Finite Group Databasebeta

← #28476057

Subgroups — #28476057

Order 15104.

318 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 43 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#21422034C211yesmaximal, characteristic, complemented
2#21422022C211yesmaximal, characteristic, complemented
3C59×#1794C211yesmaximal, characteristic
4#16837184—21no—
5C22⋊(C59⋊C16)—21no—
6C59×#585C411yescharacteristic
7C59×#421—21no—
8C59×#416—21no—
9C59×#410C411yescharacteristic
10C59×#408C2211yescharacteristic
11#14273667—41no—
12C2×C59⋊C16—41no—
132−1+4×C59—41no—
142−1+4×C59C812yes—
15H(5,𝔽2)×C59—21no—
16H(5,𝔽2)×C59—41no—
17H(5,𝔽2)×C59—41no—
18C2×#41×C59—21no—
19C2×#41×C59—41no—
20C2×#41×C59—42no—
21C2×#41×C59C2×C411yescharacteristic
22C2×M16×C59D411yescharacteristic
23C22×C8×C59D411yescharacteristic
24#117×C59—21no—
25#104×C59—42no—
26#103×C59—41no—
27#102×C59—41no—
28#99×C59—41no—
29C22⋊C8×C59—41no—
30#41×C59—22no—
31#41×C59—22no—
32#41×C59—41no—
33#41×C59—42no—
34#41×C59—41no—
35#41×C59—41no—
36#41×C59—42no—
37#41×C59—42no—
38#41×C59—44no—
39#41×C59—42no—
40#41×C59—41no—
41#41×C59—42no—
42C2×Q8×C59—21no—
43C2×Q8×C59—42no—
44C2×Q8×C59—41no—
45C2×Q8×C59C2×C811yescharacteristic
46C2×D4×C59—21no—
47C2×D4×C59—21no—
48C2×D4×C59—42no—
49C2×D4×C59—42no—
50C2×D4×C59—44no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59M1611yescharacteristic
54C22×C4×C59—21no—
55C22×C4×C59—41no—
56C22×C4×C59—42no—
57C22×C4×C59C22⋊C411yescharacteristic
58M16×C59—41no—
59C2×C8×C59—21no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C4⋊C4×C59—41no—
63C22⋊C4×C59—41no—
64C42×C59—41no—
65C59⋊C16—81no—
66C59⋊C16—81no—
67C23×C59—21no—
68C23×C59—41no—
69C23×C59—42no—
70C23×C59#10011yescharacteristic
71Q8×C59—22no—
72Q8×C59—41no—
73Q8×C59—42no—
74Q8×C59—41no—
75D4×C59—22no—
76D4×C59—41no—
77D4×C59—42no—
78D4×C59—42no—
79D4×C59—42no—
80D4×C59—44no—
81D4×C59—42no—
82D4×C59—41no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—42no—
87C2×C4×C59—41no—
88C2×C4×C59—44no—
89C2×C4×C59—42no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—42no—
93C2×C4×C59—81no—
94C2×C4×C59C22⋊C811yescharacteristic
95C2×C4×C59#9911yescharacteristic
96C8×C59—41no—
97C8×C59—41no—
98#7347—591nomaximal, Sylow
99C22×C59—21no—
100C22×C59—41no—
101C22×C59—42no—
102C22×C59—41no—
103C22×C59—81no—
104C22×C59—161no—
105C22×C59—162no—
106C22×C59#32311yescharacteristic
107C4×C59—21no—
108C4×C59—41no—
109C4×C59—42no—
110C4×C59—161no—
111C4×C59AΣL(1,𝔽16)11yescharacteristic
112C4×C59AΣL(1,𝔽16)11yescharacteristic
113#1794D5911yescharacteristic
114#1321—591no—
115#1315—591no—
116C2×C59—21no—
117C2×C59—41no—
118C2×C59—81no—
119C2×C59—82no—
120C2×C59#131611yescharacteristic
121#585Dic5911yescharacteristic
122#421—21no—
123#416—21no—
124#410Dic5911yescharacteristic
125#408C2×D5911yescharacteristic
126#349—1181no—
127C22⋊C16—1181no—
128C59#734711yescharacteristic, complemented, Sylow
1292−1+4—41no—
1302−1+4C59⋊C812yes—
131H(5,𝔽2)—21no—
132H(5,𝔽2)—41no—
133H(5,𝔽2)—41no—
134C2×#41—21no—
135C2×#41—41no—
136C2×#41—42no—
137C2×#41C2×Dic5911yescharacteristic
138C2×M16C22⋊D5911yescharacteristic
139C22×C8C22⋊D5911yescharacteristic
140#117—21no—
141M32—2361no—
142C2×C16—2361no—
143#104—42no—
144#103—41no—
145#102—41no—
146#99—41no—
147C22⋊C8—41no—
148#41—22no—
149#41—22no—
150#41—41no—
151#41—42no—
152#41—41no—
153#41—41no—
154#41—42no—
155#41—42no—
156#41—44no—
157#41—42no—
158#41—41no—
159#41—42no—
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8C2×C59⋊C811yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4#1174566411yescharacteristic
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4C22⋊Dic5911yescharacteristic
176M16—41no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—41no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C42—41no—
183C16—4721no—
184C16—4721no—
185C23—21no—
186C23—41no—
187C23—42no—
188C23#1427364411yescharacteristic
189Q8—22no—
190Q8—41no—
191Q8—42no—
192Q8—41no—
193D4—22no—
194D4—41no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—44no—
199D4—42no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4C22⋊(C59⋊C8)11yescharacteristic
213C2×C4#1427364211yescharacteristic
214C8—41no—
215C8—41no—
216C22—21no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22—161no—
222C22—162no—
223C22#1683712611yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—161no—
228C4#1683718911yescharacteristic
229C4#1683718911yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—82no—
234C2#2142202411yescharacteristic
subgroupG/H
centerC4#16837189
commutatorC2×Q8×C59C2×C8
Frattini#408C2×D59
FittingC59×#1794C2
socleC2×C59#1316
prime radicalC59×#585C4
cosocleC59×#408C22
hypercenter#1794D59
hypocenterC59#7347
LCSG ▷ C2×Q8×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ #1794

Complements

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

NH ≅ G/Ntype
1. #21422034C2semidirect
2. #21422022C2semidirect
128. C59#7347semidirect

Maximal subgroups

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

subgroupindex
#214220342
#214220222
C59×#17942
#734759