Finite Group Databasebeta

← #28502018

Subgroups — #28502018

Order 15104.

324 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 55 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#21434920C211yesmaximal, characteristic
2#21426315C212yesmaximal, complemented
3#21422089C212yesmaximal, complemented
4#21422032C211yesmaximal, characteristic
5C59×#2091C211yesmaximal, characteristic
6C2×#14273706—21no—
7#16837733—22no—
8#16837660—21no—
9#16837660C2212yes—
10#16837560—22no—
11#16837226—22no—
12#16837217—22no—
13#16837184C2212yes—
14C59×#430—21no—
15C59×#430C2211yescharacteristic
16C59×#407—21no—
17C59×#407C2211yescharacteristic
18C59×#338C2211yescharacteristic
19C22×Dic118—21no—
20C2×Dic236—21no—
21C2×Dic236D412yes—
22#14273706—22no—
23#14273706—22no—
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59—21no—
27C2×Dic59⋊C4—41no—
28C2×C22⋊Dic59—41no—
29#14273667—22no—
30#14273659—21no—
31#14273659D412yes—
32#14273652—22no—
33#14273652—22no—
34C23×C4×C59D411yescharacteristic
35C2×M16×C59—21no—
36C2×M16×C59—21no—
37C2×M16×C59D411yescharacteristic
38C2×M16×C59C2311yescharacteristic
39#108×C59—22no—
40#108×C59—42no—
41C22⋊C8×C59—22no—
42C22⋊C8×C59—42no—
43C24×C59SD1611yescharacteristic
44C22×C4×C59—21no—
45C22×C4×C59—21no—
46C22×C4×C59—41no—
47C22×C4×C59—41no—
48C22×C4×C59SD1611yescharacteristic
49C22×C4×C59C2×D411yescharacteristic
50C22×Dic59—41no—
51C22×Dic59—41no—
52C2×Dic118—22no—
53C2×Dic118—41no—
54C2×Dic118—81no—
55C2×Dic118—81no—
56M16×C59—41no—
57M16×C59—41no—
58M16×C59—41no—
59M16×C59—41no—
60M16×C59—41no—
61M16×C59—41no—
62C2×C8×C59—22no—
63C2×C8×C59—21no—
64C2×C8×C59—41no—
65C2×C8×C59C2×D412yes—
66C22⋊Dic59—42no—
67C22⋊Dic59—42no—
68C4⋊Dic59—22no—
69C4⋊Dic59—41no—
70Dic59⋊C4—42no—
71Dic59⋊C4—42no—
72Dic236—42no—
73Dic236—81no—
74C59⋊C16—42no—
75C23×C59—21no—
76C23×C59—41no—
77C23×C59—42no—
78C23×C59C2×SD1611yescharacteristic
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—41no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—81no—
86C2×C4×C59#12011yescharacteristic
87C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
88C2×Dic59—42no—
89C2×Dic59—42no—
90C2×Dic59—82no—
91C2×Dic59—82no—
92Dic118—42no—
93Dic118—42no—
94Dic118—82no—
95C8×C59—22no—
96C8×C59—42no—
97C8×C59—41no—
98C8×C59—81no—
99#12299—591nomaximal, Sylow
100C22×C59—22no—
101C22×C59—21no—
102C22×C59—41no—
103C22×C59—41no—
104C22×C59—42no—
105C22×C59—41no—
106C22×C59—81no—
107C22×C59#45011yescharacteristic
108C4×C59—21no—
109C4×C59—41no—
110C4×C59#45311yescharacteristic
111C4×C59#45811yescharacteristic
112Dic59—82no—
113Dic59—82no—
114#3195—591no—
115#2239—592no—
116#2091D5911yescharacteristic
117#1342—592no—
118#1320—591no—
119C2×C59—21no—
120C2×C59—42no—
121C2×C59—41no—
122C2×C59#161311yescharacteristic
123C2×C22⋊Q8—1181no—
124#483—1182no—
125#470—592no—
126#470—1181no—
127#451—1182no—
128#430—21no—
129#430C2×D5911yescharacteristic
130#407—21no—
131#407C2×D5911yescharacteristic
132#365—1182no—
133#362—1182no—
134#349—592no—
135#338C2×D5911yescharacteristic
136C59#1229911yescharacteristic, complemented, Sylow
137C22×Q8—1181no—
138C23×C4C22⋊D5911yescharacteristic
139C2×Q16—592no—
140C2×Q16—1181no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16C22⋊D5911yescharacteristic
144C2×M16C22×D5911yescharacteristic
145C22⋊Q8—1182no—
146C22⋊Q8—1182no—
147C22⋊Q8—2361no—
148C22⋊Q8—2361no—
149C2×C4⋊C4—1181no—
150C2×C4⋊C4—2361no—
151C2×C22⋊C4—2361no—
152M32—1182no—
153#108—22no—
154#108—42no—
155#106—592no—
156#106—1181no—
157#103—1182no—
158#103—1182no—
159C22⋊C8—22no—
160C22⋊C8—42no—
161C24#1174567111yescharacteristic
162C2×Q8—1182no—
163C2×Q8—4721no—
164C2×Q8—4721no—
165C2×Q8—2361no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—2361no—
171C22×C4—2361no—
172C22×C4#1174567111yescharacteristic
173C22×C4C2×C22⋊D5911yescharacteristic
174Q16—4721no—
175Q16—2362no—
176M16—41no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—41no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8D4×D5912yes—
186C4⋊C4—1182no—
187C4⋊C4—2361no—
188C4⋊C4—2362no—
189C4⋊C4—2362no—
190C22⋊C4—2362no—
191C22⋊C4—2362no—
192C16—2362no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1174567111yescharacteristic
197Q8—4722no—
198Q8—2362no—
199Q8—2362no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—4722no—
208C2×C4—4722no—
209C2×C4—2362no—
210C2×C4—2362no—
211C2×C4#1427369511yescharacteristic
212C2×C4#1427375211yescharacteristic
213C8—22no—
214C8—42no—
215C8—41no—
216C8—81no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1683755211yescharacteristic
225C4—21no—
226C4—41no—
227C4—4722no—
228C4—4722no—
229C4#1683756911yescharacteristic
230C4#1683759111yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2142311511yescharacteristic
subgroupG/H
centerC2#21423115
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#2091C2
socleC2×C59#1613
prime radicalC24×C59SD16
hypercenter#2091D59
hypocenterC59#12299
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2091

Complements

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

NH ≅ G/Ntype
2. #21426315C2semidirect
3. #21422089C2semidirect
136. C59#12299semidirect

Maximal subgroups

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

subgroupindex
#214349202
#214263152
#214220892
#214220322
C59×#20912
#1229959