Finite Group Databasebeta

← #28509506

Subgroups — #28509506

Order 15104.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#21426414C211yesmaximal, characteristic, complemented
2#21426411C211yesmaximal, characteristic
3#21426358C211yesmaximal, characteristic, complemented
4#21425976C211yesmaximal, characteristic, complemented
5#21422293C211yesmaximal, characteristic, complemented
6#21422285C211yesmaximal, characteristic, complemented
7C59×#1420C211yesmaximal, characteristic, complemented
8#16837834C2211yescharacteristic
9C2×#14273670—21no—
10C2×D944C2211yescharacteristic
11#16837787—21no—
12#16837518—21no—
13#16837242—21no—
14#16837241—21no—
15#16837239—21no—
16D1888—21no—
17#16837236C2211yescharacteristic
18#16837208—21no—
19C16×Dic59C2211yescharacteristic
20C2×Q32×C59C2211yescharacteristic
21C59×M64C2211yescharacteristic
22C59×#368C2211yescharacteristic
23#14273749—21no—
24C2×#11745671—41no—
25C2×D472—21no—
26#14273735—41no—
27#14273716—41no—
28Dic472D411yescharacteristic
29#14273672—21no—
30#14273670—42no—
31D944—21no—
32D944—21no—
33D944D411yescharacteristic
34C2×C59⋊C16D411yescharacteristic
35#14273653—41no—
36C8×Dic59D411yescharacteristic
37C59⋊C32—21no—
38C2×Q16×C59—21no—
39Q32×C59—21no—
40Q32×C59—21no—
41C2×C16×C59C2311yescharacteristic
42#108×C59—21no—
43C32×C59—21no—
44C2×Q8×C59—41no—
45#11745685—41no—
46C2×D236—41no—
47Q16×C59—41no—
48Q16×C59—41no—
49M16×C59—41no—
50C2×C8×C59C2×D411yescharacteristic
51#11745671—81no—
52#11745671—81no—
53(C2×D59)⋊C4—82no—
54C4×Dic59D811yescharacteristic
55C2×C59⋊C8D811yescharacteristic
56Dic236—21no—
57D472—21no—
58D472—41no—
59D472—41no—
60#11745660—41no—
61#11745659—81no—
62C8×D59—81no—
63C16×C59C2×D411yescharacteristic
64C16×C59#4111yescharacteristic
65C59⋊C16—21no—
66C22×D59—81no—
67Q8×C59—81no—
68Q8×C59—81no—
69C2×C4×C59—81no—
70C2×C4×C59C2×D811yescharacteristic
71C22⋊D59—81no—
72C2×Dic59—21no—
73D236—41no—
74D236—81no—
75D236—81no—
76C4×D59—81no—
77Dic118—41no—
78C8×C59—81no—
79C8×C59#12111yescharacteristic
80C8×C59AGL(1,ℤ/8)11yescharacteristic
81C59⋊C8D1611yescharacteristic
82C59⋊C8D1611yescharacteristic
83#13691—591nomaximal, Sylow
84C22×C59#50911yescharacteristic
85C2×D59—81no—
86C2×D59—162no—
87C2×D59—161no—
88C4×C59—161no—
89C4×C59#45911yescharacteristic
90C4×C59C2×D1611yescharacteristic
91Dic59—41no—
92Dic59—81no—
93#2264—591no—
94#2263—591no—
95#2249—591no—
96#2178—591no—
97#1422—591no—
98#1420D5911yescharacteristic, complemented
99#1419—591no—
100C2×C59—21no—
101C2×C59#220611yescharacteristic
102D59—161no—
103D59—321no—
104#508—591no—
105C2×Q32C2×D5911yescharacteristic
106C2×SD32—1181no—
107C2×D16—591no—
108#495—1181no—
109#443—1181no—
110Q64—1181no—
111SD64—1181no—
112SD64—1181no—
113D32—1181no—
114M64—591no—
115M64C2×D5911yescharacteristic
116#368C2×D5911yescharacteristic
117#359—1181no—
118C4×C16—591no—
119C59#1369111yescharacteristic, complemented, Sylow
120#135—1181no—
121C2×Q16—21no—
122C2×SD16—2361no—
123C2×D8—1181no—
124#131—2361no—
125#124—2361no—
126Q32—21no—
127Q32—21no—
128Q32—591no—
129SD32—1181no—
130SD32—2362no—
131D16—591no—
132D16—1181no—
133D16—1181no—
134C2×C16—591no—
135C2×C16C22×D5911yescharacteristic
136#108—21no—
137#104—2361no—
138C4×C8—591no—
139C32—21no—
140C32—1181no—
141#41—2361no—
142C2×Q8—41no—
143C2×D4—2361no—
144Q16—41no—
145Q16—41no—
146Q16—1181no—
147SD16—4721no—
148SD16—4721no—
149SD16—2361no—
150D8—1181no—
151D8—2361no—
152D8—2361no—
153M16—41no—
154M16—4721no—
155C2×C8—591no—
156C2×C8—4721no—
157C2×C8D4×D5911yescharacteristic
158C22⋊C4—4722no—
159C42—591no—
160C16—1181no—
161C16#1174568511yescharacteristic
162C16D4×D5911yescharacteristic
163C23—4721no—
164Q8—81no—
165Q8—81no—
166Q8—2361no—
167D4—4721no—
168D4—4721no—
169D4—4721no—
170D4—2361no—
171C2×C4—81no—
172C2×C4—1181no—
173C2×C4—4721no—
174C2×C4D8×D5911yescharacteristic
175C8—81no—
176C8—591no—
177C8—591no—
178C8#1427370011yescharacteristic
179C8#1427375411yescharacteristic
180C22—9442no—
181C22—9441no—
182C22—4721no—
183C22#1683783911yescharacteristic
184C4—161no—
185C4—4721no—
186C4—2361no—
187C4#1683759611yescharacteristic
188C4D16×D5911yescharacteristic
189C2—21no—
190C2—18881no—
191C2—9441no—
192C2#2142613011yescharacteristic
subgroupG/H
centerC2#21426130
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC59×#1420C2
socleC2×C59#2206
prime radical#21426411C2
hypercenter#1420D59
hypocenterC59#13691
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1420

Complements

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

NH ≅ G/Ntype
1. #21426414C2semidirect
3. #21426358C2semidirect
4. #21425976C2semidirect
5. #21422293C2semidirect
6. #21422285C2semidirect
7. C59×#1420C2semidirect
98. #1420D59semidirect
119. C59#13691semidirect

Maximal subgroups

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

subgroupindex
#214264142
#214264112
#214263582
#214259762
#214222932
#214222852
C59×#14202
#1369159