Finite Group Databasebeta

← #28509340

Subgroups — #28509340

Order 15104.

298 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 59 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×#16837831C211yesmaximal, characteristic, complemented
2C2×#16837241C211yesmaximal, characteristic, complemented
3C2×#16837237C211yesmaximal, characteristic, complemented
4#21426183C211yesmaximal, characteristic, complemented
5#21422276C211yesmaximal, characteristic, complemented
6(C2×D59)⋊C32C211yesmaximal, characteristic, complemented
7C59×#1415C211yesmaximal, characteristic, complemented
8C2×#14273746—21no—
9#16837831—22no—
10#16837831—22no—
11C2×Dic472C2211yescharacteristic
12C2×#14273671—21no—
13C2×D944C2211yescharacteristic
14C2×C16×D59C2211yescharacteristic
15#16837639—21no—
16#16837241—22no—
17#16837237—22no—
18C2×C59⋊C32C2211yescharacteristic
19#16837202—21no—
20C2×D16×C59C2211yescharacteristic
21C2×C32×C59C2211yescharacteristic
22C59×#366C2211yescharacteristic
23C2×#11745687—41no—
24#14273746—24no—
25#14273746—42no—
26C2×Dic236—21no—
27C2×#11745670—41no—
28C2×D472—21no—
29C2×C8×D59D411yescharacteristic
30#14273702—41no—
31Dic472—21no—
32Dic472D412yescomplemented
33#14273671—24no—
34D944—21no—
35D944—21no—
36C2×C59⋊C16D411yescharacteristic
37C16×D59—21no—
38C16×D59—21no—
39#14273647—41no—
40C59⋊C32—21no—
41C2×D8×C59—21no—
42D16×C59—21no—
43D16×C59D412yescomplemented
44C2×C16×C59C2311yescharacteristic
45#107×C59—21no—
46C32×C59—21no—
47C2×D4×C59—41no—
48C2×C22⋊D59—81no—
49C22×Dic59—81no—
50#11745687—44no—
51#11745687—82no—
52C2×D236—41no—
53C2×D236—81no—
54C2×C4×D59D811yescharacteristic
55C2×Dic118—41no—
56D8×C59—22no—
57D8×C59—41no—
58C2×C8×C59C2×D411yescharacteristic
59C4⋊C4×C59—41no—
60#11745670—44no—
61(C2×D59)⋊C4—81no—
62C2×C59⋊C8D811yescharacteristic
63Dic236—22no—
64Dic236—41no—
65D472—41no—
66D472—41no—
67C8×D59—21no—
68C8×D59—21no—
69C16×C59C2×D411yescharacteristic
70C16×C59C2×D411yescharacteristic
71C59⋊C16—21no—
72C23×C59—81no—
73C22×D59—81no—
74C22×D59D1611yescharacteristic
75D4×C59—42no—
76D4×C59—81no—
77C2×C4×C59—81no—
78C2×C4×C59C2×D811yescharacteristic
79C22⋊D59—84no—
80C2×Dic59—84no—
81C2×Dic59—81no—
82C2×Dic59D1611yescharacteristic
83D236—81no—
84D236—81no—
85D236—162no—
86C4×D59—21no—
87C4×D59—21no—
88Dic118—42no—
89Dic118—81no—
90C8×C59#12011yescharacteristic
91C8×C59AGL(1,ℤ/8)11yescharacteristic
92C59⋊C8—21no—
93#13663—591nomaximal, Sylow
94C22×C59—82no—
95C22×C59—161no—
96C22×C59—161no—
97C22×C59C2×D1611yescharacteristic
98C2×D59—21no—
99C2×D59—21no—
100C2×D59—21no—
101C2×D59—161no—
102C2×D59—161no—
103C2×D59—161no—
104C4×C59—161no—
105C4×C59#44711yescharacteristic
106C4×C59#50911yescharacteristic
107Dic59—21no—
108Dic59—82no—
109C2×#508—591no—
110C2×SD64—591no—
111C2×D32—591no—
112#2215—591no—
113#1416—591no—
114#1415D5911yescharacteristic, complemented
115C22⋊C32—591no—
