Finite Group Databasebeta

← #28502849

Subgroups — #28502849

Order 15104.

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

normal subgroups only

This group has 234 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#21426141C211yesmaximal, characteristic
2#21423694C211yesmaximal, characteristic, complemented
3D8×C59⋊C8C211yesmaximal, characteristic
4#21422159C211yesmaximal, characteristic, complemented
5#21422151C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#2224C211yesmaximal, characteristic
8C2×#14273671—21no—
9#16837798—21no—
10#16837782C2211yescharacteristic
11#16837752—21no—
12D4×C59⋊C8—21no—
13C4⋊(C59⋊C16)C2211yescharacteristic
14#16837205—21no—
15#16837155C2211yescharacteristic
16D4⋊(C59⋊C8)—21no—
17C8×C59⋊C8C2211yescharacteristic
18C59×#500C2211yescharacteristic
19C4×D8×C59C2211yescharacteristic
20C59×#366—21no—
21C59×C4⋊C16C2211yescharacteristic
22C59×#357—21no—
23C2×Dic236—21no—
24C2×Dic236—21no—
25C22×C59⋊C8—41no—
26#14273727—21no—
27#14273671—42no—
28C2×C59⋊C16—21no—
29#14273658—42no—
30C4⋊(C59⋊C8)—21no—
31#14273651—41no—
32#14273651—41no—
33C22⋊(C59⋊C8)—41no—
34C4×C59⋊C8D411yescharacteristic
35C4×C59⋊C8D411yescharacteristic
36C2×D8×C59D411yescharacteristic
37C4⋊Q8×C59—21no—
38C4×D4×C59—21no—
39C2×C16×C59—21no—
40#107×C59—21no—
41#107×C59—21no—
42#107×C59D411yescharacteristic
43#102×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×D4×C59—21no—
47C22×C4×C59—41no—
48C2×Dic118—41no—
49D8×C59—21no—
50D8×C59SD1612yes—
51C2×C8×C59C2×D411yescharacteristic
52C2×C8×C59#4111yescharacteristic
53C4⋊C4×C59—21no—
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C22⋊C4×C59—41no—
57C42×C59C2×D411yescharacteristic
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8—41no—
62C2×C59⋊C8—82no—
63C2×C59⋊C8D811yescharacteristic
64C2×C59⋊C8D811yescharacteristic
65Dic236—41no—
66Dic236—41no—
67Dic236—81no—
68C16×C59—41no—
69C59⋊C16—41no—
70C23×C59—41no—
71Q8×C59—81no—
72Q8×C59—81no—
73D4×C59—22no—
74D4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59—81no—
77C2×C4×C59—81no—
78C2×C4×C59—81no—
79C2×C4×C59#12111yescharacteristic
80C2×C4×C59C2×D811yescharacteristic
81C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
82C2×Dic59—81no—
83Dic118—81no—
84Dic118—81no—
85C8×C59—21no—
86C8×C59C2×SD1611yescharacteristic
87C8×C59#13511yescharacteristic
88C59⋊C8—22no—
89C59⋊C8—21no—
90C59⋊C8—21no—
91C59⋊C8—81no—
92#12422—591nomaximal, Sylow
93C22×C59—42no—
94C22×C59—81no—
95C22×C59—81no—
96C22×C59#45911yescharacteristic
97C4×C59—21no—
98C4×C59—81no—
99C4×C59—161no—
100C4×C59#46511yescharacteristic
101C4×C59#47111yescharacteristic
102C4×C59#50811yescharacteristic
103C4×C59#50911yescharacteristic
104Dic59—161no—
105#2224D5911yescharacteristic
106#2208—591no—
107#1707—591no—
108C8×D8—591no—
109#1367—591no—
110#1364—591no—
111#1362—591no—
112C2×C59—82no—
113C2×C59#166511yescharacteristic
114C2×C59#221011yescharacteristic
115C2×C59#221411yescharacteristic
116C2×SD32—1181no—
117#500C2×D5911yescharacteristic
118#498—1181no—
119#494—591no—
120#487—1181no—
121C4×D8C2×D5911yescharacteristic
122C8×D4—1181no—
123#366—21no—
124C4⋊C16—591no—
125C4⋊C16C2×D5911yescharacteristic
126#358—1181no—
127#357—21no—
128#335—591no—
129D4⋊C8—1181no—
130C82—591no—
131C59#1242211yescharacteristic, complemented, Sylow
132C2×Q16—1181no—
133C2×Q16—1181no—
134C2×D8C22⋊D5911yescharacteristic
135C22×C8—2361no—
136C4⋊Q8—21no—
137C4⋊Q8—1181no—
138C4×D4—21no—
139SD32—2362no—
140C2×C16—21no—
141C2×C16—1181no—
142#107—21no—
143#107—21no—
144#107C22⋊D5911yescharacteristic
145#106—2362no—
146C4⋊C8—1181no—
147#103—2361no—
148#103—2361no—
149#102—21no—
150C22⋊C8—2361no—
151C4×C8—591no—
152C4×C8—591no—
153C4×C8C22×D5911yescharacteristic
154C2×Q8—41no—
155C2×Q8—2361no—
156C2×D4—21no—
157C22×C4—41no—
158Q16—4721no—
159Q16—2361no—
160Q16—2361no—
161D8—21no—
162D8#1174567112yes—
163C2×C8—591no—
164C2×C8—591no—
165C2×C8—1181no—
166C2×C8—4722no—
167C2×C8—2361no—
168C2×C8#1174568711yescharacteristic
169C2×C8C2×C22⋊D5911yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C22⋊C4—41no—
176C42D4×D5911yescharacteristic
177C16—41no—
178C16—2361no—
179C23—41no—
180Q8—81no—
181Q8—81no—
182Q8—4721no—
183Q8—4721no—
184D4—22no—
185D4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—4721no—
191C2×C4#1427369911yescharacteristic
192C2×C4D8×D5911yescharacteristic
193C2×C4#1427375111yescharacteristic
194C8—21no—
195C8—1182no—
196C8—1181no—
197C8—1181no—
198C8—4721no—
199C8C2×#1174567111yescharacteristic
200C8#1427374711yescharacteristic
201C22—42no—
202C22—81no—
203C22—81no—
204C22#1683759511yescharacteristic
205C4—21no—
206C4—81no—
207C4—161no—
208C4—9441no—
209C4#1683763311yescharacteristic
210C4#1683766711yescharacteristic
211C4#1683783111yescharacteristic
212C4#1683783611yescharacteristic
213C2—82no—
214C2#2142343911yescharacteristic
215C2#2142614911yescharacteristic
216C2#2142617511yescharacteristic
subgroupG/H
centerC22#16837595
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2224C2
socleC22×C59#459
prime radicalC2×D8×C59D4
hypercenter#2224D59
hypocenterC59#12422
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
2. #21423694C2semidirect
4. #21422159C2semidirect
5. #21422151C2semidirect
6. #21422146C2semidirect
131. C59#12422semidirect

Maximal subgroups

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

subgroupindex
#214261412
#214236942
D8×C59⋊C82
#214221592
#214221512
#214221462
C59×#22242
#1242259