Finite Group Databasebeta

← #28488113

Subgroups — #28488113

Order 15104.

374 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 107 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#21423847C211yesmaximal, characteristic
2#21422859C211yesmaximal, characteristic
3#21422711C212yesmaximal
4#21422608C212yesmaximal
5C59×#3170C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×#14273638C2211yescharacteristic
8#16837248C2211yescharacteristic
9#16837156—21no—
10#16837156—21no—
11#16837155—21no—
12#16837153—21no—
13#16837144—24no—
14#16837139—24no—
15C59×#533C2212yes—
16C59×#500C411yescharacteristic, complemented
17C59×#494C411yescharacteristic, complemented
18C59×#487C412yescomplemented
19C59×#467—21no—
20C59×#467—21no—
21C4×Q16×C59—21no—
22C4×Q16×C59—21no—
23C2×C4×C8×C59C2211yescharacteristic
24C22×C59⋊C8—21no—
25C2×C4×Dic59—21no—
26C4⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28#14273638—21no—
29#14273638—22no—
30#14273638D412yes—
31C59⋊#95—21no—
32C59⋊#95D412yes—
33C2×Q16×C59—21no—
34C2×Q16×C59—21no—
35C22×C8×C59D411yescharacteristic
36C22×C8×C59D411yescharacteristic
37C4⋊Q8×C59C2×C412yes—
38C4⋊Q8×C59C2×C412yes—
39#125×C59—22no—
40C22⋊Q8×C59—22no—
41C22⋊Q8×C59—22no—
42C4×Q8×C59—22no—
43C4×Q8×C59—22no—
44#117×C59—22no—
45C2×C42×C59C2311yescharacteristic
46#107×C59—21no—
47#107×C59—21no—
48#103×C59—22no—
49#103×C59—22no—
50C4×C8×C59C2×C411yescharacteristic
51C4×C8×C59C2×C411yescharacteristic
52C4×C8×C59D411yescharacteristic
53C4×C8×C59D411yescharacteristic
54C2×Q8×C59—22no—
55C2×Q8×C59—22no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59#4111yescharacteristic
59C22×Dic59—22no—
60Q16×C59—42no—
61Q16×C59—42no—
62C2×C8×C59—41no—
63C2×C8×C59—41no—
64C2×C8×C59C22⋊C411yescharacteristic
65C2×C8×C59C22⋊C411yescharacteristic
66C2×C8×C59C22⋊C411yescharacteristic
67C2×C8×C59C22⋊C411yescharacteristic
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—22no—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—42no—
76C42×C59C22×C411yescharacteristic
77C42×C59C2×D411yescharacteristic
78C42×C59C2×D411yescharacteristic
79C42×C59#4111yescharacteristic
80C4×Dic59—41no—
81C4×Dic59—41no—
82C2×C59⋊C8—22no—
83C2×C59⋊C8—41no—
84C2×C59⋊C8—41no—
85C2×C59⋊C8—42no—
86C23×C59#12111yescharacteristic
87Q8×C59—42no—
88Q8×C59—42no—
89Q8×C59—82no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59C2×C22⋊C411yescharacteristic
97C2×C4×C59C4×D411yescharacteristic
98C2×C4×C59C4×D411yescharacteristic
99C2×C4×C59#12111yescharacteristic
100C2×C4×C59#12411yescharacteristic
101C2×C4×C59#12711yescharacteristic
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C8×C59—21no—
106C8×C59—21no—
107C8×C59—21no—
108C8×C59—21no—
109C59⋊C8—42no—
110C59⋊C8—82no—
111#9765—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#39011yescharacteristic
115C22×C59#47312yes—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—82no—
122C4×C59—82no—
123C4×C59#42111yescharacteristic
124C4×C59#42111yescharacteristic
125Dic59—82no—
126#3170D5911yescharacteristic
127#1746—591no—
128#1568—591no—
129#1531—592no—
130#1506—592no—
131C2×C59—22no—
132C2×C59#196311yescharacteristic
133C2×C59#197611yescharacteristic
134C2×C59#197911yescharacteristic
135#533C2×D5912yes—
136#500Dic5911yescharacteristic, complemented
137#494Dic5911yescharacteristic, complemented
138#487Dic5912yescomplemented
139#467—21no—
140#467—21no—
141C4×Q16—21no—
142C4×Q16—21no—
143#432—592no—
144C2×#97—591no—
145C2×C4×C8C2×D5911yescharacteristic
146#376—591no—
147#336—1181no—
148#336—1181no—
149#335—1181no—
150#334—1181no—
151#330—1184no—
152#328—1184no—
153C59#976511yescharacteristic, complemented, Sylow
154C2×Q16—21no—
155C2×Q16—21no—
156C22×C8—1181no—
157C22×C8C22⋊D5911yescharacteristic
158C22×C8C22⋊D5911yescharacteristic
159C4⋊Q8C2×Dic5912yes—
160C4⋊Q8C2×Dic5912yes—
161#125—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166#117—22no—
167C2×C42—1181no—
168C2×C42C22×D5911yescharacteristic
169#107—21no—
170#107—21no—
171C4⋊C8—1182no—
172#103—22no—
173#103—22no—
174C22⋊C8—1182no—
175#97—592no—
176#97—1182no—
177#97—1181no—
178C4×C8C2×Dic5911yescharacteristic
179C4×C8C2×Dic5911yescharacteristic
180C4×C8C22⋊D5911yescharacteristic
181C4×C8C22⋊D5911yescharacteristic
182#95—592no—
183#95—1181no—
184C2×Q8—22no—
185C2×Q8—22no—
186C22×C4—1182no—
187C22×C4D4×D5911yescharacteristic
188C22×C4#1174568711yescharacteristic
189C22×C4C2×C22⋊D5911yescharacteristic
190Q16—42no—
191Q16—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—1182no—
195C2×C8—2362no—
196C2×C8—2361no—
197C2×C8—2361no—
198C2×C8C22⋊Dic5911yescharacteristic
199C2×C8C22⋊Dic5911yescharacteristic
200C2×C8C22⋊Dic5911yescharacteristic
201C2×C8C22⋊Dic5911yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—42no—
210C42—2361no—
211C42—2361no—
212C42D4×D5911yescharacteristic
213C42#1174568711yescharacteristic
214C42C22×Dic5911yescharacteristic
215C42C2×C22⋊D5911yescharacteristic
216C23#1427369811yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—82no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4C2×C22⋊Dic5911yescharacteristic
230C2×C4D4×Dic5911yescharacteristic
231C2×C4D4×Dic5911yescharacteristic
232C2×C4#1427369811yescharacteristic
233C2×C4#1427371411yescharacteristic
234C2×C4#1427372311yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—4722no—
240C8—2362no—
241C22—21no—
242C22—21no—
243C22#1683730511yescharacteristic
244C22#1683767412yes—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—82no—
251C4—82no—
252C4—4722no—
253C4#1683741911yescharacteristic
254C4#1683741911yescharacteristic
255C2—22no—
256C2#2142492911yescharacteristic
257C2#2142499511yescharacteristic
258C2#2142501211yescharacteristic
subgroupG/H
centerC22#16837305
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3170C2
socleC22×C59#390
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#3170D59
hypocenterC59#9765
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
16. C59×#500C4semidirect
17. C59×#494C4semidirect
18. C59×#487C4semidirect
136. #500Dic59semidirect
137. #494Dic59semidirect
138. #487Dic59semidirect
153. C59#9765semidirect

Maximal subgroups

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

subgroupindex
#214238472
#214228592
#214227112
#214226082
C59×#31702
#976559