Finite Group Databasebeta

← #28493468

Subgroups — #28493468

Order 15104.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#21424928C211yesmaximal, characteristic
2#21422746C212yesmaximal
3#21421993C212yesmaximal, complemented
4#21421925C211yesmaximal, characteristic, complemented
5C59×#1832C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837455C2212yes—
8#16837426C2211yescharacteristic
9C2×#14273655—22no—
10#16837149—22no—
11#16837145—22no—
12C2×C4⋊C8×C59C2212yes—
13C59×#376C2211yescharacteristic
14C59×#336—22no—
15C2×#11745685—21no—
16C2×#11745664—21no—
17#14273727D411yescharacteristic
18#14273724D411yescharacteristic
19#14273715—21no—
20#14273701—22no—
21C4×D236—22no—
22C4⋊(C59⋊C8)—22no—
23C4⋊(C59⋊C8)—22no—
24#14273655—44no—
25C22⋊(C59⋊C8)—22no—
26C4×C59⋊C8—22no—
27C22×C8×C59—22no—
28C2×C42×C59D412yes—
29C2×C42×C59C2311yescharacteristic
30C4⋊C8×C59—22no—
31C4⋊C8×C59—22no—
32#95×C59—22no—
33C22×C4×C59—22no—
34C22×C4×C59C2×D411yescharacteristic
35C22×C4×C59#4112yes—
36C2×C22⋊D59—41no—
37#11745685—42no—
38#11745685—42no—
39C2×D236—21no—
40C2×D236—22no—
41C2×C4×D59—41no—
42C2×Dic118—21no—
43C2×C8×C59—42no—
44C2×C8×C59—42no—
45C2×C8×C59—42no—
46C42×C59—22no—
47C42×C59—22no—
48C42×C59C2×D411yescharacteristic
49C42×C59#4111yescharacteristic
50C42×C59#4112yes—
51(C2×D59)⋊C4—42no—
52C4⋊Dic59—22no—
53#11745664—42no—
54C2×C59⋊C8—42no—
55C2×C59⋊C8—41no—
56C23×C59#12311yescharacteristic
57C22×D59—41no—
58C2×C4×C59—42no—
59C2×C4×C59—44no—
60C2×C4×C59—42no—
61C2×C4×C59#12111yescharacteristic
62C2×C4×C59#12312yes—
63C2×C4×C59#12411yescharacteristic
64C2×C4×C59#12612yes—
65C2×C4×C59#13512yes—
66C2×C4×C59AGL(1,ℤ/8)12yes—
67C22⋊D59—82no—
68C2×Dic59—41no—
69D236—42no—
70D236—82no—
71C4×D59—81no—
72C4×D59—81no—
73Dic118—42no—
74C8×C59—82no—
75C59⋊C8—82no—
76C59⋊C8—81no—
77#10629—591nomaximal, Sylow
78C22×C59—22no—
79C22×C59#39711yescharacteristic
80C22×C59#48212yes—
81C2×D59—81no—
82C2×D59—82no—
83C4×C59—22no—
84C4×C59—22no—
85C4×C59—82no—
86C4×C59#46312yes—
87C4×C59#48811yescharacteristic
88C4×C59#49011yescharacteristic
89Dic59—81no—
90#1963—591no—
91#1832D5911yescharacteristic, complemented
92#1540—592no—
93#1301—592no—
94#1274—591no—
95C2×C59—22no—
96C2×C59#207311yescharacteristic
97C2×C59#208012yes—
98D59—161no—
99#532—591no—
100#431—592no—
101#423—591no—
102C2×C4⋊C8C2×D5912yes—
103C2×#104—1182no—
104#376C2×D5911yescharacteristic
105#336—22no—
106#332—1182no—
107#331—1182no—
108C59#1062911yescharacteristic, complemented, Sylow
109C2×#41—1181no—
110C2×M16—1181no—
111C22×C8—22no—
112C4⋊Q8—591no—
113#127—591no—
114#124—1181no—
115#121—1182no—
116C4×D4—1182no—
117C2×C42C22⋊D5912yes—
118C2×C42C22×D5911yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—1182no—
122C4⋊C8—1182no—
123#104—2364no—
124C22⋊C8—1182no—
125C4×C8—1182no—
126#95—22no—
127#41—2362no—
128#41—2362no—
129C2×Q8—1181no—
130C2×D4—1181no—
131C2×D4—1182no—
132C2×D4—2361no—
133C22×C4—22no—
134C22×C4—2361no—
135C22×C4#1174568512yes—
136C22×C4C2×C22⋊D5911yescharacteristic
137M16—2362no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—2362no—
142C2×C8—2361no—
143C4⋊C4—1182no—
144C22⋊C4—2362no—
145C42—22no—
146C42—22no—
147C42D4×D5911yescharacteristic
148C42#1174568711yescharacteristic
149C42#1174568912yes—
150C23—2361no—
151C23#1427371011yescharacteristic
152Q8—2362no—
153D4—4722no—
154D4—4722no—
155D4—2362no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—4721no—
160C2×C4—4721no—
161C2×C4—2361no—
162C2×C4#1427369811yescharacteristic
163C2×C4#1427371212yes—
164C2×C4#1427371611yescharacteristic
165C2×C4#1427372212yes—
166C2×C4#1427374712yes—
167C2×C4#1427375212yes—
168C8—82no—
169C8—4722no—
170C8—4721no—
171C22—22no—
172C22—4721no—
173C22—4722no—
174C22#1683732811yescharacteristic
175C22#1683772612yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—4721no—
180C4#1683761412yes—
181C4#1683775911yescharacteristic
182C4#1683776811yescharacteristic
183C2—22no—
184C2—9441no—
185C2#2142551011yescharacteristic
186C2#2142554712yes—
subgroupG/H
centerC2×C4#14273716
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1832C2
socleC22×C59#397
prime radical#16837952C22
hypercenter#1832D59
hypocenterC59#10629
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
3. #21421993C2semidirect
4. #21421925C2semidirect
5. C59×#1832C2semidirect
91. #1832D59semidirect
108. C59#10629semidirect

Maximal subgroups

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

subgroupindex
#214249282
#214227462
#214219932
#214219252
C59×#18322
#1062959