Finite Group Databasebeta

← #27297787

Subgroups — #27297787

Order 13568.

312 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 69 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#20287583C211yesmaximal, characteristic, complemented
2#20276938C211yesmaximal, characteristic, complemented
3#20276747C211yesmaximal, characteristic
4#20272774C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272675C211yesmaximal, characteristic, complemented
7C53×#1668C211yesmaximal, characteristic, complemented
8#16811333—21no—
9C2×#13133158—21no—
10#16811028C2211yescharacteristic
11#16811025—21no—
12#16810924—22no—
13#16810914—21no—
14C4×D424—21no—
15C4×D424C2211yescharacteristic
16C4⋊#11719908C2211yescharacteristic
17#16810424—21no—
18#16810424—22no—
19C4×#11719908C2211yescharacteristic
20C53×#493C2211yescharacteristic
21C53×#460—21no—
22C8×Q8×C53C2211yescharacteristic
23C53×#335C2211yescharacteristic
24C53×#331—21no—
25C2×D424D411yescharacteristic
26#13133217—21no—
27#13133209—22no—
28#13133195—42no—
29#13133193—41no—
30C4×D212—22no—
31C4×D212—21no—
32C4⋊C4×D53—41no—
33#13133158—42no—
34C2×#11719908—21no—
35C2×#11719908—21no—
36#13133151—21no—
37#13133151D411yescharacteristic
38#13133151D412yes—
39#13133149—22no—
40#13133139—22no—
41#13133139—21no—
42C2×SD16×C53—41no—
43C2×D8×C53—21no—
44C2×D8×C53—21no—
45#127×C53—21no—
46C4×Q8×C53D411yescharacteristic
47C4⋊C8×C53—21no—
48C4⋊C8×C53—21no—
49C4⋊C8×C53D411yescharacteristic
50#102×C53—41no—
51C4×C8×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53D811yescharacteristic
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D212—21no—
58C2×C4×D53—41no—
59C2×C4×D53—42no—
60C2×Dic106—41no—
61SD16×C53—82no—
62D8×C53—42no—
63D8×C53—41no—
64D8×C53—41no—
65C2×C8×C53—21no—
66C2×C8×C53—41no—
67C2×C8×C53C2×D411yescharacteristic
68C2×C8×C53C2×D411yescharacteristic
69C4⋊C4×C53—21no—
70C4⋊C4×C53D811yescharacteristic
71C42×C53—21no—
72C42×C53C2×D411yescharacteristic
73#11719921—42no—
74#11719921—41no—
75C4⋊Dic53—21no—
76C4⋊Dic53—22no—
77C4⋊Dic53—41no—
78#11719919—42no—
79#11719919—42no—
80D424—21no—
81D424SD1612yescomplemented
82#11719908—41no—
83#11719908—41no—
84C23×C53—81no—
85C22×D53—41no—
86Q8×C53—21no—
87Q8×C53—21no—
88D4×C53—81no—
89D4×C53—81no—
90D4×C53—81no—
91D4×C53—81no—
92D4×C53—161no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53C2×D811yescharacteristic
97C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
98C2×Dic53—42no—
99C2×Dic53—41no—
100D212—22no—
101D212—41no—
102C4×D53—81no—
103C4×D53—81no—
104C4×D53—162no—
105Dic106—161no—
106C8×C53—21no—
107C8×C53—41no—
108C8×C53—81no—
109C8×C53C2×SD1611yescharacteristic
110C8×C53#13711yescharacteristic
111#12355—531nomaximal, Sylow
112C22×C53—161no—
113C22×C53—161no—
114C22×C53—161no—
115C22×C53#44711yescharacteristic
116C2×D53—42no—
117C2×D53—81no—
118C2×D53—81no—
119C4×C53—21no—
120C4×C53—41no—
121C4×C53—41no—
122C4×C53#44811yescharacteristic
123C4×C53#45411yescharacteristic
124C4×C53#50811yescharacteristic
125C4×C53#50911yescharacteristic
126Dic53—81no—
127Dic53—82no—
128#3380—531no—
129#2240—531no—
130#2206—531no—
131#1668D5311yescharacteristic, complemented
132#1365—531no—
133Q8⋊C16—531no—
134#1331—531no—
135C2×C53—321no—
