Finite Group Databasebeta

← #28299433

Subgroups — #28299433

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊#88324C211yesmaximal, characteristic
2C2×#16811489C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5C22⋊Q8×Dic53C211yesmaximal, characteristic
6C23×#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#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—24no—
24C22×#88324—22no—
25C22×#88324—22no—
26C22×#88324—42no—
27C22×#88324—42no—
28C22⋊#88324—24no—
29C4×#88324—42no—
30C23×Dic53D411yescharacteristic
31C23×Dic53D411yescharacteristic
32Q8×Dic53—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×Q8×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C22⋊Q8×C53—22no—
53C22⋊Q8×C53C814yescomplemented
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53C2×C812yescomplemented
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C2×C812yes—
68C22×C4×C53M1612yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—41no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53SD1612yes—
80C22×Dic53Q1612yes—
81C22×Dic53C2×D411yescharacteristic
82C22×Dic53C2×D411yescharacteristic
83C2×#88324—24no—
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—48no—
88C2×#88324—42no—
89C2×#88324—42no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2×C812yescomplemented
93C22⋊C4×C53—24no—
94C22⋊Dic53—42no—
95C4⋊Dic53—42no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4×Dic53—22no—
100C4×Dic53—22no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C4×Dic53—42no—
104C4×Dic53—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53C22⋊C812yes—
110C23×C53#10312yes—
111C23×C53#10412yes—
112C23×C53C2×C22⋊C411yescharacteristic
113Q8×C53—42no—
114Q8×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53C22⋊C812yes—
128C2×C4×C53C22×C811yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—24no—
135C2×Dic53—24no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—44no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53#9912yes—
147C2×Dic53#10312yes—
148C2×Dic53#10412yes—
149C2×Dic53C2×SD1611yescharacteristic
150C2×Dic53C2×Q1611yescharacteristic
151C2×Dic53#13512yes—
152#88324—44no—
153#88324—44no—
154#7566—531nomaximal, Sylow
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53Q8⋊C814yes—
162C22×C53C2×C22⋊C811yescharacteristic
163C22×C53C2×#9911yescharacteristic
164C22×C53#41311yescharacteristic
165C22×C53C2×#10311yescharacteristic
166C22×C53#41611yescharacteristic
167C22×C53C2×#10411yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C4×C53—22no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—42no—
173C4×C53—42no—
174Dic53—21no—
175Dic53—21no—
176Dic53—22no—
177Dic53—41no—
178Dic53—81no—
179Dic53—81no—
180Dic53#32814yes—
181C4×C22⋊Q8—531no—
182C4×C22⋊C8—531no—
183C2×#336—531no—
184#1272—532no—
185#1271—532no—
186C2×C53—24no—
187C2×C53#145911yescharacteristic
188C2×C53C2×Q8⋊C811yescharacteristic
189C2×C53#147611yescharacteristic
190C2×C53C2×#32811yescharacteristic
191C2×C53#150111yescharacteristic
192C2×C53#152411yescharacteristic
193C2×C53#154211yescharacteristic
194C23×C8—1061no—
195C2×C22⋊Q8#615711yescharacteristic
196C2×C4×Q8—531no—
197C22×C42—531no—
198C2×C22⋊C8—1061no—
199C2×C4×C8—1061no—
200#391—531no—
201#389—531no—
202#387—1061no—
203#385—1061no—
204#380—531no—
205C4×C4⋊C4—531no—
206C4×C4⋊C4—1061no—
207C4×C22⋊C4—1061no—
208#336—532no—
209#336—532no—
210#336—1061no—
211C53#756611yescharacteristic, complemented, Sylow
212C22×Q8C2×#615711yescharacteristic
213C23×C4—531no—
214C23×C4—531no—
215C23×C4C2×#615711yescharacteristic
216C22×C8—1062no—
217C22×C8—1062no—
218C22×C8—1064no—
219C22×C8—2122no—
220C22×C8—2122no—
221C22⋊Q8—22no—
222C22⋊Q8#8832414yescomplemented
223C4×Q8—2122no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4C2×#615711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1061no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—1061no—
237C2×C42—1061no—
238C2×C42—2122no—
239C22⋊C8—1064no—
240C4×C8—2122no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24C22⋊#615711yescharacteristic
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8C2×#8832412yescomplemented
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4C2×#8832412yes—
269C22×C4#1171994012yes—
270C22×C4C22⋊#615711yescharacteristic
271C22×C4C22×#615711yescharacteristic
272C2×C8—1064no—
273C2×C8—1064no—
274C2×C8—2122no—
275C2×C8—2122no—
276C2×C8—2122no—
277C2×C8—2128no—
278C2×C8—2122no—
279C4⋊C4—24no—
280C4⋊C4—21no—
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2122no—
284C4⋊C4—2122no—
285C4⋊C4C2×#8832412yescomplemented
286C22⋊C4—24no—
287C22⋊C4—2122no—
288C42—1062no—
289C42—1062no—
290C42—2121no—
291C42—2121no—
292C42—2122no—
293C42—2122no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C22⋊#8832412yes—
299C23#1313328012yes—
300C23#1313328212yes—
301C23C2×C22⋊#615711yescharacteristic
302Q8—42no—
303Q8—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—532no—
317C2×C4—532no—
318C2×C4—532no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—532no—
322C2×C4—1064no—
323C2×C4—1064no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—2124no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4C22⋊#8832412yes—
339C2×C4C22×#8832411yescharacteristic
340C2×C4C2×#1171994011yescharacteristic
341C8—2124no—
342C8—2124no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22Q8⋊#8832414yes—
350C22C2×C22⋊#8832411yescharacteristic
351C22D53⋊#9911yescharacteristic
352C22#1681161011yescharacteristic
353C22C2×#1313328011yescharacteristic
354C22#1681161311yescharacteristic
355C22D53⋊#10411yescharacteristic
356C22C2×#1313328211yescharacteristic
357C4—22no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—42no—
362C4—534no—
363C4—1061no—
364C4—1061no—
365C4—1062no—
366C4—4241no—
367C4—4241no—
368C4—2121no—
369C2—24no—
370C2#2028995111yescharacteristic
371C2C2×Q8⋊#8832411yescharacteristic
372C2#2029000611yescharacteristic
373C2D53⋊#32811yescharacteristic
374C2#2029009311yescharacteristic
375C2#2029016811yescharacteristic
376C2#2029021711yescharacteristic
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#7566
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
53. C22⋊Q8×C53C8semidirect
61. C2×Q8×C53C2×C8semidirect
92. C4⋊C4×C53C2×C8semidirect
211. C53#7566semidirect
222. C22⋊Q8#88324semidirect
250. C2×Q8C2×#88324semidirect
285. C4⋊C4C2×#88324semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#883242
C2×#168114892
#202895352
#202895332
C22⋊Q8×Dic532
#756653