Finite Group Databasebeta

← #28493669

Subgroups — #28493669

Order 15104.

372 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 93 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#21424893C211yesmaximal, characteristic
2#21424270C211yesmaximal, characteristic, complemented
3#21422866C211yesmaximal, characteristic, complemented
4#21422522C211yesmaximal, characteristic
5#21422520C211yesmaximal, characteristic
6#21421926C211yesmaximal, characteristic, complemented
7C59×#1570C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9#16837455C2211yescharacteristic
10C2×#14273662—22no—
11#16837426C2211yescharacteristic
12C2×(C2×D59)⋊C8—22no—
13C2×#14273638C2211yescharacteristic
14#16837152—22no—
15#16837131—22no—
16#16837129—22no—
17C59×#431C2211yescharacteristic
18C59×#423C2211yescharacteristic
19C2×C4⋊C8×C59C2211yescharacteristic
20C59×#334—22no—
21C22×D236Q811yescharacteristic
22C22×C4×D59—21no—
23C2×#11745664—21no—
24C22×C59⋊C8D412yes—
25C4×D236—21no—
26C4×D236—21no—
27C4×D236—21no—
28C4×D236—21no—
29C2×C4⋊Dic59Q811yescharacteristic
30C2×(C2×D59)⋊C4—21no—
31#14273662—44no—
32C4⋊(C59⋊C8)—21no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)—21no—
35C22⋊(C59⋊C8)—21no—
36(C2×D59)⋊C8—44no—
37#14273638—21no—
38#14273638—21no—
39C4×C59⋊C8D412yes—
40C2×M16×C59—21no—
41C22×C8×C59—21no—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59—21no—
44C4⋊C8×C59—21no—
45C4⋊C8×C59—21no—
46C4⋊C8×C59—21no—
47C4⋊C8×C59—21no—
48C22⋊C8×C59—21no—
49C4×C8×C59D412yes—
50C23×D59—21no—
51C22×C4×C59C2×D411yescharacteristic
52C22×C4×C59C2×Q811yescharacteristic
53C22×C4×C59#4111yescharacteristic
54C22×Dic59—21no—
55C2×D236—21no—
56C2×D236—21no—
57C2×D236—42no—
58C2×C4×D59—41no—
59C2×C4×D59—41no—
60C2×C4×D59—42no—
61C2×C4×D59—42no—
62M16×C59—42no—
63C2×C8×C59—22no—
64C2×C8×C59—41no—
65C2×C8×C59—41no—
66C2×C8×C59—42no—
67C42×C59C2×D411yescharacteristic
68C42×C59C2×D411yescharacteristic
69C42×C59#4111yescharacteristic
70C42×C59#4111yescharacteristic
71(C2×D59)⋊C4—41no—
72(C2×D59)⋊C4—41no—
73C4⋊Dic59—21no—
74C4⋊Dic59—21no—
75#11745664—42no—
76C2×C59⋊C8—22no—
77C2×C59⋊C8—22no—
78C2×C59⋊C8—22no—
79C2×C59⋊C8—41no—
80C2×C59⋊C8SD1614yes—
81C23×C59C22⋊Q811yescharacteristic
82C22×D59—22no—
83C22×D59—41no—
84C22×D59—42no—
85C22×D59—41no—
86C22×D59—42no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59C22⋊Q811yescharacteristic
90C2×C4×C59C22⋊Q811yescharacteristic
91C2×C4×C59#12311yescharacteristic
92C2×C4×C59#12311yescharacteristic
93C2×C4×C59#12311yescharacteristic
94C2×C4×C59#12711yescharacteristic
95C2×C4×C59C2×SD1612yes—
96C2×C4×C59AGL(1,ℤ/8)12yes—
97C2×Dic59—41no—
98C2×Dic59—42no—
99D236—22no—
100D236—22no—
101D236—41no—
102D236—41no—
103C4×D59—82no—
104C4×D59—81no—
105C4×D59—81no—
106C8×C59—42no—
107C8×C59—81no—
108C8×C59—81no—
109C59⋊C8—24no—
110C59⋊C8—42no—
111C59⋊C8—81no—
112#10661—591nomaximal, Sylow
113C22×C59—21no—
114C22×C59—21no—
115C22×C59#39911yescharacteristic
116C22×C59#47612yes—
117C2×D59—44no—
118C2×D59—42no—
119C2×D59—82no—
120C2×D59—81no—
121C2×D59—81no—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—21no—
125C4×C59—21no—
126C4×C59#48112yes—
127C4×C59#49211yescharacteristic
128C4×C59#49611yescharacteristic
129Dic59—81no—
130#1955—591no—
131#1849—591no—
132#1570—591no—