136C2×C53#162711yescharacteristic
137C2×C53#218611yescharacteristic
138C2×C53#219211yescharacteristic
139D53—82no—
140#554—1061no—
141C2×D16—1061no—
142#500—531no—
143#499—1061no—
144#493C2×D5311yescharacteristic
145#478—1062no—
146#476—1061no—
147#460—21no—
148C8×Q8C2×D5311yescharacteristic
149C4×D8—531no—
150C4×D8—1061no—
151C4⋊C16—531no—
152#357—1061no—
153#357—1062no—
154C4×C16—531no—
155#335C2×D5311yescharacteristic
156#331—21no—
157C53#1235511yescharacteristic, complemented, Sylow
158C2×SD16—41no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—531no—
162C4⋊Q8—1061no—
163#127—21no—
164#125—1062no—
165C22⋊Q8—2122no—
166C22⋊Q8—2121no—
167C4×Q8C22⋊D5311yescharacteristic
168C4×D4—1062no—
169C4×D4—1061no—
170C2×C4⋊C4—2121no—
171D16—2122no—
172C2×C16—1061no—
173C2×C16—1061no—
174#107—531no—
175#107—532no—
176#107—1061no—
177#106—1062no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8C22⋊D5311yescharacteristic
181#102—41no—
182#102—1062no—
183#102—1061no—
184C4×C8—21no—
185C4×C8C22×D5311yescharacteristic
186C2×Q8—2121no—
187C2×Q8#1171992211yescharacteristic
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—1061no—
192C22×C4—2122no—
193C22×C4—2121no—
194SD16—82no—
195D8—42no—
196D8—41no—
197D8—41no—
198D8—532no—
199D8—1061no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8D4×D5311yescharacteristic
203C2×C8D4×D5311yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—1061no—
206C4⋊C4—1062no—
207C4⋊C4—2122no—
208C4⋊C4—2122no—
209C4⋊C4—2121no—
210C4⋊C4#1171992211yescharacteristic
211C22⋊C4—2122no—
212C22⋊C4—2121no—
213C42—21no—
214C42C2×C22⋊D5311yescharacteristic
215C16—2121no—
216C16—2121no—
217C23—81no—
218C23—2121no—
219Q8—21no—
220Q8—21no—
221Q8—8481no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—161no—
227D4—1062no—
228D4—2121no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—8482no—
232C2×C4—4241no—
233C2×C4—4241no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4#1313318511yescharacteristic
237C2×C4C2×#1171992211yescharacteristic
238C2×C4#1313324211yescharacteristic
239C8—21no—
240C8—41no—
241C8—81no—
242C8SD16×D5311yescharacteristic
243C8#1313324611yescharacteristic
244C22—161no—
245C22—161no—
246C22—161no—
247C22—4241no—
248C22—4241no—
249C22—2122no—
250C22#1681075611yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—4241no—
255C4—4242no—
256C4#1681076011yescharacteristic
257C4#1681079411yescharacteristic
258C4#1681105311yescharacteristic
259C4#1681105911yescharacteristic
260C2—321no—
261C2—4242no—
262C2#2027381611yescharacteristic
263C2#2027663611yescharacteristic
264C2#2027667011yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1668C2
socleC22×C53#447
prime radical#20276747C2
hypercenter#1668D53
hypocenterC53#12355
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. #20287583C2semidirect
2. #20276938C2semidirect
4. #20272774C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272675C2semidirect
7. C53×#1668C2semidirect
81. D424SD16semidirect
131. #1668D53semidirect
157. C53#12355semidirect

Maximal subgroups

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

subgroupindex
#202875832
#202769382
#202767472
#202727742
Q8⋊#117199082
#202726752
C53×#16682
#1235553