Finite Group Databasebeta

← #28308177

Subgroups — #28308177

Order 13568.

586 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289966C212yesmaximal, complemented
2#20289645C212yesmaximal, complemented
3#20289643C212yesmaximal, complemented
4#20277206C211yesmaximal, characteristic
5#16811643—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×#13133176C2211yescharacteristic
9C22×C4×Dic53C411yescharacteristic
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810526C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810503—22no—
15C4×C22⋊Dic53—22no—
16C22×#41×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C23×Dic53—21no—
21#13133176—41no—
22#13133176—41no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2311yescharacteristic
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53D412yes—
28C2×C4×Dic53—21no—
29C2×C4×Dic53—42no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33#13133124—22no—
34#13133124—22no—
35C2×#41×C53—22no—
36C2×#41×C53—22no—
37C2×#41×C53—42no—
38C2×#41×C53—41no—
39C2×#41×C53—41no—
40C22×Q8×C53C2×C411yescharacteristic
41C22×D4×C53—21no—
42C22×D4×C53C2×C411yescharacteristic
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C24×C53—21no—
46C24×C53C22⋊C411yescharacteristic
47#41×C53—44no—
48#41×C53—44no—
49#41×C53—44no—
50#41×C53—44no—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53M1612yes—
55C2×D4×C53—24no—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53M1612yes—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—24no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C22×Dic53C22⋊C412yes—
80C22×Dic53C2×D412yes—
81C2×#88324—44no—
82C22⋊Dic53—42no—
83C22⋊Dic53—44no—
84C22⋊Dic53—44no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—42no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53#10012yes—
102C23×C53C2×C22⋊C411yescharacteristic
103Q8×C53—42no—
104Q8×C53—42no—
105D4×C53—42no—
106D4×C53—42no—
107D4×C53—48no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—44no—
118C2×C4×C53—44no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×C4×C53C2×M1611yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—48no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136#88324—84no—
137#11507—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—44no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—42no—
147C22×C53—82no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53#40711yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53C2×#10011yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—42no—
158Dic53—42no—
159Dic53—42no—
160Dic53—42no—
161Dic53—42no—
162#2297—531no—
163#1464—532no—
164#1319—532no—
165#1318—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—42no—
169C2×C53—82no—
170C2×C53#146211yescharacteristic
171C2×C53#212611yescharacteristic
172C2×C53#213211yescharacteristic
173C22×#41#615711yescharacteristic
174C2×#117—531no—
175C22×C42—531no—
176#432—1062no—
177#391—531no—
178#391—531no—
179#390—531no—
180#390—531no—
181#386—1062no—
182C4×C22⋊C4—1062no—
183#324—534no—
184#323—1062no—
185C53#1150711yescharacteristic, complemented, Sylow
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—42no—
189C2×#41—41no—
190C2×#41—41no—
191C22×Q8C2×#615711yescharacteristic
192C22×D4—21no—
193C22×D4C2×#615711yescharacteristic
194C23×C4—21no—
195C23×C4—1061no—
196C23×C4C2×#615711yescharacteristic
197#117—2121no—
198#117—2121no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1062no—
204C2×C42—532no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—1061no—
208C2×C42—2122no—
209C4⋊C8—2122no—
210C22⋊C8—1064no—
211#97—2122no—
212#95—1062no—
213#95—1062no—
214C24—21no—
215C24C22⋊#615711yescharacteristic
216#41—44no—
217#41—44no—
218#41—44no—
219#41—44no—
220C2×Q8—21no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8#1171994012yes—
224C2×D4—24no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4#1171994012yes—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—532no—
242C22×C4—532no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22×#615711yescharacteristic
250C2×C8—2124no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C22⋊C4—2122no—
254C22⋊C4—2124no—
255C22⋊C4—2124no—
256C42—1062no—
257C42—1062no—
258C42—2121no—
259C42—2121no—
260C42—2122no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23#1313327712yes—
271C23C2×C22⋊#615711yescharacteristic
272Q8—42no—
273Q8—42no—
274D4—42no—
275D4—42no—
276D4—48no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2128no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4#1313327511yescharacteristic
302C2×C4#1313327511yescharacteristic
303C2×C4C2×#1171994011yescharacteristic
304C2×C4C2×#1171994011yescharacteristic
305C8—4244no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—44no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—82no—
316C22—81no—
317C22—81no—
318C22#1681159311yescharacteristic
319C22C2×#1313327511yescharacteristic
320C22C2×#1313327711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—2122no—
327C4—2122no—
328C4—2122no—
329C4—2122no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—82no—
334C2#2028996011yescharacteristic
335C2#2029098611yescharacteristic
336C2#2029100611yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC22×#41×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC22×#41#6157
hypocenterC53#11507
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #20289966C2semidirect
2. #20289645C2semidirect
3. #20289643C2semidirect
185. C53#11507semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896452
#202896432
#202772062
#1150753