Finite Group Databasebeta

← #28491794

Subgroups — #28491794

Order 15104.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#21424924C211yesmaximal, characteristic
2#21422655C212yesmaximal
3#21421954C212yesmaximal, complemented
4#21421929C211yesmaximal, characteristic, complemented
5C59×#2864C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837455C2212yes—
8#16837426C2211yescharacteristic
9C2×#14273655—22no—
10#16837135—24no—
11C59×#389—22no—
12C59×#387—24no—
13C59×#382C2212yes—
14C43×C59C2211yescharacteristic
15C2×#11745685—21no—
16C2×#11745664—21no—
17#14273727Q811yescharacteristic
18#14273724Q811yescharacteristic
19#14273715—21no—
20#14273701—22no—
21C4×D236—22no—
22C4⋊(C59⋊C8)—22no—
23C4⋊(C59⋊C8)—22no—
24#14273655—44no—
25C22⋊(C59⋊C8)—22no—
26C4×C59⋊C8—22no—
27C2×C4⋊C4×C59—24no—
28C2×C4⋊C4×C59—22no—
29C2×C42×C59—22no—
30C2×C42×C59D412yes—
31C2×C42×C59C2311yescharacteristic
32#95×C59—24no—
33#95×C59—22no—
34#95×C59D414yes—
35C22×C4×C59—22no—
36C22×C4×C59—24no—
37C22×C4×C59C2×D412yes—
38C22×C4×C59C2×D411yescharacteristic
39C2×C22⋊D59—41no—
40#11745685—42no—
41#11745685—42no—
42C2×D236—21no—
43C2×D236—22no—
44C2×C4×D59—41no—
45C2×Dic118—21no—
46C4⋊C4×C59—48no—
47C4⋊C4×C59—42no—
48C4⋊C4×C59—42no—
49C42×C59—22no—
50C42×C59—22no—
51C42×C59—44no—
52C42×C59C2×Q811yescharacteristic
53C42×C59#4111yescharacteristic
54C42×C59#4112yes—
55(C2×D59)⋊C4—42no—
56C4⋊Dic59—22no—
57#11745664—42no—
58C2×C59⋊C8—42no—
59C2×C59⋊C8—41no—
60C23×C59#12011yescharacteristic
61C22×D59—41no—
62C2×C4×C59—24no—
63C2×C4×C59—42no—
64C2×C4×C59—42no—
65C2×C4×C59—42no—
66C2×C4×C59—48no—
67C2×C4×C59—44no—
68C2×C4×C59C22⋊Q812yes—
69C2×C4×C59C22⋊Q811yescharacteristic
70C2×C4×C59#12412yes—
71C2×C4×C59#12411yescharacteristic
72C2×C4×C59#13512yes—
73C2×C4×C59AGL(1,ℤ/8)12yes—
74C22⋊D59—82no—
75C2×Dic59—41no—
76D236—42no—
77D236—82no—
78C4×D59—81no—
79C4×D59—81no—
80Dic118—42no—
81C59⋊C8—82no—
82C59⋊C8—81no—
83#10366—591nomaximal, Sylow
84C22×C59—22no—
85C22×C59#39311yescharacteristic
86C22×C59#44912yes—
87C2×D59—81no—
88C2×D59—82no—
89C4×C59—22no—
90C4×C59—22no—
91C4×C59—44no—
92C4×C59—84no—
93C4×C59#47812yes—
94C4×C59#48811yescharacteristic
95C4×C59#49011yescharacteristic
96Dic59—81no—
97#2864D5911yescharacteristic, complemented
98#1962—591no—
99#1517—592no—
100#1286—592no—
101#1276—591no—
102C2×C59—22no—
103C2×C59#202511yescharacteristic
104C2×C59#203112yes—
105D59—161no—
106#532—591no—
107#431—592no—
108#423—591no—
109C2×#104—1182no—
110#389—22no—
111#387—24no—
112#382C2×D5912yes—
113C43C2×D5911yescharacteristic
114#327—1184no—
115C59#1036611yescharacteristic, complemented, Sylow
116C2×#41—1181no—
117C2×M16—1181no—
118C4⋊Q8—591no—
119#127—591no—
120#124—1181no—
121#121—1182no—
122C4×D4—1182no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C22⋊D5912yes—
127C2×C42C22×D5911yescharacteristic
128C4⋊C8—1182no—
129C4⋊C8—1182no—
130#104—2364no—
131C22⋊C8—1182no—
132C4×C8—1182no—
133#95—24no—
134#95—22no—
135#95C22⋊D5914yes—
136#41—2362no—
137#41—2362no—
138C2×Q8—1181no—
139C2×D4—1182no—
140C2×D4—1181no—
141C2×D4—2361no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—2361no—
145C22×C4C2×C22⋊D5912yes—
146C22×C4C2×C22⋊D5911yescharacteristic
147M16—2362no—
148C2×C8—2362no—
149C2×C8—2361no—
150C4⋊C4—48no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—1182no—
154C22⋊C4—2362no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42Q8×D5911yescharacteristic
159C42#1174568911yescharacteristic
160C42#1174568912yes—
161C23—2361no—
162C23#1427369611yescharacteristic
163Q8—2362no—
164D4—4722no—
165D4—4722no—
166D4—2362no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—4721no—
174C2×C4—4721no—
175C2×C4—2361no—
176C2×C4#1427370312yes—
177C2×C4#1427370311yescharacteristic
178C2×C4#1427371612yes—
179C2×C4#1427371611yescharacteristic
180C2×C4#1427374712yes—
181C2×C4#1427375212yes—
182C8—4722no—
183C8—4721no—
184C22—22no—
185C22—4721no—
186C22—4722no—
187C22#1683731311yescharacteristic
188C22#1683754712yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—4721no—
194C4#1683769912yes—
195C4#1683775911yescharacteristic
196C4#1683776811yescharacteristic
197C2—22no—
198C2—9441no—
199C2#2142523911yescharacteristic
200C2#2142526512yes—
subgroupG/H
centerC22#16837313
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2864C2
socleC22×C59#393
prime radical#16837952C22
hypercenter#2864D59
hypocenterC59#10366
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
3. #21421954C2semidirect
4. #21421929C2semidirect
5. C59×#2864C2semidirect
97. #2864D59semidirect
115. C59#10366semidirect

Maximal subgroups

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

subgroupindex
#214249242
#214226552
#214219542
#214219292
C59×#28642
#1036659