Finite Group Databasebeta

← #27297792

Subgroups — #27297792

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#20287580C211yesmaximal, characteristic, complemented
2#20276939C211yesmaximal, characteristic, complemented
3#20276750C211yesmaximal, characteristic
4#20274076C211yesmaximal, characteristic, complemented
5#20272774C211yesmaximal, characteristic, complemented
6#20272695C211yesmaximal, characteristic, complemented
7C53×#1331C211yesmaximal, characteristic, complemented
8#16811330—21no—
9C2×#13133158—21no—
10#16811028C2211yescharacteristic
11#16811024—21no—
12#16811000C2211yescharacteristic
13#16810921—22no—
14#16810913—21no—
15#16810824—21no—
16C8×Dic106C2211yescharacteristic
17D8×Dic53—21no—
18C4⋊#11719908C2211yescharacteristic
19#16810425—22no—
20#16810423—21no—
21#16810366—21no—
22C4×D8×C53C2211yescharacteristic
23C4×C16×C53C2211yescharacteristic
24C53×#335C2211yescharacteristic
25C2×#11719913—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133217—21no—
29#13133214—21no—
30#13133208—22no—
31#13133196—41no—
32#13133194—42no—
33C4×Dic106D411yescharacteristic
34D4×Dic53—22no—
35C2×C4⋊Dic53—41no—
36#13133158—42no—
37C2×#11719908—21no—
38#13133151—21no—
39#13133151D412yes—
40#13133148—22no—
41C4⋊#88322D411yescharacteristic
42#13133146—21no—
43#13133139—41no—
44#13133138—22no—
45C8×Dic53—21no—
46C2×D8×C53D411yescharacteristic
47C4×D4×C53—21no—
48C2×C16×C53—21no—
49#107×C53D411yescharacteristic
50C4⋊C8×C53—21no—
51#102×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×D4×C53—21no—
54C22×C4×C53—41no—
55C22×Dic53—42no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212—41no—
59C2×Dic106—41no—
60C2×Dic106D811yescharacteristic
61D8×C53—21no—
62D8×C53SD1612yescomplemented
63C2×C8×C53—41no—
64C2×C8×C53C2×D411yescharacteristic
65C2×C8×C53C2×D411yescharacteristic
66C4⋊C4×C53—21no—
67C22⋊C4×C53—41no—
68C42×C53C2×D411yescharacteristic
69C22⋊Dic53—42no—
70C4⋊Dic53—22no—
71C4⋊Dic53—42no—
72C4⋊Dic53—41no—
73C4⋊Dic53D811yescharacteristic
74#11719919—21no—
75#11719919—42no—
76C4×Dic53—21no—
77C2×#88322—21no—
78D424—42no—
79D424—41no—
80D424—41no—
81#11719913—82no—
82C16×C53—41no—
83#11719908—41no—
84C23×C53—41no—
85C22×D53—81no—
86D4×C53—22no—
87D4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—81no—
90C2×C4×C53—81no—
91C2×C4×C53#12011yescharacteristic
92C2×C4×C53C2×D811yescharacteristic
93C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—42no—
97C2×Dic53—162no—
98D212—81no—
99D212—81no—
100D212—81no—
101D212—81no—
102D212—161no—
103Dic106—21no—
104Dic106—21no—
105Dic106—161no—
106C8×C53—21no—
107C8×C53—81no—
108C8×C53C2×SD1611yescharacteristic
109C8×C53#13711yescharacteristic
110#88322—41no—
111#12355—531nomaximal, Sylow
112C22×C53—42no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53#44711yescharacteristic
116C2×D53—161no—
117C2×D53—161no—
118C2×D53—161no—
119C4×C53—21no—
120C4×C53—81no—
121C4×C53#44811yescharacteristic
122C4×C53#45411yescharacteristic
123C4×C53#50811yescharacteristic
124C4×C53#50911yescharacteristic
125Dic53—41no—
126Dic53—41no—
127Dic53—82no—
128#3380—531no—
129#2240—531no—
130#2206—531no—
131#1668—531no—
132#1365—531no—
133Q8⋊C16—531no—
134#1331D5311yescharacteristic, complemented
