Finite Group Databasebeta

← #28299553

Subgroups — #28299553

Order 13568.

504 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290605C211yesmaximal, characteristic
2#20290569C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5C22⋊Q8×Dic53C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×Q8×Dic53C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810509—21no—
17#16810498—21no—
18#16810483C411yescharacteristic
19C4⋊C4×Dic53—21no—
20C4⋊C4×Dic53C411yescharacteristic
21C22⋊C4×Dic53—21no—
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C23×Dic53D411yescharacteristic
29C23×Dic53D411yescharacteristic
30Q8×Dic53—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×Q8×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C22⋊Q8×C53—22no—
51C22⋊Q8×C53C814yescomplemented
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C2×Q8×C53—21no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53C2×C812yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C2×C812yes—
66C22×C4×C53M1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C22×Dic53C2×D411yescharacteristic
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53C2×C812yes—
90C22⋊C4×C53—24no—
91C22⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53C22⋊C812yes—
110C23×C53C2×C22⋊C411yescharacteristic
111Q8×C53—42no—
112Q8×C53—42no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53C22⋊C812yes—
126C2×C4×C53C22×C811yescharacteristic
127C2×C4×C53C2×M1611yescharacteristic
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—44no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53#9912yes—
147C2×Dic53AGL(1,ℤ/8)11yescharacteristic
148C2×Dic53#13711yescharacteristic
149C2×Dic53#13712yes—
150#88324—82no—
151#88324—82no—
152#7600—531nomaximal, Sylow
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C2×C22⋊C811yescharacteristic
161C22×C53C2×#9911yescharacteristic
162C22×C53#41311yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53#41911yescharacteristic
165C22×C53#42111yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—42no—
171C4×C53—42no—
172Dic53—22no—
173Dic53—22no—
174Dic53—21no—
175Dic53—21no—
176Dic53—41no—
177Dic53—81no—
178Dic53—81no—
179C4×C22⋊Q8—531no—
180#1817—531no—
181#1797—531no—
182#1272—532no—
183#1271—532no—
184C2×C53—42no—
185C2×C53#145911yescharacteristic
186C2×C53#147711yescharacteristic
187C2×C53#147911yescharacteristic
188C2×C53#151111yescharacteristic
189C2×C53#151311yescharacteristic
190C2×C53#153511yescharacteristic
191C2×C53#155911yescharacteristic
192C2×C22⋊Q8#615711yescharacteristic
193C2×C4×Q8—531no—
194C22×C42—531no—
195C2×C4⋊C8—1061no—
196C2×C22⋊C8—1061no—
197C2×C22⋊C8—1061no—
198#391—531no—
199#389—531no—
200#387—1061no—
201#385—1061no—
202#380—531no—
203C4×C4⋊C4—531no—
204C4×C4⋊C4—1061no—
205C4×C22⋊C4—1061no—
206#336—532no—
207#336—532no—
208#336—1061no—
209C53#760011yescharacteristic, complemented, Sylow
210C22×Q8C2×#615711yescharacteristic
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C2×#615711yescharacteristic
214C22×C8—1062no—
215C22×C8—1062no—
216C22⋊Q8—22no—
217C22⋊Q8#8832414yescomplemented
218C4×Q8—2122no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4C2×#615711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1061no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—2122no—
234C4⋊C8—2122no—
235C22⋊C8—2122no—
236C22⋊C8—2122no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24C22⋊#615711yescharacteristic
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C2×#8832412yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1062no—
257C22×C4—1061no—
258C22×C4—1062no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4C2×#8832412yes—
265C22×C4#1171994012yes—
266C22×C4C22⋊#615711yescharacteristic
267C22×C4C22×#615711yescharacteristic
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C2×C8—2122no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2122no—
281C4⋊C4C2×#8832412yes—
282C22⋊C4—24no—
283C22⋊C4—2122no—
284C42—1062no—
285C42—1062no—
286C42—2121no—
287C42—2121no—
288C42—2122no—
289C42—2122no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C22⋊#8832412yes—
297C23C2×C22⋊#615711yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—532no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2124no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4C22⋊#8832412yes—
335C2×C4C22×#8832411yescharacteristic
336C2×C4C2×#1171994011yescharacteristic
337C8—4242no—
338C8—4242no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×C22⋊#8832411yescharacteristic
347C22D53⋊#9911yescharacteristic
348C22#1681161011yescharacteristic
349C22#1681161411yescharacteristic
350C22#1681161611yescharacteristic
351C22#1681162011yescharacteristic
352C22#1681162211yescharacteristic
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—42no—
358C4—1062no—
359C4—1062no—
360C4—1061no—
361C4—1061no—
362C4—4241no—
363C4—4241no—
364C4—2121no—
365C2—42no—
366C2#2028995111yescharacteristic
367C2#2029001011yescharacteristic
368C2#2029001711yescharacteristic
369C2#2029012411yescharacteristic
370C2#2029013311yescharacteristic
371C2#2029020211yescharacteristic
372C2#2029025111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7600
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289533C2semidirect
51. C22⋊Q8×C53C8semidirect
209. C53#7600semidirect
217. C22⋊Q8#88324semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202905692
#202895352
#202895332
C22⋊Q8×Dic532
#760053