Finite Group Databasebeta

← #28299781

Subgroups — #28299781

Order 13568.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290570C211yesmaximal, characteristic
2#20290376C211yesmaximal, characteristic
3#20289534C214yesmaximal, complemented
4#20283062C211yesmaximal, characteristic
5C22×D53⋊C8—21no—
6C2×C4⋊#88324—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133193—21no—
12C2×C42×D53C2211yescharacteristic
13#16810530—21no—
14#16810519—21no—
15#16810519C2212yes—
16#16810493C411yescharacteristic
17#16810480—21no—
18#16810480C412yes—
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×D53⋊C8—24no—
21C2×D53⋊C8—42no—
22C2×D53⋊C8—42no—
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26#13133274—42no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30#13133193—42no—
31#13133193—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C412yes—
35C2×#11719919—22no—
36C2×#11719921—22no—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40C42×D53—42no—
41#13133124—22no—
42#13133124—22no—
43C22×Q8×C53—21no—
44C4⋊Q8×C53—42no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C2×Q8×C53—42no—
50C2×Q8×C53—42no—
51C2×Q8×C53—42no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—41no—
65C2×C4×D53—41no—
66C2×C4×D53—41no—
67C2×C4×D53D812yes—
68C2×C4×D53SD1612yes—
69C2×C4×D53SD1612yes—
70C2×C4×D53Q1612yes—
71C2×#88324—24no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77D53⋊C8—48no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C42×C53—22no—
81#11719921—44no—
82C4⋊Dic53—24no—
83C4⋊Dic53—42no—
84C4⋊Dic53—42no—
85#11719919—42no—
86#11719919—42no—
87C4×Dic53—22no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53#10112yes—
96C22×D53#10212yes—
97C22×D53#10312yes—
98Q8×C53—84no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53#9912yes—
107C2×C4×C53#10212yes—
108C2×C4×C53#10312yes—
109C2×C4×C53#10412yes—
110C2×C4×C53#13112yes—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53C2×D811yescharacteristic
122C2×Dic53C2×SD1611yescharacteristic
123C2×Dic53C2×SD1611yescharacteristic
124C2×Dic53C2×Q1611yescharacteristic
125C4×D53—24no—
126C4×D53—24no—
127C4×D53—42no—
128C4×D53—42no—
129C4×D53—44no—
130#88324—44no—
131#88324—82no—
132#7665—531nomaximal, Sylow
133C22×C53#40811yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10111yescharacteristic
136C22×C53C2×#10211yescharacteristic
137C22×C53C2×#10311yescharacteristic
138C22×C53C2×#10411yescharacteristic
139C22×C53#42111yescharacteristic
140C2×D53—22no—
141C2×D53—22no—
142C2×D53—22no—
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—22no—
146C2×D53#33214yes—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—82no—
150C4×C53#32814yes—
151Dic53—21no—
152Dic53—21no—
153Dic53—21no—
154Dic53—21no—
155Dic53—41no—
156Dic53—82no—
157#2857—531no—
158#1797—531no—
159C2×#336—531no—
160#1271—534no—
161C2×C53#147211yescharacteristic
162C2×C53#148111yescharacteristic
163C2×C53#149211yescharacteristic
164C2×C53C2×#32811yescharacteristic
165C2×C53#151911yescharacteristic
166C2×C53#153111yescharacteristic
167C2×C53C2×#33211yescharacteristic
168D53—24no—
169C23×C8—1061no—
170C2×C4⋊Q8#615711yescharacteristic
171C2×C22⋊Q8—1061no—
172C22×C42—531no—
173C2×C4⋊C8—1061no—
174C2×C22⋊C8—1061no—
175#391—1061no—
176#389—532no—
177#389—1061no—
178#384—531no—
179#380—532no—
180#380—1061no—
181#336—532no—
182#336—532no—
183#336—1061no—
184C53#766511yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C23×C4—531no—
187C23×C4—531no—
188C23×C4—531no—
189C22×C8—1062no—
190C22×C8—1064no—
191C22×C8—1062no—
192C22×C8—2122no—
193C22×C8—2122no—
194C4⋊Q8—42no—
195C22⋊Q8—2122no—
196C22⋊Q8—2122no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—532no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4C2×#615712yes—
203C2×C22⋊C4—1062no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—2122no—
208C2×C42C2×#615711yescharacteristic
209C4⋊C8—2122no—
210C22⋊C8—2122no—
211#95—1062no—
212#95—1062no—
213C24—531no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C22×C4—22no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—532no—
223C22×C4—532no—
224C22×C4—531no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—2121no—
231C22×C4—2121no—
232C22×C4—2121no—
233C22×C4C22⋊#615711yescharacteristic
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22×#615711yescharacteristic
236C2×C8—1064no—
237C2×C8—2122no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2128no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—1064no—
246C4⋊C4—2122no—
247C4⋊C4—2122no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C22⋊C4—2124no—
251C42—22no—
252C42—1062no—
253C42—2121no—
254C42—2121no—
255C23—532no—
256C23—532no—
257C23—532no—
258C23—1061no—
259C23—1061no—
260C23—1061no—
261C23—1061no—
262C23C2×C22⋊#615711yescharacteristic
263Q8—84no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—531no—
274C2×C4—531no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1064no—
280C2×C4—1064no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4—2124no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313327512yes—
291C2×C4#1313327912yes—
292C2×C4#1313328012yes—
293C2×C4#1313328212yes—
294C2×C4#1313330512yes—
295C8—4242no—
296C8—2124no—
297C22—534no—
298C22—1062no—
299C22—1062no—
300C22—1062no—
301C22—1062no—
302C22—1062no—
303C22—1062no—
304C22#1681159811yescharacteristic
305C22C2×#1313327511yescharacteristic
306C22D53⋊#10111yescharacteristic
307C22C2×#1313327911yescharacteristic
308C22C2×#1313328011yescharacteristic
309C22C2×#1313328211yescharacteristic
310C22#1681162111yescharacteristic
311C4—22no—
312C4—22no—
313C4—82no—
314C4—1061no—
315C4—1061no—
316C4—1061no—
317C4—1061no—
318C4—4242no—
319C4—2121no—
320C4#1681147714yes—
321C2—1064no—
322C2#2028999511yescharacteristic
323C2#2029002511yescharacteristic
324C2#2029006211yescharacteristic
325C2C2×#1681147711yescharacteristic
326C2#2029015111yescharacteristic
327C2#2029018911yescharacteristic
328C2D53⋊#33211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#7665
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
184. C53#7665semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202903762
#202895342
#202830622
#766553