Finite Group Databasebeta

← #28503107

Subgroups — #28503107

Order 15104.

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

normal subgroups only

This group has 240 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#21426320C211yesmaximal, characteristic, complemented
2C16×D236C211yesmaximal, characteristic, complemented
3#21423486C211yesmaximal, characteristic, complemented
4#21423452C211yesmaximal, characteristic
5#21422152C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1371C211yesmaximal, characteristic, complemented
8C2×C16×D59—21no—
9#16837807C2211yescharacteristic
10#16837779C2211yescharacteristic
11#16837608—21no—
12#16837598—21no—
13C8×D236C2211yescharacteristic
14C4⋊(C59⋊C16)C2211yescharacteristic
15#16837204—21no—
16#16837204—21no—
17(C2×D59)⋊C16—21no—
18#16837149—21no—
19#16837145—21no—
20C4×C16×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745660—41no—
24C2×D472—21no—
25C2×D472—21no—
26C2×D472—41no—
27C2×C8×D59—21no—
28#14273727—21no—
29#14273724—21no—
30C4×D236D411yescharacteristic
31C2×C59⋊C16—21no—
32C16×D59—42no—
33#14273661—21no—
34#14273661—21no—
35C4⋊(C59⋊C8)D411yescharacteristic
36#14273652—41no—
37#14273649—41no—
38(C2×D59)⋊C8—21no—
39C2×C16×C59D411yescharacteristic
40C2×C16×C59D411yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D236—41no—
45C2×D236—41no—
46C2×D236—41no—
47C2×D236D811yescharacteristic
48C2×C4×D59—21no—
49C2×Dic118—41no—
50C2×C8×C59—41no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C42×C59C2×D411yescharacteristic
55(C2×D59)⋊C4—21no—
56C4⋊Dic59—41no—
57C4⋊Dic59—41no—
58C4⋊Dic59D811yescharacteristic
59C2×C59⋊C8—21no—
60D472—41no—
61D472—41no—
62D472—81no—
63D472—82no—
64#11745660—82no—
65C8×D59—41no—
66C8×D59—41no—
67C16×C59—21no—
68C16×C59—21no—
69C59⋊C16—41no—
70C22×D59—21no—
71C22×D59—81no—
72C2×C4×C59#12111yescharacteristic
73C2×C4×C59C2×D811yescharacteristic
74C2×C4×C59#13511yescharacteristic
75C2×Dic59—21no—
76C2×Dic59—81no—
77D236—21no—
78D236—81no—
79D236—81no—
80D236—81no—
81D236—81no—
82D236—161no—
83D236D1612yes—
84C4×D59—41no—
85C4×D59—41no—
86Dic118—81no—
87Dic118—81no—
88C8×C59—21no—
89C8×C59—81no—
90C8×C59—81no—
91C8×C59#13511yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C59⋊C8—41no—
94#12467—591nomaximal, Sylow
95C22×C59#46611yescharacteristic
96C2×D59—22no—
97C2×D59—41no—
98C2×D59—41no—
99C2×D59—161no—
100C2×D59—161no—
101C2×D59—161no—
102C4×C59—21no—
103C4×C59#46311yescharacteristic
104C4×C59#47111yescharacteristic
105C4×C59C2×D1611yescharacteristic
106C4×C59#50811yescharacteristic
107Dic59—41no—
108Dic59—161no—
109#2240—591no—
110D4×C16—591no—
111#1673—591no—
112#1667—591no—
113#1371D5911yescharacteristic, complemented
114#1364—591no—
115#1361—591no—
116C2×C59#168111yescharacteristic
117C2×C59#221511yescharacteristic
118C2×C59#222311yescharacteristic
119D59—42no—
120D59—321no—
121C22×C16—1181no—
122#500—591no—
123#493—591no—
124#461—1181no—
125#459—1181no—
126C8×D4—591no—
127C4⋊C16—591no—
128#357—1181no—
129#357—1181no—
130C22⋊C16—1181no—
131C4×C16C2×D5911yescharacteristic
132#335C2×D5911yescharacteristic
133#335C2×D5911yescharacteristic
134#332—1181no—
135#331—1181no—
136C59#1246711yescharacteristic, complemented, Sylow
137C2×SD16—2361no—
138C2×D8—1181no—
139C2×D8—1181no—
140C2×D8—2361no—
141C22×C8—1181no—
142C4⋊Q8—1181no—
143#127—1181no—
144C4×D4—591no—
145C2×C16—1181no—
146C2×C16—2362no—
147C2×C16D23611yescharacteristic
148C2×C16D23611yescharacteristic
149#107—1181no—
150#107—1181no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—591no—
154#103—2361no—
155#102—2361no—
156C22⋊C8—1181no—
157C4×C8C22×D5911yescharacteristic
158C2×Q8—2361no—
159C2×D4—591no—
160C2×D4—2361no—
161C2×D4—2361no—
162C2×D4—2361no—
163C22×C4—1181no—
164SD16—4722no—
165D8—4721no—
166D8—4722no—
167D8—2361no—
168D8—2361no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—1181no—
172C2×C8—2361no—
173C2×C8—2361no—
174C2×C8C2×D23611yescharacteristic
175C2×C8#1174568911yescharacteristic
176C4⋊C4—591no—
177C4⋊C4—2361no—
178C4⋊C4—2361no—
179C22⋊C4—1181no—
180C42D4×D5911yescharacteristic
181C16—21no—
182C16—21no—
183C16—2361no—
184C23—1181no—
185C23—4721no—
186Q8—4721no—
187Q8—4721no—
188D4—592no—
189D4—1181no—
190D4—9441no—
191D4—4721no—
192D4—4721no—
193D4—4721no—
194D4—4721no—
195C2×C4—1181no—
196C2×C4—4721no—
197C2×C4—2361no—
198C2×C4—2361no—
199C2×C4#1427370211yescharacteristic
200C2×C4D8×D5911yescharacteristic
201C2×C4#1427375011yescharacteristic
202C8—21no—
203C8—81no—
204C8—81no—
205C8—2361no—
206C8#1427374911yescharacteristic
207C8#1427375511yescharacteristic
208C22—1182no—
209C22—9441no—
210C22—9441no—
211C22—9441no—
212C22—2361no—
213C22—2361no—
214C22#1683763911yescharacteristic
215C4—21no—
216C4—9441no—
217C4—2361no—
218C4#1683761811yescharacteristic
219C4#1683766611yescharacteristic
220C4D16×D5911yescharacteristic
221C4#1683783511yescharacteristic
222C2—18881no—
223C2—2362no—
224C2#2142353411yescharacteristic
225C2#2142618311yescharacteristic
226C2#2142622711yescharacteristic
subgroupG/H
centerC22#16837639
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1371C2
socleC22×C59#466
prime radical#21423452C2
hypercenter#1371D59
hypocenterC59#12467
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1371

Complements

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

NH ≅ G/Ntype
1. #21426320C2semidirect
2. C16×D236C2semidirect
3. #21423486C2semidirect
5. #21422152C2semidirect
6. #21422141C2semidirect
7. C59×#1371C2semidirect
113. #1371D59semidirect
136. C59#12467semidirect

Maximal subgroups

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

subgroupindex
#214263202
C16×D2362
#214234862
#214234522
#214221522
#214221412
C59×#13712
#1246759