Finite Group Databasebeta

← #28503111

Subgroups — #28503111

Order 15104.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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
2D4×C59⋊C16C211yesmaximal, characteristic, complemented
3#21423495C211yesmaximal, characteristic, complemented
4#21422169C211yesmaximal, characteristic, complemented
5#21422155C211yesmaximal, characteristic, complemented
6#21422142C211yesmaximal, characteristic, complemented
7C59×#1667C211yesmaximal, characteristic
8C22×C59⋊C16—21no—
9#16837807C2211yescharacteristic
10#16837692—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837203—21no—
13#16837203—21no—
14C22⋊(C59⋊C16)—21no—
15C4×C59⋊C16C2211yescharacteristic
16#16837155C2211yescharacteristic
17#16837150—21no—
18C59×#493C2211yescharacteristic
19C59×#459—21no—
20C8×D4×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#331—21no—
23#14273727—21no—
24C2×C59⋊C16—21no—
25C2×C59⋊C16—42no—
26C2×C59⋊C16D411yescharacteristic
27C2×C59⋊C16D411yescharacteristic
28#14273661—21no—
29#14273661—21no—
30#14273658—41no—
31C4⋊(C59⋊C8)—21no—
32#14273648—41no—
33C2×D8×C59—21no—
34C2×D8×C59—21no—
35C2×D8×C59—41no—
36C22×C8×C59—21no—
37#127×C59—21no—
38C4×D4×C59D411yescharacteristic
39C4⋊C8×C59—21no—
40C4⋊C8×C59D411yescharacteristic
41#102×C59—41no—
42C22⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59D811yescharacteristic
48C22×C4×C59—21no—
49C2×Dic118—41no—
50D8×C59—41no—
51D8×C59—41no—
52D8×C59—81no—
53D8×C59—82no—
54C2×C8×C59—21no—
55C2×C8×C59—41no—
56C2×C8×C59—41no—
57C2×C8×C59—41no—
58C2×C8×C59C2×D411yescharacteristic
59C2×C8×C59#4111yescharacteristic
60C4⋊C4×C59D811yescharacteristic
61C22⋊C4×C59—21no—
62C42×C59C2×D411yescharacteristic
63C4⋊Dic59—41no—
64C4⋊Dic59—41no—
65C2×C59⋊C8—41no—
66C59⋊C16—21no—
67C59⋊C16—21no—
68C59⋊C16—41no—
69C23×C59—21no—
70C23×C59—81no—
71D4×C59—21no—
72D4×C59—81no—
73D4×C59—81no—
74D4×C59—81no—
75D4×C59—81no—
76D4×C59—161no—
77D4×C59SD3212yes—
78C2×C4×C59—21no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59#12111yescharacteristic
82C2×C4×C59C2×D811yescharacteristic
83C2×C4×C59#13511yescharacteristic
84C2×Dic59—81no—
85Dic118—161no—
86C8×C59—21no—
87C8×C59—41no—
88C8×C59—81no—
89C8×C59#13511yescharacteristic
90C8×C59AGL(1,ℤ/8)11yescharacteristic
91C59⋊C8—81no—
92#12468—591nomaximal, Sylow
93C22×C59—22no—
94C22×C59—41no—
95C22×C59—41no—
96C22×C59—161no—
97C22×C59—161no—
98C22×C59—161no—
99C22×C59#46611yescharacteristic
100C4×C59—21no—
101C4×C59—41no—
102C4×C59#46311yescharacteristic
103C4×C59#47111yescharacteristic
104C4×C59C2×SD3211yescharacteristic
105C4×C59#50811yescharacteristic
106Dic59—161no—
107#2240—591no—
108D4×C16—591no—
109#1675—591no—
110#1667D5911yescharacteristic
111#1372—591no—
112#1365—591no—
113#1361—591no—
114C2×C59—42no—
115C2×C59—321no—
116C2×C59#168111yescharacteristic
117C2×C59#221711yescharacteristic
118C2×C59#222111yescharacteristic
119C22×C16—1181no—
120#500—591no—
121#493C2×D5911yescharacteristic
122#476—1181no—
123#459—21no—
124C8×D4C2×D5911yescharacteristic
125C4⋊C16—591no—
126#357—1181no—
127#357—1181no—
128C22⋊C16—1181no—
129C4×C16—591no—
130#335—591no—
131#335C2×D5911yescharacteristic
132#333—1181no—
133#331—21no—
134C59#1246811yescharacteristic, complemented, Sylow
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—41no—
138C22×C8—21no—
139C4⋊Q8—1181no—
140#127—21no—
141C4×D4C22⋊D5911yescharacteristic
142C2×C16—591no—
143C2×C16—591no—
144C2×C16—1181no—
145C2×C16—2362no—
146#107—1181no—
147#107—1181no—
148#106—2361no—
149C4⋊C8—21no—
150C4⋊C8—1181no—
151C4⋊C8C22⋊D5911yescharacteristic
152#102—41no—
153#102—2361no—
154C22⋊C8—21no—
155C4×C8C22×D5911yescharacteristic
156C2×Q8—2361no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4#1174566911yescharacteristic
161C22×C4—21no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—82no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—2361no—
171C2×C8D4×D5911yescharacteristic
172C2×C8#1174568711yescharacteristic
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C4⋊C4#1174566911yescharacteristic
176C22⋊C4—21no—
177C42C2×C22⋊D5911yescharacteristic
178C16—1181no—
179C16—1181no—
180C16—2361no—
181C23—21no—
182C23—81no—
183Q8—9441no—
184D4—21no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—161no—
190D4#1427367112yes—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—4721no—
195C2×C4#1427369911yescharacteristic
196C2×C4C2×#1174566911yescharacteristic
197C2×C4#1427374711yescharacteristic
198C8—21no—
199C8—41no—
200C8—81no—
201C8—4721no—
202C8#1427374611yescharacteristic
203C8#1427375111yescharacteristic
204C22—22no—
205C22—41no—
206C22—41no—
207C22—161no—
208C22—161no—
209C22—161no—
210C22#1683763611yescharacteristic
211C4—21no—
212C4—41no—
213C4—9441no—
214C4#1683761511yescharacteristic
215C4#1683766311yescharacteristic
216C4C2×#1427367111yescharacteristic
217C4#1683783211yescharacteristic
218C2—42no—
219C2—321no—
220C2#2142353111yescharacteristic
221C2#2142619011yescharacteristic
222C2#2142621411yescharacteristic
subgroupG/H
centerC22#16837636
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
Fitting, prime radicalC59×#1667C2
socleC22×C59#466
hypercenter#1667D59
hypocenterC59#12468
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1667

Complements

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

NH ≅ G/Ntype
1. #21426319C2semidirect
2. D4×C59⋊C16C2semidirect
3. #21423495C2semidirect
4. #21422169C2semidirect
5. #21422155C2semidirect
6. #21422142C2semidirect
134. C59#12468semidirect

Maximal subgroups

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

subgroupindex
#214263192
D4×C59⋊C162
#214234952
#214221692
#214221552
#214221422
C59×#16672
#1246859