Finite Group Databasebeta

← #28483636

Subgroups — #28483636

Order 15104.

416 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 95 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#21434441C211yesmaximal, characteristic
2#21433167C211yesmaximal, characteristic
3#21422570C212yesmaximal, complemented
4#21421996C212yesmaximal, complemented
5C59×#1543C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7#16837770—21no—
8#16837766—21no—
9#16837761C414yes—
10#16837474—24no—
11#16837459C2212yes—
12#16837398—22no—
13#16837369—22no—
14C2×#14273638C2211yescharacteristic
15#16837134—24no—
16C59×#532C2211yescharacteristic
17C59×#431C2212yes—
18C59×#333—21no—
19C59×#331—21no—
20C2×#11745664—22no—
21C22×C59⋊C8—21no—
22C22×C59⋊C8—22no—
23#14273727C2×C412yes—
24#14273719—21no—
25#14273706—22no—
26C4×Dic118—22no—
27#14273686—21no—
28C4⋊(C59⋊C8)D414yes—
29#14273651—24no—
30#14273648—24no—
31C22⋊(C59⋊C8)—24no—
32C22⋊(C59⋊C8)—22no—
33#14273638—22no—
34#14273638—21no—
35#14273638C2×C412yes—
36C2×#41×C59D412yes—
37C4⋊Q8×C59D411yescharacteristic
38#127×C59D411yescharacteristic
39#124×C59C2×C412yes—
40#121×C59—22no—
41C4×D4×C59—22no—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59—22no—
44C22⋊C8×C59—22no—
45C4×C8×C59—22no—
46#41×C59—42no—
47#41×C59—42no—
48C2×Q8×C59C22⋊C412yes—
49C2×D4×C59—22no—
50C2×D4×C59—22no—
51C2×D4×C59C22⋊C412yes—
52C22×C4×C59—22no—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59#4112yes—
55C2×Dic118—22no—
56C2×C8×C59—42no—
57C4⋊C4×C59—22no—
58C22⋊C4×C59—24no—
59C42×C59C22×C411yescharacteristic
60C42×C59C2×D411yescharacteristic
61C42×C59C2×D412yes—
62C22⋊Dic59—41no—
63C4⋊Dic59—22no—
64C4⋊Dic59—22no—
65Dic59⋊C4—42no—
66C4×Dic59—41no—
67#11745664—82no—
68C2×C59⋊C8—22no—
69C2×C59⋊C8—24no—
70C2×C59⋊C8—42no—
71C2×C59⋊C8—82no—
72C23×C59—22no—
73C23×C59#12311yescharacteristic
74Q8×C59—42no—
75D4×C59—42no—
76D4×C59—82no—
77D4×C59—82no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—42no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59C2×C22⋊C411yescharacteristic
84C2×C4×C59C4×D412yes—
85C2×C4×C59#12011yescharacteristic
86C2×C4×C59#12112yes—
87C2×Dic59—42no—
88Dic118—44no—
89C8×C59—82no—
90C59⋊C8—42no—
91C59⋊C8—44no—
92#9005—591nomaximal, Sylow
93C22×C59—22no—
94C22×C59—21no—
95C22×C59—44no—
96C22×C59—42no—
97C22×C59#38611yescharacteristic
98C4×C59—22no—
99C4×C59—22no—
100C4×C59—21no—
101C4×C59—42no—
102C4×C59#45512yes—
103Dic59—82no—
104#3135—591no—
105#2980—591no—
106#1543D5911yescharacteristic
107#1497—592no—
108#1302—592no—
109C2×C59—41no—
110C2×C59—82no—
111C2×C59#188112yes—
112C2×C59#188511yescharacteristic
113#533—591no—
114#532C2×D5911yescharacteristic
115#491—1181no—
116#490—1181no—
117#489—594no—
118#435—1184no—
119#432—592no—
120#431C2×D5912yes—
121#416—1182no—
122#408—1182no—
123C2×#97—591no—
124#333—21no—
125#331—21no—
126Q8⋊C8—1184no—
127C59#900511yescharacteristic, complemented, Sylow
128C2×#41D23612yes—
129C2×M16—1182no—
130C22×C8—1182no—
131C22×C8—1181no—
132C4⋊Q8—592no—
133C4⋊Q8C22⋊D5911yescharacteristic
134#127C22⋊D5911yescharacteristic
135#125—1181no—
136#124C4×D5912yes—
137C22⋊Q8—1182no—
138#121—22no—
139C4×Q8—1182no—
140C4×D4—22no—
141#117—1181no—
142C2×C42C22×D5911yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—594no—
145#103—1184no—
146#102—1184no—
147C22⋊C8—22no—
148C22⋊C8—1184no—
149C22⋊C8—1182no—
150#97—592no—
151#97—1182no—
152#97—1181no—
153C4×C8—22no—
154#41—42no—
155#41—42no—
156C2×Q8—1182no—
157C2×Q8(C2×D59)⋊C412yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4(C2×D59)⋊C412yes—
161C22×C4—22no—
162C22×C4C2×D23611yescharacteristic
163C22×C4#1174568712yes—
164M16—4722no—
165C2×C8—42no—
166C2×C8—1182no—
167C2×C8—1184no—
168C2×C8—4722no—
169C2×C8—2362no—
170C4⋊C4—22no—
171C4⋊C4—1182no—
172C4⋊C4—1182no—
173C4⋊C4—2362no—
174C22⋊C4—24no—
175C22⋊C4—2361no—
176C42—2361no—
177C42C2×C4×D5911yescharacteristic
178C42D4×D5912yes—
179C42C2×C22⋊D5911yescharacteristic
180C23—22no—
181C23#1427371111yescharacteristic
182Q8—42no—
183Q8—2364no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—2362no—
193C2×C4C2×(C2×D59)⋊C411yescharacteristic
194C2×C4(C22⋊D59)⋊C412yes—
195C2×C4#1427369711yescharacteristic
196C2×C4#1427369912yes—
197C8—82no—
198C8—2362no—
199C8—2364no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22#1683727911yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—4722no—
210C4#1683758212yes—
211C2—41no—
212C2—82no—
213C2#2142445612yes—
214C2#2142448211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1543C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#1543D59
hypocenterC59#9005
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
3. #21422570C2semidirect
4. #21421996C2semidirect
127. C59#9005semidirect

Maximal subgroups

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

subgroupindex
#214344412
#214331672
#214225702
#214219962
C59×#15432
#900559