Finite Group Databasebeta

← #28502941

Subgroups — #28502941

Order 15104.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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#21426353C211yesmaximal, characteristic, complemented
2#21426172C211yesmaximal, characteristic
3#21425933C211yesmaximal, characteristic, complemented
4#21422171C211yesmaximal, characteristic, complemented
5#21422135C211yesmaximal, characteristic, complemented
6#21422132C211yesmaximal, characteristic, complemented
7C59×#2214C211yesmaximal, characteristic, complemented
8#16837837—21no—
9C2×#14273670—21no—
10C2×#14273670—21no—
11#16837819—21no—
12#16837779C2211yescharacteristic
13#16837223C2211yescharacteristic
14#16837223C2211yescharacteristic
15#16837214—21no—
16#16837186—21no—
17C4×C59⋊C16C2211yescharacteristic
18C59×#510—21no—
19C59×#494C2211yescharacteristic
20C59×#443C2211yescharacteristic
21C59×#364C2211yescharacteristic
22C59×#362—21no—
23C2×D472—21no—
24C2×D472—21no—
25#14273724—21no—
26#14273670—42no—
27#14273670—41no—
28#14273670—41no—
29#14273670—41no—
30#14273668—41no—
31#14273667—21no—
32#14273667—21no—
33#14273667—41no—
34C2×C59⋊C16—41no—
35C2×C59⋊C16D411yescharacteristic
36C2×C59⋊C16D411yescharacteristic
37#135×C59D411yescharacteristic
38C2×Q16×C59—21no—
39C2×Q16×C59—21no—
40#131×C59—21no—
41C4⋊Q8×C59—21no—
42Q32×C59—41no—
43SD32×C59—41no—
44M32×C59—21no—
45#108×C59D411yescharacteristic
46#104×C59—21no—
47C4×C8×C59C2311yescharacteristic
48#41×C59—21no—
49C2×Q8×C59—41no—
50C2×D236—41no—
51C2×D236—81no—
52Q16×C59—41no—
53Q16×C59—41no—
54Q16×C59—41no—
55Q16×C59—41no—
56Q16×C59D811yescharacteristic
57SD16×C59—21no—
58D8×C59D811yescharacteristic
59M16×C59—21no—
60M16×C59—41no—
61C2×C8×C59—41no—
62C2×C8×C59C2×D411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—81no—
65C42×C59#4111yescharacteristic
66D472—41no—
67D472—41no—
68D472—41no—
69D472—41no—
70C16×C59—41no—
71C59⋊C16—21no—
72C59⋊C16—41no—
73C59⋊C16—41no—
74C59⋊C16D811yescharacteristic
75C59⋊C16D811yescharacteristic
76C22×D59—81no—
77Q8×C59—21no—
78Q8×C59—81no—
79Q8×C59—81no—
80D4×C59—21no—
81D4×C59—41no—
82C2×C4×C59—21no—
83C2×C4×C59—41no—
84C2×C4×C59—81no—
85C2×C4×C59#12111yescharacteristic
86D236—81no—
87D236—81no—
88D236—162no—
89C8×C59—41no—
90C8×C59C2×D811yescharacteristic
91C8×C59C2×D811yescharacteristic
92C8×C59#13511yescharacteristic
93C8×C59AGL(1,ℤ/8)11yescharacteristic
94#12438—591nomaximal, Sylow
95C22×C59—41no—
96C22×C59#47211yescharacteristic
97C2×D59—162no—
98C2×D59—161no—
99C4×C59—22no—
100C4×C59—41no—
101C4×C59—161no—
102C4×C59#45911yescharacteristic
103C4×C59#46611yescharacteristic
104#2248—591no—
105#2214D5911yescharacteristic, complemented
106#2213—591no—
107#2170—591no—
108#1373—591no—
109#1358—591no—
110#1357—591no—
111C2×C59—21no—
112C2×C59—81no—
113C2×C59#166711yescharacteristic
114D59—321no—
115#510—21no—
116#509—1181no—
117C2×SD32—1181no—
118C2×SD32—1181no—
119#504—1181no—
120#494C2×D5911yescharacteristic
121#493—591no—
122#443C2×D5911yescharacteristic
123#364—591no—
124#364—591no—
125#364C2×D5911yescharacteristic
126#362—21no—
127#361—1181no—
128#350—1181no—
129C4×C16—591no—
130C59#1243811yescharacteristic, complemented, Sylow
131#135C22⋊D5911yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1181no—
135C2×D8—1181no—
136#131—21no—
137C4⋊Q8—21no—
138#127—1181no—
139Q32—41no—
140SD32—41no—
141SD32—2361no—
142SD32—2361no—
143SD32—2362no—
144SD32—2361no—
145D16—2361no—
146M32—21no—
147M32—1181no—
148M32—1181no—
149M32—2361no—
150C2×C16—591no—
151C2×C16—591no—
152C2×C16—2361no—
153#108C22⋊D5911yescharacteristic
154#104—21no—
155C4×C8C22×D5911yescharacteristic
156#41—21no—
157C2×Q8—41no—
158C2×D4—4721no—
159C2×D4—2361no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16#1174566911yescharacteristic
165SD16—21no—
166D8—2361no—
167D8—2361no—
168D8—2361no—
169D8—2361no—
170D8#1174566911yescharacteristic
171M16—21no—
172M16—41no—
173C2×C8—41no—
174C2×C8D4×D5911yescharacteristic
175C2×C8C2×C22⋊D5911yescharacteristic
176C4⋊C4—81no—
177C42#1174568711yescharacteristic
178C16—41no—
179C16—591no—
180C16—591no—
181C16—1181no—
182C16—2361no—
183C16—2361no—
184C23—4721no—
185Q8—21no—
186Q8—81no—
187Q8—81no—
188D4—21no—
189D4—41no—
190D4—9442no—
191D4—4721no—
192D4—4721no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4#1427369911yescharacteristic
197C8—41no—
198C8D8×D5911yescharacteristic
199C8C2×#1174566911yescharacteristic
200C8#1427374711yescharacteristic
201C8#1427375111yescharacteristic
202C22—41no—
203C22—9442no—
204C22—9441no—
205C22#1683767011yescharacteristic
206C4—22no—
207C4—41no—
208C4—161no—
209C4#1683759511yescharacteristic
210C4#1683763611yescharacteristic
211C2—21no—
212C2—81no—
213C2—18881no—
214C2#2142344911yescharacteristic
subgroupG/H
centerC2#21423449
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2214C2
socleC2×C59#1667
prime radical#21426172C2
hypercenter#2214D59
hypocenterC59#12438
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #21426353C2semidirect
3. #21425933C2semidirect
4. #21422171C2semidirect
5. #21422135C2semidirect
6. #21422132C2semidirect
7. C59×#2214C2semidirect
105. #2214D59semidirect
130. C59#12438semidirect

Maximal subgroups

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

subgroupindex
#214263532
#214261722
#214259332
#214221712
#214221352
#214221322
C59×#22142
#1243859