Finite Group Databasebeta

← #28497899

Subgroups — #28497899

Order 15104.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#21432739C211yesmaximal, characteristic
2#21425892C212yesmaximal, complemented
3#21422038C211yesmaximal, characteristic, complemented
4#21422021C212yesmaximal
5C59×#1323C211yesmaximal, characteristic, complemented
6#16837870C411yescharacteristic
7#16837462—22no—
8#16837454—22no—
9#16837429C411yescharacteristic
10#16837370C411yescharacteristic
11#16837368C2211yescharacteristic
12M16×Dic59C411yescharacteristic
13C2×C8×Dic59C2211yescharacteristic
14Dic59⋊C16—22no—
15C22⋊(C59⋊C16)C2212yes—
16C16×Dic59—22no—
17C59×#424C2211yescharacteristic
18C59×C22⋊C16C2212yes—
19C2×#11745685C2×C411yescharacteristic
20C2×#11745664C2×C411yescharacteristic
21C22×C59⋊C8C2×C411yescharacteristic
22#14273692C812yes—
23#14273689C812yes—
24(C22⋊D59)⋊C4—22no—
25#14273684—21no—
26C2×C4×Dic59C2×C411yescharacteristic
27C42×D59—21no—
28C2×C59⋊C16—22no—
29C4⋊(C59⋊C8)—22no—
30(C2×D59)⋊C8—22no—
31(C2×D59)⋊C8—22no—
32#14273637—21no—
33C8×Dic59—21no—
34C8×Dic59D412yes—
35C8×Dic59D412yes—
36C2×M16×C59C2×C411yescharacteristic
37C22×C8×C59C2311yescharacteristic
38#117×C59C2×C411yescharacteristic
39C2×C16×C59—22no—
40C4⋊C8×C59—22no—
41C22×C4×C59C22×C411yescharacteristic
42C2×C22⋊D59—21no—
43C22×Dic59M1612yes—
44#11745685—41no—
45#11745685—41no—
46C2×D236C2×C811yescharacteristic
47C2×C4×D59—21no—
48C2×C4×D59—22no—
49C2×Dic118C2×C811yescharacteristic
50M16×C59—41no—
51C2×C8×C59—21no—
52C2×C8×C59—41no—
53C2×C8×C59C2×D412yes—
54C4⋊C4×C59C2×C812yes—
55C22⋊C4×C59—21no—
56C42×C59—21no—
57(C2×D59)⋊C4—22no—
58Dic59⋊C4—22no—
59C4×Dic59—22no—
60C4×Dic59C22⋊C411yescharacteristic
61C4×Dic59C22⋊C411yescharacteristic
62C4×Dic59C2×C812yes—
63#11745664—41no—
64C2×C59⋊C8—21no—
65C2×C59⋊C8—41no—
66C2×C59⋊C8C22⋊C412yes—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59C2×M1611yescharacteristic
70C22×D59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—22no—
74C2×C4×C59C2×C22⋊C411yescharacteristic
75C2×C4×C59C22×C811yescharacteristic
76C22⋊D59—42no—
77C2×Dic59—24no—
78C2×Dic59—21no—
79C2×Dic59C22⋊C812yes—
80C2×Dic59C22⋊C812yes—
81D236—41no—
82C4×D59—41no—
83C4×D59—41no—
84C4×D59—44no—
85Dic118—41no—
86C8×C59—22no—
87C8×C59—41no—
88C59⋊C8—22no—
89C59⋊C8—41no—
90#11462—591nomaximal, Sylow
91C22×C59—22no—
92C22×C59—21no—
93C22×C59C2×C22⋊C811yescharacteristic
94C2×D59—42no—
95C2×D59—41no—
96C4×C59—21no—
97C4×C59—42no—
98C4×C59#41011yescharacteristic
99C4×C59#41311yescharacteristic
100Dic59—22no—
101Dic59—41no—
102Dic59#35014yes—
103#2922—591no—
104#2162—592no—
105#1323—591no—
106#1323D5911yescharacteristic, complemented
107#1315—592no—
108C2×C59—41no—
109C2×C59#145811yescharacteristic
110C2×C59C2×#35012yes—
111D59—81no—
112C4×#41—591no—
113#433—1182no—
114#431—1182no—
115#424—591no—
116#424C2×D5911yescharacteristic
117#408—591no—
118#408—591no—
119C4×M16—591no—
120C2×C4×C8—591no—
121C4⋊C16—1182no—
122C22⋊C16—592no—
123C22⋊C16C2×D5912yes—
124C4×C16—1182no—
125C59#1146211yescharacteristic, complemented, Sylow
126C2×#41—591no—
127C2×M16—591no—
128C2×M16C4×D5911yescharacteristic
129C22×C8—591no—
130C22×C8C22×D5911yescharacteristic
131C4×Q8—592no—
132C4×D4—592no—
133C4×D4—1182no—
134#117—1181no—
135#117C4×D5911yescharacteristic
136C2×C42—591no—
137C2×C42—1181no—
138C2×C16—22no—
139C2×C16—1182no—
140C4⋊C8—22no—
141C4⋊C8—1182no—
142C22⋊C8—1182no—
143C22⋊C8—1182no—
144#97—1181no—
145C4×C8—592no—
146C4×C8—592no—
147C4×C8—1181no—
148#41—2361no—
149#41—2361no—
150C2×Q8—591no—
151C2×D4—591no—
152C2×D4—1181no—
153C22×C4—592no—
154C22×C4—1182no—
155C22×C4—1181no—
156C22×C4C2×C4×D5911yescharacteristic
157M16—41no—
158M16—2361no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—592no—
162C2×C8—1181no—
163C2×C8—2361no—
164C2×C8D4×D5912yes—
165C4⋊C4—1182no—
166C4⋊C4C8×D5912yes—
167C22⋊C4—21no—
168C22⋊C4—1182no—
169C42—21no—
170C42—591no—
171C42—591no—
172C42—592no—
173C42—1182no—
174C16—42no—
175C16—2362no—
176C23—1181no—
177C23M16×D5911yescharacteristic
178Q8—2361no—
179D4—2361no—
180D4—2362no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—592no—
185C2×C4—592no—
186C2×C4—1184no—
187C2×C4—1181no—
188C2×C4—2364no—
189C2×C4—2361no—
190C2×C4—2361no—
191C2×C4C22⋊C4×D5911yescharacteristic
192C2×C4C2×C8×D5911yescharacteristic
193C8—22no—
194C8—41no—
195C8—1182no—
196C8—2361no—
197C22—22no—
198C22—21no—
199C22—2362no—
200C22—2361no—
201C22C22⋊C8×D5911yescharacteristic
202C4—21no—
203C4—42no—
204C4—594no—
205C4—1182no—
206C4—2361no—
207C4#1683737411yescharacteristic
208C4#1683738511yescharacteristic
209C2—41no—
210C2—4721no—
211C2#2142240911yescharacteristic
212C2#350×D5912yes—
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#1323C2
socleC22×C59C2×C22⋊C8
prime radicalC2×#11745685C2×C4
cosocleC22×C8×C59C23
hypercenter#1323D59
hypocenterC59#11462
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #21425892C2semidirect
3. #21422038C2semidirect
5. C59×#1323C2semidirect
106. #1323D59semidirect
125. C59#11462semidirect

Maximal subgroups

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

subgroupindex
#214327392
#214258922
#214220382
#214220212
C59×#13232
#1146259