Finite Group Databasebeta

← #27286687

Subgroups — #27286687

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811002C211yesmaximal, characteristic, complemented
2C2×#16810904C211yesmaximal, characteristic
3C2×#16810900C211yesmaximal, characteristic
4#20275508C211yesmaximal, characteristic
5C2×#16810370C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C22×#11719925—22no—
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16811002—22no—
12#16811002—22no—
13#16810904—22no—
14#16810904—22no—
15#16810900—22no—
16#16810900—22no—
17C2×#13133151—21no—
18C2×#13133150—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133139—21no—
21C2×#13133137—21no—
22C2×#13133129—22no—
23C2×C4×#88322C2211yescharacteristic
24#16810370—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#332—22no—
29C22×Dic106—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C2×#11719925—28no—
33C2×#11719925—44no—
34C2×#11719925—44no—
35C22×#88322—21no—
36C22×#88322D412yes—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217D412yes—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—21no—
49#13133151—24no—
50#13133150—24no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133139—24no—
54#13133137—24no—
55#13133129—44no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×Q8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53D412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—41no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212Q1612yescomplemented
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C2×Dic106—22no—
94C2×Dic106—42no—
95C2×Dic106—42no—
96C2×Dic106—41no—
97C2×C8×C53—22no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—42no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110#11719925—416no—
111#11719921—41no—
112#11719921—41no—
113C4⋊Dic53—22no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322D814yes—
125C23×C53#12111yescharacteristic
126C22×D53—22no—
127C22×D53—41no—
128C22×D53—42no—
129C22×D53—41no—
130C22×D53—42no—
131Q8×C53—44no—
132Q8×C53—81no—
133Q8×C53—81no—
134C2×C4×C53—22no—
135C2×C4×C53—42no—
136C2×C4×C53—82no—
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12711yescharacteristic
143C2×C4×C53C2×D812yes—
144C2×C4×C53C2×Q1611yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)12yes—
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—82no—
153D212—22no—
154D212—22no—
155D212—41no—
156D212—41no—
157C4×D53—82no—
158C4×D53—81no—
159C4×D53—81no—
160Dic106—44no—
161Dic106—81no—
162Dic106—81no—
163C8×C53—42no—
164C8×C53—81no—
165#88322—24no—
166#88322—42no—
167#10274—531nomaximal, Sylow
168C22×C53#39211yescharacteristic
169C22×C53#45912yes—
170C22×C53#46711yescharacteristic
171C22×C53#47011yescharacteristic
172C22×C53#47312yes—
173C2×D53—44no—
174C2×D53—42no—
175C2×D53—82no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—82no—
181C4×C53#44712yes—
182C4×C53#45111yescharacteristic
183C4×C53#45111yescharacteristic
184C4×C53#45211yescharacteristic
185C4×C53#45211yescharacteristic
186C4×C53#49311yescharacteristic
187C4×C53#49611yescharacteristic
188Dic53—82no—
189Dic53—81no—
190C2×#494—531no—
191C2×#474—531no—
192C2×#474—531no—
193#1954—531no—
194C2×#332—531no—
195C2×#332D5311yescharacteristic, complemented
196#1277—531no—
197C2×C53#166912yes—
198C2×C53#168912yes—
199C2×C53#200111yescharacteristic
200C2×C53#201111yescharacteristic
201C2×C53#202011yescharacteristic
202D53—82no—
203C22×Q16—1062no—
204C2×C4⋊Q8—531no—
205C2×C4⋊Q8C2×D5311yescharacteristic
206C2×C4×D4—531no—
207#494—1062no—
208#494—1062no—
209#474—1062no—
210#474—1062no—
211#474—1062no—
212#474—1062no—
213C2×#107—1061no—
214C2×#107—1061no—
215C2×C4⋊C8—531no—
216C2×C4⋊C8C2×D5311yescharacteristic
217C2×C4⋊C8C2×D5311yescharacteristic
218C2×#102—1061no—
