Finite Group Databasebeta

← #28509500

Subgroups — #28509500

Order 15104.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#21426414C211yesmaximal, characteristic, complemented
2#21426409C211yesmaximal, characteristic
3#21425978C211yesmaximal, characteristic, complemented
4#21422294C211yesmaximal, characteristic, complemented
5#21422289C211yesmaximal, characteristic, complemented
6#21422286C211yesmaximal, characteristic, complemented
7C59×#2249C211yesmaximal, characteristic, complemented
8#16837834C2211yescharacteristic
9#16837520—21no—
10#16837242—21no—
11#16837240—21no—
12#16837239—21no—
13#16837237—21no—
14#16837236C2211yescharacteristic
15#16837236C2211yescharacteristic
16#16837232C2211yescharacteristic
17#16837210—21no—
18C2×Q32×C59C2211yescharacteristic
19C2×SD32×C59—21no—
20C2×D16×C59C2211yescharacteristic
21C59×#495—21no—
22C4×C16×C59C2211yescharacteristic
23#14273749—21no—
24#14273733—41no—
25Dic472D411yescharacteristic
26#14273672—21no—
27D944D411yescharacteristic
28#14273663—21no—
29#14273655—41no—
30C59⋊C32—21no—
31C59⋊C32—21no—
32C2×Q16×C59—21no—
33C2×SD16×C59—41no—
34C2×D8×C59—21no—
35#124×C59—41no—
36Q32×C59—21no—
37Q32×C59—21no—
38SD32×C59—42no—
39D16×C59—21no—
40D16×C59—21no—
41C2×C16×C59D411yescharacteristic
42C2×C16×C59C2311yescharacteristic
43C4×C8×C59D411yescharacteristic
44C2×Q8×C59—41no—
45C2×D4×C59—41no—
46#11745685—41no—
47Q16×C59—41no—
48Q16×C59—41no—
49SD16×C59—81no—
50SD16×C59—81no—
51D8×C59—41no—
52D8×C59—41no—
53C2×C8×C59D811yescharacteristic
54C2×C8×C59C2×D411yescharacteristic
55C22⋊C4×C59—82no—
56C42×C59D811yescharacteristic
57#11745664—41no—
58Dic236—21no—
59D472—21no—
60#11745660—41no—
61#11745659—81no—
62C8×D59—81no—
63C16×C59—21no—
64C16×C59C2×D411yescharacteristic
65C16×C59#4111yescharacteristic
66C23×C59—81no—
67Q8×C59—81no—
68Q8×C59—81no—
69D4×C59—81no—
70D4×C59—81no—
71C2×C4×C59—21no—
72C2×C4×C59—81no—
73C2×C4×C59C2×D811yescharacteristic
74C22⋊D59—81no—
75D236—41no—
76C4×D59—81no—
77Dic118—41no—
78C8×C59D1611yescharacteristic
79C8×C59D1611yescharacteristic
80C8×C59#12111yescharacteristic
81C8×C59AGL(1,ℤ/8)11yescharacteristic
82C59⋊C8—81no—
83#13691—591nomaximal, Sylow
84C22×C59—162no—
85C22×C59—161no—
86C22×C59#50911yescharacteristic
87C2×D59—81no—
88C4×C59—41no—
89C4×C59—161no—
90C4×C59#45911yescharacteristic
91C4×C59C2×D1611yescharacteristic
92Dic59—81no—
93#2264—591no—
94#2263—591no—
95#2249D5911yescharacteristic, complemented
96#2178—591no—
97#1422—591no—
98#1420—591no—
99#1419—591no—
100C2×C59—21no—
101C2×C59—321no—
102C2×C59#220611yescharacteristic
103D59—161no—
104#508—591no—
105C2×Q32C2×D5911yescharacteristic
106C2×SD32—21no—
107C2×D16C2×D5911yescharacteristic
108#495—21no—
109#443—1181no—
110Q64—1181no—
111SD64—1181no—
112SD64—1181no—
113D32—1181no—
114M64—591no—
115M64—591no—
116#368—591no—
117#359—1181no—
118C4×C16C2×D5911yescharacteristic
119C59#1369111yescharacteristic, complemented, Sylow
120#135—1181no—
121C2×Q16—21no—
122C2×SD16—41no—
123C2×D8—21no—
124#131—2361no—
125#124—41no—
126Q32—21no—
127Q32—21no—
128Q32—591no—
129SD32—42no—
130SD32—1181no—
131D16—21no—
132D16—21no—
133D16—591no—
134C2×C16C22⋊D5911yescharacteristic
135C2×C16C22×D5911yescharacteristic
136#108—1181no—
137#104—2361no—
138C4×C8C22⋊D5911yescharacteristic
139C32—1181no—
140C32—1181no—
141#41—2361no—
142C2×Q8—41no—
143C2×D4—41no—
144Q16—41no—
145Q16—41no—
146Q16—1181no—
147SD16—81no—
148SD16—81no—
149SD16—2361no—
150D8—41no—
151D8—41no—
152D8—1181no—
153M16—4721no—
154M16—2361no—
155C2×C8—4721no—
156C2×C8#1174566911yescharacteristic
157C2×C8C2×C22⋊D5911yescharacteristic
158C22⋊C4—82no—
159C42#1174566911yescharacteristic
160C16—21no—
161C16D4×D5911yescharacteristic
162C16#1174568711yescharacteristic
163C23—81no—
164Q8—81no—
165Q8—81no—
166Q8—2361no—
167D4—81no—
168D4—81no—
169D4—4721no—
170D4—2361no—
171C2×C4—21no—
172C2×C4—81no—
173C2×C4—4721no—
174C2×C4C2×#1174566911yescharacteristic
175C8—4721no—
176C8#1427366811yescharacteristic
177C8#1427366811yescharacteristic
178C8#1427369811yescharacteristic
179C8#1427375311yescharacteristic
180C22—162no—
181C22—161no—
182C22—4721no—
183C22#1683783811yescharacteristic
184C4—41no—
185C4—161no—
186C4—4721no—
187C4#1683759411yescharacteristic
188C4C2×#1427366811yescharacteristic
189C2—21no—
190C2—321no—
191C2—9441no—
192C2#2142612811yescharacteristic
subgroupG/H
centerC2#21426128
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC59×#2249C2
socleC2×C59#2206
prime radical#21426409C2
hypercenter#2249D59
hypocenterC59#13691
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. #21426414C2semidirect
3. #21425978C2semidirect
4. #21422294C2semidirect
5. #21422289C2semidirect
6. #21422286C2semidirect
7. C59×#2249C2semidirect
95. #2249D59semidirect
119. C59#13691semidirect

Maximal subgroups

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

subgroupindex
#214264142
#214264092
#214259782
#214222942
#214222892
#214222862
C59×#22492
#1369159