Finite Group Databasebeta

← #27300938

Subgroups — #27300938

Order 13568.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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#20283786C211yesmaximal, characteristic
2#20275099C211yesmaximal, characteristic
3#20274937C212yesmaximal, complemented
4#20272845C212yesmaximal, complemented
5C53×#2132C211yesmaximal, characteristic
6#16811390C2211yescharacteristic
7#16810695—22no—
8#16810695—22no—
9#16810680—21no—
10#16810672—22no—
11#16810640—21no—
12#16810597C2211yescharacteristic
13#16810590—21no—
14#16810590C2212yes—
15#16810387—22no—
16#16810383—21no—
17C53×#532C2211yescharacteristic
18C53×#413C2212yes—
19C53×#356C412yescomplemented
20C53×#355C412yescomplemented
21#13133224—42no—
22C2×#11719917—21no—
23C2×#11719917—21no—
24C2×#11719917—22no—
25C2×#11719917D412yes—
26C22×#88322—21no—
27C22×#88322D412yes—
28#13133176—21no—
29#13133153—44no—
30C4⋊#88322—21no—
31C4⋊#88322—21no—
32#13133144—42no—
33#13133132—22no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36C22⋊#88322—22no—
37#13133128—22no—
38C2×#41×C53D411yescharacteristic
39C2×#41×C53C2311yescharacteristic
40C2×M16×C53—22no—
41C4⋊Q8×C53C2×C411yescharacteristic
42#127×C53Q811yescharacteristic
43#124×C53C2×C411yescharacteristic
44#124×C53Q811yescharacteristic
45#121×C53—22no—
46C4×D4×C53—22no—
47C2×C42×C53D411yescharacteristic
48#101×C53C2×C414yes—
49#100×C53—22no—
50#41×C53—41no—
51#41×C53—41no—
52#41×C53—41no—
53#41×C53—41no—
54C2×Q8×C53C4⋊C411yescharacteristic
55C2×Q8×C53C22×C411yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53C4⋊C411yescharacteristic
59C2×D4×C53C2×Q811yescharacteristic
60C2×D4×C53#4112yes—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65M16×C53—24no—
66M16×C53—42no—
67C2×C8×C53—42no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—22no—
70C22⋊C4×C53—22no—
71C42×C53—21no—
72C42×C53C4⋊C411yescharacteristic
73C42×C53C4⋊C411yescharacteristic
74C22⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4×Dic53—41no—
77#11719917—22no—
78#11719917—24no—
79#11719917—41no—
80#11719917—42no—
81#11719917—81no—
82C2×#88322—22no—
83C2×#88322—22no—
84C2×#88322—41no—
85C2×#88322—42no—
86C2×#88322—42no—
87C2×#88322—81no—
88C23×C53—21no—
89C23×C53C22⋊Q812yes—
90C23×C53#12311yescharacteristic
91Q8×C53—41no—
92Q8×C53—41no—
93D4×C53—41no—
94D4×C53—42no—
95D4×C53—41no—
96D4×C53—81no—
97D4×C53—82no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—81no—
108C2×C4×C53C2×C4⋊C411yescharacteristic
109C2×C4×C53C4×D412yes—
110C2×C4×C53#12011yescharacteristic
111C2×Dic53—81no—
112C8×C53—44no—
113#88322—42no—
114#88322—42no—
115#88322—42no—
116#12974—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53#38511yescharacteristic
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53B(4,𝔽2)11yescharacteristic
130C4×C53#45811yescharacteristic
131Dic53—161no—
132#2980—531no—
133#2132D5311yescharacteristic
134#1884—531no—
135#1858—532no—
136#1395—532no—
137C2×C53—21no—
138C2×C53—41no—
139C2×C53—42no—
140C2×C53—81no—
141C2×C53#186711yescharacteristic
142#568—531no—
143#532C2×D5311yescharacteristic
144#435—1062no—
145#435—1062no—
146#432—1061no—
147#430—1062no—
148#421—1061no—
149#413C2×D5312yes—
150#410—531no—
151#408—532no—
152#408—1061no—
153#356Dic5312yescomplemented
154#355Dic5312yescomplemented
155#341—1062no—
156#339—1061no—
157C53#1297411yescharacteristic, complemented, Sylow
158C2×#41D21211yescharacteristic
159C2×#41C22×D5311yescharacteristic
160#131—2122no—
161C2×M16—22no—
162C2×M16—532no—
163C2×M16—1061no—
164C2×M16—1061no—
165C2×M16—1062no—
166C22×C8—532no—
167C22×C8—1061no—
168C4⋊Q8C2×Dic5311yescharacteristic
169#127Dic10611yescharacteristic
170#124Dic10611yescharacteristic
171#124C2×Dic5311yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1061no—
175C2×C42D21211yescharacteristic
176#108—2124no—
177C4⋊C8—1061no—
178C4⋊C8—1061no—
179#104—2122no—
180#101C2×Dic5314yes—
181#100—22no—
182#99—1062no—
183C22⋊C8—1062no—
184C22⋊C8—1062no—
185C22⋊C8—1062no—
186#97—1062no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊Dic5311yescharacteristic
192C2×Q8C22×Dic5311yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊Dic5311yescharacteristic
196C2×D4C2×Dic10611yescharacteristic
197C2×D4#1171994712yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×D21211yescharacteristic
201C22×C4D4×D5312yes—
202M16—24no—
203M16—42no—
204M16—1062no—
205M16—1064no—
206M16—4241no—
207M16—2121no—
208M16—2122no—
209C2×C8—42no—
210C2×C8—1062no—
211C2×C8—1062no—
212C2×C8—4241no—
213C2×C8—2121no—
214C2×C8—2122no—
215C2×C8—2122no—
216C4⋊C4—22no—
217C4⋊C4—2121no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—2121no—
221C42—21no—
222C42—2121no—
223C42C4⋊Dic5311yescharacteristic
224C42C4⋊Dic5311yescharacteristic
225C23—21no—
226C23#1313319412yes—
227C23#1313320111yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—81no—
234D4—82no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—4241no—
246C2×C4C2×C4⋊Dic5311yescharacteristic
247C2×C4D4×Dic5312yes—
248C2×C4#1313318711yescharacteristic
249C8—44no—
250C8—2122no—
251C8—2122no—
252C8—2122no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1681049811yescharacteristic
260C4—22no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—8481no—
266C4#1681081111yescharacteristic
267C4#1681081411yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2027499811yescharacteristic
subgroupG/H
centerC4#16810811
commutatorC2×Q8×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#2132C2
socleC2×C53#1867
prime radicalC53×#532C22
cosocleC2×#41×C53C23
hypercenter#2132D53
hypocenterC53#12974
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #20274937C2semidirect
4. #20272845C2semidirect
19. C53×#356C4semidirect
20. C53×#355C4semidirect
153. #356Dic53semidirect
154. #355Dic53semidirect
157. C53#12974semidirect

Maximal subgroups

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

subgroupindex
#202837862
#202750992
#202749372
#202728452
C53×#21322
#1297453