Finite Group Databasebeta

← #28504696

Subgroups — #28504696

Order 15104.

480 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16837199C212yesmaximal
2#21424092C212yesmaximal, complemented
3#21422223C212yesmaximal, complemented
4C59×#2326C211yesmaximal, characteristic
5#16837426—22no—
6C2×#14273646C2214yes—
7#16837199—24no—
8#16837171—22no—
9C2×#124×C59C2211yescharacteristic
10C2×C4×Q8×C59C2211yescharacteristic
11C2×C4×D4×C59C411yescharacteristic
12C59×#391C2211yescharacteristic
13C59×#390C411yescharacteristic
14C59×#388—22no—
15C59×#385—22no—
16C59×#384C411yescharacteristic
17C59×#376C411yescharacteristic
18C2×#11745664—24no—
19C4⋊(C59⋊C8)—44no—
20#14273646—24no—
21#14273646—24no—
22C22×Q8×C59C2311yescharacteristic
23C22×D4×C59C2×C411yescharacteristic
24C23×C4×C59—21no—
25#124×C59—22no—
26#124×C59D412yes—
27#124×C59Q812yes—
28C4×Q8×C59—42no—
29C4×D4×C59—42no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59C2×C411yescharacteristic
33C2×C4⋊C4×C59D412yes—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—21no—
36C2×C22⋊C4×C59—22no—
37C2×C42×C59C2×C411yescharacteristic
38C2×C42×C59C2×C411yescharacteristic
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59D412yes—
41#95×C59—21no—
42#95×C59—21no—
43#95×C59—22no—
44C24×C59—21no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59C2×D411yescharacteristic
48C2×Q8×C59C2×Q811yescharacteristic
49C2×Q8×C59#4112yes—
50C2×D4×C59—21no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59C4⋊C412yes—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—42no—
58C22×C4×C59—82no—
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D412yes—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—82no—
68C22⋊C4×C59—24no—
69C22⋊C4×C59—24no—
70C22⋊C4×C59—42no—
71C22⋊C4×C59—82no—
72C42×C59—22no—
73C42×C59—24no—
74C42×C59—21no—
75C42×C59—41no—
76C42×C59C4⋊C412yes—
77#11745664—44no—
78#11745664—44no—
79C2×C59⋊C8—44no—
80C23×C59—22no—
81C23×C59—42no—
82C23×C59—42no—
83C23×C59—41no—
84C23×C59—41no—
85C23×C59C2×C22⋊C411yescharacteristic
86Q8×C59—41no—
87Q8×C59—41no—
88Q8×C59—42no—
89D4×C59—42no—
90D4×C59—42no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—84no—
109C2×C4×C59#9912yes—
110C2×C4×C59C2×C4⋊C411yescharacteristic
111C2×C4×C59#11711yescharacteristic
112C2×C4×C59#12112yes—
113C2×C4×C59C22⋊Q812yes—
114C59⋊C8—84no—
115#12803—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59—44no—
119C22×C59—42no—
120C22×C59—82no—
121C22×C59—81no—
122C22×C59—81no—
123C22×C59#38011yescharacteristic
124C22×C59C2×#9911yescharacteristic
125C22×C59#41011yescharacteristic
126C4×C59—22no—
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—22no—
130C4×C59—22no—
131C4×C59—42no—
132C4×C59—42no—
133C4×C59—41no—
134C4×C59—81no—
135C4×C59—81no—
136#2326D5911yescharacteristic
137C2×#355—592no—
138#1801—592no—
139#1394—592no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—82no—
143C2×C59#179911yescharacteristic
144C2×C59#213311yescharacteristic
145C2×C59#213411yescharacteristic
146C2×#124C2×D5911yescharacteristic
147C2×C4×Q8C2×D5911yescharacteristic
148C2×C4×D4Dic5911yescharacteristic
149#423—1182no—
150C2×#101—594no—
151#391C2×D5911yescharacteristic
152#390Dic5911yescharacteristic
153#388—22no—
154#385—22no—
155#384Dic5911yescharacteristic
156#376Dic5911yescharacteristic
157#355—1184no—
158#343—1182no—
159C59#1280311yescharacteristic, complemented, Sylow
160C22×Q8C22×D5911yescharacteristic
161C22×D4C2×Dic5911yescharacteristic
162C23×C4—21no—
163C2×M16—1184no—
164#124—22no—
165#124Dic11812yes—
166#124D23612yes—
167C4×Q8—42no—
168C4×D4—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic5911yescharacteristic
171C2×C4⋊C4C2×Dic5911yescharacteristic
172C2×C4⋊C4C22⋊D5912yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C42C2×Dic5911yescharacteristic
177C2×C42C2×Dic5911yescharacteristic
178C2×C42C2×Dic5911yescharacteristic
179C2×C42C22⋊D5912yes—
180C4⋊C8—2364no—
181#101—1184no—
182#101—1184no—
183#95—21no—
184#95—21no—
185#95—22no—
186C24—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×Dic11811yescharacteristic
190C2×Q8C2×D23611yescharacteristic
191C2×Q8#1174568512yes—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C4⋊Dic5912yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4C22⋊Dic5911yescharacteristic
202C22×C4C22⋊Dic5912yes—
203C22×C4C22⋊Dic5911yescharacteristic
204C22×C4C22×Dic5911yescharacteristic
205C22×C4C2×C22⋊D5912yes—
206M16—2364no—
207M16—2364no—
208C2×C8—2364no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—24no—
218C42—22no—
219C42—21no—
220C42—41no—
221C42C4⋊Dic5912yes—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23C2×C22⋊Dic5911yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4#1427364212yes—
252C2×C4C2×C4⋊Dic5911yescharacteristic
253C2×C4#1427368611yescharacteristic
254C2×C4#1427370112yes—
255C2×C4#1427370612yes—
256C8—4724no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22#1683726211yescharacteristic
265C22C2×#1427364211yescharacteristic
266C22#1683737611yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—81no—
276C4—81no—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#2142408411yescharacteristic
281C2#2142578511yescharacteristic
282C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#2326C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC22×Q8×C59C23
hypercenter#2326D59
hypocenterC59#12803
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #21424092C2semidirect
3. #21422223C2semidirect
159. C59#12803semidirect

Maximal subgroups

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

subgroupindex
C2×#168371992
#214240922
#214222232
C59×#23262
#1280359