Finite Group Databasebeta

← #29455446

Subgroups — #29455446

Order 15104.

4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 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#21437744C213yesmaximal, complemented
2#21433502C2112yesmaximal, complemented
3#21433488C2112yesmaximal, complemented
4#21432560C213yesmaximal, complemented
5C59×2−1+6C211yesmaximal, characteristic, complemented
6C22×Q8×D59C2211yescharacteristic, complemented
7Q8×C22⋊D59—212no—
8#16838059—212no—
9#16837973C22118yescomplemented
10#16837909C22118yescomplemented
11#16837880C2213yescomplemented
12#16837592C22148yescomplemented
13#16837589C22116yescomplemented
14#16837376C22136yescomplemented
15C59×#585—212no—
16C59×#585C22112yes—
17C2×2−1+4×C59—212no—
18C2×2−1+4×C59C2213yes—
19C22×Dic118C2313yes—
20C2×Q8×D59—212no—
21C22×C4×D59C2313yes—
22#14273725—218no—
23#14273714—218no—
24#14273708C23136yes—
25#14273706—236no—
26#14273703—218no—
27#14273698—218no—
28#14273695C23112yes—
29Q8×Dic59—212no—
30C4×C22⋊D59—236no—
31#14273686C23118yes—
32#14273642C23148yes—
332−1+4×C59—248no—
342−1+4×C59—236no—
352−1+4×C59—224no—
36H(5,𝔽2)×C59—216no—
37H(5,𝔽2)×C59—236no—
38H(5,𝔽2)×C59C23116yes—
39C2×#41×C59—248no—
40C2×#41×C59—272no—
41C2×#41×C59D4112yescomplemented
42C2×#41×C59C23118yes—
43C22×Q8×C59—216no—
44C22×Q8×C59D4112yescomplemented
45C22×Q8×C59C2311yescharacteristic
46C23×D59C2411yescharacteristic
47#41×C59—2144no—
48#41×C59—248no—
49#41×C59—248no—
50#41×C59—248no—
51#41×C59—272no—
52C2×Q8×C59—272no—
53C2×Q8×C59—248no—
54C2×Q8×C59C2×D4118yes—
55C2×Q8×C59C2×D4112yes—
56C2×D4×C59—272no—
57C2×D4×C59—248no—
58C2×D4×C59C2×D4118yes—
59C2×D4×C59C24112yes—
60C22×C4×C59—248no—
61C22×C4×C59C2×D4136yes—
62C22×C4×C59C2413yes—
63C2×C22⋊D59—212no—
64C22×Dic59C2413yes—
65Q8×D59—416no—
66C2×C4×D59—218no—
67C2×Dic118—218no—
68C22⋊Dic59—236no—
69C22⋊Dic59C24112yes—
70(C2×D59)⋊C4—212no—
71C4⋊Dic59—218no—
72Dic59⋊C4—236no—
73C4×Dic59—218no—
74C23×C59—216no—
75C23×C59C22×D4112yes—
76C23×C59C2511yescharacteristic
77C22×D59—23no—
78C22×D59—42no—
79Q8×C59—216no—
80Q8×C59—236no—
81Q8×C59#120116yes—
82D4×C59—248no—
83D4×C59—224no—
84D4×C59—236no—
85C2×C4×C59—248no—
86C2×C4×C59—272no—
87C2×C4×C59C22×D4112yes—
88C2×C4×C59C22×D4118yes—
89C22⋊D59—424no—
90C2×Dic59—212no—
91C2×Dic59—29no—
92C4×D59—412no—
93Dic118—412no—
94#60400—591nomaximal, Sylow
95C22×C59—212no—
96C22×C59—41no—
97C22×C59—412no—
98C22×C59—816no—
99C22×C59C23×D413yes—
100C2×D59—26no—
101C2×D59—43no—
102C2×D59—41no—
103C4×C59—212no—
104C4×C59C2×#120112yes—
105Dic59—412no—
106Dic59—43no—
1072−1+6D5911yescharacteristic, complemented
108#3486—593no—
109#3028—5912no—
110#3025—5912no—
111#2884—593no—
112C2×C59—23no—
113C2×C59—412no—
114C2×C59C22×#12011yescharacteristic
115D59—42no—
