Finite Group Databasebeta

← #28489677

Subgroups — #28489677

Order 15104.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434100C212yesmaximal
2#21424894C211yesmaximal, characteristic
3#21422499C212yesmaximal, complemented
4#21421927C211yesmaximal, characteristic, complemented
5C59×#3162C211yesmaximal, characteristic, complemented
6C2×C4×D236C2211yescharacteristic
7#16837730—22no—
8#16837689—22no—
9#16837604—22no—
10#16837552—22no—
11C4×#11745671—24no—
12#16837455C2212yes—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×(C2×D59)⋊C8—22no—
15Q8⋊(C59⋊C8)—24no—
16C59×#533C2212yes—
17C59×#501—22no—
18C59×#501—22no—
19C59×#427—22no—
20C2×#97×C59C2211yescharacteristic
21C22×D236D411yescharacteristic
22C22×C4×D59—21no—
23C2×#11745671—24no—
24C22×C59⋊C8—21no—
25C4×D236—22no—
26C4×D236—22no—
27C2×C4⋊Dic59D411yescharacteristic
28C2×(C2×D59)⋊C4—21no—
29#14273658—24no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)—22no—
32#14273651—24no—
33#14273647—24no—
34C22⋊(C59⋊C8)—22no—
35(C2×D59)⋊C8—44no—
36C4×C59⋊C8D414yes—
37C22×C8×C59D412yes—
38C4⋊Q8×C59D412yes—
39C4⋊Q8×C59D412yes—
40#125×C59—22no—
41C22⋊Q8×C59—22no—
42C22⋊Q8×C59—22no—
43C4×Q8×C59—22no—
44C4×Q8×C59—22no—
45#117×C59—22no—
46C2×C42×C59C2311yescharacteristic
47#107×C59—22no—
48#107×C59—22no—
49#106×C59—24no—
50#97×C59—22no—
51C23×D59—21no—
52C2×Q8×C59—22no—
53C2×Q8×C59—22no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59#4111yescharacteristic
57C22×Dic59—21no—
58C2×D236—22no—
59C2×D236—42no—
60C2×C4×D59—41no—
61C2×C4×D59—41no—
62C2×C4×D59—42no—
63C2×C4×D59—42no—
64C2×C8×C59—22no—
65C2×C8×C59—22no—
66C2×C8×C59SD1614yescomplemented
67C4⋊C4×C59—22no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—22no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—42no—
73C22⋊C4×C59—42no—
74C42×C59—42no—
75C42×C59C2×D412yes—
76C42×C59C2×D412yes—
77#11745671—48no—
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—22no—
80C2×C59⋊C8—24no—
81C2×C59⋊C8—41no—
82C2×C59⋊C8—42no—
83C23×C59#12111yescharacteristic
84C22×D59—22no—
85C22×D59—41no—
86C22×D59—42no—
87C22×D59—41no—
88C22×D59—42no—
89Q8×C59—44no—
90Q8×C59—44no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59#12012yes—
96C2×C4×C59#12011yescharacteristic
97C2×C4×C59#12112yes—
98C2×C4×C59#12711yescharacteristic
99C2×C4×C59C2×SD1612yes—
100C2×C4×C59#13712yes—
101C2×Dic59—41no—
102C2×Dic59—42no—
103D236—24no—
104D236—42no—
105C4×D59—82no—
106C4×D59—81no—
107C4×D59—81no—
108C8×C59—24no—
109C59⋊C8—44no—
110C59⋊C8—81no—
111#10036—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#39211yescharacteristic
115C22×C59#46112yes—
116C2×D59—44no—
117C2×D59—42no—
118C2×D59—82no—
119C2×D59—81no—
120C2×D59—81no—
121C4×C59—22no—
122C4×C59—22no—
123C4×C59—82no—
124C4×C59—82no—
125C4×C59#44812yes—
126C4×C59#49211yescharacteristic
127C4×C59#49711yescharacteristic
128Dic59—81no—
129#3162D5911yescharacteristic, complemented
130#3101—592no—
131#1955—591no—
132#1481—592no—
133#1275—591no—
