Finite Group Databasebeta

← #27300923

Subgroups — #27300923

Order 13568.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277896C211yesmaximal, characteristic
2#20275109C211yesmaximal, characteristic
3#20274932C212yesmaximal
4#20272841C212yesmaximal
5C2×C53×#356C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810665—22no—
8#16810632—21no—
9C2×#13133136C2211yescharacteristic
10#16810530—21no—
11#16810530C2212yes—
12#16810509—22no—
13#16810509—22no—
14#16810491—21no—
15#16810384—22no—
16#16810383—21no—
17C2×C4⋊Q8×C53C2211yescharacteristic
18C2×#101×C53C2212yes—
19C53×#356C414yescomplemented
20C2×#11719917—21no—
21Q8×Dic53—42no—
22#13133176—21no—
23#13133176—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53D412yes—
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28#13133151—42no—
29#13133149—42no—
30#13133141—42no—
31#13133136—22no—
32#13133124—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124—21no—
36#13133124—22no—
37#13133124—22no—
38C22×Q8×C53D411yescharacteristic
39C22×Q8×C53C2311yescharacteristic
40C2×M16×C53—22no—
41C4⋊Q8×C53—22no—
42C4⋊Q8×C53C2×C412yes—
43C4⋊Q8×C53Q812yes—
44C2×C4⋊C4×C53—22no—
45C2×C42×C53D411yescharacteristic
46#101×C53—22no—
47#101×C53C2×C414yes—
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C4⋊C412yes—
54C2×Q8×C53C22×C411yescharacteristic
55C2×Q8×C53C2×Q811yescharacteristic
56C2×Q8×C53#4112yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C22×Dic53—22no—
62C22×Dic53—22no—
63M16×C53—24no—
64M16×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—24no—
67C4⋊C4×C53—24no—
68C42×C53—21no—
69C42×C53C4⋊C412yes—
70C22⋊Dic53—41no—
71C4⋊Dic53—22no—
72C4⋊Dic53—22no—
73C4⋊Dic53—22no—
74C4⋊Dic53—42no—
75C4⋊Dic53—81no—
76C4×Dic53—41no—
77C4×Dic53—42no—
78C4×Dic53—42no—
79C4×Dic53—81no—
80#11719917—41no—
81#11719917—41no—
82C2×#88322—41no—
83C23×C53#12011yescharacteristic
84Q8×C53—41no—
85Q8×C53—41no—
86Q8×C53—42no—
87Q8×C53—42no—
88Q8×C53—81no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53C2×C4⋊C411yescharacteristic
99C2×C4×C53C4×D412yes—
100C2×C4×C53C22⋊Q812yes—
101C2×C4×C53#12311yescharacteristic
102C2×Dic53—44no—
103C2×Dic53—44no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C8×C53—44no—
108#88322—81no—
109#12971—531nomaximal, Sylow
110C22×C53—22no—
111C22×C53#38511yescharacteristic
112C22×C53B(4,𝔽2)11yescharacteristic
113C22×C53#45811yescharacteristic
114C4×C53—21no—
115C4×C53—22no—
116C4×C53—21no—
117C4×C53—22no—
118C4×C53—42no—
119C4×C53—42no—
120Dic53—82no—
121Dic53—82no—
122Dic53—82no—
123#2381—531no—
124C2×#356D5311yescharacteristic
125#1886—531no—
126#1857—532no—
127#1394—532no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#186711yescharacteristic
131C2×C53#220312yes—
132C2×C4⋊Q8C2×D5311yescharacteristic
133#519—531no—
134#428—1062no—
135#419—1061no—
136C2×#101—531no—
137C2×#101C2×D5312yes—
138#391—532no—
139#391—1061no—
140#387—1062no—
141#387—1062no—
142#383—1061no—
143#356Dic5314yescomplemented
144#339—1062no—
145#339—1061no—
146C53#1297111yescharacteristic, complemented, Sylow
147C22×Q8D21211yescharacteristic
148C22×Q8C22×D5311yescharacteristic
149C2×M16—22no—
150C2×M16—1061no—
151C4⋊Q8—22no—
152C4⋊Q8Dic10612yes—
153C4⋊Q8C2×Dic5312yes—
154C4×Q8—2122no—
155#117—1061no—
156#117—1062no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1061no—
160C2×C42—532no—
161C2×C42—1061no—
162C2×C42D21211yescharacteristic
163#107—2122no—
164#106—2122no—
165#103—2122no—
166#101—22no—
167#101—1062no—
168#101C2×Dic5314yes—
169#95—1061no—
170#95—1062no—
171#95—1061no—
172#95—1062no—
173#95—1062no—
174#95—1062no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic5312yes—
181C2×Q8C2×Dic10611yescharacteristic
182C2×Q8#1171994712yes—
183C2×Q8C22×Dic5311yescharacteristic
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—1062no—
187C22×C4—1062no—
188C22×C4C2×D21211yescharacteristic
189C22×C4D4×D5312yes—
190M16—24no—
191M16—42no—
192M16—2121no—
193M16—2121no—
194C2×C8—42no—
195C2×C8—2121no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—1062no—
199C4⋊C4—1062no—
200C4⋊C4—1062no—
201C4⋊C4—4241no—
202C4⋊C4—2122no—
203C22⋊C4—2121no—
204C42—21no—
205C42—4241no—
206C42—2122no—
207C42—2122no—
208C42—2121no—
209C42C4⋊Dic5312yes—
210C23#1313318711yescharacteristic
211Q8—41no—
212Q8—42no—
213Q8—41no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2124no—
226C2×C4—2124no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4C2×C4⋊Dic5311yescharacteristic
231C2×C4D4×Dic5312yes—
232C2×C4#1313319412yes—
233C2×C4#1313320111yescharacteristic
234C8—44no—
235C8—4241no—
236C22—22no—
237C22#1681049811yescharacteristic
238C22#1681081111yescharacteristic
239C22#1681081411yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—4242no—
247C4—4242no—
248C4—4242no—
249C2—21no—
250C2—21no—
251C2#2027499811yescharacteristic
252C2#2027673712yes—
subgroupG/H
centerC22#16810811
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#356C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
cosocleC22×Q8×C53C23
hypercenterC2×#356D53
hypocenterC53#12971
LCSG ▷ C2×Q8×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
19. C53×#356C4semidirect
143. #356Dic53semidirect
146. C53#12971semidirect

Maximal subgroups

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

subgroupindex
#202778962
#202751092
#202749322
#202728412
C2×C53×#3562
#1297153