Finite Group Databasebeta

← #28489304

Subgroups — #28489304

Order 15104.

544 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 85 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#21434032C212yesmaximal, complemented
2#21424165C211yesmaximal, characteristic, complemented
3#21422456C212yesmaximal, complemented
4#21421973C211yesmaximal, characteristic, complemented
5C59×#3186C211yesmaximal, characteristic
6C2×#14273686C2211yescharacteristic
7#16837723—24no—
8#16837718—22no—
9#16837713—22no—
10#16837442—22no—
11#16837407—22no—
12#16837366C2212yes—
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14#16837156—22no—
15#16837126—24no—
16C2×#121×C59C2212yes—
17C59×#480—22no—
18C59×#466—22no—
19C59×#447—22no—
20C2×C22⋊C8×C59C2211yescharacteristic
21C2×#11745664—22no—
22C22×C59⋊C8—21no—
23#14273686—22no—
24#14273686—22no—
25C2×C4⋊Dic59D411yescharacteristic
26C2×C4⋊Dic59D411yescharacteristic
27C2×C22⋊Dic59—21no—
28C2×C4×Dic59—21no—
29#14273660—24no—
30#14273658—24no—
31#14273648—24no—
32#14273648—24no—
33C22⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)D414yes—
36C22×D4×C59—22no—
37C22×D4×C59D412yes—
38C23×C4×C59C2311yescharacteristic
39C2×D8×C59—22no—
40C22×C8×C59D412yes—
41#121×C59—22no—
42#121×C59—22no—
43#121×C59—22no—
44#121×C59—22no—
45C2×C4⋊C4×C59D412yes—
46C2×C22⋊C4×C59—22no—
47#107×C59—22no—
48#102×C59—24no—
49C22⋊C8×C59—22no—
50C24×C59—22no—
51C24×C59#4111yescharacteristic
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—44no—
55C2×D4×C59—42no—
56C2×D4×C59—42no—
57C2×D4×C59—84no—
58C22×C4×C59—22no—
59C22×C4×C59—42no—
60C22×C4×C59C2×D412yes—
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D412yes—
64C22×Dic59—21no—
65C22×Dic59—21no—
66D8×C59—82no—
67C2×C8×C59—22no—
68C2×C8×C59—42no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—22no—
71C22⋊C4×C59—42no—
72C22⋊C4×C59—42no—
73C22⋊Dic59—42no—
74C4⋊Dic59—22no—
75C4⋊Dic59—22no—
76C4×Dic59—41no—
77C4×Dic59—41no—
78#11745664—44no—
79C2×C59⋊C8—24no—
80C2×C59⋊C8—42no—
81C2×C59⋊C8—41no—
82C23×C59—22no—
83C23×C59—41no—
84C23×C59—41no—
85C23×C59—44no—
86C23×C59—42no—
87C23×C59—82no—
88C23×C59—82no—
89C23×C59#12111yescharacteristic
90C23×C59#12112yes—
91D4×C59—44no—
92D4×C59—44no—
93D4×C59—84no—
94D4×C59—84no—
95C2×C4×C59—22no—
96C2×C4×C59—24no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—82no—
101C2×C4×C59#12012yes—
102C2×C4×C59#12011yescharacteristic
103C2×C4×C59#12711yescharacteristic
104C2×Dic59—42no—
105C2×Dic59—41no—
106C2×Dic59—41no—
107C2×Dic59—42no—
108C8×C59—42no—
109C59⋊C8—44no—
110C59⋊C8—81no—
111#9977—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—22no—
115C22×C59—22no—
116C22×C59—22no—
117C22×C59—82no—
118C22×C59—82no—
119C22×C59—82no—
120C22×C59—82no—
121C22×C59—164no—
122C22×C59#39211yescharacteristic
123C22×C59#47212yes—
124C4×C59—22no—
125C4×C59—21no—
126C4×C59—82no—
127C4×C59#45812yes—
128Dic59—81no—
129Dic59—81no—
130#3186D5911yescharacteristic
131#3094—592no—
132#1818—591no—
133#1471—592no—
134#1293—591no—
135C2×C59—22no—
136C2×C59—42no—
137C2×C59—162no—
138C2×C59#201711yescharacteristic
139C2×C59#201811yescharacteristic
140C2×C59#202211yescharacteristic
141C2×#121C2×D5912yes—
142C2×#117—591no—
143#482—1184no—
144#481—1182no—
145#480—22no—
146#480—1182no—
147#466—22no—
148#447—22no—
149#428—1182no—
150#418—1182no—
151#407—592no—
152C2×C22⋊C8—591no—
153C2×C22⋊C8C2×D5911yescharacteristic
154#336—1182no—
155#323—1184no—
156C59#997711yescharacteristic, complemented, Sylow
157C22×D4—22no—
158C22×D4C22⋊D5912yes—
159C23×C4C22×D5911yescharacteristic
160C2×D8—22no—
161C2×M16—1182no—
162C22×C8—1181no—
163C22×C8C22⋊D5912yes—
164#121—22no—
165#121—22no—
166#121—22no—
167#121—22no—
168#117—1182no—
169#117—1182no—
170C2×C4⋊C4—591no—
171C2×C4⋊C4—591no—
172C2×C4⋊C4C22⋊D5912yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1181no—
175C2×C42—1181no—
176#107—22no—
177#107—1184no—
178#106—1184no—
179#102—24no—
180#102—1184no—
181#102—1184no—
182C22⋊C8—22no—
183C22⋊C8—594no—
184C22⋊C8—1182no—
185C22⋊C8—1182no—
186C24—22no—
187C24#1174568711yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—84no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—1181no—
197C22×C4—1181no—
198C22×C4D4×D5911yescharacteristic
199C22×C4D4×D5912yes—
200C22×C4C2×C22⋊D5912yes—
201C22×C4C2×C22⋊D5911yescharacteristic
202D8—82no—
203M16—2364no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—1184no—
207C2×C8—2362no—
208C2×C8—2361no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—1182no—
212C4⋊C4—1182no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—2362no—
216C42—2361no—
217C42—2361no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—44no—
222C23—42no—
223C23—82no—
224C23—82no—
225C23#1427369811yescharacteristic
226C23#1427369912yes—
227D4—44no—
228D4—44no—
229D4—84no—
230D4—84no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—2362no—
238C2×C4—2362no—
239C2×C4—2361no—
240C2×C4—2361no—
241C2×C4#1427369512yes—
242C2×C4#1427369611yescharacteristic
243C2×C4#1427372311yescharacteristic
244C8—42no—
245C8—4721no—
246C8—2364no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—164no—
257C22#1683731011yescharacteristic
258C22#1683766912yes—
259C4—21no—
260C4—22no—
261C4—82no—
262C4—4721no—
263C4—4721no—
264C4#1683759212yes—
265C2—22no—
266C2—42no—
267C2—162no—
268C2#2142519911yescharacteristic
269C2#2142520311yescharacteristic
270C2#2142522511yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#3186C2
socleC22×C59#392
hypercenter#3186D59
hypocenterC59#9977
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #21434032C2semidirect
2. #21424165C2semidirect
3. #21422456C2semidirect
4. #21421973C2semidirect
156. C59#9977semidirect

Maximal subgroups

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

subgroupindex
#214340322
#214241652
#214224562
#214219732
C59×#31862
#997759