Finite Group Databasebeta

← #28500711

Subgroups — #28500711

Order 15104.

226 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 67 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#21425970C211yesmaximal, characteristic
2#21423010C211yesmaximal, characteristic, complemented
3#21423009C211yesmaximal, characteristic, complemented
4#21422161C211yesmaximal, characteristic, complemented
5#21422064C211yesmaximal, characteristic, complemented
6#21422018C211yesmaximal, characteristic, complemented
7C59×#1333C211yesmaximal, characteristic, complemented
8C2×#14273668C411yescharacteristic
9C4×D472C2211yescharacteristic
10#16837480—21no—
11#16837479—21no—
12#16837229C411yescharacteristic
13#16837203C411yescharacteristic
14#16837202C411yescharacteristic
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17D4⋊(C59⋊C8)—21no—
18#16837129—21no—
19#16837123C2211yescharacteristic
20C4×D8×C59C2211yescharacteristic
21C59×#346C2211yescharacteristic
22C59×#334C2211yescharacteristic
23C2×D472C2×C411yescharacteristic
24C2×#11745664—41no—
25C2×#11745659—41no—
26C4×D236—21no—
27#14273668—22no—
28C2×C59⋊C16C2×C411yescharacteristic
29C2×C59⋊C16C2×C411yescharacteristic
30#14273661C2×C411yescharacteristic
31C4⋊(C59⋊C8)—21no—
32#14273649—21no—
33C22⋊(C59⋊C8)—41no—
34(C2×D59)⋊C8—41no—
35C4×C59⋊C8D411yescharacteristic
36C4×C59⋊C8D411yescharacteristic
37C2×D8×C59C2×C411yescharacteristic
38C4×D4×C59—21no—
39C2×C16×C59—21no—
40#107×C59C2×C411yescharacteristic
41C4⋊C8×C59—21no—
42#102×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D4×C59—21no—
45C22×C4×C59—41no—
46C2×D236—21no—
47C2×C4×D59—41no—
48D8×C59—21no—
49D8×C59—21no—
50C2×C8×C59—41no—
51C2×C8×C59C22×C411yescharacteristic
52C2×C8×C59#4111yescharacteristic
53C4⋊C4×C59—21no—
54C22⋊C4×C59—41no—
55C42×C59C2×D411yescharacteristic
56(C2×D59)⋊C4—41no—
57C4⋊Dic59—21no—
58#11745664—82no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8—41no—
62D472—21no—
63D472—21no—
64#11745659—82no—
65C16×C59—41no—
66C59⋊C16—21no—
67C59⋊C16M1612yescomplemented
68C23×C59—41no—
69C22×D59—41no—
70D4×C59—41no—
71D4×C59—41no—
72C2×C4×C59—41no—
73C2×C4×C59—81no—
74C2×C4×C59—81no—
75C2×C4×C59C4×D411yescharacteristic
76C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
78C2×Dic59—41no—
79D236—41no—
80D236—41no—
81C4×D59—81no—
82C4×D59—81no—
83C8×C59—21no—
84C8×C59—81no—
85C8×C59C2×M1611yescharacteristic
86C8×C59#13111yescharacteristic
87C59⋊C8—41no—
88C59⋊C8—41no—
89C59⋊C8—81no—
90#12089—591nomaximal, Sylow
91C22×C59—81no—
92C22×C59—81no—
93C22×C59—81no—
94C22×C59#44211yescharacteristic
95C2×D59—81no—
96C2×D59—81no—
97C2×D59—81no—
98C4×C59—21no—
99C4×C59—21no—
100C4×C59—81no—
101C4×C59#43611yescharacteristic
102C4×C59#44411yescharacteristic
103Dic59—81no—
104#2177—591no—
105#1594—591no—
106#1594—591no—
107#1368—591no—
108#1333—591no—
109#1333D5911yescharacteristic, complemented
110#1313—591no—
111C2×C59—161no—
112C2×C59#159411yescharacteristic
113C2×C59AGL(1,ℤ/16)11yescharacteristic
114C2×C59AGL(1,ℤ/16)11yescharacteristic
115D59—161no—
116C2×D16—591no—
117C4×D8—591no—
118C4×D8C2×D5911yescharacteristic
119#436—1181no—
120#436—1181no—
121#367—591no—
122#357—591no—
123#357—591no—
124#346—591no—
125#346C2×D5911yescharacteristic
126#334—591no—
127#334C2×D5911yescharacteristic
128D4⋊C8—1181no—
129D4⋊C8—1181no—
130#322—591no—
131C59#1208911yescharacteristic, complemented, Sylow
132C2×D8—591no—
133C2×D8C4×D5911yescharacteristic
134C2×M16—2361no—
135C2×M16—2361no—
136C4×D4—21no—
137C4×D4—1181no—
138D16—1182no—
139C2×C16—21no—
140C2×C16—591no—
141C2×C16—591no—
142#107—591no—
143#107C4×D5911yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1181no—
146#102—21no—
147#102—1181no—
148C22⋊C8—2361no—
149C22⋊C8—2361no—
150C4×C8—591no—
151C4×C8—591no—
152C4×C8C22×D5911yescharacteristic
153C2×D4—21no—
154C2×D4—1181no—
155C22×C4—41no—
156C22×C4—2361no—
157D8—21no—
158D8—21no—
159D8—1181no—
160D8—1181no—
161M16—4722no—
162M16—4722no—
163C2×C8—41no—
164C2×C8—1181no—
165C2×C8—1181no—
166C2×C8—2361no—
167C2×C8C2×C4×D5911yescharacteristic
168C2×C8#1174568711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1181no—
171C22⋊C4—41no—
172C22⋊C4—2361no—
173C42D4×D5911yescharacteristic
174C16—41no—
175C16—592no—
176C16—1181no—
177C23—41no—
178C23—2361no—
179D4—41no—
180D4—41no—
181D4—2361no—
182D4—2361no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—4721no—
187C2×C4—4721no—
188C2×C4—2361no—
189C2×C4(C22⋊D59)⋊C411yescharacteristic
190C2×C4#1427375111yescharacteristic
191C2×C4#1427375411yescharacteristic
192C8—21no—
193C8—81no—
194C8—4721no—
195C8—2361no—
196C8—2361no—
197C8M16×D5911yescharacteristic
198C8#1427373311yescharacteristic
199C22—81no—
200C22—81no—
201C22—81no—
202C22—4721no—
203C22—4721no—
204C22—4721no—
205C22#1683751111yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—4721no—
210C4#1683747711yescharacteristic
211C4#1683752111yescharacteristic
212C2—161no—
213C2—9441no—
214C2#2142300711yescharacteristic
215C2#2142599111yescharacteristic
216C2#2142599711yescharacteristic
subgroupG/H
centerC22#16837511
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1333C2
socleC22×C59#442
prime radicalC2×#14273668C4
cosocleC4×C8×C59C23
hypercenter#1333D59
hypocenterC59#12089
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #21423010C2semidirect
3. #21423009C2semidirect
4. #21422161C2semidirect
5. #21422064C2semidirect
6. #21422018C2semidirect
7. C59×#1333C2semidirect
67. C59⋊C16M16semidirect
109. #1333D59semidirect
131. C59#12089semidirect

Maximal subgroups

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

subgroupindex
#214259702
#214230102
#214230092
#214221612
#214220642
#214220182
C59×#13332
#1208959