116C2×C59—162no—
117C2×C59#218411yescharacteristic
118C2×C59#226211yescharacteristic
119C2×C59#226311yescharacteristic
120D59—41no—
121D59—321no—
122C2×#135—1181no—
123#508—1182no—
124#508—1182no—
125C2×Q32—591no—
126C2×SD32—1181no—
127C2×D16—591no—
128C2×D16C2×D5911yescharacteristic
129C22×C16—591no—
130#466—1181no—
131SD64—1182no—
132D32—1182no—
133C2×C32—591no—
134C2×C32C2×D5911yescharacteristic
135#366C2×D5911yescharacteristic
136#357—1181no—
137C59#1366311yescharacteristic, complemented, Sylow
138C2×#41—2361no—
139#135—1184no—
140#135—2362no—
141C2×Q16—1181no—
142C2×SD16—2361no—
143C2×D8—21no—
144C2×D8—1181no—
145C22×C8—591no—
146#121—2361no—
147Q32—592no—
148Q32—1181no—
149SD32—1184no—
150D16—21no—
151D16—1181no—
152D16—1181no—
153D16D23612yescomplemented
154C2×C16—591no—
155C2×C16—1181no—
156C2×C16—1181no—
157C2×C16C22×D5911yescharacteristic
158#107—21no—
159#102—2361no—
160C32—21no—
161C32—1181no—
162#41—4722no—
163#41—2364no—
164C2×Q8—2361no—
165C2×D4—41no—
166C2×D4—4721no—
167C2×D4—4721no—
168C2×D4—2361no—
169C22×C4—591no—
170C22×C4—4721no—
171Q16—1182no—
172Q16—2361no—
173SD16—2364no—
174D8—22no—
175D8—41no—
176D8—2361no—
177D8—2361no—
178C2×C8—591no—
179C2×C8—1181no—
180C2×C8—1181no—
181C2×C8D4×D5911yescharacteristic
182C4⋊C4—41no—
183C22⋊C4—4721no—
184C16—1181no—
185C16C2×D23611yescharacteristic
186C16D4×D5911yescharacteristic
187C23—81no—
188C23—591no—
189C23—4721no—
190Q8—4721no—
191Q8—2362no—
192D4—42no—
193D4—81no—
194D4—9442no—
195D4—4724no—
196D4—4721no—
197D4—4721no—
198C2×C4—81no—
199C2×C4—591no—
200C2×C4—1181no—
201C2×C4—1181no—
202C2×C4—4724no—
203C2×C4—4721no—
204C2×C4D8×D5911yescharacteristic
205C8—1181no—
206C8#1427369711yescharacteristic
207C8#1427375411yescharacteristic
208C22—82no—
209C22—161no—
210C22—161no—
211C22—1181no—
212C22—1181no—
213C22—1181no—
214C22—9441no—
215C22—9441no—
216C22—9441no—
217C22D16×D5911yescharacteristic
218C4—161no—
219C4—1181no—
220C4—4722no—
221C4#1683753711yescharacteristic
222C4#1683783911yescharacteristic
223C2—162no—
224C2—18881no—
225C2—2361no—
226C2#2142600911yescharacteristic
227C2#2142640411yescharacteristic
228C2#2142640711yescharacteristic
subgroupG/H
centerC22D16×D59
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC59×#1415C2
socleC22×C59C2×D16
hypercenter#1415D59
hypocenterC59#13663
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
1. C2×#16837831C2semidirect
2. C2×#16837241C2semidirect
3. C2×#16837237C2semidirect
4. #21426183C2semidirect
5. #21422276C2semidirect
6. (C2×D59)⋊C32C2semidirect
7. C59×#1415C2semidirect
32. Dic472D4semidirect
43. D16×C59D4semidirect
114. #1415D59semidirect
137. C59#13663semidirect
153. D16D236semidirect

Maximal subgroups

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

subgroupindex
C2×#168378312
C2×#168372412
C2×#168372372
#214261832
#214222762
(C2×D59)⋊C322
C59×#14152
#1366359