Finite Group Databasebeta

← #27300075

Subgroups — #27300075

Order 13568.

680 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16810416C212yesmaximal
2#20274703C212yesmaximal, complemented
3#20272840C212yesmaximal, complemented
4C4×#127×C53C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6#16810483—22no—
7#16810416—24no—
8#16810390—22no—
9C2×#127×C53C2211yescharacteristic
10C2×C4×D4×C53—22no—
11C2×C4×D4×C53C411yescharacteristic, complemented
12C2×C4×D4×C53C2211yescharacteristic
13C53×#390—22no—
14C53×#390C411yescharacteristic, complemented
15C53×#390C2211yescharacteristic
16C53×#384C411yescharacteristic, complemented
17C43×C53C411yescharacteristic, complemented
18C2×C4⋊Dic53—42no—
19C2×C22⋊Dic53—24no—
20#13133132—24no—
21#13133132—24no—
22#13133124—42no—
23C22×D4×C53—22no—
24C22×D4×C53C2×C411yescharacteristic
25C22×D4×C53C2311yescharacteristic
26C23×C4×C53—21no—
27C23×C4×C53D412yes—
28#127×C53—22no—
29#127×C53D412yes—
30#127×C53Q812yes—
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C4×D4×C53—48no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53D412yes—
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C24×C53—21no—
49C24×C53C2×D412yes—
50C2×D4×C53—24no—
51C2×D4×C53—24no—
52C2×D4×C53—21no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—44no—
56C2×D4×C53—44no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53C4⋊C412yescomplemented
60C2×D4×C53C2×D411yescharacteristic
61C2×D4×C53C2×Q811yescharacteristic
62C2×D4×C53#4112yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—44no—
69C22×C4×C53—44no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C412yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×Dic53—44no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—44no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—44no—
83C22⋊C4×C53—42no—
84C42×C53—24no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—42no—
89C42×C53—41no—
90C42×C53C4⋊C412yescomplemented
91C22⋊Dic53—44no—
92C22⋊Dic53—44no—
93C4⋊Dic53—84no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#12112yes—
105C23×C53C22⋊Q812yes—
106D4×C53—41no—
107D4×C53—41no—
108D4×C53—42no—
109D4×C53—42no—
110D4×C53—48no—
111D4×C53—48no—
112D4×C53—42no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—44no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×C4×C53—82no—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53C2×C4⋊C411yescharacteristic
135C2×C4×C53#11711yescharacteristic
136C2×Dic53—84no—
137C2×Dic53—84no—
138C2×Dic53—84no—
139#12797—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—44no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53#38011yescharacteristic
155C22×C53C2×#9911yescharacteristic
156C22×C53#41011yescharacteristic
157C4×C53—24no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—42no—
162C4×C53—42no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53#35312yescomplemented
166Dic53—164no—
167C4×#127D5311yescharacteristic
168C2×#353—532no—
169#1799—532no—
170#1393—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—42no—
175C2×C53—82no—
176C2×C53#179911yescharacteristic
177C2×C53C2×#35311yescharacteristic
178C2×C53#212611yescharacteristic
179C2×#127C2×D5311yescharacteristic
180C2×C4×D4—22no—
181C2×C4×D4Dic5311yescharacteristic, complemented
182C2×C4×D4C2×D5311yescharacteristic
183C2×#99—534no—
184#390—22no—
185#390Dic5311yescharacteristic, complemented
186#390C2×D5311yescharacteristic
187#384Dic5311yescharacteristic, complemented
188#380—1062no—
189C43Dic5311yescharacteristic, complemented
190#353—1064no—
191#342—1062no—
192C53#1279711yescharacteristic, complemented, Sylow
193C22×D4—22no—
194C22×D4C2×Dic5311yescharacteristic
195C22×D4C22×D5311yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D5312yes—
198#127—22no—
199#127Dic10612yes—
200#127D21212yes—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—48no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—2122no—
208C2×C4⋊C4C2×Dic5311yescharacteristic
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1064no—
214C2×C22⋊C4C22⋊D5312yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×Dic5311yescharacteristic
218C2×C42C2×Dic5311yescharacteristic
219C2×C42C2×Dic5311yescharacteristic
220#99—1064no—
221#99—1064no—
222#95—2122no—
223C24—21no—
224C24C2×C22⋊D5312yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4C4⋊Dic5312yescomplemented
235C2×D4C2×Dic10611yescharacteristic
236C2×D4C2×D21211yescharacteristic
237C2×D4#1171994512yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—2124no—
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5312yes—
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22×Dic5311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—4244no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—2124no—
261C22⋊C4—2124no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42C4⋊Dic5312yescomplemented
269C23—22no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic5311yescharacteristic
279C23#1313319112yes—
280C23#1313319612yes—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—42no—
285D4—48no—
286D4—48no—
287D4—42no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—4244no—
308C2×C4—4244no—
309C2×C4—4244no—
310C2×C4#1313313211yescharacteristic
311C2×C4#1313313211yescharacteristic
312C2×C4C2×C4⋊Dic5311yescharacteristic
313C2×C4#1313317611yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1681048311yescharacteristic
329C22C2×#1313313211yescharacteristic
330C22#1681059711yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—8484no—
340C4#1681041612yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—42no—
345C2—82no—
346C2#2027470311yescharacteristic
347C2C2×#1681041611yescharacteristic
348C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#127×C53C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenterC4×#127D53
hypocenterC53#12797
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. #20274703C2semidirect
3. #20272840C2semidirect
11. C2×C4×D4×C53C4semidirect
14. C53×#390C4semidirect
16. C53×#384C4semidirect
17. C43×C53C4semidirect
59. C2×D4×C53C4⋊C4semidirect
90. C42×C53C4⋊C4semidirect
165. C4×C53#353semidirect
181. C2×C4×D4Dic53semidirect
185. #390Dic53semidirect
187. #384Dic53semidirect
189. C43Dic53semidirect
192. C53#12797semidirect
234. C2×D4C4⋊Dic53semidirect
268. C42C4⋊Dic53semidirect
340. C4#16810416semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
#202747032
#202728402
C4×#127×C532
#1279753