Finite Group Databasebeta

← #28501180

Subgroups — #28501180

Order 15104.

354 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 59 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#21435535C211yesmaximal, characteristic
2#21426013C212yesmaximal, complemented
3#21422110C212yesmaximal, complemented
4#21422038C211yesmaximal, characteristic, complemented
5C59×#2047C211yesmaximal, characteristic, complemented
6#16838007—21no—
7C2×#14273670—22no—
8#16837703—22no—
9#16837655—21no—
10#16837638C2212yes—
11#16837598—22no—
12#16837206—22no—
13C22⋊(C59⋊C16)C2212yes—
14C22×Q16×C59C2211yescharacteristic
15C59×#424C2211yescharacteristic
16C59×#419—21no—
17C2×#101×C59—21no—
18C59×#341C2211yescharacteristic
19C2×D472—21no—
20C2×D472D412yes—
21#14273724—21no—
22#14273719—21no—
23#14273712—42no—
24#14273701—22no—
25#14273700—42no—
26C4×D236—22no—
27#14273670—44no—
28C2×C59⋊C16—22no—
29#14273661D412yes—
30#14273659—21no—
31#14273649—22no—
32#14273649—22no—
33C22×Q8×C59—21no—
34C2×Q16×C59—22no—
35C2×Q16×C59—42no—
36C2×M16×C59—21no—
37C2×M16×C59D411yescharacteristic
38C22×C8×C59C2311yescharacteristic
39#117×C59D411yescharacteristic
40C4⋊C8×C59—22no—
41#103×C59—42no—
42#101×C59—42no—
43C2×Q8×C59—22no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C22×C4×C59—41no—
50C22×C4×C59C2×D411yescharacteristic
51C2×C22⋊D59—42no—
52C2×D236—22no—
53C2×D236—42no—
54C2×C4×D59—42no—
55Q16×C59—24no—
56Q16×C59—42no—
57M16×C59—41no—
58M16×C59—41no—
59M16×C59—41no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C2×C8×C59—41no—
63C2×C8×C59C2×D412yes—
64C4⋊C4×C59D812yes—
65C22⋊C4×C59—21no—
66C42×C59—21no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69C4⋊Dic59—22no—
70Dic59⋊C4—42no—
71D472—42no—
72D472—81no—
73C59⋊C16—42no—
74C23×C59AGL(1,ℤ/8)11yescharacteristic
75C22×D59—42no—
76Q8×C59—42no—
77Q8×C59—42no—
78Q8×C59—81no—
79Q8×C59—81no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—22no—
83C2×C4×C59—42no—
84C2×C4×C59—81no—
85C2×C4×C59—81no—
86C2×C4×C59#12011yescharacteristic
87C2×C4×C59C2×D811yescharacteristic
88C22⋊D59—162no—
89C2×Dic59—42no—
90D236—42no—
91D236—42no—
92D236—162no—
93C4×D59—162no—
94C8×C59—22no—
95C8×C59—41no—
96C8×C59—81no—
97#12167—591nomaximal, Sylow
98C22×C59—21no—
99C22×C59—22no—
100C22×C59#44711yescharacteristic
101C2×D59—82no—
102C2×D59—82no—
103C2×D59—82no—
104C4×C59—21no—
105C4×C59—42no—
106C4×C59—82no—
107C4×C59#45411yescharacteristic
108C4×C59B(4,𝔽2)11yescharacteristic
109Dic59—82no—
110#3259—591no—
111#2185—592no—
112#2047D5911yescharacteristic, complemented
113#1349—592no—
114#1323—591no—
115C2×C59—41no—
116C2×C59#159811yescharacteristic
117C2×C59#219412yes—
118D59—162no—
119C22×Q16C2×D5911yescharacteristic
120#540—1181no—
121C2×SD32—1182no—
122#478—1182no—
123#469—1181no—
124#466—592no—
125#459—1182no—
126#424C2×D5911yescharacteristic
127#419—21no—
128C2×#101—21no—
129#358—1182no—
130C22⋊C16—592no—
131#341C2×D5911yescharacteristic
132C59#1216711yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C2×Q16—22no—
135C2×Q16—42no—
136C2×D8—592no—
137C2×D8—1181no—
138C2×M16—21no—
139C2×M16C22⋊D5911yescharacteristic
140C22×C8C22×D5911yescharacteristic
141#127—1181no—
142#125—1181no—
143#123—2362no—
144#121—1182no—
145#121—2362no—
146C4×D4—1182no—
147#117C22⋊D5911yescharacteristic
148SD32—2364no—
149C2×C16—1182no—
150#107—592no—
151#106—1181no—
152C4⋊C8—22no—
153#103—42no—
154#102—1182no—
155#102—1182no—
156#101—42no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—1182no—
164C2×D4—2362no—
165C2×D4—2362no—
166C22×C4—41no—
167C22×C4—2362no—
168C22×C4C2×C22⋊D5911yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—4721no—
172D8—2362no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8D4×D5912yes—
180C4⋊C4—1182no—
181C4⋊C4—2362no—
182C4⋊C4#1174566912yes—
183C22⋊C4—21no—
184C22⋊C4—2362no—
185C22⋊C4—2362no—
186C42—21no—
187C16—2362no—
188C23—2362no—
189C23#1427375211yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—9442no—
195D4—9442no—
196D4—2362no—
197D4—2362no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—9442no—
205C2×C4—2362no—
206C2×C4#1427369511yescharacteristic
207C2×C4C2×#1174566911yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—22no—
212C22—21no—
213C22—4722no—
214C22—4722no—
215C22—4722no—
216C22#1683753511yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—4722no—
221C4#1683757411yescharacteristic
222C4#1683758811yescharacteristic
223C2—41no—
224C2—9442no—
225C2#2142303011yescharacteristic
226C2#2142606512yes—
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2047C2
socleC22×C59#447
prime radical#21435535C2
hypercenter#2047D59
hypocenterC59#12167
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #2047

Complements

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

NH ≅ G/Ntype
2. #21426013C2semidirect
3. #21422110C2semidirect
4. #21422038C2semidirect
5. C59×#2047C2semidirect
112. #2047D59semidirect
132. C59#12167semidirect

Maximal subgroups

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

subgroupindex
#214355352
#214260132
#214221102
#214220382
C59×#20472
#1216759