Finite Group Databasebeta

← #28502839

Subgroups — #28502839

Order 15104.

218 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 65 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#21426249C211yesmaximal, characteristic
2#21426143C211yesmaximal, characteristic
3#21423690C211yesmaximal, characteristic
4#21422902C211yesmaximal, characteristic
5#21422159C211yesmaximal, characteristic
6#21422155C211yesmaximal, characteristic
7C59×#1362C211yesmaximal, characteristic
8C2×#14273673—21no—
9#16837807C2211yescharacteristic
10#16837798—21no—
11#16837754—21no—
12#16837526—21no—
13C4×Dic236C2211yescharacteristic
14#16837228—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837207—21no—
17#16837206—21no—
18#16837132—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236D411yescharacteristic
24#14273727—21no—
25C4×Dic118—21no—
26#14273673—42no—
27C2×C59⋊C16—21no—
28#14273661—21no—
29#14273661—21no—
30#14273661D411yescharacteristic
31#14273658—41no—
32#14273658—41no—
33Dic59⋊C8—41no—
34#14273652—21no—
35#14273650—42no—
36C4×C59⋊C8D411yescharacteristic
37C4×C59⋊C8D411yescharacteristic
38C8×Dic59—41no—
39C2×Q16×C59—21no—
40C2×Q16×C59—21no—
41C4⋊Q8×C59—21no—
42C2×C16×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×Dic118—21no—
47C2×Dic118—41no—
48Q16×C59—41no—
49Q16×C59—41no—
50Q16×C59—81no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C42×C59C2×D411yescharacteristic
57C4⋊Dic59—21no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60Dic59⋊C4—41no—
61C4×Dic59—41no—
62C2×C59⋊C8—21no—
63C2×C59⋊C8D811yescharacteristic
64C2×C59⋊C8D811yescharacteristic
65Dic236—21no—
66Dic236SD1612yes—
67C16×C59—41no—
68C59⋊C16—41no—
69Q8×C59—81no—
70Q8×C59—81no—
71C2×C4×C59—81no—
72C2×C4×C59#12111yescharacteristic
73C2×C4×C59C2×D811yescharacteristic
74C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
75C2×Dic59—41no—
76C2×Dic59—41no—
77C2×Dic59—81no—
78Dic118—22no—
79Dic118—41no—
80Dic118—82no—
81C8×C59—21no—
82C8×C59—81no—
83C8×C59C2×SD1611yescharacteristic
84C8×C59#13511yescharacteristic
85C59⋊C8—21no—
86C59⋊C8—41no—
87C59⋊C8Q3212yescomplemented
88#12421—591nomaximal, Sylow
89C22×C59#45911yescharacteristic
90C4×C59—21no—
91C4×C59—161no—
92C4×C59#46511yescharacteristic
93C4×C59#47111yescharacteristic
94C4×C59C2×Q3211yescharacteristic
95C4×C59#51011yescharacteristic
96Dic59—42no—
97Dic59—81no—
98Dic59—161no—
99#2227—591no—
100#2209—591no—
101#1707—591no—
102C8×Q16—591no—
103#1367—591no—
104#1365—591no—
105#1362D5911yescharacteristic
106C2×C59#166511yescharacteristic
107C2×C59#220911yescharacteristic
108C2×C59#221211yescharacteristic
109C2×Q32—1181no—
110#500—591no—
111#498—1181no—
112#494C2×D5911yescharacteristic
113#487—1181no—
114C8×Q8—1181no—
115C4×Q16—591no—
116#366—1181no—
117C4⋊C16—591no—
118C4⋊C16C2×D5911yescharacteristic
119#358—1181no—
120#358—1181no—
121#335C2×D5911yescharacteristic
122Q8⋊C8—1181no—
123C82—591no—
124C59#1242111yescharacteristic, complemented, Sylow
125C2×Q16—21no—
126C2×Q16—21no—
127C2×Q16—591no—
128C4⋊Q8—21no—
129C4⋊Q8—1181no—
130C4×Q8—1181no—
131Q32—2362no—
132C2×C16—21no—
133C2×C16—1181no—
134#107—591no—
135#107—1181no—
136#107—1181no—
137#106—2361no—
138#106—2361no—
139C4⋊C8—21no—
140C4⋊C8—2361no—
141#103—1181no—
142#103—2362no—
143C4×C8—591no—
144C4×C8—591no—
145C4×C8—2361no—
146C4×C8C22×D5911yescharacteristic
147C2×Q8—41no—
148C2×Q8—1181no—
149C2×Q8—2361no—
150Q16—41no—
151Q16—41no—
152Q16—81no—
153Q16—592no—
154Q16—1181no—
155C2×C8—41no—
156C2×C8—591no—
157C2×C8—591no—
158C2×C8—1181no—
159C2×C8#1174568511yescharacteristic
160C2×C8D4×D5911yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—1181no—
164C4⋊C4—2361no—
165C4⋊C4—2361no—
166C4⋊C4—2361no—
167C42—2361no—
168C42D4×D5911yescharacteristic
169C16—41no—
170C16—2361no—
171Q8—81no—
172Q8—81no—
173Q8—1182no—
174Q8—4722no—
175Q8—2361no—
176C2×C4—81no—
177C2×C4—4721no—
178C2×C4—2361no—
179C2×C4—2361no—
180C2×C4#1427370011yescharacteristic
181C2×C4D8×D5911yescharacteristic
182C2×C4#1427375411yescharacteristic
183C8—21no—
184C8—81no—
185C8—592no—
186C8—1181no—
187C8—2361no—
188C8SD16×D5911yescharacteristic
189C8#1427374611yescharacteristic
190C22#1683759611yescharacteristic
191C4—21no—
192C4—161no—
193C4—9441no—
194C4—4721no—
195C4—2362no—
196C4#1683763011yescharacteristic
197C4#1683766411yescharacteristic
198C4Q32×D5911yescharacteristic
199C4#1683784111yescharacteristic
200C2#2142343611yescharacteristic
201C2#2142614511yescharacteristic
202C2#2142616811yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socle, prime radicalC22×C59#459
hypercenter#1362D59
hypocenterC59#12421
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
87. C59⋊C8Q32semidirect
124. C59#12421semidirect

Maximal subgroups

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

subgroupindex
#214262492
#214261432
#214236902
#214229022
#214221592
#214221552
C59×#13622
#1242159