Finite Group Databasebeta

← #28503112

Subgroups — #28503112

Order 15104.

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

normal subgroups only

This group has 234 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#21426319C211yesmaximal, characteristic, complemented
2#21425915C211yesmaximal, characteristic, complemented
3#21423496C211yesmaximal, characteristic, complemented
4#21423448C211yesmaximal, characteristic
5#21422169C211yesmaximal, characteristic, complemented
6#21422153C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic, complemented
8C2×C16×D59—21no—
9#16837807C2211yescharacteristic
10#16837693—21no—
11#16837594—21no—
12C8×D236C2211yescharacteristic
13#16837203—21no—
14#16837203—21no—
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837150—21no—
19#16837146—21no—
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745669—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#14273659—41no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)D411yescharacteristic
35#14273649—41no—
36#14273648—41no—
37(C2×D59)⋊C8—21no—
38C2×D8×C59—21no—
39C2×D8×C59—21no—
40#127×C59—21no—
41C2×C16×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D236D811yescharacteristic
48C2×C4×D59—21no—
49C2×Dic118—41no—
50D8×C59—41no—
51D8×C59—41no—
52D8×C59—81no—
53C2×C8×C59—41no—
54C2×C8×C59C2×D411yescharacteristic
55C2×C8×C59#4111yescharacteristic
56C42×C59C2×D411yescharacteristic
57#11745669—82no—
58(C2×D59)⋊C4—21no—
59C4⋊Dic59—41no—
60C4⋊Dic59—41no—
61C4⋊Dic59D811yescharacteristic
62C2×C59⋊C8—21no—
63C2×C59⋊C8—41no—
64C8×D59—41no—
65C8×D59—41no—
66C16×C59—41no—
67C59⋊C16—21no—
68C59⋊C16—21no—
69C23×C59—81no—
70C22×D59—21no—
71D4×C59—81no—
72D4×C59—81no—
73D4×C59—81no—
74D4×C59—81no—
75D4×C59—161no—
76C2×C4×C59#12111yescharacteristic
77C2×C4×C59C2×D811yescharacteristic
78C2×C4×C59#13511yescharacteristic
79C2×Dic59—21no—
80C2×Dic59—81no—
81D236—21no—
82D236SD3212yes—
83C4×D59—41no—
84C4×D59—41no—
85Dic118—161no—
86C8×C59—21no—
87C8×C59—81no—
88C8×C59#13511yescharacteristic
89C8×C59AGL(1,ℤ/8)11yescharacteristic
90C59⋊C8—41no—
91C59⋊C8—81no—
92#12468—591nomaximal, Sylow
93C22×C59—161no—
94C22×C59—161no—
95C22×C59—161no—
96C22×C59#46611yescharacteristic
97C2×D59—22no—
98C2×D59—41no—
99C2×D59—41no—
100C4×C59—21no—
101C4×C59#46311yescharacteristic
102C4×C59#47111yescharacteristic
103C4×C59C2×SD3211yescharacteristic
104C4×C59#50811yescharacteristic
105Dic59—41no—
106Dic59—161no—
107#2240—591no—
108D4×C16—591no—
109#1675—591no—
110#1667—591no—
111#1372—591no—
112#1365—591no—
113#1361D5911yescharacteristic, complemented
114C2×C59—321no—
115C2×C59#168111yescharacteristic
116C2×C59#221711yescharacteristic
117C2×C59#222111yescharacteristic
118D59—42no—
119C22×C16—1181no—
120#500—591no—
121#493C2×D5911yescharacteristic
122#476—1181no—
123#459—1181no—
124C8×D4—591no—
125C4⋊C16C2×D5911yescharacteristic
126#357—1181no—
127#357—1181no—
128C22⋊C16—1181no—
129C4×C16—591no—
130#335—591no—
131#335C2×D5911yescharacteristic
132#333—1181no—
133#331—1181no—
134C59#1246811yescharacteristic, complemented, Sylow
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2361no—
138C22×C8—1181no—
139C4⋊Q8—1181no—
140#127—21no—
141C4×D4—591no—
142C2×C16—21no—
143C2×C16—591no—
144C2×C16—591no—
145C2×C16—2362no—
146#107—1181no—
147#107—1181no—
148#106—2361no—
149C4⋊C8—21no—
150C4⋊C8—591no—
151C4⋊C8—1181no—
152#102—2361no—
153#102—2361no—
154C22⋊C8—1181no—
155C4×C8C22×D5911yescharacteristic
156C2×Q8—2361no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—591no—
161C22×C4—1181no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—4722no—
166C2×C8—41no—
167C2×C8—1181no—
168C2×C8—2361no—
169C2×C8—2361no—
170C2×C8—2361no—
171C2×C8#1174568511yescharacteristic
172C2×C8D4×D5911yescharacteristic
173C4⋊C4—591no—
174C4⋊C4—2361no—
175C4⋊C4—2361no—
176C22⋊C4—1181no—
177C42D4×D5911yescharacteristic
178C16—41no—
179C16—1181no—
180C16—1181no—
181C23—81no—
182C23—1181no—
183Q8—9441no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—592no—
190D4—1181no—
191C2×C4—1181no—
192C2×C4—4721no—
193C2×C4—2361no—
194C2×C4—2361no—
195C2×C4#1427370011yescharacteristic
196C2×C4D8×D5911yescharacteristic
197C2×C4#1427374811yescharacteristic
198C8—21no—
199C8—81no—
200C8—4721no—
201C8—2361no—
202C8#1427374811yescharacteristic
203C8#1427375111yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—1182no—
208C22—2361no—
209C22—2361no—
210C22#1683763711yescharacteristic
211C4—21no—
212C4—9441no—
213C4—2361no—
214C4#1683761611yescharacteristic
215C4#1683766411yescharacteristic
216C4SD32×D5911yescharacteristic
217C4#1683783111yescharacteristic
218C2—321no—
219C2—2362no—
220C2#2142353211yescharacteristic
221C2#2142619111yescharacteristic
222C2#2142621511yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#466
prime radical#21423448C2
hypercenter#1361D59
hypocenterC59#12468
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
1. #21426319C2semidirect
2. #21425915C2semidirect
3. #21423496C2semidirect
5. #21422169C2semidirect
6. #21422153C2semidirect
7. C59×#1361C2semidirect
113. #1361D59semidirect
134. C59#12468semidirect

Maximal subgroups

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

subgroupindex
#214263192
#214259152
#214234962
#214234482
#214221692
#214221532
C59×#13612
#1246859