133#1570D5911yescharacteristic, complemented
134#1486—591no—
135#1486—591no—
136#1275—591no—
137C2×C59—22no—
138C2×C59#209111yescharacteristic
139C2×C59#209411yescharacteristic
140C2×C59#209811yescharacteristic
141D59—82no—
142C2×C4×D4—591no—
143#431—591no—
144#431C2×D5911yescharacteristic
145C2×#108—1182no—
146#423—591no—
147#423C2×D5911yescharacteristic
148C2×C4⋊C8C2×D5911yescharacteristic
149C2×C22⋊C8—1182no—
150C2×#97—591no—
151#334—22no—
152#334—1182no—
153D4⋊C8—1182no—
154D4⋊C8—1182no—
155C59#1066111yescharacteristic, complemented, Sylow
156C22×D4—591no—
157C23×C4—1181no—
158C2×M16—21no—
159C2×M16—1181no—
160C22×C8—21no—
161C22×C8—592no—
162C4×D4—1181no—
163C4×D4—1181no—
164C4×D4—1181no—
165C4×D4—1181no—
166C2×C4⋊C4—591no—
167C2×C22⋊C4—1181no—
168C2×C42C22×D5911yescharacteristic
169#108—2364no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—1181no—
176C4⋊C8—1181no—
177C4⋊C8—1181no—
178C22⋊C8—21no—
179C22⋊C8—1181no—
180C22⋊C8—2364no—
181#97—1181no—
182#97—1181no—
183C4×C8—592no—
184C4×C8C22⋊D5912yes—
185C24—1181no—
186C2×D4—1181no—
187C2×D4—1181no—
188C2×D4—2362no—
189C22×C4—1181no—
190C22×C4—2361no—
191C22×C4—2361no—
192C22×C4—2362no—
193C22×C4—2362no—
194C22×C4#1174568511yescharacteristic
195C22×C4D4×D5911yescharacteristic
196C22×C4Q8×D5911yescharacteristic
197M16—42no—
198M16—2362no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—594no—
204C2×C8—1182no—
205C2×C8—1182no—
206C2×C8—1182no—
207C2×C8—2361no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C22⋊C4—2361no—
211C22⋊C4—2361no—
212C42#1174568511yescharacteristic
213C42D4×D5911yescharacteristic
214C42#1174568911yescharacteristic
215C42C2×C22⋊D5911yescharacteristic
216C23—1182no—
217C23—2361no—
218C23—2362no—
219C23—2362no—
220C23—2361no—
221C23#1427370511yescharacteristic
222D4—1182no—
223D4—1182no—
224D4—2361no—
225D4—2361no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4722no—
229C2×C4—4721no—
230C2×C4—4721no—
231C2×C4—2362no—
232C2×C4—2361no—
233C2×C4#1427370311yescharacteristic
234C2×C4#1427370511yescharacteristic
235C2×C4#1427371011yescharacteristic
236C2×C4#1427371211yescharacteristic
237C2×C4#1427371211yescharacteristic
238C2×C4#1427372311yescharacteristic
239C2×C4SD16×D5912yes—
240C2×C4#1427375412yes—
241C8—42no—
242C8—81no—
243C8—81no—
244C8—1184no—
245C8—4721no—
246C8—2362no—
247C22—21no—
248C22—21no—
249C22—4722no—
250C22—4721no—
251C22—4721no—
252C22—2364no—
253C22—2362no—
254C22#1683733711yescharacteristic
255C22#1683769112yes—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—4721no—
261C4#1683772212yes—
262C4#1683777311yescharacteristic
263C4#1683778811yescharacteristic
264C2—22no—
265C2—4722no—
266C2#2142561111yescharacteristic
267C2#2142562611yescharacteristic
268C2#2142565211yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1570C2
socleC22×C59#399
prime radicalC22×D236Q8
hypercenter#1570D59
hypocenterC59#10661
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #21424270C2semidirect
3. #21422866C2semidirect
6. #21421926C2semidirect
7. C59×#1570C2semidirect
133. #1570D59semidirect
155. C59#10661semidirect

Maximal subgroups

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

subgroupindex
#214248932
#214242702
#214228662
#214225222
#214225202
#214219262
C59×#15702
#1066159