Finite Group Databasebeta

← #28299629

Subgroups — #28299629

Order 13568.

630 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290643C211yesmaximal, characteristic, complemented
2#20290500C211yesmaximal, characteristic, complemented
3#20289531C214yesmaximal, complemented
4#127×Dic53C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×D4×Dic53—21no—
12C2×D4×Dic53—21no—
13C2×D4×Dic53C412yes—
14#16810526—21no—
15#16810526—21no—
16#16810526C2212yes—
17#16810493C411yescharacteristic
18C42×Dic53C2211yescharacteristic
19C2×#127×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133272—42no—
24C23×Dic53—22no—
25D4×Dic53—44no—
26D4×Dic53—44no—
27D4×Dic53—42no—
28D4×Dic53—42no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C412yes—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41C22×D4×C53—21no—
42C22×D4×C53—21no—
43C22×D4×C53C2×C412yes—
44#127×C53—24no—
45C2×C42×C53C2×C411yescharacteristic
46C24×C53—22no—
47C2×D4×C53—24no—
48C2×D4×C53—24no—
49C2×D4×C53—22no—
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×C53M1614yes—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53—44no—
62C22×Dic53—44no—
63C22×Dic53—44no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×#88324—24no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C42×C53—22no—
75C22⋊Dic53—44no—
76C22⋊Dic53—42no—
77C22⋊Dic53—42no—
78C22⋊Dic53—44no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53D812yes—
88C4×Dic53SD1612yes—
89C23×C53—24no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—44no—
98D4×C53—44no—
99D4×C53—48no—
100D4×C53—48no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53#10212yes—
108C2×C4×C53#10412yes—
109C2×C4×C53C2×M1612yes—
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—88no—
122C2×Dic53#10012yes—
123C2×Dic53#10212yes—
124C2×Dic53C2×D811yescharacteristic
125C2×Dic53C2×SD1611yescharacteristic
126C2×Dic53AGL(1,ℤ/8)12yes—
127#88324—44no—
128#88324—82no—
129#7621—531nomaximal, Sylow
130C22×C53—44no—
131C22×C53—44no—
132C22×C53—44no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53#40711yescharacteristic
138C22×C53#41011yescharacteristic
139C22×C53C2×#10011yescharacteristic
140C22×C53C2×#10211yescharacteristic
141C22×C53#41811yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C22×C53#42111yescharacteristic
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—22no—
147Dic53—22no—
148Dic53—21no—
149Dic53—21no—
150Dic53—22no—
151Dic53—22no—
152Dic53—82no—
153C4×#127—531no—
154#1839—531no—
155#1767—531no—
156#1270—534no—
157C2×C53—84no—
158C2×C53#146611yescharacteristic
159C2×C53#148611yescharacteristic
160C2×C53#149011yescharacteristic
161C2×C53#150211yescharacteristic
162C2×C53#151411yescharacteristic
163C2×C53#153011yescharacteristic
164C2×C53#154511yescharacteristic
165C2×#127#615711yescharacteristic
166C2×C4×D4—532no—
167C2×C4×D4—1061no—
168C2×C4×D4—1061no—
169C2×C4⋊C8—1061no—
170C2×#97—1061no—
171#390—532no—
172#390—1061no—
173#390—1061no—
174#384—531no—
175C43—531no—
176#336—532no—
177#336—532no—
178#336—1061no—
179#336—1061no—
180C53#762111yescharacteristic, complemented, Sylow
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C2×#615712yes—
184C23×C4—1062no—
185C22×C8—1062no—
186C22×C8—1062no—
187#127—24no—
188C4×D4—2122no—
189C4×D4—2122no—
190C4×D4—2124no—
191C4×D4—2124no—
192C2×C4⋊C4—532no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42C2×#615711yescharacteristic
205C4⋊C8—1064no—
206#97—2122no—
207C24—22no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4#1171994014yes—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—1062no—
223C22×C4—2124no—
224C22×C4—2124no—
225C22×C4—2124no—
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—1064no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C4⋊C4—2122no—
238C22⋊C4—2122no—
239C22⋊C4—2122no—
240C22⋊C4—2124no—
241C22⋊C4—2124no—
242C42—22no—
243C42—532no—
244C42—532no—
245C42—1062no—
246C42—1062no—
247C42—1062no—
248C42—2121no—
249C42—2121no—
250C23—24no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23C2×C22⋊#615711yescharacteristic
258D4—44no—
259D4—48no—
260D4—44no—
261D4—48no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—532no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—4248no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4#1313327912yes—
286C2×C4#1313328112yes—
287C2×C4C2×#1171994012yes—
288C8—4242no—
289C8—2124no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22#1681159311yescharacteristic
298C22#1681160311yescharacteristic
299C22D53⋊#10011yescharacteristic
300C22C2×#1313327911yescharacteristic
301C22#1681161511yescharacteristic
302C22C2×#1313328111yescharacteristic
303C22#1681162111yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—1061no—
308C4—1061no—
309C4—1062no—
310C4—1062no—
311C4—1062no—
312C4—4242no—
313C2—84no—
314C2#2028997511yescharacteristic
315C2#2029004011yescharacteristic
316C2#2029005411yescharacteristic
317C2#2029009511yescharacteristic
318C2#2029013511yescharacteristic
319C2#2029018511yescharacteristic
320C2#2029022311yescharacteristic
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#7621
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. #20290643C2semidirect
2. #20290500C2semidirect
3. #20289531C2semidirect
180. C53#7621semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202905002
#202895312
#127×Dic532
#762153