134C2×C59—22no—
135C2×C59#201011yescharacteristic
136C2×C59#201311yescharacteristic
137C2×C59#201711yescharacteristic
138D59—82no—
139#533C2×D5912yes—
140C2×C4×D4—591no—
141#501—22no—
142#501—22no—
143#483—1182no—
144#476—1182no—
145#461—1182no—
146#450—1182no—
147C4×SD16—1184no—
148#431—592no—
149#427—22no—
150C2×C4⋊C8—591no—
151C2×C22⋊C8—1182no—
152C2×#97C2×D5911yescharacteristic
153Q8⋊C8—1184no—
154C59#1003611yescharacteristic, complemented, Sylow
155C22×D4—591no—
156C23×C4—1181no—
157C2×SD16—1184no—
158C22×C8—1181no—
159C22×C8C22⋊D5912yes—
160C4⋊Q8C22⋊D5912yes—
161C4⋊Q8C22⋊D5912yes—
162#125—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×D4—1182no—
168C4×D4—1182no—
169#117—22no—
170C2×C4⋊C4—591no—
171C2×C22⋊C4—1181no—
172C2×C42C22×D5911yescharacteristic
173#107—22no—
174#107—22no—
175#106—24no—
176#106—1184no—
177C4⋊C8—1182no—
178C4⋊C8—1182no—
179#103—1184no—
180#102—1184no—
181C22⋊C8—1182no—
182C22⋊C8—2364no—
183#97—22no—
184C4×C8—594no—
185C24—1181no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×D4—1182no—
189C2×D4—2362no—
190C22×C4—1181no—
191C22×C4—2361no—
192C22×C4—2361no—
193C22×C4—2362no—
194C22×C4—2362no—
195C22×C4D4×D5911yescharacteristic
196C22×C4#1174568711yescharacteristic
197C22×C4C2×C22⋊D5911yescharacteristic
198SD16—2368no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—1184no—
202C2×C8—2361no—
203C2×C8—2362no—
204C2×C8#1174567114yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—1182no—
212C22⋊C4—42no—
213C22⋊C4—2362no—
214C42—42no—
215C42D4×D5912yes—
216C42C2×C22⋊D5912yes—
217C23—1182no—
218C23—2361no—
219C23—2362no—
220C23—2361no—
221C23—2362no—
222C23#1427369811yescharacteristic
223Q8—44no—
224Q8—44no—
225D4—1184no—
226D4—2362no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—4721no—
232C2×C4—4721no—
233C2×C4—4722no—
234C2×C4—2361no—
235C2×C4—2362no—
236C2×C4#1427369512yes—
237C2×C4#1427369611yescharacteristic
238C2×C4#1427369912yes—
239C2×C4#1427372311yescharacteristic
240C2×C4C2×#1174567112yes—
241C2×C4#1427375812yes—
242C8—24no—
243C8—4721no—
244C8—2364no—
245C22—21no—
246C22—21no—
247C22—4722no—
248C22—4721no—
249C22—4721no—
250C22—2364no—
251C22—2362no—
252C22#1683731011yescharacteristic
253C22#1683760412yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—4721no—
259C4#1683754112yes—
260C4#1683777611yescharacteristic
261C4#1683779411yescharacteristic
262C2—22no—
263C2—4722no—
264C2#2142516011yescharacteristic
265C2#2142517911yescharacteristic
266C2#2142519911yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3162C2
socleC22×C59#392
prime radicalC22×D236D4
hypercenter#3162D59
hypocenterC59#10036
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #21422499C2semidirect
4. #21421927C2semidirect
5. C59×#3162C2semidirect
66. C2×C8×C59SD16semidirect
129. #3162D59semidirect
154. C59#10036semidirect
204. C2×C8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214341002
#214248942
#214224992
#214219272
C59×#31622
#1003659