋊C8C2×D5311yescharacteristic
201C2×C4×C8—1061no—
202#390—534no—
203#390—1061no—
204#390—1061no—
205C43—531no—
206#336—1061no—
207#332—22no—
208#332—1062no—
209#331—1062no—
210#331—1062no—
211C53#969811yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C22×D4—532no—
214C22×D4—1061no—
215C22×D4—1061no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—1061no—
219C22×C8—1061no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224#127—1061no—
225#127—1061no—
226#127—1061no—
227#127—1061no—
228#124—2124no—
229#124—2124no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C4×D5312yes—
232C2×C22⋊C4—1064no—
233C2×C22⋊C4—1064no—
234C2×C42—532no—
235C2×C42—531no—
236C2×C42—531no—
237C2×C42—532no—
238C2×C42C22×D5311yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—532no—
243C4⋊C8—1061no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246C4⋊C8—2124no—
247C4⋊C8C22⋊D5312yes—
248C4×C8—2122no—
249C24—1062no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—1062no—
255C2×D4—1062no—
256C2×D4—2124no—
257C2×D4—2121no—
258C2×D4—2121no—
259C2×D4—2121no—
260C2×D4—2121no—
261C2×D4—2122no—
262C2×D4—2122no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—22no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4D4×D5311yescharacteristic
272C22×C4D4×D5311yescharacteristic
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—1062no—
279C2×C8—2121no—
280C2×C8—2122no—
281C2×C8—2121no—
282C2×C8—2121no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—2128no—
288C22⋊C4—2128no—
289C42—534no—
290C42—534no—
291C42—1061no—
292C42—1061no—
293C42—1061no—
294C42—1061no—
295C42—2122no—
296C42#1171994511yescharacteristic
297C42D4×D5311yescharacteristic
298C42#1171994711yescharacteristic
299C42C2×C22⋊D5311yescharacteristic
300C23—2122no—
301C23—2124no—
302C23—2124no—
303C23—2122no—
304C23—2122no—
305C23C22⋊C4×D5311yescharacteristic
306Q8—82no—
307Q8—81no—
308Q8—81no—
309D4—4242no—
310D4—4242no—
311D4—4241no—
312D4—4241no—
313D4—4241no—
314D4—4241no—
315D4—4242no—
316D4—4241no—
317D4—4241no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—534no—
322C2×C4—534no—
323C2×C4—1062no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4#1313317711yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4#1313318911yescharacteristic
330C2×C4#1313319011yescharacteristic
331C2×C4#1313320311yescharacteristic
332C2×C4#1313321311yescharacteristic
333C2×C4D8×D5312yes—
334C2×C4SD16×D5312yes—
335C2×C4#1313323811yescharacteristic
336C2×C4#1313324011yescharacteristic
337C2×C4#1313324411yescharacteristic
338C2×C4#1313324911yescharacteristic
339C8—42no—
340C8—81no—
341C8—4241no—
342C8—2122no—
343C22—4242no—
344C22—4242no—
345C22—4242no—
346C22—4244no—
347C22—4244no—
348C22#1681052311yescharacteristic
349C22#1681060612yes—
350C22#102×D5312yes—
351C22#1681061811yescharacteristic
352C22#1681062311yescharacteristic
353C4—21no—
354C4—21no—
355C4—82no—
356C4—1064no—
357C4—1064no—
358C4#1681084011yescharacteristic
359C4#1681084111yescharacteristic
360C4#1681084711yescharacteristic
361C4#1681084811yescharacteristic
362C4#1681096812yes—
363C4#1681099411yescharacteristic
364C4#1681099811yescharacteristic
365C2—8482no—
366C2#2027336312yes—
367C2#2027339112yes—
368C2#2027551911yescharacteristic
369C2#2027556311yescharacteristic
370C2#2027564411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9698
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20274866C2semidirect
3. C4⋊C8×Dic53C2semidirect
4. C2×#16810369C2semidirect
7. C2×C53×#332C2semidirect
184. C2×#332D53semidirect
211. C53#9698semidirect

Maximal subgroups

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

subgroupindex
#202830732
#202748662
C4⋊C8×Dic532
C2×#168103692
C2×#168103662
C2×C53×#3322
#969853