135C2×C53—82no—
136C2×C53#162711yescharacteristic
137C2×C53#218611yescharacteristic
138C2×C53#219211yescharacteristic
139D53—321no—
140#554—1061no—
141C2×D16—1061no—
142#500—531no—
143#499—1061no—
144#493—531no—
145#478—1062no—
146#476—1061no—
147#460—1061no—
148C8×Q8—531no—
149C4×D8—1061no—
150C4×D8C2×D5311yescharacteristic
151C4⋊C16—531no—
152#357—1062no—
153#357—1061no—
154C4×C16C2×D5311yescharacteristic
155#335C2×D5311yescharacteristic
156#331—1061no—
157C53#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2121no—
159C2×D8—1061no—
160C2×D8—1061no—
161C2×D8D21211yescharacteristic
162C4⋊Q8—1061no—
163#127—1061no—
164#125—1062no—
165C22⋊Q8—2122no—
166C22⋊Q8—2121no—
167C4×Q8—531no—
168C4×D4—21no—
169C4×D4—1062no—
170C2×C4⋊C4—2121no—
171D16—2122no—
172C2×C16—21no—
173C2×C16—1061no—
174#107—532no—
175#107—1061no—
176#107D21211yescharacteristic
177#106—1062no—
178C4⋊C8—21no—
179C4⋊C8—531no—
180C4⋊C8—1061no—
181#102—21no—
182#102—1062no—
183#102—2121no—
184C4×C8—1061no—
185C4×C8C22×D5311yescharacteristic
186C2×Q8—531no—
187C2×Q8—2121no—
188C2×D4—21no—
189C2×D4—2121no—
190C2×D4—2121no—
191C2×D4—2121no—
192C22×C4—41no—
193C22×C4—2122no—
194SD16—4242no—
195D8—21no—
196D8—2122no—
197D8—2121no—
198D8—2121no—
199D8#1171991312yescomplemented
200C2×C8—41no—
201C2×C8—1061no—
202C2×C8C2×D21211yescharacteristic
203C2×C8D4×D5311yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—531no—
206C4⋊C4—1061no—
207C4⋊C4—1062no—
208C4⋊C4—2122no—
209C4⋊C4—2122no—
210C4⋊C4—2121no—
211C22⋊C4—41no—
212C22⋊C4—2122no—
213C42—1061no—
214C42D4×D5311yescharacteristic
215C16—41no—
216C16—2121no—
217C23—41no—
218C23—4241no—
219Q8—8481no—
220Q8—1061no—
221Q8—1061no—
222D4—22no—
223D4—41no—
224D4—8481no—
225D4—4241no—
226D4—4241no—
227D4—4241no—
228D4—4241no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—8482no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—2122no—
236C2×C4#1313318711yescharacteristic
237C2×C4D8×D5311yescharacteristic
238C2×C4#1313324411yescharacteristic
239C8—21no—
240C8—81no—
241C8—2121no—
242C8C2×#1171991311yescharacteristic
243C8#1313325011yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—8481no—
248C22—8481no—
249C22—8481no—
250C22#1681075811yescharacteristic
251C4—21no—
252C4—81no—
253C4—4242no—
254C4—2121no—
255C4—2121no—
256C4#1681076411yescharacteristic
257C4#1681079711yescharacteristic
258C4#1681105411yescharacteristic
259C4#1681105711yescharacteristic
260C2—82no—
261C2—16961no—
262C2#2027382111yescharacteristic
263C2#2027663911yescharacteristic
264C2#2027667311yescharacteristic
subgroupG/H
centerC22#16810758
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1331C2
socleC22×C53#447
prime radical#20276750C2
hypercenter#1331D53
hypocenterC53#12355
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
1. #20287580C2semidirect
2. #20276939C2semidirect
4. #20274076C2semidirect
5. #20272774C2semidirect
6. #20272695C2semidirect
7. C53×#1331C2semidirect
62. D8×C53SD16semidirect
134. #1331D53semidirect
157. C53#12355semidirect
199. D8#11719913semidirect

Maximal subgroups

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

subgroupindex
#202875802
#202769392
#202767502
#202740762
#202727742
#202726952
C53×#13312
#1235553