116#585—212no—
117#585C2×D59112yes—
118C2×2−1+4—212no—
119C2×2−1+4C2×D5913yes—
120C23×Q8—591no—
121D4×Q8—11812no—
122#548—11812no—
123#536—5918no—
124#525—5918no—
125#519—593no—
126#458—5948no—
127B(4,𝔽2)—5916no—
128#410—5936no—
129C59#6040011yescharacteristic, complemented, Sylow
1302−1+4—248no—
1312−1+4—236no—
1322−1+4—224no—
133H(5,𝔽2)—216no—
134H(5,𝔽2)—236no—
135H(5,𝔽2)C22×D59116yes—
136C2×#41—248no—
137C2×#41—272no—
138C2×#41C22⋊D59112yescomplemented
139C2×#41C22×D59118yes—
140C22×Q8—216no—
141C22×Q8—593no—
142C22×Q8—11812no—
143C22×Q8C22⋊D59112yescomplemented
144C22×Q8C22×D5911yescharacteristic
145C23×C4—593no—
146C4⋊Q8—11818no—
147#124—11818no—
148#123—5936no—
149C22⋊Q8—11818no—
150C22⋊Q8—11836no—
151#121—11818no—
152#120—5912no—
153C4×Q8—11812no—
154C4×D4—11836no—
155#117—5918no—
156#99—5948no—
157C24—591no—
158#41—2144no—
159#41—248no—
160#41—248no—
161#41—248no—
162#41—272no—
163C2×Q8—272no—
164C2×Q8—248no—
165C2×Q8—11818no—
166C2×Q8—23616no—
167C2×Q8C2×C22⋊D59118yes—
168C2×Q8C2×C22⋊D59112yes—
169C2×D4—272no—
170C2×D4—248no—
171C2×D4—11812no—
172C2×D4C2×C22⋊D59118yes—
173C2×D4C23×D59112yes—
174C22×C4—248no—
175C22×C4—593no—
176C22×C4—11818no—
177C22×C4C2×C22⋊D59136yes—
178C22×C4C23×D5913yes—
179C4⋊C4—11818no—
180C4⋊C4—11836no—
181C22⋊C4—5912no—
182C22⋊C4—11836no—
183C22⋊C4—11812no—
184C42—11818no—
185C23—216no—
186C23—1183no—
187C23—2362no—
188C23C22×C22⋊D59112yes—
189C23C24×D5911yescharacteristic
190Q8—216no—
191Q8—236no—
192Q8—23612no—
193Q8#14273696116yes—
194D4—248no—
195D4—224no—
196D4—236no—
197D4—23624no—
198C2×C4—248no—
199C2×C4—272no—
200C2×C4—11812no—
201C2×C4—1189no—
202C2×C4—23612no—
203C2×C4C22×C22⋊D59112yes—
204C2×C4C22×C22⋊D59118yes—
205C22—212no—
206C22—41no—
207C22—412no—
208C22—816no—
209C22—1186no—
210C22—2363no—
211C22—2361no—
212C22C23×C22⋊D5913yes—
213C4—212no—
214C4—23612no—
215C4—2363no—
216C4C2×#14273696112yes—
217C2—23no—
218C2—412no—
219C2—2362no—
220C2C22×#1427369611yescharacteristic
subgroupG/H
centerC2C22×#14273696
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC59×2−1+6C2
socleC2×C59C22×#120
hypercenter2−1+6D59
hypocenterC59#60400
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #21437744C2semidirect
2. #21433502C2semidirect
3. #21433488C2semidirect
4. #21432560C2semidirect
5. C59×2−1+6C2semidirect
6. C22×Q8×D59C22semidirect
9. #16837973C22semidirect
10. #16837909C22semidirect
11. #16837880C22semidirect
12. #16837592C22semidirect
13. #16837589C22semidirect
14. #16837376C22semidirect
41. C2×#41×C59D4semidirect
44. C22×Q8×C59D4semidirect
107. 2−1+6D59semidirect
129. C59#60400semidirect
138. C2×#41C22⋊D59semidirect
143. C22×Q8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214377442
#214335022
#214334882
#214325602
C59×2−1+62
#6040059