Finite Group Databasebeta

← #28490013

Subgroups — #28490013

Order 15104.

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

normal subgroups only

This group has 600 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
1C2×#16837761C212yesmaximal, complemented
2#21424915C211yesmaximal, characteristic, complemented
3C2×#16837142C212yesmaximal, complemented
4#21421977C211yesmaximal, characteristic, complemented
5C59×#2819C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837761—24no—
8#16837761—24no—
9C2×#14273651—22no—
10C2×#14273650—22no—
11C2×#14273648—22no—
12C2×#14273638C2211yescharacteristic
13C2×#14273638C2212yes—
14#16837142—24no—
15C2×#124×C59C2212yes—
16C59×#394—22no—
17C59×#393—22no—
18C59×#392—22no—
19C59×#382C2211yescharacteristic
20C22×Dic118—21no—
21C22×C59⋊C8—22no—
22C22×C59⋊C8—21no—
23#14273727—22no—
24#14273727—21no—
25#14273727D412yescomplemented
26C2×C4⋊Dic59—22no—
27#14273651—28no—
28#14273650—44no—
29#14273648—28no—
30#14273638—22no—
31#14273638—22no—
32#14273638D414yescomplemented
33C22×Q8×C59D412yes—
34C22×D4×C59—22no—
35C22×D4×C59D412yes—
36#124×C59—22no—
37#124×C59—22no—
38#124×C59—22no—
39#124×C59—22no—
40C2×C4⋊C4×C59D412yes—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—24no—
44C2×C42×C59C2311yescharacteristic
45#95×C59—22no—
46#95×C59—22no—
47C24×C59—22no—
48C2×Q8×C59—22no—
49C2×Q8×C59—22no—
50C2×Q8×C59—44no—
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—44no—
54C2×D4×C59—82no—
55C2×D4×C59—84no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59C2×D412yes—
59C22×C4×C59C2×D411yescharacteristic
60C22×Dic59—41no—
61C2×Dic118—22no—
62C2×Dic118—42no—
63C2×Dic118—42no—
64C2×Dic118—41no—
65C4⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C22⋊C4×C59—42no—
68C22⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—84no—
71C42×C59C2×D411yescharacteristic
72C42×C59C2×D412yes—
73C42×C59#4111yescharacteristic
74C4⋊Dic59—24no—
75C4⋊Dic59—42no—
76C2×C59⋊C8—22no—
77C2×C59⋊C8—24no—
78C2×C59⋊C8—41no—
79C2×C59⋊C8—44no—
80C2×C59⋊C8—41no—
81C23×C59—42no—
82C23×C59—42no—
83C23×C59—42no—
84C23×C59—44no—
85C23×C59—44no—
86C23×C59#12011yescharacteristic
87Q8×C59—44no—
88Q8×C59—82no—
89D4×C59—44no—
90D4×C59—82no—
91D4×C59—88no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—24no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—44no—
98C2×C4×C59—44no—
99C2×C4×C59#12012yes—
100C2×C4×C59#12111yescharacteristic
101C2×C4×C59#12112yes—
102C2×C4×C59#12711yescharacteristic
103C2×Dic59—42no—
104C2×Dic59—81no—
105C2×Dic59—81no—
106Dic118—44no—
107Dic118—82no—
108C59⋊C8—44no—
109C59⋊C8—42no—
110#10085—591nomaximal, Sylow
111C22×C59—84no—
112C22×C59—84no—
113C22×C59—82no—
114C22×C59—82no—
115C22×C59—82no—
116C22×C59#39211yescharacteristic
117C22×C59#45311yescharacteristic
118C22×C59#45411yescharacteristic
119C22×C59#45514yes—
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—82no—
125C4×C59—82no—
126Dic59—82no—
127C2×#489—592no—
128#2819D5911yescharacteristic
129#1960—591no—
130C2×#329—592no—
131#1295—591no—
132C2×C59—162no—
133C2×C59#165912yes—
134C2×C59#166212yes—
135C2×C59#200412yes—
136C2×C59#200611yescharacteristic
137C2×C4⋊Q8—591no—
138C2×#124C2×D5912yes—
139#489—1184no—
140#489—1184no—
141C2×#103—1182no—
142C2×#103—1182no—
143C2×#102—1182no—
144C2×#97—592no—
145C2×#97—591no—
146#394—22no—
147#393—22no—
148#392—22no—
149#382C2×D5911yescharacteristic
150#329—1184no—
151C59#1008511yescharacteristic, complemented, Sylow
152C22×Q8—1181no—
153C22×Q8C22⋊D5912yes—
154C22×D4—22no—
155C22×D4C22⋊D5912yes—
156C22×C8—1182no—
157C22×C8—1181no—
158C4⋊Q8—592no—
159C4⋊Q8—1182no—
160C4⋊Q8—1181no—
161#124—22no—
162#124—22no—
163#124—22no—
164#124—22no—
165C2×C4⋊C4—1182no—
166C2×C4⋊C4C22⋊D5912yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C42C22×D5911yescharacteristic
171#103—1188no—
172#103—2364no—
173#102—1188no—
174#97—594no—
175#97—1182no—
176#97—1182no—
177#95—22no—
178#95—22no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—44no—
183C2×Q8—1182no—
184C2×Q8—2362no—
185C2×Q8—2362no—
186C2×Q8—2361no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—44no—
190C2×D4—82no—
191C2×D4—84no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—2361no—
195C22×C4C2×C22⋊D5912yes—
196C22×C4C2×C22⋊D5911yescharacteristic
197C2×C8—1182no—
198C2×C8—1184no—
199C2×C8—2361no—
200C2×C8—2364no—
201C2×C8—2361no—
202C4⋊C4—42no—
203C4⋊C4—1184no—
204C4⋊C4—2362no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—84no—
210C42D4×D5911yescharacteristic
211C42D4×D5912yes—
212C42#1174568711yescharacteristic
213C23—42no—
214C23—42no—
215C23—42no—
216C23—44no—
217C23—44no—
218C23#1427369611yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—4722no—
222Q8—2364no—
223D4—44no—
224D4—82no—
225D4—88no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—4721no—
234C2×C4—4721no—
235C2×C4—2362no—
236C2×C4#1427369512yes—
237C2×C4#1427369811yescharacteristic
238C2×C4#1427369912yes—
239C2×C4#1427372311yescharacteristic
240C8—2364no—
241C8—2362no—
242C22—84no—
243C22—84no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22#1683731011yescharacteristic
248C22#1683757011yescharacteristic
249C22#1683757511yescharacteristic
250C22#1683758014yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—82no—
257C4—4722no—
258C2—162no—
259C2#2142340212yes—
260C2#2142341912yes—
261C2#2142512312yes—
262C2#2142513811yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#2819C2
socleC23×C59#120
hypercenter#2819D59
hypocenterC59#10085
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. C2×#16837761C2semidirect
2. #21424915C2semidirect
3. C2×#16837142C2semidirect
4. #21421977C2semidirect
25. #14273727D4semidirect
32. #14273638D4semidirect
151. C59#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#168377612
#214249152
C2×#168371422
#214219772
C59×#28192
#1008559