Finite Group Databasebeta

← #28509331

Subgroups — #28509331

Order 15104.

294 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 61 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×Q32×D59C211yesmaximal, characteristic, complemented
2C2×#16837240C211yesmaximal, characteristic, complemented
3#21426179C211yesmaximal, characteristic
4#21422278C211yesmaximal, characteristic, complemented
5#21422277C211yesmaximal, characteristic, complemented
6(C2×D59)⋊C32C211yesmaximal, characteristic, complemented
7C2×C59×SD64C211yesmaximal, characteristic, complemented
8C2×Q16×D59—21no—
9C2×Dic472C2211yescharacteristic
10C2×#14273673—21no—
11Q32×D59—22no—
12Q32×D59—22no—
13C2×C16×D59C2211yescharacteristic
14#16837635—21no—
15#16837240—22no—
16C2×C59⋊C32C2211yescharacteristic
17#16837228C2211yescharacteristic
18#16837203—21no—
19C2×Q32×C59C2211yescharacteristic
20C2×D16×C59C2211yescharacteristic
21C59×SD64—22no—
22C2×C32×C59C2211yescharacteristic
23C2×Q8×D59—41no—
24C2×Dic236—21no—
25C2×#11745672—41no—
26Q16×D59—24no—
27Q16×D59—42no—
28C2×C8×D59D411yescharacteristic
29#14273698—41no—
30Dic472—21no—
31Dic472D412yescomplemented
32#14273673—24no—
33C2×C59⋊C16D411yescharacteristic
34C16×D59—21no—
35C16×D59—21no—
36#14273661—21no—
37#14273648—41no—
38C59⋊C32—21no—
39C2×Q16×C59—21no—
40C2×D8×C59—21no—
41Q32×C59—21no—
42Q32×C59D412yescomplemented
43D16×C59—21no—
44D16×C59—21no—
45C2×C16×C59C2311yescharacteristic
46C32×C59—21no—
47C2×Q8×C59—41no—
48C2×D4×C59—41no—
49C2×C22⋊D59—81no—
50Q8×D59—44no—
51Q8×D59—82no—
52C2×C4×D59—81no—
53C2×C4×D59D811yescharacteristic
54C2×Dic118—41no—
55C2×Dic118—81no—
56Q16×C59—22no—
57Q16×C59—41no—
58D8×C59—41no—
59D8×C59—41no—
60C2×C8×C59C2×D411yescharacteristic
61C22⋊Dic59—81no—
62#11745672—44no—
63C4⋊Dic59—41no—
64C2×C59⋊C8D811yescharacteristic
65Dic236—22no—
66Dic236—41no—
67C8×D59—21no—
68C8×D59—21no—
69C16×C59C2×D411yescharacteristic
70C16×C59C2×D411yescharacteristic
71C59⋊C16—21no—
72C23×C59—81no—
73C22×D59D1611yescharacteristic
74Q8×C59—42no—
75Q8×C59—81no—
76D4×C59—81no—
77D4×C59—81no—
78C2×C4×C59—81no—
79C2×C4×C59C2×D811yescharacteristic
80C22⋊D59—162no—
81C2×Dic59—81no—
82C2×Dic59—81no—
83C2×Dic59D1611yescharacteristic
84C4×D59—21no—
85C4×D59—21no—
86C4×D59—84no—
87Dic118—42no—
88Dic118—84no—
89Dic118—81no—
90C8×C59#12011yescharacteristic
91C8×C59AGL(1,ℤ/8)11yescharacteristic
92C59⋊C8—21no—
93#13662—591nomaximal, Sylow
94C22×C59—161no—
95C22×C59—161no—
96C22×C59—161no—
97C22×C59C2×D1611yescharacteristic
98C2×D59—21no—
99C2×D59—21no—
100C2×D59SD6412yescomplemented
101C4×C59—82no—
102C4×C59#44711yescharacteristic
103C4×C59#50911yescharacteristic
104Dic59—21no—
105Dic59—82no—
106Dic59—161no—
107C22×Q32—591no—
108C2×SD64—591no—
109C2×SD64D5911yescharacteristic, complemented
110#2215—591no—
111#1416—591no—
112#1416—591no—
