Finite Group Databasebeta

← #28502417

Subgroups — #28502417

Order 15104.

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

normal subgroups only

This group has 296 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#21436961C211yesmaximal, characteristic
2#21426324C211yesmaximal, characteristic, complemented
3#21426141C211yesmaximal, characteristic
4#21423467C211yesmaximal, characteristic, complemented
5#21422155C211yesmaximal, characteristic, complemented
6#21422076C211yesmaximal, characteristic, complemented
7C59×#1331C211yesmaximal, characteristic
8#16838109—21no—
9C2×#14273671—21no—
10#16837807C2211yescharacteristic
11#16837803—21no—
12#16837782C2211yescharacteristic
13#16837700—22no—
14#16837692—21no—
15#16837688—21no—
16C8×Dic118C2211yescharacteristic
17D8×Dic59—21no—
18C4⋊(C59⋊C16)C2211yescharacteristic
19#16837207—22no—
20#16837205—21no—
21#16837147—21no—
22C4×D8×C59C2211yescharacteristic
23C4×C16×C59C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×Dic236—21no—
26C2×Dic236—21no—
27#14273727—21no—
28#14273727—21no—
29#14273718—22no—
30#14273706—41no—
31#14273704—42no—
32C4×Dic118D411yescharacteristic
33D4×Dic59—22no—
34C2×C4⋊Dic59—41no—
35#14273671—42no—
36C2×C59⋊C16—21no—
37#14273661—21no—
38#14273661D412yes—
39#14273659—41no—
40#14273658—22no—
41C4⋊(C59⋊C8)D411yescharacteristic
42Dic59⋊C8—21no—
43#14273652—41no—
44#14273648—22no—
45C8×Dic59—21no—
46C2×D8×C59D411yescharacteristic
47C4×D4×C59—21no—
48C2×C16×C59—21no—
49#107×C59D411yescharacteristic
50C4⋊C8×C59—21no—
51#102×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×D4×C59—21no—
54C22×C4×C59—41no—
55C22×Dic59—42no—
56C2×Dic118—41no—
57C2×Dic118—41no—
58C2×Dic118D811yescharacteristic
59D8×C59—21no—
60D8×C59SD1612yes—
61C2×C8×C59—41no—
62C2×C8×C59C2×D411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—21no—
65C22⋊C4×C59—41no—
66C42×C59C2×D411yescharacteristic
67C22⋊Dic59—42no—
68C4⋊Dic59—22no—
69C4⋊Dic59—42no—
70C4⋊Dic59—41no—
71C4⋊Dic59—41no—
72C4⋊Dic59—41no—
73C4⋊Dic59D811yescharacteristic
74Dic59⋊C4—21no—
75Dic59⋊C4—42no—
76C4×Dic59—21no—
77C2×C59⋊C8—21no—
78Dic236—42no—
79Dic236—41no—
80Dic236—41no—
81C16×C59—41no—
82C59⋊C16—41no—
83C23×C59—41no—
84D4×C59—22no—
85D4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—81no—
88C2×C4×C59—81no—
89C2×C4×C59#12011yescharacteristic
90C2×C4×C59C2×D811yescharacteristic
91C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
92C2×Dic59—21no—
93C2×Dic59—21no—
94C2×Dic59—42no—
95C2×Dic59—81no—
96C2×Dic59—162no—
97Dic118—21no—
98Dic118—21no—
99Dic118—81no—
100Dic118—81no—
101Dic118—161no—
102C8×C59—21no—
103C8×C59—81no—
104C8×C59C2×SD1611yescharacteristic
105C8×C59#13711yescharacteristic
106C59⋊C8—41no—
107#12359—591nomaximal, Sylow
108C22×C59—42no—
109C22×C59—81no—
110C22×C59—81no—
111C22×C59#44711yescharacteristic
112C4×C59—21no—
113C4×C59—81no—
114C4×C59#44811yescharacteristic
115C4×C59#45411yescharacteristic
116C4×C59#50811yescharacteristic
117C4×C59#51011yescharacteristic
118Dic59—41no—
119Dic59—41no—
120Dic59—82no—
121Dic59—161no—
122#3380—591no—
