Finite Group Databasebeta

← #27297789

Subgroups — #27297789

Order 13568.

312 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 75 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#20276940C211yesmaximal, characteristic, complemented
2#20276750C211yesmaximal, characteristic
3#20274075C211yesmaximal, characteristic, complemented
4#20272773C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272676C211yesmaximal, characteristic, complemented
7C53×#3380C211yesmaximal, characteristic, complemented
8C2×#13133158—21no—
9#16811000C2211yescharacteristic
10#16810823—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810423—21no—
13#16810423—22no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810366—21no—
17C53×#554—21no—
18C53×#500C2211yescharacteristic
19C53×#499—21no—
20C53×#478—22no—
21C53×#476—21no—
22C8×Q8×C53C2211yescharacteristic
23C4×D8×C53—21no—
24C4×D8×C53C2211yescharacteristic
25C2×#11719924—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133214—21no—
29#13133158—42no—
30C2×#11719908—21no—
31C2×#11719908—21no—
32C4⋊#88322—21no—
33#13133137—41no—
34C2×D8×C53D411yescharacteristic
35C4⋊Q8×C53—21no—
36#125×C53—22no—
37C22⋊Q8×C53—42no—
38C22⋊Q8×C53—41no—
39C4×Q8×C53D411yescharacteristic
40C4×D4×C53—22no—
41C4×D4×C53—21no—
42C2×C4⋊C4×C53—41no—
43#107×C53—21no—
44#107×C53D411yescharacteristic
45#107×C53D412yes—
46#106×C53—22no—
47C4⋊C8×C53—21no—
48C4⋊C8×C53D411yescharacteristic
49#102×C53—22no—
50#102×C53—21no—
51C4×C8×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—41no—
54C2×Q8×C53D811yescharacteristic
55C2×D4×C53—21no—
56C22×C4×C53—42no—
57C22×C4×C53—41no—
58C2×D212—41no—
59C2×D212—41no—
60C2×D212—41no—
61D8×C53—21no—
62D8×C53SD1612yescomplemented
63C2×C8×C53—21no—
64C2×C8×C53C2×D411yescharacteristic
65C2×C8×C53C2×D411yescharacteristic
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53D811yescharacteristic
73C22⋊C4×C53—42no—
74C22⋊C4×C53—41no—
75C42×C53—21no—
76C42×C53C2×D411yescharacteristic
77#11719924—82no—
78C2×#88322—41no—
79D424—42no—
80D424—41no—
81D424—41no—
82#11719908—41no—
83#11719908—41no—
84C23×C53—41no—
85C22×D53—81no—
86Q8×C53—21no—
87Q8×C53—21no—
88Q8×C53—161no—
89D4×C53—22no—
90D4×C53—41no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53—162no—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53C2×D811yescharacteristic
100C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
101D212—81no—
102D212—81no—
103D212—81no—
104D212—81no—
105D212—161no—
106C8×C53—21no—
107C8×C53—41no—
108C8×C53C2×SD1611yescharacteristic
109C8×C53#13711yescharacteristic
110#88322—81no—
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—41no—
121C4×C53—41no—
122C4×C53—81no—
123C4×C53—82no—
124C4×C53#44811yescharacteristic
125C4×C53#45411yescharacteristic
126C4×C53#50811yescharacteristic
127C4×C53#50911yescharacteristic
128#3380D5311yescharacteristic, complemented
129#2240—531no—
130#2206—531no—
131#1668—531no—
132#1365—531no—
133Q8⋊C16—531no—
134#1331—531no—
135C2×C53—82no—
136C2×C53#162711yescharacteristic
137C2×C53#218611yescharacteristic
138C2×C53#219211yescharacteristic
139D53—321no—
140#554—21no—
141C2×D16—1061no—
142#500C2×D5311yescharacteristic
143#499—21no—
144#493—531no—
145#478—22no—
146#476—21no—
147#460—1061no—
148C8×Q8C2×D5311yescharacteristic
149C4×D8—21no—
150C4×D8C2×D5311yescharacteristic
151C4⋊C16—531no—
152#357—1061no—
153#357—1062no—
154C4×C16—531no—
155#335—531no—
156#331—1061no—
157C53#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2121no—
159C2×D8—1061no—
160C2×D8—1061no—
161C2×D8C22⋊D5311yescharacteristic
162C4⋊Q8—21no—
163#127—1061no—
164#125—22no—
165C22⋊Q8—42no—
166C22⋊Q8—41no—
167C4×Q8C22⋊D5311yescharacteristic
168C4×D4—22no—
169C4×D4—21no—
170C2×C4⋊C4—41no—
171D16—2122no—
172C2×C16—1061no—
173C2×C16—1061no—
174#107—21no—
175#107C22⋊D5311yescharacteristic
176#107C22⋊D5312yes—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—1061no—
180C4⋊C8C22⋊D5311yescharacteristic
181#102—22no—
182#102—21no—
183#102—2121no—
184C4×C8—21no—
185C4×C8C22×D5311yescharacteristic
186C2×Q8—41no—
187C2×Q8#1171992211yescharacteristic
188C2×D4—21no—
189C2×D4—2121no—
190C2×D4—2121no—
191C2×D4—2121no—
192C22×C4—42no—
193C22×C4—41no—
194SD16—4242no—
195D8—21no—
196D8—2122no—
197D8—2121no—
198D8—2121no—
199D8#1171992412yescomplemented
200C2×C8—21no—
201C2×C8—2121no—
202C2×C8C2×C22⋊D5311yescharacteristic
203C2×C8C2×C22⋊D5311yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4#1171992211yescharacteristic
211C22⋊C4—42no—
212C22⋊C4—41no—
213C42—21no—
214C42C2×C22⋊D5311yescharacteristic
215C16—2121no—
216C16—2121no—
217C23—41no—
218C23—4241no—
219Q8—21no—
220Q8—21no—
221Q8—161no—
222D4—22no—
223D4—41no—
224D4—8481no—
225D4—4241no—
226D4—4241no—
227D4—4241no—
228D4—4241no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—162no—
236C2×C4#1313318611yescharacteristic
237C2×C4C2×#1171992211yescharacteristic
238C2×C4#1313324311yescharacteristic
239C8—21no—
240C8—41no—
241C8—4241no—
242C8C2×#1171992411yescharacteristic
243C8#1313324811yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—8481no—
248C22—8481no—
249C22—8481no—
250C22#1681075711yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4#1681076211yescharacteristic
257C4#1681079611yescharacteristic
258C4#1681105311yescharacteristic
259C4#1681105811yescharacteristic
260C2—82no—
261C2—16961no—
262C2#2027381811yescharacteristic
263C2#2027663811yescharacteristic
264C2#2027667211yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3380C2
socleC22×C53#447
prime radical#20276750C2
hypercenter#3380D53
hypocenterC53#12355
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #20276940C2semidirect
3. #20274075C2semidirect
4. #20272773C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272676C2semidirect
7. C53×#3380C2semidirect
62. D8×C53SD16semidirect
128. #3380D53semidirect
157. C53#12355semidirect
199. D8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202769402
#202767502
#202740752
#202727732
Q8⋊#117199082
#202726762
C53×#33802
#1235553