Finite Group Databasebeta

← #28299721

Subgroups — #28299721

Order 13568.

500 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290606C211yesmaximal, characteristic
2#20290571C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5C22⋊Q8×Dic53C211yesmaximal, characteristic
6C2×#13133284—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—
25#13133284—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×C53—22no—
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×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53SD1612yes—
78C22×Dic53Q1612yes—
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×C53M1612yes—
90C22⋊C4×C53—24no—
91C22⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4×Dic53—22no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100C4×Dic53—42no—
101C4×Dic53—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53#10312yes—
109C23×C53C2×C22⋊C411yescharacteristic
110Q8×C53—42no—
111Q8×C53—42no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
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×C53C2×M1611yescharacteristic
126C2×C4×C53C2×M1611yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—44no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53#9912yes—
145C2×Dic53#10312yes—
146C2×Dic53C2×SD1611yescharacteristic
147C2×Dic53C2×Q1611yescharacteristic
148C2×Dic53#13512yes—
149#88324—82no—
150#88324—82no—
151#7648—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#40711yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53#41311yescharacteristic
162C22×C53C2×#10311yescharacteristic
163C22×C53#41611yescharacteristic
164C22×C53#42111yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—41no—
169C4×C53—42no—
170C4×C53—42no—
171Dic53—22no—
172Dic53—22no—
173Dic53—21no—
174Dic53—22no—
175Dic53—21no—
176Dic53—81no—
177Dic53—81no—
178C4×C22⋊Q8—531no—
179#1817—531no—
180#1797—531no—
181#1272—532no—
182#1271—532no—
183C2×C53—42no—
184C2×C53#146511yescharacteristic
185C2×C53#148711yescharacteristic
186C2×C53#148811yescharacteristic
187C2×C53#151611yescharacteristic
188C2×C53#152011yescharacteristic
189C2×C53#153511yescharacteristic
190C2×C53#154211yescharacteristic
191C2×C22⋊Q8#615711yescharacteristic
192C2×C4×Q8—531no—
193C22×C42—531no—
194C2×C4⋊C8—1061no—
195C2×C22⋊C8—1061no—
196C2×C22⋊C8—1061no—
197#391—531no—
198#389—531no—
199#387—1061no—
200#385—1061no—
201#380—531no—
202C4×C4⋊C4—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—1061no—
205#336—532no—
206#336—532no—
207#336—1061no—
208C53#764811yescharacteristic, complemented, Sylow
209C22×Q8C2×#615711yescharacteristic
210C23×C4—531no—
211C23×C4—531no—
212C23×C4C2×#615711yescharacteristic
213C22×C8—1062no—
214C22×C8—1062no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C4×Q8—2122no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C2×#615711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1061no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—1061no—
231C2×C42—1061no—
232C2×C42—2122no—
233C4⋊C8—2122no—
234C22⋊C8—2122no—
235C22⋊C8—2122no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24C22⋊#615711yescharacteristic
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#1171994012yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4C22⋊#615711yescharacteristic
266C22×C4C22×#615711yescharacteristic
267C2×C8—2122no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2122no—
279C4⋊C4#1171994012yes—
280C22⋊C4—24no—
281C22⋊C4—2122no—
282C42—1062no—
283C42—1062no—
284C42—2121no—
285C42—2121no—
286C42—2122no—
287C42—2122no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1313328012yes—
295C23C2×C22⋊#615711yescharacteristic
296Q8—42no—
297Q8—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—532no—
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—2122no—
325C2×C4—2122no—
326C2×C4—2124no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4C2×#1171994011yescharacteristic
334C2×C4C2×#1171994011yescharacteristic
335C8—4242no—
336C8—4242no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1681159311yescharacteristic
345C22D53⋊#9911yescharacteristic
346C22#1681161011yescharacteristic
347C22C2×#1313328011yescharacteristic
348C22#1681161311yescharacteristic
349C22#1681162011yescharacteristic
350C22#1681162211yescharacteristic
351C4—21no—
352C4—21no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—1062no—
357C4—1062no—
358C4—1061no—
359C4—1062no—
360C4—1061no—
361C4—4241no—
362C4—4241no—
363C2—42no—
364C2#2028997111yescharacteristic
365C2#2029004311yescharacteristic
366C2#2029004611yescharacteristic
367C2#2029014211yescharacteristic
368C2#2029015311yescharacteristic
369C2#2029020211yescharacteristic
370C2#2029021711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7648
LCSG ▷ C22×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
208. C53#7648semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202905712
#202895352
#202895332
C22⋊Q8×Dic532
#764853