Finite Group Databasebeta

← #28502850

Subgroups — #28502850

Order 15104.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426230C211yesmaximal, characteristic
2#21426142C211yesmaximal, characteristic
3#21423694C211yesmaximal, characteristic, complemented
4#21422890C211yesmaximal, characteristic
5#21422159C211yesmaximal, characteristic, complemented
6#21422144C211yesmaximal, characteristic, complemented
7C59×#1364C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9#16837798—21no—
10#16837782C2211yescharacteristic
11#16837752—21no—
12C4×D472C2211yescharacteristic
13#16837467—21no—
14#16837227—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837207—21no—
17#16837202—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×D472D411yescharacteristic
26C2×C8×D59—41no—
27#14273727—21no—
28C4×D236—21no—
29#14273672—42no—
30C2×C59⋊C16—21no—
31#14273661D411yescharacteristic
32#14273658—42no—
33#14273651—41no—
34#14273651—41no—
35#14273649—21no—
36(C2×D59)⋊C8—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
39C4⋊Q8×C59—21no—
40C2×C16×C59—21no—
41#107×C59—21no—
42#107×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×D236—21no—
47C2×C4×D59—41no—
48C2×Dic118—41no—
49C2×C8×C59—41no—
50C2×C8×C59C2×D411yescharacteristic
51C2×C8×C59#4111yescharacteristic
52C4⋊C4×C59—41no—
53C4⋊C4×C59—41no—
54C42×C59C2×D411yescharacteristic
55(C2×D59)⋊C4—41no—
56C4⋊Dic59—21no—
57C4⋊Dic59—41no—
58C4⋊Dic59—41no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8D811yescharacteristic
61C2×C59⋊C8D811yescharacteristic
62Dic236—41no—
63Dic236—41no—
64Dic236—81no—
65D472—21no—
66D472SD1612yes—
67C8×D59—82no—
68C16×C59—41no—
69C59⋊C16—41no—
70C22×D59—41no—
71Q8×C59—81no—
72Q8×C59—81no—
73C2×C4×C59—81no—
74C2×C4×C59#12111yescharacteristic
75C2×C4×C59C2×D811yescharacteristic
76C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
77C2×Dic59—41no—
78C2×Dic59—81no—
79D236—22no—
80D236—41no—
81C4×D59—81no—
82C4×D59—81no—
83Dic118—81no—
84Dic118—81no—
85C8×C59—21no—
86C8×C59—81no—
87C8×C59C2×SD1611yescharacteristic
88C8×C59#13511yescharacteristic
89C59⋊C8—22no—
90C59⋊C8—21no—
91C59⋊C8—21no—
92#12422—591nomaximal, Sylow
93C22×C59#45911yescharacteristic
94C2×D59—42no—
95C2×D59—81no—
96C2×D59—81no—
97C4×C59—21no—
98C4×C59—161no—
99C4×C59#46511yescharacteristic
100C4×C59#47111yescharacteristic
101C4×C59#50811yescharacteristic
102C4×C59#50911yescharacteristic
103Dic59—81no—
104Dic59—161no—
105#2224—591no—
106#2208—591no—
107#1707—591no—
108C8×D8—591no—
109#1367—591no—
110#1364D5911yescharacteristic, complemented
111#1362—591no—
112C2×C59#166511yescharacteristic
113C2×C59#221011yescharacteristic
114C2×C59#221411yescharacteristic
115D59—82no—
116C2×SD32—1181no—
117#500C2×D5911yescharacteristic
118#498—1181no—
119#494—591no—
120#487—1181no—
121C4×D8—591no—
122C8×D4—1181no—
123#366—1181no—
124C4⋊C16—591no—
125C4⋊C16C2×D5911yescharacteristic
126#358—1181no—
127#357—1181no—
128#335C2×D5911yescharacteristic
129D4⋊C8—1181no—
130C82—591no—
131C59#1242211yescharacteristic, complemented, Sylow
132C2×Q16—1181no—
133C2×Q16—1181no—
134C2×D8—591no—
135C22×C8—2361no—
136C4⋊Q8—21no—
137C4⋊Q8—1181no—
138C4×D4—1181no—
139SD32—2362no—
140C2×C16—21no—
141C2×C16—1181no—
142#107—21no—
143#107—21no—
144#107—591no—
145#106—2362no—
146C4⋊C8—21no—
147#103—2361no—
148#103—2361no—
149#102—1181no—
150C22⋊C8—2361no—
151C4×C8—591no—
152C4×C8—591no—
153C4×C8C22×D5911yescharacteristic
154C2×Q8—41no—
155C2×Q8—2361no—
156C2×D4—1181no—
157C22×C4—2361no—
158Q16—4721no—
159Q16—2361no—
160Q16—2361no—
161D8—592no—
162D8—1181no—
163C2×C8—41no—
164C2×C8—591no—
165C2×C8—591no—
166C2×C8—1181no—
167C2×C8—4722no—
168C2×C8#1174568511yescharacteristic
169C2×C8D4×D5911yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1181no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C22⋊C4—2361no—
176C42D4×D5911yescharacteristic
177C16—41no—
178C16—2361no—
179C23—2361no—
180Q8—81no—
181Q8—81no—
182Q8—4721no—
183Q8—4721no—
184D4—1182no—
185D4—2361no—
186C2×C4—81no—
187C2×C4—4721no—
188C2×C4—4721no—
189C2×C4—4721no—
190C2×C4—2361no—
191C2×C4#1427370011yescharacteristic
192C2×C4D8×D5911yescharacteristic
193C2×C4#1427375411yescharacteristic
194C8—21no—
195C8—81no—
196C8—1182no—
197C8—1181no—
198C8—1181no—
199C8SD16×D5911yescharacteristic
200C8#1427375011yescharacteristic
201C22—4721no—
202C22—4721no—
203C22—2362no—
204C22#1683759611yescharacteristic
205C4—21no—
206C4—161no—
207C4—9441no—
208C4—4721no—
209C4#1683763411yescharacteristic
210C4#1683766811yescharacteristic
211C4#1683783311yescharacteristic
212C4#1683783611yescharacteristic
213C2—4722no—
214C2#2142344011yescharacteristic
215C2#2142615011yescharacteristic
216C2#2142617611yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59#459
prime radicalC2×D472D4
hypercenter#1364D59
hypocenterC59#12422
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
3. #21423694C2semidirect
5. #21422159C2semidirect
6. #21422144C2semidirect
7. C59×#1364C2semidirect
110. #1364D59semidirect
131. C59#12422semidirect

Maximal subgroups

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

subgroupindex
#214262302
#214261422
#214236942
#214228902
#214221592
#214221442
C59×#13642
#1242259