Finite Group Databasebeta

← #28505520

Subgroups — #28505520

Order 15104.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#21427277C211yesmaximal, characteristic
2#21424490C211yesmaximal, characteristic
3#21424313C212yesmaximal
4#21422222C212yesmaximal
5C2×C59×#356C211yesmaximal, characteristic
6#16837880C2211yescharacteristic
7#16837444—22no—
8#16837411—21no—
9C2×#14273646C2211yescharacteristic
10#16837309—21no—
11#16837309C2212yes—
12#16837288—22no—
13#16837288—22no—
14#16837270—21no—
15#16837163—22no—
16#16837162—21no—
17C2×C4⋊Q8×C59C2211yescharacteristic
18C2×#101×C59C2212yes—
19C59×#356C414yescomplemented
20C2×#11745664—21no—
21Q8×Dic59—42no—
22#14273686—21no—
23#14273686—22no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59D412yes—
26C2×C4×Dic59—21no—
27C2×C4×Dic59D412yes—
28#14273661—42no—
29#14273659—42no—
30#14273651—42no—
31#14273646—22no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C22×Q8×C59D411yescharacteristic
39C22×Q8×C59C2311yescharacteristic
40C2×M16×C59—22no—
41C4⋊Q8×C59—22no—
42C4⋊Q8×C59C2×C412yes—
43C4⋊Q8×C59Q812yes—
44C2×C4⋊C4×C59—22no—
45C2×C42×C59D411yescharacteristic
46#101×C59—22no—
47#101×C59C2×C414yes—
48C2×Q8×C59—21no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59—41no—
52C2×Q8×C59—41no—
53C2×Q8×C59C4⋊C412yes—
54C2×Q8×C59C22×C411yescharacteristic
55C2×Q8×C59C2×Q811yescharacteristic
56C2×Q8×C59#4112yes—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D412yes—
61C22×Dic59—22no—
62C22×Dic59—22no—
63M16×C59—24no—
64M16×C59—42no—
65C2×C8×C59—42no—
66C4⋊C4×C59—24no—
67C4⋊C4×C59—24no—
68C42×C59—21no—
69C42×C59C4⋊C412yes—
70C22⋊Dic59—41no—
71C4⋊Dic59—22no—
72C4⋊Dic59—22no—
73C4⋊Dic59—22no—
74C4⋊Dic59—42no—
75C4⋊Dic59—81no—
76C4×Dic59—41no—
77C4×Dic59—42no—
78C4×Dic59—42no—
79C4×Dic59—81no—
80#11745664—41no—
81#11745664—41no—
82C2×C59⋊C8—41no—
83C23×C59#12011yescharacteristic
84Q8×C59—41no—
85Q8×C59—41no—
86Q8×C59—42no—
87Q8×C59—42no—
88Q8×C59—81no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—42no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59C2×C4⋊C411yescharacteristic
99C2×C4×C59C4×D412yes—
100C2×C4×C59C22⋊Q812yes—
101C2×C4×C59#12311yescharacteristic
102C2×Dic59—44no—
103C2×Dic59—44no—
104C2×Dic59—42no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107C8×C59—44no—
108C59⋊C8—81no—
109#12971—591nomaximal, Sylow
110C22×C59—22no—
111C22×C59#38511yescharacteristic
112C22×C59B(4,𝔽2)11yescharacteristic
113C22×C59#45811yescharacteristic
114C4×C59—21no—
115C4×C59—22no—
116C4×C59—21no—
117C4×C59—22no—
118C4×C59—42no—
119C4×C59—42no—
120Dic59—82no—
121Dic59—82no—
122Dic59—82no—
123#2381—591no—
124C2×#356D5911yescharacteristic
125#1886—591no—
126#1857—592no—
127#1394—592no—
128C2×C59—21no—
129C2×C59—21no—
130C2×C59#186711yescharacteristic
131C2×C59#220312yes—
132C2×C4⋊Q8C2×D5911yescharacteristic
133#519—591no—
134#428—1182no—
135#419—1181no—
136C2×#101—591no—
137C2×#101C2×D5912yes—
138#391—592no—
139#391—1181no—
140#387—1182no—
141#387—1182no—
142#383—1181no—
143#356Dic5914yescomplemented
144#339—1182no—
145#339—1181no—
146C59#1297111yescharacteristic, complemented, Sylow
147C22×Q8D23611yescharacteristic
148C22×Q8C22×D5911yescharacteristic
149C2×M16—22no—
150C2×M16—1181no—
151C4⋊Q8—22no—
152C4⋊Q8Dic11812yes—
153C4⋊Q8C2×Dic5912yes—
154C4×Q8—2362no—
155#117—1181no—
156#117—1182no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—1181no—
160C2×C42—592no—
161C2×C42—1181no—
162C2×C42D23611yescharacteristic
163#107—2362no—
164#106—2362no—
165#103—2362no—
166#101—22no—
167#101—1182no—
168#101C2×Dic5914yes—
169#95—1181no—
170#95—1182no—
171#95—1181no—
172#95—1182no—
173#95—1182no—
174#95—1182no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic5912yes—
181C2×Q8C2×Dic11811yescharacteristic
182C2×Q8#1174568712yes—
183C2×Q8C22×Dic5911yescharacteristic
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—1182no—
187C22×C4—1182no—
188C22×C4C2×D23611yescharacteristic
189C22×C4D4×D5912yes—
190M16—24no—
191M16—42no—
192M16—2361no—
193M16—2361no—
194C2×C8—42no—
195C2×C8—2361no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—1182no—
199C4⋊C4—1182no—
200C4⋊C4—1182no—
201C4⋊C4—4721no—
202C4⋊C4—2362no—
203C22⋊C4—2361no—
204C42—21no—
205C42—4721no—
206C42—2362no—
207C42—2362no—
208C42—2361no—
209C42C4⋊Dic5912yes—
210C23#1427369711yescharacteristic
211Q8—41no—
212Q8—42no—
213Q8—41no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2364no—
226C2×C4—2364no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4C2×C4⋊Dic5911yescharacteristic
231C2×C4D4×Dic5912yes—
232C2×C4#1427370412yes—
233C2×C4#1427371111yescharacteristic
234C8—44no—
235C8—4721no—
236C22—22no—
237C22#1683727711yescharacteristic
238C22#1683759011yescharacteristic
239C22#1683759311yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—4722no—
247C4—4722no—
248C4—4722no—
249C2—21no—
250C2—21no—
251C2#2142437911yescharacteristic
252C2#2142611812yes—
subgroupG/H
centerC22#16837590
commutatorC2×Q8×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC2×C59×#356C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
cosocleC22×Q8×C59C23
hypercenterC2×#356D59
hypocenterC59#12971
LCSG ▷ C2×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. C59×#356C4semidirect
143. #356Dic59semidirect
146. C59#12971semidirect

Maximal subgroups

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

subgroupindex
#214272772
#214244902
#214243132
#214222222
C2×C59×#3562
#1297159