Finite Group Databasebeta

← #28502389

Subgroups — #28502389

Order 15104.

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#21436961C211yesmaximal, characteristic, complemented
2#21426320C211yesmaximal, characteristic, complemented
3#21426131C211yesmaximal, characteristic
4#21423457C211yesmaximal, characteristic, complemented
5#21422155C211yesmaximal, characteristic, complemented
6#21422076C211yesmaximal, characteristic, complemented
7C59×#1331C211yesmaximal, characteristic, complemented
8#16838109—21no—
9C2×#14273668—21no—
10#16837807C2211yescharacteristic
11#16837803—21no—
12#16837779C2211yescharacteristic
13#16837700—22no—
14#16837692—21no—
15#16837603—21no—
16C8×Dic118C2211yescharacteristic
17D8×Dic59—21no—
18C4⋊(C59⋊C16)C2211yescharacteristic
19#16837204—22no—
20#16837202—21no—
21#16837145—21no—
22C4×D8×C59C2211yescharacteristic
23C4×C16×C59C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×#11745660—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273727—21no—
29#14273724—21no—
30#14273718—22no—
31#14273706—41no—
32#14273704—42no—
33C4×Dic118D411yescharacteristic
34D4×Dic59—22no—
35C2×C4⋊Dic59—41no—
36#14273668—42no—
37C2×C59⋊C16—21no—
38#14273661—21no—
39#14273661D412yes—
40#14273658—22no—
41C4⋊(C59⋊C8)D411yescharacteristic
42Dic59⋊C8—21no—
43#14273649—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×D236—41no—
57C2×D236—41no—
58C2×D236—41no—
59C2×Dic118—41no—
60C2×Dic118D811yescharacteristic
61D8×C59—21no—
62D8×C59SD1612yescomplemented
63C2×C8×C59—41no—
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—21no—
67C22⋊C4×C59—41no—
68C42×C59C2×D411yescharacteristic
69C22⋊Dic59—42no—
70C4⋊Dic59—22no—
71C4⋊Dic59—42no—
72C4⋊Dic59—41no—
73C4⋊Dic59D811yescharacteristic
74Dic59⋊C4—21no—
75Dic59⋊C4—42no—
76C4×Dic59—21no—
77C2×C59⋊C8—21no—
78D472—42no—
79D472—41no—
80D472—41no—
81#11745660—82no—
82C16×C59—41no—
83C59⋊C16—41no—
84C23×C59—41no—
85C22×D59—81no—
86D4×C59—22no—
87D4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—81no—
90C2×C4×C59—81no—
91C2×C4×C59#12011yescharacteristic
92C2×C4×C59C2×D811yescharacteristic
93C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—42no—
97C2×Dic59—162no—
98D236—81no—
99D236—81no—
100D236—81no—
101D236—81no—
102D236—161no—
103Dic118—21no—
104Dic118—21no—
105Dic118—161no—
106C8×C59—21no—
107C8×C59—81no—
108C8×C59C2×SD1611yescharacteristic
109C8×C59#13711yescharacteristic
110C59⋊C8—41no—
111#12355—591nomaximal, Sylow
112C22×C59—42no—
113C22×C59—81no—
114C22×C59—81no—
115C22×C59#44711yescharacteristic
116C2×D59—161no—
117C2×D59—161no—
118C2×D59—161no—
119C4×C59—21no—
120C4×C59—81no—
121C4×C59#44811yescharacteristic
122C4×C59#45411yescharacteristic
123C4×C59#50811yescharacteristic
124C4×C59#50911yescharacteristic
125Dic59—41no—
126Dic59—41no—
127Dic59—82no—
128#3380—591no—
129#2240—591no—
130#2206—591no—
131#1668—591no—
132#1365—591no—
133Q8⋊C16—591no—
134#1331D5911yescharacteristic, complemented
135C2×C59—82no—
136C2×C59#162711yescharacteristic
137C2×C59#218611yescharacteristic
138C2×C59#219211yescharacteristic
139D59—321no—
