Finite Group Databasebeta

← #28491783

Subgroups — #28491783

Order 15104.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#21432474C211yesmaximal, characteristic
2#21425073C211yesmaximal, characteristic
3#21422661C212yesmaximal
4#21421955C212yesmaximal
5C59×#1276C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7C2×#14273727—21no—
8#16837426C2211yescharacteristic
9#16837308—24no—
10#16837268—21no—
11#16837268C2212yes—
12#16837162—21no—
13#16837162—21no—
14#16837140—24no—
15C59×#431C2212yes—
16C43×C59C2211yescharacteristic
17C22×Dic118—22no—
18C2×#11745664—21no—
19#14273727—42no—
20#14273727—42no—
21#14273727Q812yescomplemented
22#14273719—21no—
23#14273706—22no—
24C4×Dic118—22no—
25#14273686—21no—
26C2×C4⋊Dic59—22no—
27C2×C4⋊Dic59—22no—
28C4⋊(C59⋊C8)—22no—
29C59⋊#95—24no—
30C59⋊#95—22no—
31C59⋊#95D414yes—
32C2×C42×C59—22no—
33C2×C42×C59D411yescharacteristic
34C2×C42×C59D411yescharacteristic
35C2×C42×C59C2311yescharacteristic
36C4⋊C8×C59—22no—
37C22⋊C8×C59—22no—
38C4×C8×C59—22no—
39C22×C4×C59—21no—
40C22×C4×C59—21no—
41C22×C4×C59C2×D412yes—
42C22×C4×C59C2×D411yescharacteristic
43C22×Dic59—24no—
44C2×Dic118—22no—
45C2×Dic118—44no—
46C2×Dic118—48no—
47C2×C8×C59—42no—
48C42×C59—21no—
49C42×C59—21no—
50C42×C59—44no—
51C42×C59SD1612yes—
52C42×C59Q1612yes—
53C42×C59C2×Q811yescharacteristic
54C42×C59#4112yes—
55C42×C59#4111yescharacteristic
56C22⋊Dic59—41no—
57C4⋊Dic59—22no—
58C4⋊Dic59—22no—
59C4⋊Dic59—44no—
60C4⋊Dic59—42no—
61C4⋊Dic59—42no—
62Dic59⋊C4—42no—
63C4×Dic59—41no—
64#11745664—41no—
65#11745664—41no—
66C2×C59⋊C8—41no—
67C23×C59#12011yescharacteristic
68C2×C4×C59—24no—
69C2×C4×C59—22no—
70C2×C4×C59—22no—
71C2×C4×C59—42no—
72C2×C4×C59—42no—
73C2×C4×C59C22⋊Q812yes—
74C2×C4×C59C22⋊Q811yescharacteristic
75C2×C4×C59#12412yes—
76C2×C4×C59#12411yescharacteristic
77C2×C4×C59C2×SD1611yescharacteristic
78C2×C4×C59C2×Q1611yescharacteristic
79C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C59#13711yescharacteristic
81C2×Dic59—48no—
82C2×Dic59—42no—
83C2×Dic59—44no—
84Dic118—88no—
85Dic118—82no—
86C8×C59—82no—
87C59⋊C8—81no—
88#10363—591nomaximal, Sylow
89C22×C59—22no—
90C22×C59#39311yescharacteristic
91C22×C59#45011yescharacteristic
92C22×C59#45111yescharacteristic
93C4×C59—24no—
94C4×C59—22no—
95C4×C59—22no—
96C4×C59—42no—
97C4×C59#47511yescharacteristic
98C4×C59#47711yescharacteristic
99C4×C59#48711yescharacteristic
100C4×C59#48911yescharacteristic
101Dic59—84no—
102Dic59—82no—
103#2862—591no—
104#1994—591no—
105#1518—592no—
106#1286—592no—
107#1276D5911yescharacteristic
108C2×C59—21no—
109C2×C59—21no—
110C2×C59#202411yescharacteristic
111C2×C59#203112yes—
112#533—591no—
113C2×C4⋊Q8—1181no—
114#431C2×D5912yes—
115#423—591no—
116#391—1184no—
117#382—592no—
118#382—1181no—
119C43C2×D5911yescharacteristic
120#339—1181no—
121#339—1181no—
122#328—1184no—
123C59#1036311yescharacteristic, complemented, Sylow
124C22×Q8—1182no—
125C2×M16—1181no—
126C4⋊Q8—592no—
127C4⋊Q8—2362no—
128C4⋊Q8—2362no—
129#125—1181no—
130C22⋊Q8—1182no—
131C4×Q8—1182no—
132#117—1181no—
133C2×C4⋊C4—1182no—
134C2×C4⋊C4—1182no—
135C2×C42—22no—
136C2×C42D23611yescharacteristic
137C2×C42D23611yescharacteristic
138C2×C42C22×D5911yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—1182no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—594no—
144#95—1184no—
145#95—1182no—
146C2×Q8—1182no—
147C2×Q8—2364no—
148C2×Q8—2368no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1184no—
152C22×C4C2×D23611yescharacteristic
153C22×C4D4×D5912yes—
154M16—2361no—
155M16—2361no—
156C2×C8—42no—
157C2×C8—2361no—
158C4⋊C4—1182no—
159C4⋊C4—1182no—
160C4⋊C4—2364no—
161C4⋊C4—2362no—
162C4⋊C4—2362no—
163C4⋊C4—2362no—
164C22⋊C4—2361no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2361no—
169C42#1174566012yes—
170C42Dic23612yes—
171C42#1174568512yes—
172C42#1174568711yescharacteristic
173C42Q8×D5911yescharacteristic
174C23#1427369711yescharacteristic
175Q8—4722no—
176Q8—4728no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—2368no—
183C2×C4—2364no—
184C2×C4—2362no—
185C2×C4#1427370512yes—
186C2×C4#1427370711yescharacteristic
187C2×C4#1427371312yes—
188C2×C4#1427371511yescharacteristic
189C2×C4C2×#1174566011yescharacteristic
190C2×C4C2×Dic23611yescharacteristic
191C2×C4#1427375511yescharacteristic
192C2×C4#1427376011yescharacteristic
193C8—82no—
194C8—4721no—
195C22—22no—
196C22#1683731511yescharacteristic
197C22#1683755511yescharacteristic
198C22#1683756011yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—4724no—
204C4—4722no—
205C4#1683768811yescharacteristic
206C4#1683769811yescharacteristic
207C4#1683775311yescharacteristic
208C4#1683776211yescharacteristic
209C2—21no—
210C2—21no—
211C2#2142523811yescharacteristic
212C2#2142526912yes—
subgroupG/H
centerC22#16837315
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1276C2
socleC22×C59#393
prime radicalC23×C59#120
hypercenter#1276D59
hypocenterC59#10363
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
21. #14273727Q8semidirect
123. C59#10363semidirect

Maximal subgroups

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

subgroupindex
#214324742
#214250732
#214226612
#214219552
C59×#12762
#1036359