Finite Group Databasebeta

← #27286696

Subgroups — #27286696

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810974C211yesmaximal, characteristic, complemented
2C2×#16810914C211yesmaximal, characteristic
3C2×#16810904C211yesmaximal, characteristic
4#20275509C211yesmaximal, characteristic
5C2×#16810371C211yesmaximal, characteristic, complemented
6C2×#16810368C211yesmaximal, characteristic, complemented
7C53×#1277C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810974—22no—
12#16810974—22no—
13#16810914—22no—
14#16810914—22no—
15#16810904—22no—
16#16810904—22no—
17C2×#13133151—21no—
18C2×#13133149—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133142—22no—
21C2×#13133139—21no—
22C2×#13133139—21no—
23C2×#13133129—22no—
24#16810371—22no—
25#16810368—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106—21no—
31C22×D212D411yescharacteristic
32C22×C4×D53—21no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217D412yes—
41#13133217D412yes—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53D411yescharacteristic
51C2×#11719921—21no—
52#13133151—24no—
53#13133149—24no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56#13133142—28no—
57#13133139—24no—
58#13133139—24no—
59#13133129—44no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C22×C8×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67C4⋊C8×C53D412yes—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C23×D53—21no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4111yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212Q1612yes—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—42no—
86C2×C4×D53—42no—
87C2×Dic106—22no—
88C2×Dic106—22no—
89C2×Dic106—42no—
90C2×Dic106—42no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106—82no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—22no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C2×#88322—41no—
120C2×#88322—42no—
121C23×C53#12111yescharacteristic
122C22×D53—22no—
123C22×D53—41no—
124C22×D53—42no—
125C22×D53—41no—
126C22×D53—42no—
127C2×C4×C53—22no—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12711yescharacteristic
134C2×C4×C53C2×Q1611yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53#13512yes—
137C2×C4×C53AGL(1,ℤ/8)12yes—
138C2×C4×C53#13711yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—42no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146D212—22no—
147D212—22no—
148D212—41no—
149D212—41no—
150C4×D53—82no—
151C4×D53—81no—
152C4×D53—81no—
153Dic106—44no—
154Dic106—82no—
155Dic106—81no—
156Dic106—81no—
157Dic106—82no—
158C8×C53—42no—
159C8×C53—42no—
160C8×C53—42no—
161#88322—81no—
162#10275—531nomaximal, Sylow
163C22×C53#39211yescharacteristic
164C22×C53#46312yes—
165C22×C53#46711yescharacteristic
166C22×C53#47011yescharacteristic
167C22×C53#47112yes—
168C2×D53—44no—
169C2×D53—42no—
170C2×D53—82no—
171C2×D53—81no—
172C2×D53—81no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53#44912yes—
176C4×C53#45111yescharacteristic
177C4×C53#45111yescharacteristic
178C4×C53#45211yescharacteristic
179C4×C53#45211yescharacteristic
180C4×C53#49511yescharacteristic
181C4×C53#49611yescharacteristic
182Dic53—82no—
183Dic53—81no—
184Dic53—82no—
185C2×#487—531no—
186C2×#476—531no—
187C2×#474—531no—
188#1954—531no—
189C2×#333—531no—
190C2×#332—531no—
191#1277D5311yescharacteristic, complemented
192C2×C53#168312yes—
193C2×C53#170312yes—
194C2×C53#200111yescharacteristic
195C2×C53#201211yescharacteristic
196C2×C53#201911yescharacteristic
197D53—82no—
198C2×C4⋊Q8—531no—
199C2×C4⋊Q8—531no—
200C2×C4×D4—531no—
201#487—1062no—
202#487—1062no—
203#476—1062no—
204#476—1062no—
205#474—1062no—
206#474—1062no—
207C2×#107—1061no—
208C2×#106—1061no—
209C2×C4⋊C8—531no—
