Finite Group Databasebeta

← #28488666

Subgroups — #28488666

Order 15104.

412 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 123 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#21434419C211yesmaximal, characteristic
2#21424234C211yesmaximal, characteristic, complemented
3#21422552C212yesmaximal, complemented
4#21422548C212yesmaximal
5C2×C59×#322C211yesmaximal, characteristic, complemented
6#16837959C2211yescharacteristic
7#16837952C2211yescharacteristic
8#16837757—22no—
9#16837753C412yes—
10#16837749C412yes—
11#16837426C2212yes—
12#16837399—22no—
13#16837167—22no—
14#16837134—24no—
15#16837131—24no—
16C2×C4×C8×C59C2212yes—
17C2×C4×C8×C59C2211yescharacteristic
18C59×#322—22no—
19C2×#11745685—21no—
20C2×#11745664—22no—
21#14273727C2×C411yescharacteristic
22#14273727C2×C412yes—
23#14273724C2×C411yescharacteristic
24#14273719—21no—
25#14273715—21no—
26#14273706—22no—
27#14273701—22no—
28C4×Dic118—22no—
29C4×D236—22no—
30#14273686—21no—
31C4⋊(C59⋊C8)—22no—
32C4⋊(C59⋊C8)—22no—
33#14273652—24no—
34#14273649—24no—
35C22×C8×C59—22no—
36C22×C8×C59D412yes—
37C2×C42×C59C2311yescharacteristic
38C4×C8×C59—22no—
39C4×C8×C59C2×C412yes—
40C4×C8×C59D414yes—
41C4×C8×C59D412yes—
42C22×C4×C59C2×D411yescharacteristic
43C22×C4×C59#4112yes—
44C2×C22⋊D59—41no—
45#11745685—44no—
46C2×D236—21no—
47C2×D236—22no—
48C2×C4×D59—41no—
49C2×Dic118—21no—
50C2×Dic118—22no—
51C2×C8×C59—24no—
52C2×C8×C59—24no—
53C2×C8×C59—44no—
54C2×C8×C59C22⋊C414yes—
55C42×C59C22×C411yescharacteristic
56C42×C59C2×D411yescharacteristic
57C42×C59C2×D412yes—
58C22⋊Dic59—41no—
59(C2×D59)⋊C4—42no—
60C4⋊Dic59—22no—
61C4⋊Dic59—22no—
62C4⋊Dic59—22no—
63Dic59⋊C4—42no—
64C4×Dic59—41no—
65#11745664—44no—
66C2×C59⋊C8—42no—
67C23×C59#12411yescharacteristic
68C22×D59—41no—
69C2×C4×C59—24no—
70C2×C4×C59C2×C22⋊C411yescharacteristic
71C2×C4×C59C4×D412yes—
72C2×C4×C59#12112yes—
73C2×C4×C59#12711yescharacteristic
74C2×C4×C59#13514yes—
75C22⋊D59—82no—
76C2×Dic59—42no—
77C2×Dic59—41no—
78D236—42no—
79D236—44no—
80C4×D59—82no—
81Dic118—42no—
82Dic118—44no—
83C8×C59—24no—
84C8×C59—44no—
85C59⋊C8—82no—
86#9863—591nomaximal, Sylow
87C22×C59—22no—
88C22×C59#39011yescharacteristic
89C22×C59#48812yes—
90C2×D59—81no—
91C2×D59—82no—
92C4×C59—22no—
93C4×C59#41612yes—
94C4×C59#44414yes—
95C4×C59#49512yes—
96Dic59—81no—
97Dic59—82no—
98#3131—591no—
99#1837—591no—
100#1493—592no—
101#1492—592no—
102C2×#322D5911yescharacteristic, complemented
103C2×C59—22no—
104C2×C59#195811yescharacteristic
105C2×C59#197512yes—
106D59—161no—
107#533—591no—
108#532—591no—
109#488—1182no—
110#487—592no—
111#486—592no—
112#423—592no—
113#416—1182no—
114C2×C4×C8C2×D5912yes—
115C2×C4×C8C2×D5911yescharacteristic
116#341—1182no—
117Q8⋊C8—1184no—
118D4⋊C8—1184no—
119#322—22no—
120C59#986311yescharacteristic, complemented, Sylow
121C2×#41—1181no—
122C2×M16—1182no—
123C22×C8—22no—
124C22×C8D23612yes—
125C4⋊Q8—591no—
126C4⋊Q8—592no—
127#127—591no—
128#125—1181no—
129#124—1181no—
130C22⋊Q8—1182no—
131#121—1182no—
132C4×Q8—1182no—
133C4×D4—1182no—
134#117—1181no—
135C2×C42C22×D5911yescharacteristic
136C4⋊C8—1182no—
137C4⋊C8—1182no—
138#103—1184no—
139#102—1184no—
140C4×C8—22no—
141C4×C8C4×D5912yes—
142C4×C8D23614yes—
143C4×C8C22⋊D5912yes—
144#41—2364no—
145C2×Q8—1181no—
146C2×Q8—1182no—
147C2×D4—1181no—
148C2×D4—1182no—
149C2×D4—2361no—
150C22×C4—2361no—
151C22×C4C2×D23611yescharacteristic
152C22×C4#1174568512yes—
153M16—2364no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—44no—
157C2×C8—2362no—
158C2×C8(C2×D59)⋊C414yes—
159C4⋊C4—1182no—
160C4⋊C4—1182no—
161C4⋊C4—1182no—
162C4⋊C4—2362no—
163C22⋊C4—2361no—
164C22⋊C4—2362no—
165C42—2361no—
166C42C2×C4×D5911yescharacteristic
167C42C2×D23612yes—
168C42C2×C22⋊D5911yescharacteristic
169C23—2361no—
170C23#1427371511yescharacteristic
171Q8—2362no—
172Q8—2364no—
173D4—4722no—
174D4—2362no—
175D4—2364no—
176C2×C4—24no—
177C2×C4—4722no—
178C2×C4—2362no—
179C2×C4—2361no—
180C2×C4C2×(C2×D59)⋊C411yescharacteristic
181C2×C4C4×D23612yes—
182C2×C4#1427370112yes—
183C2×C4#1427372411yescharacteristic
184C2×C4#1427374914yes—
185C8—24no—
186C8—44no—
187C8—4722no—
188C22—22no—
189C22—4721no—
190C22—4722no—
191C22#1683730411yescharacteristic
192C22#1683775712yes—
193C4—22no—
194C4—4721no—
195C4—4722no—
196C4#1683739912yes—
197C4#1683752514yes—
198C4#1683778612yes—
199C2—22no—
200C2—9441no—
201C2#2142490711yescharacteristic
202C2#2142498912yes—
subgroupG/H
centerC2×C4#14273724
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#322C2
socleC22×C59#390
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenterC2×#322D59
hypocenterC59#9863
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322

Complements

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

NH ≅ G/Ntype
2. #21424234C2semidirect
3. #21422552C2semidirect
5. C2×C59×#322C2semidirect
102. C2×#322D59semidirect
120. C59#9863semidirect

Maximal subgroups

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

subgroupindex
#214344192
#214242342
#214225522
#214225482
C2×C59×#3222
#986359