113C22⋊C32—591no—
114C2×C59—321no—
115C2×C59#218411yescharacteristic
116C2×C59C2×SD6411yescharacteristic
117C2×C59#226411yescharacteristic
118D59—22no—
119C22×Q16—1181no—
120C2×Q32—591no—
121C2×Q32—1181no—
122C2×Q32—1182no—
123C2×Q32—1182no—
124C2×Q32C2×D5911yescharacteristic
125C2×D16C2×D5911yescharacteristic
126C22×C16—591no—
127#466—1181no—
128SD64—22no—
129SD64—1182no—
130C2×C32—591no—
131C2×C32C2×D5911yescharacteristic
132#366—591no—
133#357—1181no—
134C59#1366211yescharacteristic, complemented, Sylow
135C22×Q8—2361no—
136C2×Q16—21no—
137C2×Q16—1184no—
138C2×Q16—1181no—
139C2×Q16—2361no—
140C2×Q16—2362no—
141C2×D8—21no—
142C22×C8—591no—
143#121—2361no—
144Q32—21no—
145Q32—592no—
146Q32—1184no—
147Q32—1181no—
148Q32C22⋊D5912yescomplemented
149D16—21no—
150D16—21no—
151C2×C16—591no—
152C2×C16—1181no—
153C2×C16—1181no—
154C2×C16C22×D5911yescharacteristic
155#107—1181no—
156#102—2361no—
157C32—21no—
158C32—1181no—
159C2×Q8—41no—
160C2×Q8—4721no—
161C2×Q8—4722no—
162C2×Q8—2364no—
163C2×Q8—2361no—
164C2×D4—41no—
165C2×D4—4721no—
166C22×C4—591no—
167C22×C4—4721no—
168Q16—22no—
169Q16—41no—
170Q16—1182no—
171Q16—2364no—
172Q16—2361no—
173D8—41no—
174D8—41no—
175C2×C8—591no—
176C2×C8—1181no—
177C2×C8—1181no—
178C2×C8D4×D5911yescharacteristic
179C4⋊C4—2361no—
180C22⋊C4—4721no—
181C16—1181no—
182C16D4×D5911yescharacteristic
183C16C2×C22⋊D5911yescharacteristic
184C23—81no—
185C23—591no—
186Q8—42no—
187Q8—81no—
188Q8—4724no—
189Q8—4721no—
190Q8—2362no—
191D4—81no—
192D4—81no—
193D4—9442no—
194C2×C4—81no—
195C2×C4—591no—
196C2×C4—1181no—
197C2×C4—1181no—
198C2×C4—4724no—
199C2×C4—4721no—
200C2×C4—4721no—
201C2×C4D8×D5911yescharacteristic
202C8—1181no—
203C8#1427369511yescharacteristic
204C8#1427375111yescharacteristic
205C22—161no—
206C22—161no—
207C22—161no—
208C22—592no—
209C22—1181no—
210C22—1181no—
211C22D16×D5911yescharacteristic
212C4—82no—
213C4—1181no—
214C4—9441no—
215C4—4722no—
216C4#1683753411yescharacteristic
217C4#1683783611yescharacteristic
218C2—321no—
219C2—1182no—
220C2#2142600611yescharacteristic
221C2SD64×D5911yescharacteristic
222C2#2142641211yescharacteristic
subgroupG/H
centerC22D16×D59
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC2×C59×SD64C2
socleC22×C59C2×D16
prime radical#21426179C2
hypercenterC2×SD64D59
hypocenterC59#13662
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
1. C2×Q32×D59C2semidirect
2. C2×#16837240C2semidirect
4. #21422278C2semidirect
5. #21422277C2semidirect
6. (C2×D59)⋊C32C2semidirect
7. C2×C59×SD64C2semidirect
31. Dic472D4semidirect
42. Q32×C59D4semidirect
100. C2×D59SD64semidirect
109. C2×SD64D59semidirect
134. C59#13662semidirect
148. Q32C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
C2×Q32×D592
C2×#168372402
#214261792
#214222782
#214222772
(C2×D59)⋊C322
C2×C59×SD642
#1366259