Finite Group Databasebeta

← #28500684

Subgroups — #28500684

Order 15104.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4×#14273672C211yesmaximal, characteristic
2#21422992C211yesmaximal, characteristic, complemented
3#21422980C211yesmaximal, characteristic
4#21422059C211yesmaximal, characteristic, complemented
5#21422056C211yesmaximal, characteristic
6#21422012C211yesmaximal, characteristic, complemented
7C59×#1371C211yesmaximal, characteristic, complemented
8C2×#14273672C411yescharacteristic
9#16837529—21no—
10C4×Dic236C2211yescharacteristic
11C4×D472C2211yescharacteristic
12#16837479—21no—
13#16837230C411yescharacteristic
14#16837207C411yescharacteristic
15#16837204C411yescharacteristic
16C4×C59⋊C16C2211yescharacteristic
17#16837132—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C4×C16×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236C2×C411yescharacteristic
24C2×D472C2×C411yescharacteristic
25C2×#11745659—41no—
26C4×Dic118—21no—
27C4×D236—21no—
28#14273672—22no—
29C2×C59⋊C16—21no—
30#14273661C2×C411yescharacteristic
31#14273661C2×C411yescharacteristic
32Dic59⋊C8—41no—
33#14273652—21no—
34#14273649—21no—
35(C2×D59)⋊C8—41no—
36#14273637—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
39C2×C16×C59C2×C411yescharacteristic
40C2×C16×C59C2×C411yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D236—21no—
45C2×C4×D59—41no—
46C2×Dic118—21no—
47C2×C8×C59—41no—
48C2×C8×C59—41no—
49C2×C8×C59C22×C411yescharacteristic
50C2×C8×C59#4111yescharacteristic
51C42×C59C2×D411yescharacteristic
52(C2×D59)⋊C4—41no—
53C4⋊Dic59—21no—
54C4⋊Dic59—21no—
55Dic59⋊C4—41no—
56C4×Dic59—41no—
57C2×C59⋊C8—21no—
58C2×C59⋊C8D811yescharacteristic
59C2×C59⋊C8D811yescharacteristic
60Dic236—21no—
61Dic236—21no—
62D472—21no—
63D472—21no—
64#11745659—82no—
65C16×C59—21no—
66C16×C59M1612yescomplemented
67C59⋊C16—22no—
68C22×D59—41no—
69C2×C4×C59C4×D411yescharacteristic
70C2×C4×C59C2×D811yescharacteristic
71C2×C4×C59#13511yescharacteristic
72C2×Dic59—41no—
73C2×Dic59—41no—
74C2×Dic59—41no—
75D236—41no—
76D236—41no—
77C4×D59—81no—
78C4×D59—81no—
79Dic118—41no—
80Dic118—41no—
81C8×C59—21no—
82C8×C59—81no—
83C8×C59—81no—
84C8×C59C2×M1611yescharacteristic
85C8×C59#13111yescharacteristic
86C59⋊C8—21no—
87C59⋊C8—21no—
88C59⋊C8—41no—
89#12084—591nomaximal, Sylow
90C22×C59C4×D811yescharacteristic
91C2×D59—81no—
92C2×D59—81no—
93C2×D59—81no—
94C4×C59—21no—
95C4×C59#43611yescharacteristic
96C4×C59#44311yescharacteristic
97C4×C59#50911yescharacteristic
98C4×C59#51011yescharacteristic
99Dic59—81no—
100Dic59—81no—
101Dic59—81no—
102C4×SD32—591no—
103#1591—591no—
104#1589—591no—
105#1371D5911yescharacteristic, complemented
106#1332—591no—
107#1331—591no—
108#1311—591no—
109C2×C59#158911yescharacteristic
110C2×C59#217511yescharacteristic
111C2×C59#217911yescharacteristic
112D59—161no—
113C2×SD32—591no—
114#446—1181no—
115C4×Q16—591no—
116C4×D8—591no—
117#436—1181no—
118#367—591no—
119#358—591no—
120#357—591no—
121C4×C16—591no—
122C4×C16C2×D5911yescharacteristic
123#335C2×D5911yescharacteristic
124#335C2×D5911yescharacteristic
125Q8⋊C8—1181no—
126D4⋊C8—1181no—
127C82—591no—
128C59#1208411yescharacteristic, complemented, Sylow
129C2×Q16—591no—
130C2×D8—591no—
131C2×M16—2361no—
132C4×Q8—1181no—
133C4×D4—1181no—
134SD32—1182no—
135C2×C16—1181no—
136C2×C16C4×D5911yescharacteristic
137C2×C16C4×D5911yescharacteristic
138#107—591no—
139#107—591no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—2361no—
143#103—1181no—
144#102—1181no—
145C22⋊C8—2361no—
146#97—2361no—
147C4×C8—591no—
148C4×C8—591no—
149C4×C8C22×D5911yescharacteristic
150C2×Q8—1181no—
151C2×D4—1181no—
152C22×C4—2361no—
153Q16—1181no—
154Q16—1181no—
155D8—1181no—
156D8—1181no—
157M16—4722no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—591no—
161C2×C8—591no—
162C2×C8—1181no—
163C2×C8C2×C4×D5911yescharacteristic
164C2×C8#1174568911yescharacteristic
165C4⋊C4—1181no—
166C4⋊C4—1181no—
167C4⋊C4—2361no—
168C22⋊C4—2361no—
169C42—2361no—
170C42D4×D5911yescharacteristic
171C16—21no—
172C16—1182no—
173C16#1174565912yescomplemented
174C23—2361no—
175Q8—2361no—
176Q8—2361no—
177D4—2361no—
178D4—2361no—
179C2×C4—4721no—
180C2×C4—4721no—
181C2×C4—2361no—
182C2×C4—2361no—
183C2×C4—2361no—
184C2×C4#1427368911yescharacteristic
185C2×C4D8×D5911yescharacteristic
186C2×C4#1427375011yescharacteristic
187C8—21no—
188C8—81no—
189C8—81no—
190C8—1181no—
191C8—1181no—
192C8—2361no—
193C8C2×#1174565911yescharacteristic
194C8#1427373511yescharacteristic
195C22—4721no—
196C22—4721no—
197C22—4721no—
198C22#1683748411yescharacteristic
199C4—21no—
200C4—4721no—
201C4—4721no—
202C4—4721no—
203C4#1683747911yescharacteristic
204C4#1683751811yescharacteristic
205C4#1683783611yescharacteristic
206C4#1683784411yescharacteristic
207C2—9441no—
208C2#2142298011yescharacteristic
209C2#2142595911yescharacteristic
210C2#2142598111yescharacteristic
subgroupG/H
centerC2×C4D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1371C2
socleC22×C59C4×D8
prime radicalC2×D472C2×C4
cosocleC4×C8×C59C23
hypercenter#1371D59
hypocenterC59#12084
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1371

Complements

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

NH ≅ G/Ntype
2. #21422992C2semidirect
4. #21422059C2semidirect
6. #21422012C2semidirect
7. C59×#1371C2semidirect
66. C16×C59M16semidirect
105. #1371D59semidirect
128. C59#12084semidirect
173. C16#11745659semidirect

Maximal subgroups

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

subgroupindex
C4×#142736722
#214229922
#214229802
#214220592
#214220562
#214220122
C59×#13712
#1208459