Finite Group Databasebeta

← #28483651

Subgroups — #28483651

Order 15104.

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

normal subgroups only

This group has 398 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#21434444C211yesmaximal, characteristic
2#21433202C211yesmaximal, characteristic, complemented
3#21422566C212yesmaximal
4#21421996C212yesmaximal, complemented
5C59×#1544C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837772C412yes—
8#16837768C412yes—
9#16837764—22no—
10#16837533—24no—
11#16837463—22no—
12#16837459C2212yes—
13#16837396—22no—
14C2×#14273638C2211yescharacteristic
15#16837131—24no—
16C59×#533C2211yescharacteristic
17C59×#431C2212yes—
18C59×#332—22no—
19C2×#11745685—21no—
20C22×C59⋊C8—21no—
21#14273727C2×C411yescharacteristic
22#14273724C2×C411yescharacteristic
23#14273715—21no—
24#14273701—22no—
25C4×D236—22no—
26C4⋊(C59⋊C8)—24no—
27C4⋊(C59⋊C8)D414yes—
28#14273650—24no—
29#14273647—24no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33#14273638—21no—
34#14273638C2×C412yes—
35C4×C59⋊C8—22no—
36C4⋊Q8×C59D412yes—
37#125×C59C2×C412yes—
38C22⋊Q8×C59—22no—
39C4×Q8×C59—22no—
40#117×C59D412yes—
41C2×C42×C59C2311yescharacteristic
42C4⋊C8×C59—22no—
43C22⋊C8×C59—22no—
44C4×C8×C59—22no—
45C2×Q8×C59—22no—
46C22×C4×C59C2×D411yescharacteristic
47C22×C4×C59#4112yes—
48C2×C22⋊D59—41no—
49#11745685—44no—
50C2×D236—21no—
51C2×D236—22no—
52C2×C4×D59—41no—
53C2×Dic118—21no—
54C2×C8×C59—42no—
55C4⋊C4×C59—24no—
56C4⋊C4×C59—22no—
57C4⋊C4×C59C22⋊C414yes—
58C22⋊C4×C59—22no—
59C42×C59—22no—
60C42×C59C22×C411yescharacteristic
61C42×C59C2×D411yescharacteristic
62C42×C59C2×D412yes—
63(C2×D59)⋊C4—42no—
64C4⋊Dic59—22no—
65C2×C59⋊C8—22no—
66C2×C59⋊C8—24no—
67C2×C59⋊C8—42no—
68C23×C59#12311yescharacteristic
69C22×D59—41no—
70Q8×C59—82no—
71C2×C4×C59—22no—
72C2×C4×C59—24no—
73C2×C4×C59—42no—
74C2×C4×C59C2×C22⋊C411yescharacteristic
75C2×C4×C59C4×D412yes—
76C2×C4×C59#12011yescharacteristic
77C2×C4×C59#12112yes—
78C22⋊D59—82no—
79C2×Dic59—41no—
80D236—44no—
81D236—42no—
82C4×D59—82no—
83Dic118—42no—
84C8×C59—82no—
85C59⋊C8—42no—
86C59⋊C8—44no—
87#9008—591nomaximal, Sylow
88C22×C59—22no—
89C22×C59—21no—
90C22×C59#38611yescharacteristic
91C2×D59—82no—
92C2×D59—81no—
93C4×C59—22no—
94C4×C59—22no—
95C4×C59—21no—
96C4×C59—44no—
97C4×C59#45412yes—
98Dic59—81no—
99#3135—591no—
100#2987—591no—
101#1544D5911yescharacteristic, complemented
102#1496—592no—
103#1302—592no—
104C2×C59—41no—
105C2×C59#188112yes—
106C2×C59#188811yescharacteristic
107D59—161no—
108#533C2×D5911yescharacteristic
109#532—591no—
110#491—592no—
111#490—592no—
112#489—1182no—
113#446—1184no—
114#433—1182no—
115#432—592no—
116#431C2×D5912yes—
117#416—1182no—
118C2×#97—591no—
119#332—22no—
120D4⋊C8—1184no—
121C59#900811yescharacteristic, complemented, Sylow
122C2×#41—1181no—
123C22×C8—1181no—
124C4⋊Q8—591no—
125C4⋊Q8C22⋊D5912yes—
126#127—591no—
127#125C4×D5912yes—
128#124—1181no—
129C22⋊Q8—22no—
130#121—1182no—
131C4×Q8—22no—
132C4×D4—1182no—
133#117D23612yes—
134C2×C42C22×D5911yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—594no—
137C4⋊C8—1184no—
138#103—1184no—
139#102—1184no—
140C22⋊C8—22no—
141C22⋊C8—1182no—
142#97—592no—
143#97—1182no—
144#97—1182no—
145#97—1181no—
146C4×C8—22no—
147C4×C8—1182no—
148#41—2364no—
149C2×Q8—22no—
150C2×Q8—1181no—
151C2×D4—1181no—
152C2×D4—1182no—
153C2×D4—2361no—
154C22×C4—2361no—
155C22×C4C2×D23611yescharacteristic
156C22×C4#1174568712yes—
157C2×C8—42no—
158C2×C8—1182no—
159C2×C8—1184no—
160C2×C8—2362no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1182no—
164C4⋊C4(C2×D59)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—2362no—
167C42—22no—
168C42C2×C4×D5911yescharacteristic
169C42D4×D5912yes—
170C42C2×C22⋊D5911yescharacteristic
171C23—2361no—
172C23#1427371111yescharacteristic
173Q8—82no—
174Q8—2362no—
175D4—4722no—
176D4—2364no—
177D4—2362no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—4722no—
182C2×C4—2361no—
183C2×C4C2×(C2×D59)⋊C411yescharacteristic
184C2×C4(C22⋊D59)⋊C412yes—
185C2×C4#1427369711yescharacteristic
186C2×C4#1427369912yes—
187C8—82no—
188C8—2362no—
189C8—2364no—
190C22—22no—
191C22—21no—
192C22—4722no—
193C22—4721no—
194C22#1683727911yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—4721no—
200C4#1683757712yes—
201C2—41no—
202C2—9441no—
203C2#2142445612yes—
204C2#2142449711yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1544C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1544D59
hypocenterC59#9008
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
2. #21433202C2semidirect
4. #21421996C2semidirect
5. C59×#1544C2semidirect
101. #1544D59semidirect
121. C59#9008semidirect

Maximal subgroups

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

subgroupindex
#214344442
#214332022
#214225662
#214219962
C59×#15442
#900859