219C2×#102—1061no—
220C2×C22⋊C8—1062no—
221C2×C4×C8—531no—
222#332—22no—
223#332—1062no—
224C53#1027411yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×Q8—1061no—
227C22×D4—531no—
228C23×C4—1061no—
229C2×Q16—1068no—
230C2×Q16—2124no—
231C2×Q16—2124no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—532no—
235C22×C8—1061no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—532no—
240C4⋊Q8—1061no—
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243C4⋊Q8D21212yes—
244C4×D4—1061no—
245C4×D4—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C22⋊C4—1061no—
254C2×C42C22×D5311yescharacteristic
255#107—1064no—
256#107—1064no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—532no—
261C4⋊C8—1061no—
262C4⋊C8C22⋊D5312yes—
263#102—1064no—
264#102—1064no—
265C22⋊C8—2124no—
266C4×C8—1061no—
267C4×C8—1061no—
268C24—1061no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—1062no—
274C2×Q8—2122no—
275C2×Q8—2122no—
276C2×Q8—2121no—
277C2×D4—532no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C22×C4—41no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2122no—
289C22×C4—2122no—
290C22×C4#1171994511yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C22×C4D4×D5311yescharacteristic
293Q16—21216no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—534no—
300C2×C8—1062no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8—2121no—
304C2×C8—2121no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—1062no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1062no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42C2×D21211yescharacteristic
318C42D4×D5311yescharacteristic
319C42D4×D5311yescharacteristic
320C42C2×C22⋊D5311yescharacteristic
321C23—1062no—
322C23—2121no—
323C23—2122no—
324C23—2121no—
325C23—2122no—
326C23#1313319011yescharacteristic
327Q8—44no—
328Q8—81no—
329Q8—81no—
330Q8—4241no—
331Q8—4241no—
332Q8—2124no—
333D4—1062no—
334D4—1062no—
335D4—2121no—
336D4—2121no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—4242no—
341C2×C4—4241no—
342C2×C4—4242no—
343C2×C4—4241no—
344C2×C4—2121no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4#1313318511yescharacteristic
348C2×C4#1313318711yescharacteristic
349C2×C4#1313318711yescharacteristic
350C2×C4#1313319011yescharacteristic
351C2×C4#1313319111yescharacteristic
352C2×C4#1313321311yescharacteristic
353C2×C4D8×D5312yes—
354C2×C4Q16×D5311yescharacteristic
355C2×C4#1313323811yescharacteristic
356C2×C4#1313324412yes—
357C2×C4#1313324611yescharacteristic
358C2×C4#1313324911yescharacteristic
359C8—42no—
360C8—81no—
361C8—1064no—
362C8—2122no—
363C22—4242no—
364C22—4241no—
365C22—4241no—
366C22—2122no—
367C22—2124no—
368C22#1681053211yescharacteristic
369C22#1681081712yes—
370C22#1681086311yescharacteristic
371C22#1681088011yescharacteristic
372C22#1681089712yes—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—4241no—
377C4—4242no—
378C4#1681075812yes—
379C4#1681077711yescharacteristic
380C4#1681078011yescharacteristic
381C4#1681078211yescharacteristic
382C4#1681078811yescharacteristic
383C4#1681099811yescharacteristic
384C4#1681100911yescharacteristic
385C2—4242no—
386C2#2027407912yes—
387C2#2027420312yes—
388C2#2027572411yescharacteristic
389C2#2027578711yescharacteristic
390C2#2027583511yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#332D53
hypocenterC53#10274
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. C2×#16811002C2semidirect
5. C2×#16810370C2semidirect
6. #20272549C2semidirect
7. C2×C53×#332C2semidirect
88. C2×D212Q16semidirect
195. C2×#332D53semidirect
224. C53#10274semidirect

Maximal subgroups

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

subgroupindex
C2×#168110022
C2×#168109042
C2×#168109002
#202755082
C2×#168103702
#202725492
C2×C53×#3322
#1027453