Finite Group Databasebeta

← #27275845

Subgroups — #27275845

Order 13568.

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

normal subgroups only

This group has 998 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#20273396C212yesmaximal, complemented
3#20273140C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic, complemented
5C2×C53×#547C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810372—24no—
11D4⋊#88322—24no—
12#16810347—22no—
13C22×#41×C53C412yes—
14C53×#547—24no—
15C53×#547—24no—
16C2×C4⋊Q8×C53C2211yescharacteristic
17C2×#123×C53—22no—
18C2×C22⋊Q8×C53C412yes—
19C2×#121×C53—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×C4×D4×C53C2211yescharacteristic
22C22×C4⋊C4×C53—21no—
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×C412yes—
34C22×D4×C53—22no—
35C22×D4×C53C2×C411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C23×C4×C53—22no—
38C23×C4×C53—21no—
39C23×C4×C53C2×C412yes—
40C4⋊Q8×C53—22no—
41C4⋊Q8×C53D414yes—
42#123×C53—216no—
43C22⋊Q8×C53—24no—
44C22⋊Q8×C53—24no—
45#121×C53—28no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C4×D4×C53D414yes—
50C2×C4⋊C4×C53—28no—
51C2×C4⋊C4×C53—22no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53D412yes—
59C2×C42×C53C2311yescharacteristic
60C24×C53—21no—
61C24×C53C22⋊C412yes—
62#41×C53—416no—
63#41×C53—48no—
64#41×C53—48no—
65C2×Q8×C53—22no—
66C2×Q8×C53—44no—
67C2×Q8×C53C22⋊C414yes—
68C2×D4×C53—24no—
69C2×D4×C53—28no—
70C2×D4×C53—21no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—48no—
76C2×D4×C53C22⋊C412yes—
77C22×C4×C53—28no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—24no—
83C22×C4×C53—44no—
84C22×C4×C53—48no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C22⋊C412yes—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C4⋊C4×C53—28no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C22⋊C412yes—
98C22⋊C4×C53—28no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—24no—
101C42×C53C2×D412yes—
102C42×C53C2×D412yes—
103#11719917—42no—
104#11719917—42no—
105C2×#88322—42no—
106C2×#88322—42no—
107C23×C53—24no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53C2×C22⋊C411yescharacteristic
118Q8×C53—44no—
119Q8×C53—44no—
120D4×C53—24no—
121D4×C53—22no—
122D4×C53—416no—
123D4×C53—42no—
124D4×C53—42no—
125D4×C53#10014yescomplemented
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—24no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—44no—
132C2×C4×C53—44no—
133C2×C4×C53—416no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—44no—
137C2×C4×C53—44no—
138C2×C4×C53—44no—
139C2×C4×C53—42no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53#12012yes—
143C2×C4×C53#12012yes—
144C2×C4×C53#13512yes—
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147#88322—82no—
148#88322—82no—
149#8393—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—24no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53#37911yescharacteristic
163C22×C53#41611yescharacteristic
164C22×C53#41711yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—44no—
169C4×C53—42no—
170C4×C53C2×#10012yes—
171C4×C53#44912yes—
172C4×C53#45212yes—
173C2×#547D5311yescharacteristic
174#1800—531no—
175#1548—532no—
176#1486—532no—
177#1460—531no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—44no—
181C2×C53—82no—
182C2×C53#178311yescharacteristic
183C2×C53#178811yescharacteristic
184C2×C53#179611yescharacteristic
185C22×#41Dic5312yes—
186#547—24no—
187#547—24no—
188C2×C4⋊Q8C2×D5311yescharacteristic
189C2×#123—22no—
190C2×C22⋊Q8Dic5312yes—
191C2×#121—21no—
192C2×C4×D4C2×D5311yescharacteristic
193C2×C4×D4C2×D5311yescharacteristic
194C22×C4⋊C4—21no—
195#431—532no—
196#423—532no—
197C2×#100—1061no—
198#343—1061no—
199#333—1064no—
200D4⋊C8—1064no—
201#323—1062no—
202C53#839311yescharacteristic, complemented, Sylow
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—48no—
206C22×Q8C2×Dic5312yes—
207C22×D4—22no—
208C22×D4C2×Dic5311yescharacteristic
209C22×D4C22⋊D5311yescharacteristic
210C23×C4—21no—
211C23×C4—22no—
212C23×C4C2×Dic5312yes—
213C2×M16—1062no—
214C4⋊Q8—22no—
215C4⋊Q8C22⋊D5314yes—
216#123—216no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219#121—28no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C22⋊D5314yes—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C2×Dic5311yescharacteristic
229C2×C4⋊C4C22⋊D5311yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C22⋊D5312yes—
233C2×C42C22×D5311yescharacteristic
234C4⋊C8—1062no—
235C4⋊C8—1062no—
236C4⋊C8—1062no—
237#100—2122no—
238C22⋊C8—1062no—
239C4×C8—1062no—
240C24—21no—
241C24C22⋊Dic5312yes—
242#41—416no—
243#41—48no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—44no—
247C2×Q8C22⋊Dic5314yes—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—48no—
256C2×D4C22⋊Dic5312yes—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22⋊Dic5312yes—
269C22×C4C22×Dic5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272M16—2122no—
273M16—2122no—
274C2×C8—2122no—
275C2×C8—2122no—
276C4⋊C4—28no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4C22⋊Dic5312yes—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C42C2×C22⋊D5312yes—
286C42C2×C22⋊D5312yes—
287C23—24no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic5311yescharacteristic
298Q8—44no—
299Q8—44no—
300D4—24no—
301D4—22no—
302D4—416no—
303D4—42no—
304D4—42no—
305D4#1313313414yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4C2×C22⋊Dic5311yescharacteristic
321C2×C4C2×C22⋊Dic5311yescharacteristic
322C2×C4#1313318612yes—
323C2×C4#1313318612yes—
324C2×C4#1313323712yes—
325C2×C4#1313324211yescharacteristic
326C2×C4#1313324711yescharacteristic
327C8—4242no—
328C8—4242no—
329C22—21no—
330C22—21no—
331C22—24no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#1681047911yescharacteristic
342C22#1681061911yescharacteristic
343C22#1681062411yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4C2×#1313313412yes—
350C4#1681076812yes—
351C4#1681078512yes—
352C2—21no—
353C2—21no—
354C2—44no—
355C2—82no—
356C2#2027463611yescharacteristic
357C2#2027465511yescharacteristic
358C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#547C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenterC2×#547D53
hypocenterC53#8393
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20274707C2semidirect
2. #20273396C2semidirect
3. #20273140C2semidirect
4. #20273036C2semidirect
125. D4×C53#100semidirect
202. C53#8393semidirect
305. D4#13133134semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202733962
#202731402
#202730362
C2×C53×#5472
#839353