Finite Group Databasebeta

← #356×Dic53 #27299276

Subgroups — #356×Dic53 #27299276

Order 13568.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4⋊Q8×Dic53C211yesmaximal, characteristic
2C2×#16810422C211yesmaximal, characteristic
3#101×Dic53C212yesmaximal
4#20272841C212yesmaximal
5C2×C53×#356C211yesmaximal, characteristic
6C2×Q8×Dic53C411yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133136C2212yes—
9M16×Dic53—22no—
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810522—22no—
13#16810493C411yescharacteristic
14C4⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic
16#16810422C414yescomplemented
17#16810391—22no—
18C2×C4⋊Q8×C53C2211yescharacteristic
19C2×#101×C53C2212yes—
20C53×#356C414yescomplemented
21C2×#11719917—22no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24Q8×Dic53—41no—
25Q8×Dic53—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53D412yes—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37#13133136—22no—
38#13133136C2×C414yes—
39#13133127—42no—
40C8×Dic53—42no—
41#13133124—22no—
42#13133124—22no—
43C22×Q8×C53C2×C411yescharacteristic
44C22×Q8×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C4⋊Q8×C53—22no—
47C4⋊Q8×C53C2×C412yes—
48C4⋊Q8×C53C2×C412yes—
49C2×C4⋊C4×C53—22no—
50C2×C42×C53C2×C411yescharacteristic
51#101×C53—22no—
52#101×C53C2×C414yes—
53C2×Q8×C53—21no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C2×Q8×C53C4212yes—
59C2×Q8×C53C22×C411yescharacteristic
60C2×Q8×C53C22×C411yescharacteristic
61C2×Q8×C53#4112yes—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C412yes—
70C22×Dic53C22⋊C411yescharacteristic
71M16×C53—24no—
72M16×C53—42no—
73C2×C8×C53—42no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—24no—
76C42×C53—21no—
77C42×C53C4212yes—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4⋊Dic53—44no—
82C4⋊Dic53—41no—
83C4×Dic53—24no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—42no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92#11719917—24no—
93#11719917—42no—
94C2×#88322—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—41no—
97Q8×C53—41no—
98Q8×C53—42no—
99Q8×C53—42no—
100Q8×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53C2×C4211yescharacteristic
112C2×C4×C53#11711yescharacteristic
113C2×C4×C53C4×D412yes—
114C2×C4×C53C4×D412yes—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53#9911yescharacteristic
126C2×Dic53#9911yescharacteristic
127C8×C53—44no—
128#88322—44no—
129C4×#356—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53C4×C22⋊C411yescharacteristic
133C22×C53C2×#9911yescharacteristic
134C22×C53#41011yescharacteristic
135C4×C53—24no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—42no—
141Dic53—21no—
142Dic53—42no—
143Dic53—42no—
144Dic53—41no—
145Dic53—81no—
146Dic53#35612yesdirect product factor
147C4×C4⋊Q8—531no—
148C2×#356—531no—
149C2×#356D5311yescharacteristic
150C4×#101—532no—
151#1394—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53C4×#9911yescharacteristic
155C2×C53C2×#35611yescharacteristic
156C2×C53#213211yescharacteristic
157C2×C4⋊Q8C2×D5311yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×Q8—531no—
160C2×#101—532no—
161C2×#101C2×D5312yes—
162C4×M16—1062no—
163#391—531no—
164#391—531no—
165#389—1062no—
166#384—531no—
167C4×C4⋊C4—1062no—
168C43—531no—
169#356—534no—
170#356Dic5314yesdirect product factor
171#343—1062no—
172C53C4×#35611yescharacteristic, complemented, Sylow
173C22×Q8C4×D5311yescharacteristic
174C22×Q8C22×D5311yescharacteristic
175C2×M16—22no—
176C2×M16—1062no—
177C4⋊Q8—22no—
178C4⋊Q8C4×D5312yes—
179C4⋊Q8C2×Dic5312yes—
180C4×Q8—2121no—
181C4×Q8—2121no—
182C4×Q8—2121no—
183C4×Q8—2121no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—532no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1062no—
190C2×C42—531no—
191C2×C42—532no—
192C2×C42—531no—
193C2×C42—1061no—
194C2×C42—1061no—
195C2×C42—1061no—
196C2×C42C4×D5311yescharacteristic
197#101—22no—
198#101—534no—
199#101—1062no—
200#101C2×Dic5314yes—
201#97—2122no—
202C4×C8—2122no—
203#95—1062no—
204#95—1062no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8C4×Dic5312yes—
211C2×Q8C2×C4×D5311yescharacteristic
212C2×Q8#1171994712yes—
213C2×Q8C22×Dic5311yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—531no—
217C22×C4—532no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4C2×C4×D5311yescharacteristic
222C22×C4D4×D5312yes—
223M16—24no—
224M16—42no—
225M16—1064no—
226M16—2122no—
227C2×C8—42no—
228C2×C8—2122no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—2121no—
232C4⋊C4—2122no—
233C4⋊C4—2121no—
234C4⋊C4—2124no—
235C4⋊C4—2122no—
236C42—21no—
237C42—1062no—
238C42—1062no—
239C42—1061no—
240C42—1061no—
241C42—1064no—
242C42—2121no—
243C42—2122no—
244C42—2121no—
245C42—2121no—
246C42C4×Dic5312yes—
247C23C22⋊C4×D5311yescharacteristic
248Q8—41no—
249Q8—42no—
250Q8—41no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—1062no—
269C2×C4—2122no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2122no—
275C2×C4C2×C4×Dic5311yescharacteristic
276C2×C4#1313317311yescharacteristic
277C2×C4#1313317712yes—
278C2×C4D4×Dic5312yes—
279C8—44no—
280C8—2124no—
281C22—21no—
282C22—21no—
283C22C22⋊C4×Dic5311yescharacteristic
284C22#99×D5311yescharacteristic
285C22#1681059511yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—532no—
293C4—1061no—
294C4—4241no—
295C4—2122no—
296C4—2121no—
297C4—2122no—
298C2—21no—
299C2—21no—
300C2#99×Dic5311yescharacteristic
301C2#356×D5311yescharacteristic
302C2#2027639911yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC2×C53×#356C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC2×#356D53
hypocenterC53C4×#356
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
16. #16810422C4semidirect
20. C53×#356C4semidirect
146. Dic53#356direct (N × H)
170. #356Dic53direct (N × H)
172. C53C4×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic532
C2×#168104222
#101×Dic532
#202728412
C2×C53×#3562
C4×#35653