Finite Group Databasebeta

← #28502832

Subgroups — #28502832

Order 15104.

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

normal subgroups only

This group has 234 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#21426244C211yesmaximal, characteristic
2#21426138C211yesmaximal, characteristic
3#21423690C211yesmaximal, characteristic, complemented
4#21422890C211yesmaximal, characteristic
5#21422159C211yesmaximal, characteristic, complemented
6#21422155C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9#16837807C2211yescharacteristic
10#16837798—21no—
11#16837754—21no—
12C4×D472C2211yescharacteristic
13#16837467—21no—
14#16837230—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837206—21no—
17#16837204—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×D472D411yescharacteristic
24C2×C8×D59—41no—
25#14273727—21no—
26C4×D236—21no—
27#14273670—42no—
28C2×C59⋊C16—21no—
29#14273661—21no—
30#14273661—21no—
31#14273661D411yescharacteristic
32#14273658—41no—
33#14273658—41no—
34#14273650—42no—
35#14273649—21no—
36(C2×D59)⋊C8—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
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×D236—21no—
47C2×C4×D59—41no—
48C2×Dic118—41no—
49Q16×C59—41no—
50Q16×C59—41no—
51Q16×C59—81no—
52C2×C8×C59—41no—
53C2×C8×C59C2×D411yescharacteristic
54C2×C8×C59#4111yescharacteristic
55C4⋊C4×C59—41no—
56C4⋊C4×C59—41no—
57C42×C59C2×D411yescharacteristic
58(C2×D59)⋊C4—41no—
59C4⋊Dic59—21no—
60C4⋊Dic59—41no—
61C4⋊Dic59—41no—
62C2×C59⋊C8—21no—
63C2×C59⋊C8D811yescharacteristic
64C2×C59⋊C8D811yescharacteristic
65D472—21no—
66D472SD1612yes—
67C8×D59—82no—
68C16×C59—41no—
69C59⋊C16—41no—
70C22×D59—41no—
71Q8×C59—81no—
72Q8×C59—81no—
73C2×C4×C59—81no—
74C2×C4×C59#12111yescharacteristic
75C2×C4×C59C2×D811yescharacteristic
76C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
77C2×Dic59—41no—
78C2×Dic59—81no—
79D236—22no—
80D236—41no—
81C4×D59—81no—
82C4×D59—81no—
83Dic118—82no—
84C8×C59—21no—
85C8×C59—81no—
86C8×C59C2×SD1611yescharacteristic
87C8×C59#13511yescharacteristic
88C59⋊C8—21no—
89C59⋊C8—41no—
90C59⋊C8SD3212yescomplemented
91#12420—591nomaximal, Sylow
92C22×C59#45911yescharacteristic
93C2×D59—42no—
94C2×D59—81no—
95C2×D59—81no—
96C4×C59—21no—
97C4×C59—161no—
98C4×C59#46511yescharacteristic
99C4×C59#47111yescharacteristic
100C4×C59C2×SD3211yescharacteristic
101C4×C59#51011yescharacteristic
102Dic59—81no—
103Dic59—161no—
104#2226—591no—
105#2208—591no—
106#1707—591no—
107C8×D8—591no—
108#1367—591no—
109#1365—591no—
110#1362D5911yescharacteristic, complemented
111C2×C59#166511yescharacteristic
112C2×C59#220811yescharacteristic
113C2×C59#221211yescharacteristic
114D59—82no—
115C2×SD32—1181no—
116#500—591no—
117#498—1181no—
118#494C2×D5911yescharacteristic
119#487—1181no—
120C4×D8—591no—
121C8×D4—1181no—
122#367—1181no—
123C4⋊C16—591no—
124C4⋊C16C2×D5911yescharacteristic
125#358—1181no—
126#357—1181no—
127#335C2×D5911yescharacteristic
128D4⋊C8—1181no—
129C82—591no—
130C59#1242011yescharacteristic, complemented, Sylow
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8—591no—
134C22×C8—2361no—
135C4⋊Q8—21no—
136C4⋊Q8—1181no—
137C4×D4—1181no—
138SD32—2362no—
139C2×C16—21no—
140C2×C16—1181no—
141#107—591no—
142#107—1181no—
143#107—1181no—
144#106—2361no—
145#106—2361no—
146C4⋊C8—21no—
147#103—2362no—
148#102—1181no—
149C22⋊C8—2361no—
150C4×C8—591no—
151C4×C8—591no—
152C4×C8C22×D5911yescharacteristic
153C2×Q8—41no—
154C2×Q8—2361no—
155C2×D4—1181no—
156C22×C4—2361no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—592no—
161D8—1181no—
162C2×C8—41no—
163C2×C8—591no—
164C2×C8—591no—
165C2×C8—1181no—
166C2×C8—4722no—
167C2×C8#1174568511yescharacteristic
168C2×C8D4×D5911yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—1181no—
172C4⋊C4—2361no—
173C4⋊C4—2361no—
174C22⋊C4—2361no—
175C42D4×D5911yescharacteristic
176C16—41no—
177C16—2361no—
178C23—2361no—
179Q8—81no—
180Q8—81no—
181Q8—4722no—
182D4—1182no—
183D4—2361no—
184C2×C4—81no—
185C2×C4—4721no—
186C2×C4—4721no—
187C2×C4—4721no—
188C2×C4—2361no—
189C2×C4#1427370011yescharacteristic
190C2×C4D8×D5911yescharacteristic
191C2×C4#1427375411yescharacteristic
192C8—21no—
193C8—81no—
194C8—592no—
195C8—1181no—
196C8—2361no—
197C8SD16×D5911yescharacteristic
198C8#1427374611yescharacteristic
199C22—4721no—
200C22—4721no—
201C22—2362no—
202C22#1683759611yescharacteristic
203C4—21no—
204C4—161no—
205C4—9441no—
206C4—4721no—
207C4#1683763011yescharacteristic
208C4#1683766411yescharacteristic
209C4SD32×D5911yescharacteristic
210C4#1683784411yescharacteristic
211C2—4722no—
212C2#2142343611yescharacteristic
213C2#2142614011yescharacteristic
214C2#2142616811yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59#459
prime radicalC2×D472D4
hypercenter#1362D59
hypocenterC59#12420
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
3. #21423690C2semidirect
5. #21422159C2semidirect
6. #21422155C2semidirect
7. C59×#1362C2semidirect
90. C59⋊C8SD32semidirect
110. #1362D59semidirect
130. C59#12420semidirect

Maximal subgroups

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

subgroupindex
#214262442
#214261382
#214236902
#214228902
#214221592
#214221552
C59×#13622
#1242059