Finite Group Databasebeta

← #28483097

Subgroups — #28483097

Order 15104.

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

normal subgroups only

This group has 336 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#21425078C211yesmaximal, characteristic
2#21422864C212yesmaximal
3#21422691C211yesmaximal, characteristic
4#21421988C212yesmaximal
5C59×#2987C211yesmaximal, characteristic
6#16837426C2211yescharacteristic
7C4×#11745664C2212yes—
8#16837248C2211yescharacteristic
9#16837161—22no—
10#16837154—22no—
11#16837152—22no—
12#16837144C411yescharacteristic, complemented
13#16837144C412yescomplemented
14#16837144C411yescharacteristic, complemented
15C59×#533C2211yescharacteristic
16C59×#446—22no—
17C59×#446—22no—
18C59×#433—22no—
19C59×#432C2212yes—
20C2×#11745664—22no—
21C2×#11745664—21no—
22C2×C4×Dic59—21no—
23C4⋊(C59⋊C8)—22no—
24#14273638C2×C412yes—
25#14273638C2×C412yes—
26C4×C59⋊C8—22no—
27C59⋊#95—22no—
28C4⋊Q8×C59Q812yes—
29#125×C59C2×C411yescharacteristic
30#125×C59C2×C411yescharacteristic
31C22⋊Q8×C59—22no—
32C4×Q8×C59—22no—
33#117×C59D412yes—
34C2×C42×C59C2311yescharacteristic
35C4⋊C8×C59—22no—
36C4⋊C8×C59—22no—
37C4⋊C8×C59D412yes—
38C4⋊C8×C59D412yes—
39C22⋊C8×C59—22no—
40#97×C59—22no—
41#97×C59—22no—
42C4×C8×C59—22no—
43C2×Q8×C59—22no—
44C22×C4×C59C2×D411yescharacteristic
45C22×C4×C59#4112yes—
46C22×Dic59—41no—
47C2×C8×C59—22no—
48C2×C8×C59—22no—
49C4⋊C4×C59—22no—
50C4⋊C4×C59—22no—
51C4⋊C4×C59—22no—
52C4⋊C4×C59C4⋊C412yes—
53C4⋊C4×C59C4⋊C412yes—
54C22⋊C4×C59—22no—
55C42×C59—22no—
56C42×C59C22×C411yescharacteristic
57C42×C59C2×D412yes—
58C42×C59C2×Q811yescharacteristic
59C4×Dic59—22no—
60C4×Dic59—41no—
61#11745664—44no—
62#11745664—42no—
63C2×C59⋊C8—22no—
64C2×C59⋊C8—22no—
65C2×C59⋊C8—41no—
66C23×C59#12311yescharacteristic
67Q8×C59—82no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—22no—
71C2×C4×C59—22no—
72C2×C4×C59—42no—
73C2×C4×C59C2×C4⋊C411yescharacteristic
74C2×C4×C59C4×D412yes—
75C2×C4×C59#12011yescharacteristic
76C2×C4×C59C22⋊Q812yes—
77C2×Dic59—42no—
78C2×Dic59—82no—
79C8×C59—42no—
80C8×C59—42no—
81C59⋊C8—24no—
82C59⋊C8—42no—
83C59⋊C8—81no—
84#8919—591nomaximal, Sylow
85C22×C59—21no—
86C22×C59—22no—
87C22×C59#38511yescharacteristic
88C4×C59—21no—
89C4×C59—22no—
90C4×C59—22no—
91C4×C59—42no—
92C4×C59—42no—
93C4×C59#45411yescharacteristic
94C4×C59#45611yescharacteristic
95Dic59—82no—
96#2987D5911yescharacteristic
97#1995—591no—
98#1569—592no—
99#1526—591no—
100#1300—592no—
101C2×C59—41no—
102C2×C59#185711yescharacteristic
103C2×C59#186112yes—
104#533C2×D5911yescharacteristic
105#446—22no—
106#446—22no—
107#433—22no—
108#432C2×D5912yes—
109#423—591no—
110C4×M16—592no—
111#376—591no—
112#339—1182no—
113#335—1182no—
114#334—1182no—
115#330—591no—
116#330—592no—
117#330—591no—
118C59#891911yescharacteristic, complemented, Sylow
119C2×M16—1182no—
120C2×M16—1181no—
121C4⋊Q8Dic11812yes—
122#125C4×D5911yescharacteristic
123#125C4×D5911yescharacteristic
124C22⋊Q8—22no—
125C4×Q8—22no—
126#117C22⋊D5912yes—
127C2×C42—1181no—
128C2×C42C22×D5911yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—1182no—
132C4⋊C8C22⋊D5912yes—
133C4⋊C8C22⋊D5912yes—
134C22⋊C8—22no—
135#97—22no—
136#97—22no—
137#97—592no—
138#97—592no—
139C4×C8—22no—
140C4×C8—1182no—
141#95—1182no—
142C2×Q8—22no—
143C22×C4—2361no—
144C22×C4#1174568512yes—
145C22×C4C2×C22⋊D5911yescharacteristic
146M16—2364no—
147M16—2362no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—1182no—
151C2×C8—1182no—
152C2×C8—2361no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4Dic59⋊C412yes—
157C4⋊C4Dic59⋊C412yes—
158C22⋊C4—22no—
159C42—22no—
160C42—1182no—
161C42—2361no—
162C42C2×Dic11811yescharacteristic
163C42C2×C4×D5911yescharacteristic
164C42C2×C22⋊D5912yes—
165C23#1427371011yescharacteristic
166Q8—82no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—4722no—
173C2×C4—2362no—
174C2×C4C2×Dic59⋊C411yescharacteristic
175C2×C4C4×C22⋊D5912yes—
176C2×C4#1427369611yescharacteristic
177C2×C4#1427370612yes—
178C8—42no—
179C8—42no—
180C8—1184no—
181C8—4721no—
182C8—2362no—
183C22—21no—
184C22—22no—
185C22#1683727511yescharacteristic
186C4—21no—
187C4—22no—
188C4—22no—
189C4—42no—
190C4—42no—
191C4—4722no—
192C4#1683757511yescharacteristic
193C4#1683758511yescharacteristic
194C2—41no—
195C2#2142431111yescharacteristic
196C2#2142433612yes—
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2987C2
socleC22×C59#385
prime radicalC23×C59#123
cosocleC2×C42×C59C23
hypercenter#2987D59
hypocenterC59#8919
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
12. #16837144C4semidirect
13. #16837144C4semidirect
14. #16837144C4semidirect
118. C59#8919semidirect

Maximal subgroups

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

subgroupindex
#214250782
#214228642
#214226912
#214219882
C59×#29872
#891959