Finite Group Databasebeta

← #28507779

Subgroups — #28507779

Order 15104.

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

normal subgroups only

This group has 370 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#21434620C211yesmaximal, characteristic
2#21425817C212yesmaximal
3#21424967C211yesmaximal, characteristic
4#21422202C212yesmaximal
5C59×#1377C211yesmaximal, characteristic
6C22×Dic236C2211yescharacteristic
7#16837959—21no—
8#16837811—24no—
9#16837440—21no—
10#16837440C2212yes—
11C2×#14273659C2211yescharacteristic
12C2×#14273652—21no—
13#16837308—21no—
14#16837207—24no—
15C2×M32×C59C2212yes—
16C2×#97×C59C2211yescharacteristic
17C22×Dic118—21no—
18C2×Dic236—22no—
19C2×Dic236—22no—
20C2×Dic236—21no—
21C2×Dic236Q812yescomplemented
22#14273727—22no—
23#14273719—22no—
24#14273706—42no—
25C4×Dic118—42no—
26#14273686—22no—
27C2×C4⋊Dic59—21no—
28#14273661—22no—
29#14273661D414yes—
30#14273659—22no—
31#14273659—22no—
32#14273659—22no—
33#14273652—42no—
34C59⋊#95—42no—
35C22×C8×C59D411yescharacteristic
36C22×C8×C59C2311yescharacteristic
37C2×C42×C59D411yescharacteristic
38M32×C59—22no—
39M32×C59—22no—
40C2×C16×C59—22no—
41#97×C59—22no—
42C22×C4×C59D812yes—
43C22×C4×C59C2×D411yescharacteristic
44C22×Dic59—41no—
45C22×Dic59—41no—
46C2×Dic118—22no—
47C2×Dic118—41no—
48C2×Dic118—41no—
49C2×Dic118—41no—
50C2×Dic118—41no—
51C2×Dic118—42no—
52C2×Dic118—41no—
53C2×C8×C59—21no—
54C2×C8×C59—21no—
55C2×C8×C59SD1612yes—
56C2×C8×C59C2×D412yes—
57C2×C8×C59C2×Q811yescharacteristic
58C2×C8×C59#4112yes—
59C2×C8×C59#4111yescharacteristic
60C42×C59—21no—
61C42×C59—21no—
62C22⋊Dic59—42no—
63C4⋊Dic59—24no—
64C4⋊Dic59—42no—
65C4⋊Dic59—41no—
66C4⋊Dic59—41no—
67Dic59⋊C4—44no—
68C4×Dic59—42no—
69Dic236—24no—
70Dic236—42no—
71C16×C59—42no—
72C23×C59C2×D811yescharacteristic
73C2×C4×C59—22no—
74C2×C4×C59—42no—
75C2×C4×C59#12011yescharacteristic
76C2×C4×C59C22⋊Q811yescharacteristic
77C2×C4×C59#12411yescharacteristic
78C2×C4×C59C2×SD1611yescharacteristic
79C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C59#13711yescharacteristic
81C2×Dic59—44no—
82C2×Dic59—42no—
83C2×Dic59—81no—
84C2×Dic59—81no—
85C2×Dic59—82no—
86C2×Dic59—81no—
87Dic118—42no—
88Dic118—42no—
89Dic118—84no—
90Dic118—81no—
91Dic118—81no—
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#39311yescharacteristic
102C4×C59#44811yescharacteristic
103C4×C59#47611yescharacteristic
104C4×C59#48911yescharacteristic
105Dic59—84no—
106Dic59—82no—
107Dic59—161no—
108#3162—591no—
109#2141—592no—
110#1971—591no—
111#1386—592no—
112#1377D5911yescharacteristic
113C2×C59—21no—
114C2×C59—21no—
115C2×C59#202311yescharacteristic
116C2×C59#219112yes—
117C22×Q16—591no—
118#533—1181no—
119C2×M32C2×D5912yes—
120#501—1184no—
121#427—592no—
122#427—1181no—
123C2×#106—591no—
124C2×#103—1181no—
125C2×#97C2×D5911yescharacteristic
126#391—1181no—
127#358—1184no—
128C59#1336811yescharacteristic, complemented, Sylow
129C22×Q8—1181no—
130C2×Q16—592no—
131C2×Q16—1182no—
132C2×Q16—1182no—
133C2×Q16—1181no—
134C22×C8D23611yescharacteristic
135C22×C8C22×D5911yescharacteristic
136C4⋊Q8—1182no—
137#125—1182no—
138C22⋊Q8—2362no—
139C4×Q8—2362no—
140#117—1182no—
141C2×C4⋊C4—1181no—
142C2×C42D23611yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—594no—
147#107—1182no—
148#106—1182no—
149#106—1182no—
150#106—1182no—
151#103—2362no—
152#97—22no—
153#95—2362no—
154C2×Q8—1182no—
155C2×Q8—2362no—
156C2×Q8—2361no—
157C2×Q8—2361no—
158C2×Q8—2361no—
159C2×Q8—2361no—
160C2×Q8—2361no—
161C22×C4—2361no—
162C22×C4—2361no—
163C22×C4D47212yes—
164C22×C4C2×D23611yescharacteristic
165Q16—1184no—
166Q16—2362no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1174566012yes—
170C2×C8#1174568512yes—
171C2×C8D4×D5912yes—
172C2×C8#1174568711yescharacteristic
173C2×C8Q8×D5911yescharacteristic
174C4⋊C4—1184no—
175C4⋊C4—2364no—
176C4⋊C4—2362no—
177C4⋊C4—2361no—
178C4⋊C4—2361no—
179C22⋊C4—2362no—
180C42—21no—
181C42—21no—
182C42—2362no—
183C16—42no—
184C23C2×D47211yescharacteristic
185Q8—4724no—
186Q8—4721no—
187Q8—4721no—
188Q8—2362no—
189Q8—2362no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—4722no—
193C2×C4—4721no—
194C2×C4—4721no—
195C2×C4—4721no—
196C2×C4—2362no—
197C2×C4—2364no—
198C2×C4#1427369711yescharacteristic
199C2×C4#1427370711yescharacteristic
200C2×C4#1427371511yescharacteristic
201C2×C4C2×#1174566011yescharacteristic
202C2×C4#1427375511yescharacteristic
203C2×C4#1427376011yescharacteristic
204C8—22no—
205C8#1427370512yes—
206C8#1427371312yes—
207C22—22no—
208C22#1683753811yescharacteristic
209C22#1683768311yescharacteristic
210C22#1683774911yescharacteristic
211C4—42no—
212C4—9441no—
213C4—4722no—
214C4—4724no—
215C4#1683731511yescharacteristic
216C4#1683754311yescharacteristic
217C4#1683769311yescharacteristic
218C4#1683776211yescharacteristic
219C2—21no—
220C2—21no—
221C2#2142523311yescharacteristic
222C2#2142604912yes—
subgroupG/H
centerC22#16837538
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#1377C2
socleC22×C59#447
prime radicalC23×C59C2×D8
hypercenter#1377D59
hypocenterC59#13368
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
21. C2×Dic236Q8semidirect
128. C59#13368semidirect

Maximal subgroups

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

subgroupindex
#214346202
#214258172
#214249672
#214222022
C59×#13772
#1336859