Finite Group Databasebeta

← #28509505

Subgroups — #28509505

Order 15104.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#21426408C211yesmaximal, characteristic
2#21426358C211yesmaximal, characteristic, complemented
3#21425977C211yesmaximal, characteristic, complemented
4#21422293C211yesmaximal, characteristic, complemented
5#21422289C211yesmaximal, characteristic, complemented
6#21422287C211yesmaximal, characteristic, complemented
7C59×#2264C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9C2×D944C2211yescharacteristic
10#16837787—21no—
11#16837519—21no—
12#16837239—21no—
13#16837237—21no—
14#16837236C2211yescharacteristic
15#16837232C2211yescharacteristic
16#16837209—21no—
17C16×Dic59C2211yescharacteristic
18C59×#508C2211yescharacteristic
19C2×Q32×C59C2211yescharacteristic
20C59×Q64—21no—
21C59×SD64—21no—
22C59×M64C2211yescharacteristic
23C2×#11745671—41no—
24C2×D472—21no—
25#14273734—41no—
26#14273716—41no—
27#14273670—42no—
28D944—21no—
29D944—21no—
30C2×C59⋊C16D411yescharacteristic
31#14273663—21no—
32#14273654—41no—
33C8×Dic59D411yescharacteristic
34C59⋊C32—21no—
35#135×C59—21no—
36C2×Q16×C59—21no—
37Q32×C59—21no—
38Q32×C59—21no—
39Q32×C59D411yescharacteristic
40SD32×C59—21no—
41D16×C59D411yescharacteristic
42C2×C16×C59C2311yescharacteristic
43C32×C59—21no—
44#41×C59—41no—
45C2×Q8×C59—41no—
46C2×D236—41no—
47Q16×C59—21no—
48Q16×C59—41no—
49Q16×C59—41no—
50SD16×C59—41no—
51D8×C59—21no—
52C2×C8×C59C2×D411yescharacteristic
53#11745671—81no—
54#11745671—81no—
55(C2×D59)⋊C4—82no—
56C4×Dic59D811yescharacteristic
57#11745664—41no—
58#11745664—81no—
59C2×C59⋊C8—81no—
60C2×C59⋊C8D811yescharacteristic
61D472—41no—
62D472—41no—
63C16×C59C2×D411yescharacteristic
64C16×C59#4111yescharacteristic
65C59⋊C16—21no—
66C22×D59—81no—
67Q8×C59—41no—
68Q8×C59—81no—
69Q8×C59—81no—
70D4×C59—41no—
71D4×C59—81no—
72C2×C4×C59—81no—
73C2×C4×C59—81no—
74C2×C4×C59C2×D811yescharacteristic
75C2×Dic59—21no—
76D236—81no—
77D236—81no—
78C8×C59#12111yescharacteristic
79C8×C59AGL(1,ℤ/8)11yescharacteristic
80C59⋊C8—81no—
81C59⋊C8D1611yescharacteristic
82C59⋊C8D1611yescharacteristic
83#13691—591nomaximal, Sylow
84C22×C59—81no—
85C22×C59#50911yescharacteristic
86C2×D59—162no—
87C2×D59—161no—
88C4×C59—81no—
89C4×C59—161no—
90C4×C59#45911yescharacteristic
91C4×C59C2×D1611yescharacteristic
92Dic59—41no—
93#2264D5911yescharacteristic, complemented
94#2263—591no—
95#2249—591no—
96#2178—591no—
97#1422—591no—
98#1420—591no—
99#1419—591no—
100C2×C59—21no—
101C2×C59—161no—
102C2×C59#220611yescharacteristic
103D59—321no—
104#508C2×D5911yescharacteristic
105C2×Q32C2×D5911yescharacteristic
106C2×SD32—1181no—
107C2×D16—591no—
108#495—1181no—
109#443—1181no—
110Q64—21no—
111SD64—21no—
112SD64—1181no—
113D32—1181no—
114M64—591no—
115M64C2×D5911yescharacteristic
116#368—591no—
117#359—1181no—
118C4×C16—591no—
119C59#1369111yescharacteristic, complemented, Sylow
120#135—21no—
121C2×Q16—21no—
122C2×SD16—2361no—
123C2×D8—1181no—
124#131—2361no—
125#124—2361no—
126Q32—21no—
127Q32—21no—
128Q32C22⋊D5911yescharacteristic
129SD32—21no—
130SD32—2362no—
131D16—1181no—
132D16—1181no—
133D16C22⋊D5911yescharacteristic
134C2×C16—591no—
135C2×C16C22×D5911yescharacteristic
136#108—1181no—
137#104—2361no—
138C4×C8—591no—
139C32—21no—
140C32—1181no—
141#41—41no—
142C2×Q8—41no—
143C2×D4—2361no—
144Q16—21no—
145Q16—41no—
146Q16—41no—
147SD16—41no—
148SD16—4721no—
149SD16—4721no—
150D8—21no—
151D8—2361no—
152D8—2361no—
153M16—4721no—
154M16—2361no—
155C2×C8—591no—
156C2×C8—4721no—
157C2×C8D4×D5911yescharacteristic
158C22⋊C4—4722no—
159C42—591no—
160C16—1181no—
161C16#1174568711yescharacteristic
162C16C2×C22⋊D5911yescharacteristic
163C23—4721no—
164Q8—41no—
165Q8—81no—
166Q8—81no—
167D4—41no—
168D4—81no—
169D4—4721no—
170D4—4721no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—1181no—
174C2×C4D8×D5911yescharacteristic
175C8—591no—
176C8—591no—
177C8—4721no—
178C8#1427369911yescharacteristic
179C8#1427375111yescharacteristic
180C22—81no—
181C22—9442no—
182C22—9441no—
183C22#1683783611yescharacteristic
184C4—81no—
185C4—161no—
186C4—2361no—
187C4#1683759511yescharacteristic
188C4D16×D5911yescharacteristic
189C2—21no—
190C2—161no—
191C2—18881no—
192C2#2142612911yescharacteristic
subgroupG/H
centerC2#21426129
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC59×#2264C2
socleC2×C59#2206
prime radical#21426408C2
hypercenter#2264D59
hypocenterC59#13691
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
2. #21426358C2semidirect
3. #21425977C2semidirect
4. #21422293C2semidirect
5. #21422289C2semidirect
6. #21422287C2semidirect
7. C59×#2264C2semidirect
93. #2264D59semidirect
119. C59#13691semidirect

Maximal subgroups

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

subgroupindex
#214264082
#214263582
#214259772
#214222932
#214222892
#214222872
C59×#22642
#1369159