Finite Group Databasebeta

← #28498124

Subgroups — #28498124

Order 15104.

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

normal subgroups only

This group has 284 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#21432733C211yesmaximal, characteristic
2#21425905C212yesmaximal, complemented
3#21422047C211yesmaximal, characteristic, complemented
4#21422031C212yesmaximal
5C59×#1327C211yesmaximal, characteristic, complemented
6C2×#14273684C411yescharacteristic
7#16837462—22no—
8#16837462—22no—
9#16837426C411yescharacteristic
10#16837365C411yescharacteristic
11#16837365C2211yescharacteristic
12M16×Dic59C411yescharacteristic
13M16×Dic59C2211yescharacteristic
14#16837222—22no—
15#16837184C2212yes—
16#16837179—22no—
17C59×#423C2211yescharacteristic
18C59×#349C2212yes—
19C22×C4×D59C2×C411yescharacteristic
20C2×#11745664C2×C411yescharacteristic
21C2×#11745664C2×C411yescharacteristic
22#14273684—22no—
23#14273684—21no—
24#14273684—21no—
25C2×Dic59⋊C4—21no—
26C2×(C2×D59)⋊C4—21no—
27C2×C4×Dic59C2×C411yescharacteristic
28#14273667—22no—
29C4⋊(C59⋊C8)—22no—
30(C2×D59)⋊C8—22no—
31(C2×D59)⋊C8—22no—
32#14273637—21no—
33#14273637D412yes—
34C8×Dic59—21no—
35C8×Dic59D412yes—
36C2×M16×C59C2×C411yescharacteristic
37C2×M16×C59C2311yescharacteristic
38C2×C42×C59C2×C411yescharacteristic
39M32×C59—22no—
40C4⋊C8×C59—22no—
41C23×D59M1611yescharacteristic
42C22×C4×C59—21no—
43C22×C4×C59C22×C411yescharacteristic
44C22×Dic59—21no—
45C22×Dic59M1611yescharacteristic
46C2×C4×D59—21no—
47C2×C4×D59—21no—
48C2×C4×D59—41no—
49C2×C4×D59—41no—
50M16×C59—41no—
51M16×C59—41no—
52C2×C8×C59—21no—
53C2×C8×C59C2×D412yes—
54C42×C59—21no—
55C42×C59M1612yes—
56(C2×D59)⋊C4—22no—
57(C2×D59)⋊C4—22no—
58Dic59⋊C4—22no—
59Dic59⋊C4—22no—
60C4×Dic59—21no—
61C4×Dic59C22⋊C411yescharacteristic
62C4×Dic59C22⋊C411yescharacteristic
63#11745664—41no—
64#11745664—41no—
65C2×C59⋊C8—21no—
66C2×C59⋊C8C22⋊C412yes—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59C2×M1611yescharacteristic
70C22×D59—21no—
71C22×D59—41no—
72C22×D59—81no—
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—22no—
76C2×C4×C59—81no—
77C2×C4×C59C2×C22⋊C411yescharacteristic
78C2×C4×C59C2×M1611yescharacteristic
79C2×Dic59—21no—
80C2×Dic59—21no—
81C2×Dic59—21no—
82C2×Dic59—41no—
83C2×Dic59—81no—
84C4×D59—42no—
85C4×D59—41no—
86C4×D59—41no—
87C8×C59—22no—
88C8×C59—41no—
89C59⋊C8—22no—
90C59⋊C8—41no—
91#11522—591nomaximal, Sylow
92C22×C59—21no—
93C22×C59—41no—
94C22×C59#40711yescharacteristic
95C2×D59—42no—
96C2×D59—41no—
97C2×D59—82no—
98C4×C59—21no—
99C4×C59—42no—
100C4×C59#41011yescharacteristic
101C4×C59#41311yescharacteristic
102Dic59—41no—
103Dic59—41no—
104Dic59—41no—
105#2921—591no—
106#2165—592no—
107#1327—591no—
108#1327D5911yescharacteristic, complemented
109#1320—592no—
110C2×C59—21no—
111C2×C59—41no—
112C2×C59#146411yescharacteristic
113D59—81no—
114C2×#117—591no—
115#433—1182no—
116#433—1182no—
117#423—591no—
118#423C2×D5911yescharacteristic
119#407—591no—
120#407—591no—
121C4×M16—591no—
122C4×M16—591no—
123#364—1182no—
124#349—592no—
125#349C2×D5912yes—
126#347—1182no—
127C59#1152211yescharacteristic, complemented, Sylow
128C23×C4—591no—
129C2×M16—591no—
130C2×M16—591no—
131C2×M16C4×D5911yescharacteristic
132C2×M16C22×D5911yescharacteristic
133#117—1181no—
134#117—1181no—
135#117—1182no—
136C2×C4⋊C4—1181no—
137C2×C22⋊C4—1181no—
138C2×C42—591no—
139C2×C42C4×D5911yescharacteristic
140M32—22no—
141M32—1182no—
142C4⋊C8—22no—
143C4⋊C8—1182no—
144C22⋊C8—1182no—
145C22⋊C8—1182no—
146#97—592no—
147#97—1181no—
148C4×C8—592no—
149C4×C8—1181no—
150C24—591no—
151C22×C4—21no—
152C22×C4—591no—
153C22×C4—1181no—
154C22×C4—1181no—
155C22×C4—1181no—
156C22×C4—2361no—
157C22×C4—2361no—
158C22×C4C2×C4×D5911yescharacteristic
159M16—41no—
160M16—41no—
161M16—2361no—
162M16—2361no—
163C2×C8—21no—
164C2×C8—592no—
165C2×C8—1181no—
166C2×C8D4×D5912yes—
167C4⋊C4—1182no—
168C4⋊C4—1182no—
169C22⋊C4—1182no—
170C22⋊C4—1182no—
171C42—21no—
172C42—591no—
173C42—591no—
174C42—1181no—
175C42#1174565912yes—
176C16—42no—
177C16—2362no—
178C23—1181no—
179C23—4721no—
180C23—2361no—
181C23M16×D5911yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—81no—
186C2×C4—1181no—
187C2×C4—1181no—
188C2×C4—1181no—
189C2×C4—4721no—
190C2×C4—2362no—
191C2×C4—2361no—
192C2×C4—2361no—
193C2×C4—2361no—
194C2×C4C22⋊C4×D5911yescharacteristic
195C2×C4C2×#1174565911yescharacteristic
196C8—22no—
197C8—41no—
198C8—1182no—
199C8—2361no—
200C22—21no—
201C22—41no—
202C22—4722no—
203C22—2362no—
204C22—2361no—
205C22#1683736411yescharacteristic
206C4—21no—
207C4—42no—
208C4—2361no—
209C4—2361no—
210C4—2361no—
211C4#1683737411yescharacteristic
212C4#1683738511yescharacteristic
213C2—21no—
214C2—41no—
215C2—4721no—
216C2#2142242911yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1327C2
socleC2×C59#1464
prime radicalC23×D59M16
cosocleC2×M16×C59C23
hypercenter#1327D59
hypocenterC59#11522
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #21425905C2semidirect
3. #21422047C2semidirect
5. C59×#1327C2semidirect
108. #1327D59semidirect
127. C59#11522semidirect

Maximal subgroups

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

subgroupindex
#214327332
#214259052
#214220472
#214220312
C59×#13272
#1152259