Finite Group Databasebeta

← #28508374

Subgroups — #28508374

Order 15104.

416 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 79 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#21432089C211yesmaximal, characteristic
2#21424311C212yesmaximal
3#21422223C211yesmaximal, characteristic
4#21422222C212yesmaximal
5C59×#1857C211yesmaximal, characteristic
6#16837445—22no—
7C2×#14273646C2212yes—
8#16837323—22no—
9#16837322—21no—
10#16837309C2212yes—
11#16837308—21no—
12#16837306—22no—
13#16837161—22no—
14C59×#519C2211yescharacteristic
15C59×#428—21no—
16C2×#101×C59C2211yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118—21no—
20C22×Dic118—21no—
21C2×#11745664—22no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59D412yes—
24C2×Dic59⋊C4—22no—
25C2×C4×Dic59D412yes—
26#14273660—42no—
27#14273658—42no—
28#14273646—22no—
29#14273646Q814yescomplemented
30C59⋊#95—21no—
31C59⋊#95—22no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C22×Q8×C59C2311yescharacteristic
35C2×M16×C59—21no—
36C4×Q8×C59—41no—
37C4×Q8×C59—41no—
38#117×C59—21no—
39#117×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C42×C59D411yescharacteristic
42#107×C59—41no—
43#106×C59—41no—
44#101×C59—21no—
45#101×C59Q812yescomplemented
46#95×C59—22no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59C2×Q811yescharacteristic
50C2×Q8×C59C2×Q812yes—
51C2×Q8×C59#4111yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59C2×D412yes—
55C22×C4×C59C2×D411yescharacteristic
56C22×Dic59—22no—
57C22×Dic59—22no—
58C2×Dic118—24no—
59C2×Dic118—81no—
60C2×Dic118—81no—
61C2×Dic118—82no—
62C2×Dic118—81no—
63M16×C59—22no—
64M16×C59—41no—
65C2×C8×C59—41no—
66C4⋊C4×C59—21no—
67C4⋊C4×C59—21no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—41no—
71C22⋊C4×C59—41no—
72C42×C59—42no—
73C42×C59—41no—
74C42×C59—41no—
75C4⋊Dic59—42no—
76C4⋊Dic59—81no—
77Dic59⋊C4—82no—
78C4×Dic59—42no—
79#11745664—24no—
80#11745664—42no—
81C2×C59⋊C8—42no—
82C23×C59#12011yescharacteristic
83Q8×C59—41no—
84Q8×C59—42no—
85Q8×C59—41no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59C22⋊Q812yes—
98C2×C4×C59C22⋊Q811yescharacteristic
99C2×C4×C59C4⋊Q811yescharacteristic
100C2×C4×C59C4⋊Q812yes—
101C2×Dic59—24no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107Dic118—84no—
108Dic118—84no—
109C8×C59—42no—
110C59⋊C8—44no—
111#13464—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#39511yescharacteristic
115C22×C59B(4,𝔽2)11yescharacteristic
116C22×C59#45811yescharacteristic
117C4×C59—22no—
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—22no—
121C4×C59—42no—
122C4×C59—81no—
123C4×C59—81no—
124C4×C59—81no—
125Dic59—44no—
126Dic59—82no—
127#2811—591no—
128#1857—592no—
129#1857D5911yescharacteristic
130#1394—592no—
131#1394—591no—
132C2×C59—21no—
133C2×C59—21no—
134C2×C59#206211yescharacteristic
135C2×C59#220211yescharacteristic
136C2×C59#220511yescharacteristic
137#519C2×D5911yescharacteristic
138#428—21no—
139#428—1182no—
140C2×#101—592no—
141C2×#101C2×D5911yescharacteristic
142#395—1182no—
143#395—1181no—
144#391—592no—
145#391—1182no—
146#391—1181no—
147#391C2×D5911yescharacteristic
148#339—21no—
149#339—1182no—
150C59#1346411yescharacteristic, complemented, Sylow
151C22×Q8—1181no—
152C22×Q8—1181no—
153C22×Q8C22×D5911yescharacteristic
154C2×M16—21no—
155C2×M16—1182no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—21no—
160C2×C4⋊C4—592no—
161C2×C4⋊C4—1182no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4C22⋊D5911yescharacteristic
164C2×C42—592no—
165C2×C42C22⋊D5911yescharacteristic
166#107—41no—
167#107—2362no—
168#106—41no—
169#106—2362no—
170#101—21no—
171#101—594no—
172#101—1182no—
173#101Dic11812yescomplemented
174#95—22no—
175#95—1182no—
176#95—1181no—
177#95—1182no—
178#95—1182no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—1184no—
182C2×Q8—4721no—
183C2×Q8—4721no—
184C2×Q8—4722no—
185C2×Q8—4721no—
186C2×Q8C2×Dic11811yescharacteristic
187C2×Q8#1174568511yescharacteristic
188C2×Q8Q8×D5912yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—1182no—
192C22×C4—1182no—
193C22×C4D4×D5912yes—
194C22×C4C2×C22⋊D5911yescharacteristic
195M16—22no—
196M16—41no—
197M16—1184no—
198M16—2362no—
199C2×C8—41no—
200C2×C8—2362no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—4722no—
207C4⋊C4—4721no—
208C4⋊C4—2362no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—2362no—
214C23#1427369511yescharacteristic
215Q8—41no—
216Q8—41no—
217Q8—42no—
218Q8—4724no—
219Q8—4724no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1184no—
232C2×C4—2362no—
233C2×C4—2362no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4—2362no—
237C2×C4#1427370512yes—
238C2×C4#1427370611yescharacteristic
239C2×C4#1427372511yescharacteristic
240C2×C4#1427372612yes—
241C8—42no—
242C8—2364no—
243C22—21no—
244C22—21no—
245C22#1683732211yescharacteristic
246C22#1683758811yescharacteristic
247C22#1683759111yescharacteristic
248C4—21no—
249C4—22no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—4722no—
257C4—2364no—
258C2—21no—
259C2—21no—
260C2#2142544911yescharacteristic
261C2#2142611211yescharacteristic
262C2#2142612511yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC22×Q8×C59C23
FrattiniC22×Q8C22×D59
FittingC59×#1857C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#1857D59
hypocenterC59#13464
LCSG ▷ C22×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
29. #14273646Q8semidirect
45. #101×C59Q8semidirect
150. C59#13464semidirect
173. #101Dic118semidirect

Maximal subgroups

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

subgroupindex
#214320892
#214243112
#214222232
#214222222
C59×#18572
#1346459