Finite Group Databasebeta

← #28509541

Subgroups — #28509541

Order 15104.

190 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C2×#16837242C211yesmaximal, characteristic, complemented
2C2×#16837235C211yesmaximal, characteristic
3#21426188C211yesmaximal, characteristic
4#21426184C211yesmaximal, characteristic
5#21422297C211yesmaximal, characteristic, complemented
6#21422278C211yesmaximal, characteristic, complemented
7C59×#1416C211yesmaximal, characteristic, complemented
8C2×Dic472C2211yescharacteristic
9C2×C16×D59C2211yescharacteristic
10#16837644—21no—
11#16837640—21no—
12#16837242—22no—
13#16837235—22no—
14C2×C59⋊C32C2211yescharacteristic
15#16837228C2211yescharacteristic
16#16837206—21no—
17#16837205—21no—
18C2×Q32×C59C2211yescharacteristic
19C2×C32×C59C2211yescharacteristic
20C59×#366C2211yescharacteristic
21C2×Dic236—21no—
22C2×C8×D59D411yescharacteristic
23#14273707—41no—
24#14273703—41no—
25Dic472—21no—
26Dic472—21no—
27C2×C59⋊C16D411yescharacteristic
28C16×D59—21no—
29C16×D59—21no—
30#14273661—21no—
31#14273651—41no—
32#14273650—41no—
33C59⋊C32D412yes—
34C2×Q16×C59—21no—
35Q32×C59—21no—
36Q32×C59—21no—
37C2×C16×C59C2311yescharacteristic
38#107×C59—21no—
39C32×C59—21no—
40C2×Q8×C59—41no—
41C2×C4×D59D811yescharacteristic
42C2×Dic118—41no—
43Q16×C59—41no—
44Q16×C59—41no—
45C2×C8×C59C2×D411yescharacteristic
46C4⋊C4×C59—41no—
47(C2×D59)⋊C4—81no—
48(C2×D59)⋊C4—81no—
49C4⋊Dic59—41no—
50C4⋊Dic59—81no—
51Dic59⋊C4—81no—
52C2×C59⋊C8D811yescharacteristic
53Dic236—41no—
54Dic236—41no—
55C8×D59—21no—
56C8×D59—21no—
57C16×C59C2×D411yescharacteristic
58C16×C59#4111yescharacteristic
59C59⋊C16—21no—
60C22×D59D1611yescharacteristic
61Q8×C59—81no—
62Q8×C59—81no—
63C2×C4×C59—81no—
64C2×C4×C59—81no—
65C2×C4×C59C2×D811yescharacteristic
66C2×Dic59—81no—
67C2×Dic59—81no—
68C2×Dic59D1611yescharacteristic
69C4×D59—21no—
70C4×D59—21no—
71Dic118—81no—
72Dic118—81no—
73C8×C59#12111yescharacteristic
74C8×C59#13511yescharacteristic
75C59⋊C8—21no—
76#13698—591nomaximal, Sylow
77C22×C59C2×D1611yescharacteristic
78C2×D59—21no—
79C2×D59—21no—
80C2×D59—21no—
81C4×C59—161no—
82C4×C59—161no—
83C4×C59#46611yescharacteristic
84C4×C59#50811yescharacteristic
85Dic59—21no—
86Dic59—161no—
87Dic59—161no—
88C2×Q64—591no—
89C2×M64—591no—
90#2216—591no—
91#2216—591no—
92#1424—591no—
93#1416—591no—
94#1416D5911yescharacteristic, complemented
95C2×C59#221511yescharacteristic
96C2×C59#226411yescharacteristic
97C2×C59#226411yescharacteristic
98D59—41no—
99C2×Q32—591no—
100C2×Q32C2×D5911yescharacteristic
101C22×C16—591no—
102#467—1181no—
103#467—1181no—
104Q64—1182no—
105M64—1182no—
106C2×C32—591no—
107C2×C32C2×D5911yescharacteristic
108#366—591no—
109#366C2×D5911yescharacteristic
110#358—1181no—
111#358—1181no—
112C59#1369811yescharacteristic, complemented, Sylow
113C2×Q16—21no—
114C2×Q16—1181no—
115C22×C8—591no—
116C22⋊Q8—2361no—
117C22⋊Q8—2361no—
118Q32—21no—
119Q32—21no—
120Q32—1181no—
121Q32—1181no—
122C2×C16—591no—
123C2×C16—1181no—
124C2×C16—1181no—
125C2×C16C22×D5911yescharacteristic
126#107—21no—
127#107—1181no—
128#103—2361no—
129#103—2361no—
130C32—21no—
131C32—592no—
132C2×Q8—41no—
133C2×Q8—2361no—
134C22×C4—591no—
135Q16—41no—
136Q16—41no—
137Q16—2361no—
138Q16—2361no—
139C2×C8—591no—
140C2×C8—1181no—
141C2×C8—1181no—
142C2×C8D4×D5911yescharacteristic
143C4⋊C4—41no—
144C4⋊C4—4721no—
145C4⋊C4—4721no—
146C4⋊C4—2361no—
147C22⋊C4—4721no—
148C22⋊C4—4721no—
149C16—1181no—
150C16#1174568511yescharacteristic
151C16D4×D5911yescharacteristic
152C23—591no—
153Q8—81no—
154Q8—81no—
155Q8—4721no—
156Q8—4721no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—591no—
160C2×C4—1181no—
161C2×C4—1181no—
162C2×C4—4721no—
163C2×C4—4721no—
164C2×C4D8×D5911yescharacteristic
165C8—1181no—
166C8#1427370011yescharacteristic
167C8#1427374811yescharacteristic
168C22—1181no—
169C22—1181no—
170C22—1181no—
171C22D16×D5911yescharacteristic
172C4—161no—
173C4—161no—
174C4—1181no—
175C4—9441no—
176C4—9441no—
177C4#1683763711yescharacteristic
178C4#1683783311yescharacteristic
179C2—2361no—
180C2#2142618111yescharacteristic
181C2#2142641211yescharacteristic
182C2#2142641511yescharacteristic
subgroupG/H
centerC22D16×D59
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC59×#1416C2
socleC22×C59C2×D16
prime radicalC22×D59D16
hypercenter#1416D59
hypocenterC59#13698
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1416

Complements

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

NH ≅ G/Ntype
1. C2×#16837242C2semidirect
5. #21422297C2semidirect
6. #21422278C2semidirect
7. C59×#1416C2semidirect
94. #1416D59semidirect
112. C59#13698semidirect

Maximal subgroups

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

subgroupindex
C2×#168372422
C2×#168372352
#214261882
#214261842
#214222972
#214222782
C59×#14162
#1369859