Finite Group Databasebeta

← #28476209

Subgroups — #28476209

Order 15104.

244 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 37 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#21425200C211yesmaximal, characteristic
2#21422217C211yesmaximal, characteristic, complemented
3C59×#1315C211yesmaximal, characteristic, complemented
4#16838202C411yescharacteristic, complemented
5#16838200—21no—
6#16837376—21no—
7#16837368—21no—
8C59⋊#338C411yescharacteristic, complemented
9C59×#408C2211yescharacteristic
10C59×C22⋊C16—21no—
11C2×#11745685—21no—
12#14273760—41no—
13#14273758—41no—
14#14273757—41no—
15#14273755—41no—
16#14273753—41no—
17#14273752—41no—
18#14273749—41no—
19#14273747—42no—
20C2×#11745664—21no—
21#14273686—41no—
22#14273642—22no—
23#14273642—41no—
24(C2×D59)⋊C8—42no—
25C2×#41×C59D411yescharacteristic
26C2×M16×C59C2×C411yescharacteristic
27C22×C8×C59D411yescharacteristic
28C2×C16×C59—41no—
29C22⋊C8×C59—41no—
30#41×C59—41no—
31#41×C59—41no—
32C2×Q8×C59SD1611yescharacteristic
33C2×D4×C59—21no—
34C2×D4×C59D811yescharacteristic
35C22×C4×C59—21no—
36C22×C4×C59C22⋊C411yescharacteristic
37C2×C22⋊D59—41no—
38#11745685—41no—
39#11745685—41no—
40#11745685—41no—
41#11745685—41no—
42C2×D236—21no—
43C2×C4×D59—41no—
44C2×Dic118—21no—
45M16×C59—41no—
46C2×C8×C59—21no—
47C2×C8×C59—21no—
48C2×C8×C59—41no—
49C22⋊Dic59—42no—
50#11745672—42no—
51#11745671—42no—
52#11745670—42no—
53#11745669—42no—
54C4⋊Dic59—81no—
55C4×Dic59—81no—
56#11745664—41no—
57#11745664—41no—
58C2×C59⋊C8—22no—
59Dic236—41no—
60D472—41no—
61#11745660—42no—
62C16×C59—81no—
63C23×C59—21no—
64C23×C59#10411yescharacteristic
65C22×D59—41no—
66Q8×C59—41no—
67D4×C59—41no—
68D4×C59—81no—
69C2×C4×C59—21no—
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—41no—
73C2×C4×C59—41no—
74C2×C4×C59#9911yescharacteristic
75C2×C4×C59#10211yescharacteristic
76C22⋊D59—81no—
77C22⋊D59—81no—
78C2×Dic59—41no—
79C2×Dic59—82no—
80D236—41no—
81D236—41no—
82C4×D59—81no—
83C4×D59—81no—
84Dic118—41no—
85Dic118—41no—
86C8×C59—41no—
87C8×C59—41no—
88C59⋊C8—42no—
89#7406—591nomaximal, Sylow
90C22×C59—21no—
91C22×C59—41no—
92C22×C59—41no—
93C22×C59—81no—
94C22×C59#32711yescharacteristic
95C2×D59—82no—
96C2×D59—81no—
97C4×C59—21no—
98C4×C59—41no—
99C4×C59AΣL(1,𝔽16)11yescharacteristic
100C4×C59#35311yescharacteristic
101Dic59—81no—
102Dic59—162no—
103#2017—591no—
104#1391—591no—
105#1315D5911yescharacteristic, complemented
106C2×C59—21no—
107C2×C59—41no—
108C2×C59—81no—
109C2×C59#134311yescharacteristic
110D59—161no—
111#575—591no—
112#575—1181no—
113#410—1181no—
114#408—1181no—
115#408C2×D5911yescharacteristic
116C22⋊C16—21no—
117#338—591no—
118C59#740611yescharacteristic, complemented, Sylow
119C2×#41—1181no—
120C2×#41D23611yescharacteristic
121#137—2361no—
122#137—2361no—
123#137—2361no—
124AGL(1,ℤ/8)—2361no—
125AGL(1,ℤ/8)—2361no—
126AGL(1,ℤ/8)—2361no—
127#135—2361no—
128#135—2362no—
129C2×M16—1181no—
130C2×M16C4×D5911yescharacteristic
131C22×C8C22⋊D5911yescharacteristic
132#117—2361no—
133C2×C16—41no—
134#99—1182no—
135#99—2361no—
136C22⋊C8—41no—
137C22⋊C8—2362no—
138#41—41no—
139#41—41no—
140#41—2361no—
141#41—2361no—
142#41—2361no—
143#41—2361no—
144C2×Q8—1181no—
145C2×Q8#1174566011yescharacteristic
146C2×D4—21no—
147C2×D4—1181no—
148C2×D4—2361no—
149C2×D4D47211yescharacteristic
150C22×C4—21no—
151C22×C4—2361no—
152C22×C4(C2×D59)⋊C411yescharacteristic
153Q16—2361no—
154Q16—2362no—
155SD16—2362no—
156SD16—2362no—
157SD16—2362no—
158D8—2361no—
159D8—2362no—
160M16—41no—
161M16—2361no—
162M16—2361no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—1182no—
167C4⋊C4—4721no—
168C22⋊C4—2362no—
169C42—4721no—
170C16—81no—
171C23—21no—
172C23—2361no—
173C23#1427365311yescharacteristic
174Q8—41no—
175Q8—2361no—
176Q8—2361no—
177D4—41no—
178D4—81no—
179D4—4721no—
180D4—4721no—
181D4—2361no—
182D4—2361no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—4721no—
189C2×C4—4722no—
190C2×C4—4721no—
191C2×C4—2361no—
192C2×C4#1427364111yescharacteristic
193C2×C4#1427364911yescharacteristic
194C8—41no—
195C8—41no—
196C8—2362no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—4722no—
202C22—4721no—
203C22#1683713711yescharacteristic
204C4—21no—
205C4—41no—
206C4—9442no—
207C4—4721no—
208C4#1683719011yescharacteristic
209C4#1683719411yescharacteristic
210C2—21no—
211C2—41no—
212C2—81no—
213C2—9441no—
214C2#2142209411yescharacteristic
subgroupG/H
centerC4#16837194
commutatorC2×M16×C59C2×C4
Frattini#408C2×D59
FittingC59×#1315C2
socleC2×C59#1343
prime radical#21425200C2
cosocleC59×#408C22
hypercenter#1315D59
hypocenterC59#7406
LCSG ▷ C2×M16×C59 ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #21422217C2semidirect
3. C59×#1315C2semidirect
4. #16838202C4semidirect
8. C59⋊#338C4semidirect
105. #1315D59semidirect
118. C59#7406semidirect

Maximal subgroups

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

subgroupindex
#214252002
#214222172
C59×#13152
#740659