Finite Group Databasebeta

← #28502826

Subgroups — #28502826

Order 15104.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426248C211yesmaximal, characteristic, complemented
2#21426136C211yesmaximal, characteristic
3#21423688C211yesmaximal, characteristic
4#21422160C211yesmaximal, characteristic, complemented
5#21422155C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C8×Q16×C59C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9#16837807C2211yescharacteristic
10#16837779C2211yescharacteristic
11#16837775—21no—
12#16837749—21no—
13#16837227—21no—
14C4⋊(C59⋊C16)C2211yescharacteristic
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837206—21no—
17#16837202—21no—
18C8×Q8×C59—21no—
19C4×Q16×C59C2211yescharacteristic
20C59×#335C2211yescharacteristic
21C59×Q8⋊C8—21no—
22C82×C59C2211yescharacteristic
23C2×#11745660—42no—
24C2×D472—21no—
25C2×D472—21no—
26#14273727—21no—
27#14273724—21no—
28#14273670—42no—
29C2×C59⋊C16—21no—
30C2×C59⋊C16—21no—
31#14273661—21no—
32#14273661—21no—
33#14273649—41no—
34#14273649—41no—
35C2×Q16×C59D411yescharacteristic
36C4×Q8×C59—21no—
37#107×C59D411yescharacteristic
38C4⋊C8×C59—21no—
39C4⋊C8×C59—41no—
40#103×C59—21no—
41C4×C8×C59—41no—
42C4×C8×C59D411yescharacteristic
43C4×C8×C59D411yescharacteristic
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—21no—
46C2×D236—41no—
47C2×D236—41no—
48C2×D236—41no—
49C2×Dic118—41no—
50Q16×C59—21no—
51Q16×C59SD1612yescomplemented
52C2×C8×C59—21no—
53C2×C8×C59—41no—
54C2×C8×C59D811yescharacteristic
55C2×C8×C59D811yescharacteristic
56C2×C8×C59C2×D411yescharacteristic
57C2×C8×C59#4111yescharacteristic
58C4⋊C4×C59—21no—
59C4⋊C4×C59—41no—
60C42×C59—41no—
61C42×C59C2×D411yescharacteristic
62C4⋊Dic59—41no—
63C4⋊Dic59—41no—
64D472—41no—
65D472—41no—
66D472—81no—
67#11745660—84no—
68C59⋊C16—41no—
69C59⋊C16—41no—
70C22×D59—81no—
71Q8×C59—22no—
72Q8×C59—41no—
73C2×C4×C59—41no—
74C2×C4×C59—41no—
75C2×C4×C59#12111yescharacteristic
76C2×C4×C59C2×D811yescharacteristic
77C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
78C2×Dic59—81no—
79D236—82no—
80D236—81no—
81D236—81no—
82D236—81no—
83D236—81no—
84Dic118—82no—
85C8×C59—22no—
86C8×C59—21no—
87C8×C59—21no—
88C8×C59—21no—
89C8×C59—81no—
90C8×C59C2×SD1611yescharacteristic
91C8×C59#13511yescharacteristic
92#12419—591nomaximal, Sylow
93C22×C59#45911yescharacteristic
94C2×D59—161no—
95C2×D59—161no—
96C2×D59—161no—
97C4×C59—21no—
98C4×C59—42no—
99C4×C59—81no—
100C4×C59#46511yescharacteristic
101C4×C59#47111yescharacteristic
102C4×C59#50811yescharacteristic
103C4×C59#51011yescharacteristic
104Dic59—161no—
105#2227—591no—
106#2207—591no—
107#1706—591no—
108C8×Q16D5911yescharacteristic, complemented
109#1367—591no—
110#1365—591no—
111#1361—591no—
112C2×C59#166511yescharacteristic
113C2×C59#221111yescharacteristic
114C2×C59#221311yescharacteristic
115D59—321no—
116C2×SD32—1181no—
117#500—591no—
118#493—591no—
119#492—1181no—
120#486—1181no—
121C8×Q8—21no—
122C4×Q16C2×D5911yescharacteristic
123#366—1181no—
124C4⋊C16—591no—
125C4⋊C16—591no—
126#358—1181no—
127#357—1181no—
128#335C2×D5911yescharacteristic
129Q8⋊C8—21no—
130C82C2×D5911yescharacteristic
131C59#1241911yescharacteristic, complemented, Sylow
132C2×Q16D23611yescharacteristic
133C2×SD16—2362no—
134C2×D8—1181no—
135C2×D8—1181no—
136C4⋊Q8—1181no—
137#127—1181no—
138C4×Q8—21no—
139SD32—2362no—
140C2×C16—1181no—
141C2×C16—1181no—
142#107—1181no—
143#107—1181no—
144#107D23611yescharacteristic
145C4⋊C8—21no—
146C4⋊C8—41no—
147#103—21no—
148#102—2361no—
149#102—2361no—
150C4×C8—41no—
151C4×C8C22⋊D5911yescharacteristic
152C4×C8C22⋊D5911yescharacteristic
153C4×C8C22×D5911yescharacteristic
154C2×Q8—21no—
155C2×Q8—2361no—
156C2×D4—2361no—
157C2×D4—2361no—
158C2×D4—2361no—
159Q16—21no—
160Q16#1174566012yescomplemented
161SD16—4724no—
162D8—4721no—
163D8—2361no—
164D8—2361no—
165C2×C8—21no—
166C2×C8—41no—
167C2×C8#1174566911yescharacteristic
168C2×C8#1174566911yescharacteristic
169C2×C8C2×D23611yescharacteristic
170C2×C8#1174568511yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—41no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C42—41no—
176C42C2×C22⋊D5911yescharacteristic
177C16—2361no—
178C16—2361no—
179C23—4721no—
180Q8—22no—
181Q8—41no—
182Q8—4722no—
183D4—4722no—
184D4—4721no—
185D4—4721no—
186D4—4721no—
187D4—4721no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—4721no—
191C2×C4#1427370111yescharacteristic
192C2×C4C2×#1174566911yescharacteristic
193C2×C4#1427375211yescharacteristic
194C8—22no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—81no—
199C8C2×#1174566011yescharacteristic
200C8#1427374911yescharacteristic
201C22—9441no—
202C22—9441no—
203C22—9441no—
204C22#1683759711yescharacteristic
205C4—21no—
206C4—42no—
207C4—81no—
208C4—9441no—
209C4#1683763111yescharacteristic
210C4#1683766511yescharacteristic
211C4#1683783211yescharacteristic
212C4#1683784311yescharacteristic
213C2—18881no—
214C2#2142343711yescharacteristic
215C2#2142615811yescharacteristic
216C2#2142617211yescharacteristic
subgroupG/H
centerC22#16837597
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC8×Q16×C59C2
socleC22×C59#459
prime radical#16837779C22
hypercenterC8×Q16D59
hypocenterC59#12419
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ C8×Q16

Complements

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

NH ≅ G/Ntype
1. #21426248C2semidirect
4. #21422160C2semidirect
5. #21422155C2semidirect
7. C8×Q16×C59C2semidirect
51. Q16×C59SD16semidirect
108. C8×Q16D59semidirect
131. C59#12419semidirect
160. Q16#11745660semidirect

Maximal subgroups

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

subgroupindex
#214262482
#214261362
#214236882
#214221602
#214221552
#214221412
C8×Q16×C592
#1241959