Finite Group Databasebeta

← #28507778

Subgroups — #28507778

Order 15104.

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

normal subgroups only

This group has 370 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#21434620C211yesmaximal, characteristic
2#21425816C212yesmaximal
3#21422203C212yesmaximal
4#21422180C211yesmaximal, characteristic
5C59×#1971C211yesmaximal, characteristic
6#16837959—21no—
7C2×#14273667C2212yes—
8#16837811—24no—
9#16837440—21no—
10#16837440C2212yes—
11#16837206—24no—
12C22×Q16×C59C2211yescharacteristic
13C2×#106×C59C2211yescharacteristic
14C2×#103×C59—21no—
15C2×#97×C59C2211yescharacteristic
16C59×#391—21no—
17#14273727—22no—
18#14273719—22no—
19#14273706—42no—
20C4×Dic118—42no—
21#14273686—22no—
22#14273667—22no—
23#14273667—22no—
24C2×C59⋊C16—22no—
25#14273661—22no—
26#14273661D414yes—
27#14273659—22no—
28#14273659—22no—
29C22×Q8×C59—21no—
30C2×Q16×C59—22no—
31C2×Q16×C59—22no—
32C2×Q16×C59—21no—
33C2×Q16×C59Q812yescomplemented
34C22×C8×C59D411yescharacteristic
35C22×C8×C59C2311yescharacteristic
36C2×C4⋊C4×C59—21no—
37C2×C42×C59D411yescharacteristic
38#106×C59—22no—
39#103×C59—42no—
40#97×C59—22no—
41#95×C59—42no—
42C2×Q8×C59—22no—
43C2×Q8×C59—41no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59—41no—
50C22×C4×C59D812yes—
51C22×C4×C59C2×D411yescharacteristic
52C2×Dic118—42no—
53Q16×C59—24no—
54Q16×C59—42no—
55C2×C8×C59—21no—
56C2×C8×C59—21no—
57C2×C8×C59SD1612yes—
58C2×C8×C59C2×D412yes—
59C2×C8×C59C2×Q811yescharacteristic
60C2×C8×C59#4112yes—
61C2×C8×C59#4111yescharacteristic
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C42×C59—21no—
65C42×C59—21no—
66C22⋊Dic59—42no—
67C4⋊Dic59—24no—
68C4⋊Dic59—42no—
69Dic59⋊C4—44no—
70C4×Dic59—42no—
71C59⋊C16—42no—
72C23×C59C2×D811yescharacteristic
73Q8×C59—42no—
74Q8×C59—42no—
75Q8×C59—81no—
76Q8×C59—81no—
77C2×C4×C59—22no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59—81no—
81C2×C4×C59—81no—
82C2×C4×C59—82no—
83C2×C4×C59—81no—
84C2×C4×C59#12011yescharacteristic
85C2×C4×C59C22⋊Q811yescharacteristic
86C2×C4×C59#12411yescharacteristic
87C2×C4×C59C2×SD1611yescharacteristic
88C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C59#13711yescharacteristic
90C2×Dic59—44no—
91Dic118—84no—
92C8×C59—22no—
93C8×C59C22⋊Q812yes—
94C8×C59#12412yes—
95#13368—591nomaximal, Sylow
96C22×C59—22no—
97C22×C59#44711yescharacteristic
98C22×C59#47411yescharacteristic
99C22×C59#48611yescharacteristic
100C4×C59—42no—
101C4×C59—82no—
102C4×C59—161no—
103C4×C59#39311yescharacteristic
104C4×C59#44811yescharacteristic
105C4×C59#47611yescharacteristic
106C4×C59#48911yescharacteristic
107Dic59—84no—
108#3162—591no—
109#2141—592no—
110#1971D5911yescharacteristic
111#1386—592no—
112#1377—591no—
113C2×C59—21no—
114C2×C59—21no—
115C2×C59#202311yescharacteristic
116C2×C59#219112yes—
117C22×Q16C2×D5911yescharacteristic
118#533—1181no—
119C2×M32—592no—
120#501—1184no—
121#427—592no—
122#427—1181no—
123C2×#106C2×D5911yescharacteristic
124C2×#103—21no—
125C2×#97C2×D5911yescharacteristic
126#391—21no—
127#358—1184no—
128C59#1336811yescharacteristic, complemented, Sylow
129C22×Q8—21no—
130C2×Q16—21no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16Dic11812yescomplemented
134C22×C8C22⋊D5911yescharacteristic
135C22×C8C22×D5911yescharacteristic
136C4⋊Q8—1182no—
137#125—1182no—
138C22⋊Q8—2362no—
139C4×Q8—2362no—
140#117—1182no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D5911yescharacteristic
143M32—1182no—
144M32—1182no—
145C2×C16—1182no—
146#107—594no—
147#107—1182no—
148#106—22no—
149#106—1182no—
150#106—1182no—
151#103—42no—
152#97—22no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—2362no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#1174566912yes—
164C22×C4C2×C22⋊D5911yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1174567012yes—
170C2×C8C2×Dic11811yescharacteristic
171C2×C8#1174568511yescharacteristic
172C2×C8D4×D5912yes—
173C2×C8#1174568712yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—1184no—
177C4⋊C4—2364no—
178C4⋊C4—2362no—
179C22⋊C4—2362no—
180C42—21no—
181C42—21no—
182C42—2362no—
183C16—2362no—
184C23C2×#1174566911yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—4724no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—2364no—
198C2×C4#1427369511yescharacteristic
199C2×C4#1427370611yescharacteristic
200C2×C4#1427371411yescharacteristic
201C2×C4C2×#1174567011yescharacteristic
202C2×C4#1427375211yescharacteristic
203C2×C4#1427375711yescharacteristic
204C8—22no—
205C8#1427370412yes—
206C8#1427371312yes—
207C22—22no—
208C22#1683753511yescharacteristic
209C22#1683768211yescharacteristic
210C22#1683774811yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—4724no—
215C4#1683731411yescharacteristic
216C4#1683754011yescharacteristic
217C4#1683769211yescharacteristic
218C4#1683776111yescharacteristic
219C2—21no—
220C2—21no—
221C2#2142523211yescharacteristic
222C2#2142604612yes—
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#1971C2
socleC22×C59#447
prime radicalC23×C59C2×D8
hypercenter#1971D59
hypocenterC59#13368
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
33. C2×Q16×C59Q8semidirect
128. C59#13368semidirect
133. C2×Q16Dic118semidirect

Maximal subgroups

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

subgroupindex
#214346202
#214258162
#214222032
#214221802
C59×#19712
#1336859