Finite Group Databasebeta

← #27297612

Subgroups — #27297612

Order 13568.

322 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 71 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#20287721C211yesmaximal, characteristic, complemented
2#20276970C211yesmaximal, characteristic
3#20276752C211yesmaximal, characteristic
4#20272763C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272678C211yesmaximal, characteristic, complemented
7C53×#1668C211yesmaximal, characteristic
8#16811352—21no—
9C2×#13133161—21no—
10C2×#13133161—22no—
11#16811003—21no—
12#16811003C2211yescharacteristic
13#16810834—22no—
14#16810834—21no—
15C4×Dic212—21no—
16C4×Dic212C2211yescharacteristic
17C4⋊#11719908C2211yescharacteristic
18#16810424—21no—
19C4×#11719908C2211yescharacteristic
20C53×#493C2211yescharacteristic
21C53×#460—21no—
22C8×Q8×C53C2211yescharacteristic
23C53×#335C2211yescharacteristic
24C53×#331—21no—
25C2×Dic212—21no—
26C2×Dic212—22no—
27C2×Dic212D411yescharacteristic
28C2×Dic212D412yes—
29#13133217—22no—
30#13133217—21no—
31#13133208—42no—
32#13133208—41no—
33Q8×Dic53—41no—
34C4×Dic106—22no—
35C4×Dic106—21no—
36#13133161—44no—
37#13133161—42no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133151D411yescharacteristic
41#13133142—22no—
42#13133142—21no—
43C2×SD16×C53—41no—
44C2×D8×C53—21no—
45C2×D8×C53—21no—
46#127×C53—21no—
47C4×Q8×C53D411yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53—21no—
50C4⋊C8×C53D411yescharacteristic
51#102×C53—41no—
52C4×C8×C53—21no—
53C4×C8×C53C2311yescharacteristic
54C2×Q8×C53D811yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×Dic106—21no—
59C2×Dic106—22no—
60SD16×C53—82no—
61D8×C53—42no—
62D8×C53—41no—
63D8×C53—41no—
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53C2×D411yescharacteristic
67C2×C8×C53C2×D411yescharacteristic
68C4⋊C4×C53—21no—
69C4⋊C4×C53D811yescharacteristic
70C42×C53—21no—
71C42×C53C2×D411yescharacteristic
72C4⋊Dic53—21no—
73C4⋊Dic53—42no—
74C4⋊Dic53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—42no—
78#11719919—41no—
79C4×Dic53—42no—
80C4×Dic53—41no—
81Dic212—21no—
82Dic212—21no—
83Dic212—42no—
84Dic212—41no—
85Dic212—41no—
86Dic212—82no—
87#11719908—22no—
88#11719908—41no—
89C23×C53—81no—
90Q8×C53—21no—
91Q8×C53SD3212yescomplemented
92D4×C53—81no—
93D4×C53—81no—
94D4×C53—81no—
95D4×C53—81no—
96D4×C53—161no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53C2×D811yescharacteristic
101C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
102C2×Dic53—42no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105Dic106—41no—
106Dic106—42no—
107Dic106—42no—
108Dic106—41no—
109C8×C53—21no—
110C8×C53—41no—
111C8×C53—81no—
112C8×C53#13511yescharacteristic
113C8×C53AGL(1,ℤ/8)11yescharacteristic
114#12330—531nomaximal, Sylow
115C22×C53—161no—
116C22×C53—161no—
117C22×C53—161no—
118C22×C53#44711yescharacteristic
119C4×C53—22no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53#44911yescharacteristic
123C4×C53#45311yescharacteristic
124C4×C53C2×SD3211yescharacteristic
125C4×C53#51011yescharacteristic
126Dic53—81no—
127Dic53—82no—
128Dic53—81no—
129#3395—531no—
130#2248—531no—
131#2207—531no—
132#1668D5311yescharacteristic
133#1362—531no—
134Q8⋊C16—531no—
135#1332—531no—
136C2×C53—321no—
137C2×C53#162111yescharacteristic
138C2×C53#218511yescharacteristic
139C2×C53#219311yescharacteristic
140#557—1061no—
141C2×SD32—1061no—
142C2×SD32—1062no—
143#494—531no—
144#494—1061no—
145#493C2×D5311yescharacteristic
146#462—1062no—
147#462—1061no—
148#460—21no—
149C8×Q8C2×D5311yescharacteristic
150C4×Q16—531no—
151C4×Q16—1061no—
152C4⋊C16—531no—
153#357—1061no—
154C4×C16—531no—
155#335C2×D5311yescharacteristic
156#331—21no—
157C53#1233011yescharacteristic, complemented, Sylow
158C2×Q16—531no—
159C2×Q16—532no—
160C2×Q16—1062no—
161C2×Q16—1061no—
162C2×SD16—41no—
163C2×D8—21no—
164C2×D8—21no—
165C4⋊Q8—1062no—
166C4⋊Q8—1061no—
167#127—21no—
168#125—2122no—
169#125—2121no—
170C4×Q8—1062no—
171C4×Q8—1061no—
172C4×Q8—2121no—
173C4×Q8C22⋊D5311yescharacteristic
174SD32—2124no—
175SD32—2122no—
176C2×C16—1061no—
177C2×C16—1061no—
178#107—531no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8C22⋊D5311yescharacteristic
182#103—1062no—
183#103—1061no—
184#102—41no—
185C4×C8—21no—
186C4×C8C22×D5311yescharacteristic
187C2×Q8—1061no—
188C2×Q8—1062no—
189C2×Q8#1171992211yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193Q16—1061no—
194Q16—1061no—
195Q16—4242no—
196Q16—2122no—
197Q16—2121no—
198Q16—2121no—
199SD16—82no—
200D8—42no—
201D8—41no—
202D8—41no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8D4×D5311yescharacteristic
206C2×C8D4×D5311yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—1061no—
209C4⋊C4—2122no—
210C4⋊C4—2122no—
211C4⋊C4—2122no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4#1171992211yescharacteristic
216C42—21no—
217C42—2122no—
218C42—2121no—
219C42C2×C22⋊D5311yescharacteristic
220C16—1062no—
221C16—2121no—
222C23—81no—
223Q8—21no—
224Q8—2121no—
225Q8—2122no—
226Q8—2121no—
227Q8—2122no—
228Q8#1313316112yescomplemented
229D4—81no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—161no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—2121no—
237C2×C4—2122no—
238C2×C4—2121no—
239C2×C4#1313318511yescharacteristic
240C2×C4C2×#1171992211yescharacteristic
241C2×C4#1313324211yescharacteristic
242C8—21no—
243C8—41no—
244C8—81no—
245C8#1313323811yescharacteristic
246C8#1313324411yescharacteristic
247C22—161no—
248C22—161no—
249C22—161no—
250C22#1681075611yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—4241no—
255C4—4242no—
256C4—4241no—
257C4#1681076711yescharacteristic
258C4#1681078911yescharacteristic
259C4C2×#1313316111yescharacteristic
260C4#1681106311yescharacteristic
261C2—321no—
262C2#2027377811yescharacteristic
263C2#2027663111yescharacteristic
264C2#2027667711yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1668C2
socleC22×C53#447
prime radicalC53×#493C22
hypercenter#1668D53
hypocenterC53#12330
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
1. #20287721C2semidirect
4. #20272763C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272678C2semidirect
91. Q8×C53SD32semidirect
157. C53#12330semidirect
228. Q8#13133161semidirect

Maximal subgroups

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

subgroupindex
#202877212
#202769702
#202767522
#202727632
Q8⋊#117199082
#202726782
C53×#16682
#1233053