210C2×C4⋊C8C2×D5311yescharacteristic
211C2×C4⋊C8C2×D5311yescharacteristic
212C2×#103—1062no—
213C2×#102—1061no—
214C2×#102—1061no—
215C2×C22⋊C8—1062no—
216C2×C4×C8C2×D5311yescharacteristic
217#333—1062no—
218#332—1062no—
219C53#1027511yescharacteristic, complemented, Sylow
220C22×Q8—1061no—
221C22×Q8—1061no—
222C22×D4—531no—
223C23×C4—1061no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—1061no—
227C22×C8D21212yes—
228C4⋊Q8—532no—
229C4⋊Q8—532no—
230C4⋊Q8—1061no—
231C4⋊Q8—1061no—
232C4⋊Q8—1061no—
233C4⋊Q8—1061no—
234C4⋊Q8—1061no—
235C4⋊Q8—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4—1061no—
239C4×D4—1061no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42C22×D5311yescharacteristic
247#107—1064no—
248#106—1064no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253C4⋊C8D21212yes—
254C4⋊C8D21212yes—
255#103—1068no—
256#102—1064no—
257#102—1064no—
258C22⋊C8—2124no—
259C4×C8—21no—
260C4×C8—21no—
261C24—1061no—
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—4242no—
265C2×Q8—2122no—
266C2×Q8—2122no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×D4—532no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4C2×D21211yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4#1171994911yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—2121no—
295C2×C8—2122no—
296C4⋊C4—1062no—
297C4⋊C4—1062no—
298C4⋊C4—1062no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1062no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42C2×D21211yescharacteristic
309C42C2×D21211yescharacteristic
310C42D4×D5311yescharacteristic
311C42D4×D5311yescharacteristic
312C23—1062no—
313C23—2121no—
314C23—2122no—
315C23—2121no—
316C23—2122no—
317C23#1313319211yescharacteristic
318Q8—4242no—
319Q8—4242no—
320Q8—4241no—
321Q8—4241no—
322Q8—2124no—
323D4—1062no—
324D4—1062no—
325D4—2121no—
326D4—2121no—
327C2×C4—22no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—4241no—
331C2×C4—4242no—
332C2×C4—4241no—
333C2×C4—2121no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4#1313318711yescharacteristic
338C2×C4#1313318711yescharacteristic
339C2×C4#1313318711yescharacteristic
340C2×C4#1313319211yescharacteristic
341C2×C4#1313319211yescharacteristic
342C2×C4#1313321411yescharacteristic
343C2×C4Q16×D5311yescharacteristic
344C2×C4#1313323611yescharacteristic
345C2×C4#1313323912yes—
346C2×C4#1313324512yes—
347C2×C4#1313324611yescharacteristic
348C2×C4#1313324611yescharacteristic
349C8—42no—
350C8—42no—
351C8—42no—
352C8—4241no—
353C22—4241no—
354C22—4242no—
355C22—4241no—
356C22—2122no—
357C22—2124no—
358C22#1681053311yescharacteristic
359C22#1681083912yes—
360C22#1681086511yescharacteristic
361C22#1681088211yescharacteristic
362C22#1681088712yes—
363C4—21no—
364C4—21no—
365C4—4241no—
366C4—4242no—
367C4—4242no—
368C4#1681077012yes—
369C4#1681078011yescharacteristic
370C4#1681078011yescharacteristic
371C4#1681078611yescharacteristic
372C4#1681078611yescharacteristic
373C4#1681100711yescharacteristic
374C4#1681101211yescharacteristic
375C2—4242no—
376C2#2027416712yes—
377C2#2027429112yes—
378C2#2027572611yescharacteristic
379C2#2027579511yescharacteristic
380C2#2027583211yescharacteristic
subgroupG/H
centerC23#13133192
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1277C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1277D53
hypocenterC53#10275
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
1. C2×#16810974C2semidirect
5. C2×#16810371C2semidirect
6. C2×#16810368C2semidirect
7. C53×#1277C2semidirect
191. #1277D53semidirect
219. C53#10275semidirect

Maximal subgroups

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

subgroupindex
C2×#168109742
C2×#168109142
C2×#168109042
#202755092
C2×#168103712
C2×#168103682
C53×#12772
#1027553