Finite Group Databasebeta

← #27275805

Subgroups — #27275805

Order 13568.

1050 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274707C211yesmaximal, characteristic, complemented
2#20273383C212yesmaximal, complemented
3#20273144C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic, complemented
5C2×C53×#550C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810367—24no—
11Q8⋊#88322—24no—
12#16810347—22no—
13C22×#41×C53C412yes—
14C53×#550—24no—
15C53×#550—24no—
16C2×#127×C53—21no—
17C2×#127×C53C2211yescharacteristic
18C2×#121×C53—22no—
19C2×#121×C53C412yes—
20C2×C4×Q8×C53C2211yescharacteristic
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C2211yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133134—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C2×#41×C53—24no—
31C2×#41×C53—24no—
32C2×#41×C53—48no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53—22no—
35C22×D4×C53—22no—
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—22no—
41C23×C4×C53C2×C412yes—
42#127×C53—28no—
43#127×C53—22no—
44#127×C53D414yes—
45#121×C53—216no—
46#121×C53—24no—
47#121×C53—24no—
48C4×Q8×C53—22no—
49C4×Q8×C53D414yes—
50C4×D4×C53—28no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53D412yes—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C24×C53—22no—
61C24×C53C22⋊C412yes—
62#41×C53—416no—
63#41×C53—48no—
64#41×C53—48no—
65C2×Q8×C53—24no—
66C2×Q8×C53—21no—
67C2×Q8×C53C22⋊C412yes—
68C2×D4×C53—28no—
69C2×D4×C53—28no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—24no—
73C2×D4×C53—24no—
74C2×D4×C53—22no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—48no—
78C2×D4×C53—44no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—82no—
83C2×D4×C53—84no—
84C2×D4×C53C22⋊C414yes—
85C22×C4×C53—21no—
86C22×C4×C53—28no—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—48no—
91C22×C4×C53—44no—
92C22×C4×C53C22⋊C412yes—
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C4⋊C4×C53—21no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53C22⋊C412yes—
100C22⋊C4×C53—28no—
101C22⋊C4×C53—24no—
102C42×C53—24no—
103C42×C53C2×D412yes—
104C42×C53C2×D412yes—
105#11719917—42no—
106#11719917—42no—
107C2×#88322—42no—
108C2×#88322—42no—
109C23×C53—22no—
110C23×C53—24no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—44no—
114C23×C53—44no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53C2×C22⋊C411yescharacteristic
120Q8×C53—24no—
121Q8×C53—22no—
122Q8×C53#10014yescomplemented
123D4×C53—44no—
124D4×C53—416no—
125D4×C53—44no—
126D4×C53—42no—
127D4×C53—44no—
128D4×C53—42no—
129D4×C53—44no—
130D4×C53—84no—
131D4×C53—88no—
132C2×C4×C53—24no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—41no—
138C2×C4×C53—416no—
139C2×C4×C53—44no—
140C2×C4×C53—42no—
141C2×C4×C53—44no—
142C2×C4×C53—42no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53#12012yes—
146C2×C4×C53#12012yes—
147C2×C4×C53#13512yes—
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150#88322—82no—
151#88322—82no—
152#8385—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—42no—
156C22×C53—44no—
157C22×C53—42no—
158C22×C53—44no—
159C22×C53—42no—
160C22×C53—44no—
161C22×C53—82no—
162C22×C53—82no—
163C22×C53—84no—
164C22×C53—84no—
165C22×C53#37911yescharacteristic
166C22×C53#41611yescharacteristic
167C22×C53#41711yescharacteristic
168C4×C53—22no—
169C4×C53—24no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53C2×#10012yes—
174C4×C53#44912yes—
175C4×C53#45212yes—
176C2×#550D5311yescharacteristic
177#1800—531no—
178#1545—532no—
179#1487—532no—
180#1460—531no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—82no—
184C2×C53—84no—
185C2×C53#178311yescharacteristic
186C2×C53#178811yescharacteristic
187C2×C53#179511yescharacteristic
188C22×#41Dic5312yes—
189#550—24no—
190#550—24no—
191C2×#127—21no—
192C2×#127C2×D5311yescharacteristic
193C2×#121—22no—
194C2×#121Dic5312yes—
195C2×C4×Q8C2×D5311yescharacteristic
196C2×C4×D4—21no—
197C2×C4×D4C2×D5311yescharacteristic
198#431—532no—
199#423—532no—
200C2×#100—1061no—
201#343—1061no—
202#331—1064no—
203Q8⋊C8—1064no—
204#323—1062no—
205C53#838511yescharacteristic, complemented, Sylow
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—48no—
209C22×Q8C2×Dic5311yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2×Dic5312yes—
215C22×D4C22⋊D5311yescharacteristic
216C23×C4—22no—
217C23×C4C2×Dic5312yes—
218C2×M16—1062no—
219#127—28no—
220#127—22no—
221#127C22⋊D5314yes—
222#121—216no—
223#121—24no—
224#121—24no—
225C4×Q8—22no—
226C4×Q8C22⋊D5314yes—
227C4×D4—28no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊D5312yes—
235C2×C42—21no—
236C2×C42C22×D5311yescharacteristic
237C4⋊C8—1062no—
238C4⋊C8—1062no—
239C4⋊C8—1062no—
240#100—2122no—
241C22⋊C8—1062no—
242C4×C8—1062no—
243C24—22no—
244C24C22⋊Dic5312yes—
245#41—416no—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—21no—
250C2×Q8C22⋊Dic5312yes—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—84no—
267C2×D4C22⋊Dic5314yes—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4C22⋊Dic5312yes—
276C22×C4C22×Dic5311yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279M16—2122no—
280M16—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—21no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4C22⋊Dic5312yes—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C42—24no—
290C42C2×C22⋊D5312yes—
291C42C2×C22⋊D5312yes—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—22no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C2×C22⋊Dic5311yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8#1313313414yescomplemented
306D4—44no—
307D4—416no—
308D4—42no—
309D4—44no—
310D4—42no—
311D4—44no—
312D4—44no—
313D4—84no—
314D4—88no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4C2×C22⋊Dic5311yescharacteristic
327C2×C4C2×C22⋊Dic5311yescharacteristic
328C2×C4#1313318612yes—
329C2×C4#1313318612yes—
330C2×C4#1313323712yes—
331C2×C4#1313324211yescharacteristic
332C2×C4#1313324711yescharacteristic
333C8—4242no—
334C8—4242no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—84no—
346C22—84no—
347C22#1681047911yescharacteristic
348C22#1681061911yescharacteristic
349C22#1681062411yescharacteristic
350C4—24no—
351C4—21no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4C2×#1313313412yes—
356C4#1681076812yes—
357C4#1681078512yes—
358C2—21no—
359C2—21no—
360C2—84no—
361C2—82no—
362C2#2027463611yescharacteristic
363C2#2027465511yescharacteristic
364C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#550C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenterC2×#550D53
hypocenterC53#8385
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #20274707C2semidirect
2. #20273383C2semidirect
3. #20273144C2semidirect
4. #20273036C2semidirect
122. Q8×C53#100semidirect
205. C53#8385semidirect
305. Q8#13133134semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202733832
#202731442
#202730362
C2×C53×#5502
#838553