Finite Group Databasebeta

← #28299833

Subgroups — #28299833

Order 13568.

572 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290501C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3#20289531C214yesmaximal, complemented
4#20283071C211yesmaximal, characteristic
5C2×C4⋊#88324—21no—
6C2×#13133272—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133204—21no—
13C2×#13133204—21no—
14#16810526—21no—
15#16810526—21no—
16#16810526C412yes—
17#16810526C2212yes—
18C42×Dic53C2211yescharacteristic
19C2×#127×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C4⋊#88324—24no—
23#13133272—42no—
24C4×#88324—24no—
25C22×Dic106—21no—
26#13133217—42no—
27#13133204—44no—
28#13133204—42no—
29#13133204—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C412yes—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42C22×D4×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53C2×C412yes—
45#127×C53—42no—
46C2×C42×C53C2×C411yescharacteristic
47C24×C53—22no—
48C2×D4×C53—24no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×Dic106—42no—
68C2×#88324—24no—
69C2×#88324—24no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C42×C53—22no—
75C22⋊Dic53—44no—
76C22⋊Dic53—44no—
77C22⋊Dic53—42no—
78C22⋊Dic53—42no—
79C22⋊Dic53—44no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C4×Dic53D812yes—
88C4×Dic53SD1612yes—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—84no—
98D4×C53—84no—
99D4×C53—84no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53#10012yes—
105C2×C4×C53#10212yes—
106C2×C4×C53#10412yes—
107C2×C4×C53#10412yes—
108C2×C4×C53#13112yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
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×Dic53#10212yes—
122C2×Dic53C2×D811yescharacteristic
123C2×Dic53C2×SD1611yescharacteristic
124C2×Dic53AGL(1,ℤ/8)12yes—
125Dic106—84no—
126#88324—44no—
127#88324—44no—
128#7679—531nomaximal, Sylow
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—82no—
136C22×C53#40811yescharacteristic
137C22×C53#41011yescharacteristic
138C22×C53C2×#10011yescharacteristic
139C22×C53C2×#10211yescharacteristic
140C22×C53#41811yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C4×C53—22no—
144C4×C53—22no—
145C4×C53#32914yes—
146Dic53—22no—
147Dic53—21no—
148Dic53—21no—
149Dic53—22no—
150Dic53—41no—
151Dic53—82no—
152#2858—531no—
153#1767—531no—
154#1744—531no—
155#1270—534no—
156C2×C53—162no—
157C2×C53#146911yescharacteristic
158C2×C53#148411yescharacteristic
159C2×C53#149211yescharacteristic
160C2×C53#150211yescharacteristic
161C2×C53#150511yescharacteristic
162C2×C53C2×#32911yescharacteristic
163C2×C53#154511yescharacteristic
164C2×C4⋊Q8—531no—
165C2×#127#615711yescharacteristic
166C2×#124—1061no—
167C2×#124—1061no—
168C2×C4⋊C8—1061no—
169C2×#97—1061no—
170C2×C4×C8—1061no—
171#390—532no—
172#390—532no—
173#390—1061no—
174#390—1061no—
175C43—531no—
176#336—532no—
177#336—532no—
178#336—1061no—
179C53#767911yescharacteristic, complemented, Sylow
180C22×Q8—1061no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C2×#615712yes—
184C22×C8—1062no—
185C22×C8—1062no—
186C4⋊Q8—2122no—
187#127—42no—
188#124—2122no—
189#124—2122no—
190#124—2124no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—1061no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42C2×#615711yescharacteristic
204C4⋊C8—1064no—
205#97—2122no—
206C4×C8—1064no—
207C24—22no—
208C2×Q8—2122no—
209C2×Q8—2122no—
210C2×Q8—2122no—
211C2×D4—24no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—1062no—
225C22×C4C22⋊#615711yescharacteristic
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22×#615711yescharacteristic
228C2×C8—1064no—
229C2×C8—1064no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C22⋊C4—2124no—
237C22⋊C4—2122no—
238C22⋊C4—2122no—
239C22⋊C4—2124no—
240C22⋊C4—2124no—
241C42—22no—
242C42—532no—
243C42—532no—
244C42—1062no—
245C42—1062no—
246C42—1062no—
247C42—1062no—
248C42—2121no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23C2×C22⋊#615711yescharacteristic
257Q8—4244no—
258D4—84no—
259D4—84no—
260D4—84no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—532no—
266C2×C4—531no—
267C2×C4—531no—
268C2×C4—532no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313327712yes—
282C2×C4#1313327912yes—
283C2×C4#1313328112yes—
284C2×C4#1313328112yes—
285C2×C4#1313330412yes—
286C8—2124no—
287C8—2124no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22#1681159711yescharacteristic
296C22#1681160311yescharacteristic
297C22C2×#1313327711yescharacteristic
298C22C2×#1313327911yescharacteristic
299C22#1681161511yescharacteristic
300C22C2×#1313328111yescharacteristic
301C22C2×#1313328111yescharacteristic
302C4—22no—
303C4—22no—
304C4—1062no—
305C4—1061no—
306C4—1061no—
307C4—1062no—
308C4—4242no—
309C4—2121no—
310C4#1681147814yes—
311C2—162no—
312C2#2028998511yescharacteristic
313C2#2029003511yescharacteristic
314C2#2029006111yescharacteristic
315C2#2029009511yescharacteristic
316C2#2029010411yescharacteristic
317C2C2×#1681147811yescharacteristic
318C2#2029022211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#127#6157
hypocenterC53#7679
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290501C2semidirect
2. #20290430C2semidirect
3. #20289531C2semidirect
179. C53#7679semidirect

Maximal subgroups

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

subgroupindex
#202905012
#202904302
#202895312
#202830712
#767953