140#554—1181no—
141C2×D16—1181no—
142#500—591no—
143#499—1181no—
144#493—591no—
145#478—1182no—
146#476—1181no—
147#460—1181no—
148C8×Q8—591no—
149C4×D8—1181no—
150C4×D8C2×D5911yescharacteristic
151C4⋊C16—591no—
152#357—1182no—
153#357—1181no—
154C4×C16C2×D5911yescharacteristic
155#335C2×D5911yescharacteristic
156#331—1181no—
157C59#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2361no—
159C2×D8—1181no—
160C2×D8—1181no—
161C2×D8D23611yescharacteristic
162C4⋊Q8—1181no—
163#127—1181no—
164#125—1182no—
165C22⋊Q8—2362no—
166C22⋊Q8—2361no—
167C4×Q8—591no—
168C4×D4—21no—
169C4×D4—1182no—
170C2×C4⋊C4—2361no—
171D16—2362no—
172C2×C16—21no—
173C2×C16—1181no—
174#107—592no—
175#107—1181no—
176#107D23611yescharacteristic
177#106—1182no—
178C4⋊C8—21no—
179C4⋊C8—591no—
180C4⋊C8—1181no—
181#102—21no—
182#102—1182no—
183#102—2361no—
184C4×C8—1181no—
185C4×C8C22×D5911yescharacteristic
186C2×Q8—591no—
187C2×Q8—2361no—
188C2×D4—21no—
189C2×D4—2361no—
190C2×D4—2361no—
191C2×D4—2361no—
192C22×C4—41no—
193C22×C4—2362no—
194SD16—4722no—
195D8—21no—
196D8—2362no—
197D8—2361no—
198D8—2361no—
199D8#1174566012yescomplemented
200C2×C8—41no—
201C2×C8—1181no—
202C2×C8C2×D23611yescharacteristic
203C2×C8D4×D5911yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—591no—
206C4⋊C4—1181no—
207C4⋊C4—1182no—
208C4⋊C4—2362no—
209C4⋊C4—2362no—
210C4⋊C4—2361no—
211C22⋊C4—41no—
212C22⋊C4—2362no—
213C42—1181no—
214C42D4×D5911yescharacteristic
215C16—41no—
216C16—2361no—
217C23—41no—
218C23—4721no—
219Q8—1181no—
220Q8—1181no—
221Q8—9441no—
222D4—22no—
223D4—41no—
224D4—9441no—
225D4—4721no—
226D4—4721no—
227D4—4721no—
228D4—4721no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—1181no—
233C2×C4—1181no—
234C2×C4—9442no—
235C2×C4—2362no—
236C2×C4#1427369711yescharacteristic
237C2×C4D8×D5911yescharacteristic
238C2×C4#1427375411yescharacteristic
239C8—21no—
240C8—81no—
241C8—2361no—
242C8C2×#1174566011yescharacteristic
243C8#1427376011yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—9441no—
248C22—9441no—
249C22—9441no—
250C22#1683753711yescharacteristic
251C4—21no—
252C4—81no—
253C4—4722no—
254C4—2361no—
255C4—2361no—
256C4#1683754311yescharacteristic
257C4#1683757611yescharacteristic
258C4#1683783311yescharacteristic
259C4#1683783611yescharacteristic
260C2—82no—
261C2—18881no—
262C2#2142320211yescharacteristic
263C2#2142602011yescharacteristic
264C2#2142605411yescharacteristic
subgroupG/H
centerC22#16837537
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1331C2
socleC22×C59#447
prime radical#21426131C2
hypercenter#1331D59
hypocenterC59#12355
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
1. #21436961C2semidirect
2. #21426320C2semidirect
4. #21423457C2semidirect
5. #21422155C2semidirect
6. #21422076C2semidirect
7. C59×#1331C2semidirect
62. D8×C59SD16semidirect
134. #1331D59semidirect
157. C59#12355semidirect
199. D8#11745660semidirect

Maximal subgroups

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

subgroupindex
#214369612
#214263202
#214261312
#214234572
#214221552
#214220762
C59×#13312
#1235559