Finite Group Databasebeta

← #28503126

Subgroups — #28503126

Order 15104.

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426323C211yesmaximal, characteristic, complemented
2#21425915C211yesmaximal, characteristic
3#21423486C211yesmaximal, characteristic
4#21423458C211yesmaximal, characteristic
5#21422169C211yesmaximal, characteristic, complemented
6#21422150C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8C2×C16×D59—21no—
9#16837807C2211yescharacteristic
10#16837679—21no—
11#16837608—21no—
12C8×D236C2211yescharacteristic
13#16837206—21no—
14#16837206—21no—
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837149—21no—
19#16837148—21no—
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745660—41no—
24C2×C8×D59—21no—
25#14273727—21no—
26C4×D236D411yescharacteristic
27C2×C59⋊C16D411yescharacteristic
28C2×C59⋊C16D411yescharacteristic
29C16×D59—42no—
30#14273661—21no—
31#14273661—21no—
32#14273660—41no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)D411yescharacteristic
35#14273652—41no—
36#14273647—41no—
37(C2×D59)⋊C8—21no—
38C2×Q16×C59—21no—
39C2×Q16×C59—21no—
40C4⋊Q8×C59—21no—
41C2×C16×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×Q8×C59—41no—
45C2×D236D811yescharacteristic
46C2×C4×D59—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
57(C2×D59)⋊C4—21no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C4⋊Dic59D811yescharacteristic
61C2×C59⋊C8—21no—
62C2×C59⋊C8—41no—
63#11745660—82no—
64C8×D59—41no—
65C8×D59—41no—
66C16×C59—41no—
67C59⋊C16—21no—
68C59⋊C16—21no—
69C22×D59—21no—
70Q8×C59—81no—
71Q8×C59—81no—
72C2×C4×C59—81no—
73C2×C4×C59#12111yescharacteristic
74C2×C4×C59C2×D811yescharacteristic
75C2×C4×C59#13511yescharacteristic
76C2×Dic59—21no—
77C2×Dic59—81no—
78D236—21no—
79D236SD3212yes—
80C4×D59—41no—
81C4×D59—41no—
82Dic118—81no—
83Dic118—81no—
84C8×C59—21no—
85C8×C59—81no—
86C8×C59#13511yescharacteristic
87C8×C59AGL(1,ℤ/8)11yescharacteristic
88C59⋊C8—41no—
89C59⋊C8—81no—
90#12470—591nomaximal, Sylow
91C22×C59#46611yescharacteristic
92C2×D59—22no—
93C2×D59—41no—
94C2×D59—41no—
95C4×C59—21no—
96C4×C59—161no—
97C4×C59#46311yescharacteristic
98C4×C59#47111yescharacteristic
99C4×C59C2×SD3211yescharacteristic
100C4×C59#50811yescharacteristic
101Dic59—41no—
102Dic59—161no—
103#2241—591no—
104D4×C16—591no—
105#1673—591no—
106#1669—591no—
107#1372—591no—
108#1364—591no—
109#1362D5911yescharacteristic, complemented
110C2×C59#168111yescharacteristic
111C2×C59#221711yescharacteristic
112C2×C59#222211yescharacteristic
113D59—42no—
114C22×C16—1181no—
115#500—591no—
116#494C2×D5911yescharacteristic
117#474—1181no—
118#461—1181no—
119C8×D4—591no—
120C4⋊C16C2×D5911yescharacteristic
121#358—1181no—
122#358—1181no—
123C22⋊C16—1181no—
124C4×C16—591no—
125#335—591no—
126#335C2×D5911yescharacteristic
127#332—1181no—
128#332—1181no—
129C59#1247011yescharacteristic, complemented, Sylow
130C2×Q16—21no—
131C2×Q16—21no—
132C2×SD16—2361no—
133C22×C8—1181no—
134C4⋊Q8—21no—
135C4⋊Q8—1181no—
136C4×D4—591no—
137C2×C16—21no—
138C2×C16—591no—
139C2×C16—591no—
140C2×C16—2362no—
141#107—1181no—
142#107—1181no—
143#107—2361no—
144C4⋊C8—21no—
145C4⋊C8—591no—
146C4⋊C8—1181no—
147#103—2361no—
148#102—2361no—
149C22⋊C8—1181no—
150C4×C8C22×D5911yescharacteristic
151C2×Q8—41no—
152C2×Q8—2361no—
153C2×D4—591no—
154C22×C4—1181no—
155Q16—41no—
156Q16—41no—
157Q16—81no—
158SD16—4722no—
159C2×C8—41no—
160C2×C8—1181no—
161C2×C8—2361no—
162C2×C8—2361no—
163C2×C8—2361no—
164C2×C8#1174568511yescharacteristic
165C2×C8D4×D5911yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—591no—
169C4⋊C4—2361no—
170C4⋊C4—2361no—
171C22⋊C4—1181no—
172C42D4×D5911yescharacteristic
173C16—41no—
174C16—1181no—
175C16—1181no—
176C23—1181no—
177Q8—81no—
178Q8—81no—
179Q8—4721no—
180Q8—4721no—
181D4—592no—
182D4—1181no—
183C2×C4—81no—
184C2×C4—1181no—
185C2×C4—4721no—
186C2×C4—2361no—
187C2×C4—2361no—
188C2×C4#1427370011yescharacteristic
189C2×C4D8×D5911yescharacteristic
190C2×C4#1427374811yescharacteristic
191C8—21no—
192C8—81no—
193C8—4721no—
194C8—2361no—
195C8#1427374811yescharacteristic
196C8#1427375111yescharacteristic
197C22—1182no—
198C22—2361no—
199C22—2361no—
200C22#1683763711yescharacteristic
201C4—21no—
202C4—161no—
203C4—9441no—
204C4—2361no—
205C4#1683761611yescharacteristic
206C4#1683766411yescharacteristic
207C4SD32×D5911yescharacteristic
208C4#1683783511yescharacteristic
209C2—2362no—
210C2#2142353211yescharacteristic
211C2#2142619511yescharacteristic
212C2#2142622011yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59#466
prime radicalC2×D236D8
hypercenter#1362D59
hypocenterC59#12470
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
1. #21426323C2semidirect
5. #21422169C2semidirect
6. #21422150C2semidirect
7. C59×#1362C2semidirect
109. #1362D59semidirect
129. C59#12470semidirect

Maximal subgroups

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

subgroupindex
#214263232
#214259152
#214234862
#214234582
#214221692
#214221502
C59×#13622
#1247059