Finite Group Databasebeta

← #28486221

Subgroups — #28486221

Order 15104.

436 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 99 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#21434472C211yesmaximal, characteristic
2#21422809C212yesmaximal
3#21422734C211yesmaximal, characteristic, complemented
4#21421967C212yesmaximal, complemented
5C59×#2987C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7#16837768—22no—
8#16837750C414yes—
9#16837459C2212yes—
10#16837406—22no—
11C4×#11745664C2211yescharacteristic
12#16837145—24no—
13#16837144—22no—
14C59×#533C2211yescharacteristic
15C59×#446—24no—
16C59×#433—22no—
17C59×#432C2212yes—
18C22×D236—21no—
19C22×D236—22no—
20C2×#11745664—21no—
21#14273724—22no—
22#14273724—21no—
23#14273724C2×C412yes—
24C4⋊(C59⋊C8)—22no—
25#14273647—28no—
26C22⋊(C59⋊C8)—22no—
27#14273638—21no—
28#14273638—22no—
29C4×C59⋊C8C2×C412yes—
30C4⋊Q8×C59C2×C412yes—
31#125×C59D412yes—
32C22⋊Q8×C59—22no—
33C4×Q8×C59—22no—
34#117×C59D412yes—
35C2×C42×C59C2311yescharacteristic
36C4⋊C8×C59—24no—
37C4⋊C8×C59D414yes—
38C22⋊C8×C59—22no—
39#97×C59—22no—
40#97×C59—22no—
41C4×C8×C59—22no—
42C23×D59—41no—
43C2×Q8×C59—22no—
44C22×C4×C59C2×D411yescharacteristic
45C22×C4×C59#4112yes—
46C2×D236—22no—
47C2×D236—24no—
48C2×D236—44no—
49C2×D236—41no—
50C2×D236—42no—
51C2×D236—84no—
52C2×C8×C59—24no—
53C4⋊C4×C59—22no—
54C4⋊C4×C59—24no—
55C4⋊C4×C59C22⋊C414yes—
56C22⋊C4×C59—22no—
57C42×C59—22no—
58C42×C59C22×C411yescharacteristic
59C42×C59C2×D411yescharacteristic
60C42×C59C2×D412yes—
61#11745664—42no—
62C2×C59⋊C8—22no—
63C2×C59⋊C8—42no—
64C23×C59#12311yescharacteristic
65C22×D59—42no—
66C22×D59—82no—
67C22×D59—84no—
68Q8×C59—82no—
69C2×C4×C59—22no—
70C2×C4×C59—24no—
71C2×C4×C59—42no—
72C2×C4×C59C2×C22⋊C411yescharacteristic
73C2×C4×C59C4×D412yes—
74C2×C4×C59#12011yescharacteristic
75C2×C4×C59#12112yes—
76D236—44no—
77D236—84no—
78D236—84no—
79D236—82no—
80C8×C59—44no—
81C59⋊C8—42no—
82C59⋊C8—82no—
83#9424—591nomaximal, Sylow
84C22×C59—21no—
85C22×C59—22no—
86C22×C59#38611yescharacteristic
87C2×D59—82no—
88C2×D59—84no—
89C2×D59—164no—
90C4×C59—21no—
91C4×C59—22no—
92C4×C59—22no—
93C4×C59—44no—
94C4×C59#45412yes—
95#3138—591no—
96#2987D5911yescharacteristic, complemented
97#1555—592no—
98#1537—591no—
99#1291—592no—
100C2×C59—41no—
101C2×C59#188811yescharacteristic
102C2×C59#190312yes—
103D59—162no—
104#533C2×D5911yescharacteristic
105C2×#127—591no—
106#490—1182no—
107#486—594no—
108#446—24no—
109#433—22no—
110#432—592no—
111#432C2×D5912yes—
112#418—1182no—
113C4×M16—591no—
114#331—1184no—
115#330—1182no—
116C59#942411yescharacteristic, complemented, Sylow
117C22×D4—1182no—
118C22×D4—1181no—
119C2×M16—1181no—
120C4⋊Q8C4×D5912yes—
121#127—592no—
122#127—1181no—
123#127—1182no—
124#125D23612yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117C22⋊D5912yes—
128C2×C42C22×D5911yescharacteristic
129C4⋊C8—24no—
130C4⋊C8—1182no—
131C4⋊C8C22⋊D5914yes—
132#102—1188no—
133C22⋊C8—22no—
134C22⋊C8—1182no—
135#97—22no—
136#97—22no—
137#97—1181no—
138#97—1182no—
139C4×C8—22no—
140C4×C8—592no—
141C24—2361no—
142C2×Q8—22no—
143C2×D4—1182no—
144C2×D4—1184no—
145C2×D4—4724no—
146C2×D4—2364no—
147C2×D4—2361no—
148C2×D4—2362no—
149C22×C4#1174568512yes—
150C22×C4C2×C22⋊D5911yescharacteristic
151M16—2362no—
152C2×C8—24no—
153C2×C8—1182no—
154C2×C8—2362no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4(C2×D59)⋊C414yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D5911yescharacteristic
161C42C2×D23611yescharacteristic
162C42C2×C22⋊D5912yes—
163C23—4724no—
164C23—4722no—
165C23—2362no—
166C23#1427371011yescharacteristic
167Q8—82no—
168D4—4724no—
169D4—4724no—
170D4—4722no—
171D4—2364no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×(C2×D59)⋊C411yescharacteristic
176C2×C4C4×C22⋊D5912yes—
177C2×C4#1427369611yescharacteristic
178C2×C4#1427370112yes—
179C8—44no—
180C8—4722no—
181C8—2362no—
182C22—21no—
183C22—22no—
184C22—9444no—
185C22—4722no—
186C22—4724no—
187C22#1683728111yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1683757512yes—
193C2—41no—
194C2—9442no—
195C2#2142449911yescharacteristic
196C2#2142457612yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2987C2
socleC22×C59#386
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenter#2987D59
hypocenterC59#9424
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
3. #21422734C2semidirect
4. #21421967C2semidirect
5. C59×#2987C2semidirect
96. #2987D59semidirect
116. C59#9424semidirect

Maximal subgroups

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

subgroupindex
#214344722
#214228092
#214227342
#214219672
C59×#29872
#942459