Finite Group Databasebeta

← #28503225

Subgroups — #28503225

Order 15104.

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

normal subgroups only

This group has 234 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#21426238C211yesmaximal, characteristic, complemented
2#21426136C211yesmaximal, characteristic
3#21423688C211yesmaximal, characteristic
4#21422166C211yesmaximal, characteristic, complemented
5#21422152C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C59×#1591C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9#16837807C2211yescharacteristic
10#16837779C2211yescharacteristic
11#16837775—21no—
12#16837749—21no—
13#16837229—21no—
14C4⋊(C59⋊C16)C2211yescharacteristic
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837206—21no—
17#16837202—21no—
18C59×#446—21no—
19C4×Q16×C59C2211yescharacteristic
20C59×#335C2211yescharacteristic
21C59×Q8⋊C8—21no—
22C82×C59C2211yescharacteristic
23C2×#11745660—41no—
24C2×#11745660—41no—
25C2×D472—21no—
26C2×D472—21no—
27#14273727—21no—
28#14273724—21no—
29#14273670—42no—
30C2×C59⋊C16—21no—
31C2×C59⋊C16—21no—
32#14273661—21no—
33#14273661—21no—
34#14273649—42no—
35C2×Q16×C59D411yescharacteristic
36C4×Q8×C59—21no—
37#107×C59D411yescharacteristic
38C4⋊C8×C59—21no—
39C4⋊C8×C59—41no—
40#103×C59—21no—
41#97×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×C59—21no—
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—82no—
68#11745660—82no—
69C59⋊C16—41no—
70C59⋊C16—41no—
71C22×D59—81no—
72Q8×C59—41no—
73Q8×C59—41no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59#12111yescharacteristic
77C2×C4×C59C2×D811yescharacteristic
78C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
79C2×Dic59—81no—
80D236—82no—
81D236—81no—
82D236—81no—
83D236—81no—
84D236—81no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59—21no—
89C8×C59—41no—
90C8×C59—81no—
91C8×C59SD3212yescomplemented
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C8×C59#13711yescharacteristic
94#12484—591nomaximal, Sylow
95C22×C59#45911yescharacteristic
96C2×D59—161no—
97C2×D59—161no—
98C2×D59—161no—
99C4×C59—21no—
100C4×C59—81no—
101C4×C59—81no—
102C4×C59#46811yescharacteristic
103C4×C59#47111yescharacteristic
104C4×C59C2×SD3211yescharacteristic
105C4×C59#50911yescharacteristic
106Dic59—161no—
107#2225—591no—
108#2207—591no—
109#1706—591no—
110#1591D5911yescharacteristic, complemented
111#1370—591no—
112#1364—591no—
113#1361—591no—
114C2×C59#168511yescharacteristic
115C2×C59#220711yescharacteristic
116C2×C59#221211yescharacteristic
117D59—321no—
118C2×SD32—1181no—
119#500—591no—
120#493—591no—
121#492—1181no—
122#486—1181no—
123#446—21no—
124C4×Q16C2×D5911yescharacteristic
125#367—1181no—
126C4⋊C16—591no—
127C4⋊C16—591no—
128#358—1181no—
129#357—1181no—
130#335C2×D5911yescharacteristic
131Q8⋊C8—21no—
132C82C2×D5911yescharacteristic
133C59#1248411yescharacteristic, complemented, Sylow
134C2×Q16D23611yescharacteristic
135C2×SD16—2361no—
136C2×SD16—2361no—
137C2×D8—1181no—
138C2×D8—1181no—
139C4⋊Q8—1181no—
140#127—1181no—
141C4×Q8—21no—
142SD32—2362no—
143C2×C16—1181no—
144C2×C16—1181no—
145#107—1181no—
146#107—1181no—
147#107D23611yescharacteristic
148C4⋊C8—21no—
149C4⋊C8—41no—
150#103—21no—
151#102—2362no—
152#97—41no—
153C4×C8C22⋊D5911yescharacteristic
154C4×C8C22⋊D5911yescharacteristic
155C4×C8C22×D5911yescharacteristic
156C2×Q8—21no—
157C2×Q8—2361no—
158C2×D4—2361no—
159C2×D4—2361no—
160C2×D4—2361no—
161Q16—21no—
162Q16—21no—
163SD16—4722no—
164SD16—4722no—
165D8—4721no—
166D8—2361no—
167D8—2361no—
168C2×C8—21no—
169C2×C8—41no—
170C2×C8#1174566911yescharacteristic
171C2×C8#1174566911yescharacteristic
172C2×C8C2×D23611yescharacteristic
173C2×C8#1174568511yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—2361no—
177C4⋊C4—2361no—
178C42—41no—
179C42C2×C22⋊D5911yescharacteristic
180C16—2361no—
181C16—2361no—
182C23—4721no—
183Q8—41no—
184Q8—41no—
185Q8—4721no—
186Q8—4721no—
187D4—4722no—
188D4—4721no—
189D4—4721no—
190D4—4721no—
191D4—4721no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—4721no—
195C2×C4#1427370111yescharacteristic
196C2×C4C2×#1174566911yescharacteristic
197C2×C4#1427375211yescharacteristic
198C8—21no—
199C8—21no—
200C8—41no—
201C8—81no—
202C8#1427367012yescomplemented
203C8#1427375511yescharacteristic
204C8#1427376011yescharacteristic
205C22—9441no—
206C22—9441no—
207C22—9441no—
208C22#1683759711yescharacteristic
209C4—21no—
210C4—81no—
211C4—81no—
212C4—9441no—
213C4#1683764811yescharacteristic
214C4#1683766511yescharacteristic
215C4C2×#1427367011yescharacteristic
216C4#1683783711yescharacteristic
217C2—18881no—
218C2#2142356111yescharacteristic
219C2#2142613611yescharacteristic
220C2#2142616511yescharacteristic
subgroupG/H
centerC22#16837597
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1591C2
socleC22×C59#459
prime radical#16837779C22
hypercenter#1591D59
hypocenterC59#12484
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1591

Complements

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

NH ≅ G/Ntype
1. #21426238C2semidirect
4. #21422166C2semidirect
5. #21422152C2semidirect
7. C59×#1591C2semidirect
91. C8×C59SD32semidirect
110. #1591D59semidirect
133. C59#12484semidirect
202. C8#14273670semidirect

Maximal subgroups

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

subgroupindex
#214262382
#214261362
#214236882
#214221662
#214221522
#214221412
C59×#15912
#1248459