123#2241—591no—
124#2208—591no—
125#1670—591no—
126#1365—591no—
127Q8⋊C16—591no—
128#1331D5911yescharacteristic
129C2×C59—82no—
130C2×C59#162711yescharacteristic
131C2×C59#218911yescharacteristic
132C2×C59#219411yescharacteristic
133#554—1181no—
134C2×SD32—1181no—
135#500—591no—
136#499—1181no—
137#494—591no—
138#478—1182no—
139#476—1181no—
140#475—1181no—
141C8×Q8—591no—
142C4×D8—1181no—
143C4×D8C2×D5911yescharacteristic
144C4⋊C16—591no—
145#358—1182no—
146#358—1181no—
147C4×C16C2×D5911yescharacteristic
148#335C2×D5911yescharacteristic
149#332—1181no—
150C59#1235911yescharacteristic, complemented, Sylow
151C2×Q16—1181no—
152C2×Q16—1181no—
153C2×D8D23611yescharacteristic
154C4⋊Q8—1181no—
155C4⋊Q8—1181no—
156#125—1182no—
157C22⋊Q8—2362no—
158C22⋊Q8—2361no—
159C4×Q8—591no—
160C4×D4—21no—
161C4×D4—1182no—
162C2×C4⋊C4—2361no—
163SD32—2362no—
164C2×C16—21no—
165C2×C16—1181no—
166#107—592no—
167#107—1181no—
168#107D23611yescharacteristic
169#106—1182no—
170#106—2361no—
171C4⋊C8—21no—
172C4⋊C8—591no—
173C4⋊C8—1181no—
174#103—2361no—
175#102—21no—
176#102—1182no—
177C4×C8—1181no—
178C4×C8C22×D5911yescharacteristic
179C2×Q8—591no—
180C2×Q8—2361no—
181C2×Q8—2361no—
182C2×D4—21no—
183C22×C4—41no—
184C22×C4—2362no—
185Q16—2362no—
186Q16—2361no—
187Q16—2361no—
188D8—21no—
189D8#1174566012yes—
190C2×C8—41no—
191C2×C8—1181no—
192C2×C8C2×D23611yescharacteristic
193C2×C8D4×D5911yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—591no—
196C4⋊C4—1181no—
197C4⋊C4—1182no—
198C4⋊C4—2362no—
199C4⋊C4—2362no—
200C4⋊C4—2361no—
201C4⋊C4—2361no—
202C4⋊C4—2361no—
203C22⋊C4—41no—
204C22⋊C4—2362no—
205C42—1181no—
206C42D4×D5911yescharacteristic
207C16—41no—
208C16—2361no—
209C23—41no—
210Q8—1181no—
211Q8—1181no—
212Q8—9441no—
213Q8—4721no—
214Q8—4721no—
215D4—22no—
216D4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—1181no—
221C2×C4—1181no—
222C2×C4—9442no—
223C2×C4—4721no—
224C2×C4—2362no—
225C2×C4#1427369711yescharacteristic
226C2×C4D8×D5911yescharacteristic
227C2×C4#1427375411yescharacteristic
228C8—21no—
229C8—81no—
230C8—2361no—
231C8C2×#1174566011yescharacteristic
232C8#1427376011yescharacteristic
233C22—42no—
234C22—81no—
235C22—81no—
236C22#1683753711yescharacteristic
237C4—21no—
238C4—81no—
239C4—9441no—
240C4—4722no—
241C4—2361no—
242C4—2361no—
243C4#1683754311yescharacteristic
244C4#1683757611yescharacteristic
245C4#1683783111yescharacteristic
246C4#1683784111yescharacteristic
247C2—82no—
248C2#2142320211yescharacteristic
249C2#2142603711yescharacteristic
250C2#2142606811yescharacteristic
subgroupG/H
centerC22#16837537
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1331C2
socleC22×C59#447
prime radicalC2×D8×C59D4
hypercenter#1331D59
hypocenterC59#12359
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #21426324C2semidirect
4. #21423467C2semidirect
5. #21422155C2semidirect
6. #21422076C2semidirect
150. C59#12359semidirect

Maximal subgroups

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

subgroupindex
#214369612
#214263242
#214261412
#214234672
#214221552
#214220762
C59×#13312
#1235959