Finite Group Databasebeta

← #28500210

Subgroups — #28500210

Order 15104.

268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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#21434756C211yesmaximal, characteristic
2#21425911C211yesmaximal, characteristic
3#21425860C211yesmaximal, characteristic
4#21422157C211yesmaximal, characteristic, complemented
5#21422134C211yesmaximal, characteristic, complemented
6#21422134C211yesmaximal, characteristic, complemented
7C59×#1366C211yesmaximal, characteristic, complemented
8#16838015C412yes—
9M32×D59—21no—
10#16837782C2211yescharacteristic
11#16837772C412yes—
12#16837661—21no—
13#16837644—21no—
14#16837508—21no—
15#16837496—21no—
16#16837356C2211yescharacteristic
17#16837223C2211yescharacteristic
18#16837223C2211yescharacteristic
19#16837222—21no—
20#16837183—21no—
21#16837179—21no—
22C59×#499C2211yescharacteristic
23C59×#364C2211yescharacteristic
24C59×#364C2211yescharacteristic
25C2×Dic236—21no—
26C2×Dic236—21no—
27C2×#11745659D411yescharacteristic
28C2×C8×D59D411yescharacteristic
29#14273727C2×C412yes—
30#14273722—42no—
31#14273718—42no—
32#14273707—22no—
33#14273692—22no—
34#14273684C2×C411yescharacteristic
35#14273667—21no—
36#14273667—21no—
37#14273666—41no—
38C16×D59—41no—
39#14273650—22no—
40#14273650—22no—
41#14273638C2×C411yescharacteristic
42#14273637D411yescharacteristic
43C8×Dic59D411yescharacteristic
44#125×C59C2×C412yes—
45M32×C59—21no—
46M32×C59—21no—
47#107×C59—21no—
48#106×C59—21no—
49C4×C8×C59C2311yescharacteristic
50C2×C4×D59C22⋊C411yescharacteristic
51C2×Dic118—22no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59C2×D411yescharacteristic
54C4⋊C4×C59—22no—
55C4⋊C4×C59—42no—
56C42×C59C22×C411yescharacteristic
57(C2×D59)⋊C4—21no—
58(C2×D59)⋊C4—42no—
59C4⋊Dic59—42no—
60C4⋊Dic59—42no—
61Dic59⋊C4—21no—
62Dic59⋊C4—42no—
63C4×Dic59—42no—
64C4×Dic59C22⋊C411yescharacteristic
65C2×C59⋊C8C22⋊C411yescharacteristic
66C2×C59⋊C8C22⋊C411yescharacteristic
67Dic236—41no—
68Dic236—41no—
69Dic236—41no—
70Dic236—41no—
71#11745659—21no—
72#11745659—21no—
73C8×D59—21no—
74C8×D59—21no—
75C16×C59—41no—
76C16×C59—41no—
77C59⋊C16—41no—
78C59⋊C16—41no—
79C22×D59#10011yescharacteristic
80C2×C4×C59—21no—
81C2×C4×C59—42no—
82C2×C4×C59C2×C22⋊C411yescharacteristic
83C2×Dic59—21no—
84C2×Dic59—42no—
85C2×Dic59#10011yescharacteristic
86C4×D59—21no—
87C4×D59—21no—
88Dic118—42no—
89Dic118—42no—
90C8×C59#13511yescharacteristic
91C8×C59#13511yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C8×C59#13711yescharacteristic
94C59⋊C8—21no—
95C59⋊C8—21no—
96#11990—591nomaximal, Sylow
97C22×C59C2×#10011yescharacteristic
98C2×D59—21no—
99C2×D59—41no—
100C4×C59—41no—
101C4×C59—82no—
102C4×C59#41611yescharacteristic
103C4×C59#41711yescharacteristic
104Dic59—21no—
105Dic59—41no—
106Dic59—82no—
107#3177—591no—
108#2166—591no—
109#2153—591no—
110#1366—591no—
111#1366D5911yescharacteristic, complemented
112#1358—591no—
113#1358—591no—
114C2×C59—21no—
115C2×C59#155111yescharacteristic
116D59—41no—
117#541—592no—
118C2×M32—1181no—
119#499C2×D5911yescharacteristic
120#494—591no—
121#491—592no—
122#470—1181no—
123#467—1181no—
124#441—1181no—
125C4×Q16—1181no—
126#405—591no—
127#364—591no—
128#364—591no—
129#364—1181no—
130#364C2×D5911yescharacteristic
131#364C2×D5911yescharacteristic
132#349—1181no—
133#347—1181no—
134C59#1199011yescharacteristic, complemented, Sylow
135C2×Q16—1181no—
136C2×Q16—1181no—
137C2×M16—591no—
138C22×C8—591no—
139C4⋊Q8—592no—
140#126—2362no—
141#125—2362no—
142#125C4×D5912yes—
143C22⋊Q8—1182no—
144C4×Q8—1182no—
145#117—591no—
146M32—21no—
147M32—21no—
148M32—1181no—
149M32—1181no—
150M32—2361no—
151C2×C16—2361no—
152#107—21no—
153#106—21no—
154#103—1182no—
155#103—1182no—
156#97—591no—
157#97—591no—
158C4×C8—591no—
159C4×C8C22×D5911yescharacteristic
160C2×Q8—1182no—
161C22×C4—591no—
162Q16—2361no—
163Q16—2361no—
164Q16—2361no—
165Q16—2361no—
166M16—1181no—
167M16—1181no—
168C2×C8—591no—
169C2×C8—591no—
170C2×C8—1181no—
171C2×C8—1181no—
172C2×C8D4×D5911yescharacteristic
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—1181no—
177C4⋊C4—2362no—
178C4⋊C4—2362no—
179C4⋊C4—2362no—
180C22⋊C4—1181no—
181C22⋊C4—2362no—
182C42—591no—
183C42—2362no—
184C42C2×C4×D5911yescharacteristic
185C16—41no—
186C16—41no—
187C16—2361no—
188C16—2361no—
189C23—591no—
190Q8—2362no—
191Q8—2362no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—591no—
195C2×C4—1181no—
196C2×C4—1181no—
197C2×C4—1181no—
198C2×C4—2362no—
199C2×C4C22⋊C4×D5911yescharacteristic
200C8—1181no—
201C8—1181no—
202C8#1427374811yescharacteristic
203C8#1427375011yescharacteristic
204C8#1427375411yescharacteristic
205C8#1427375911yescharacteristic
206C22—1181no—
207C22—2361no—
208C22#100×D5911yescharacteristic
209C4—41no—
210C4—82no—
211C4—1181no—
212C4—4722no—
213C4—2361no—
214C4#1683739711yescharacteristic
215C4#1683740211yescharacteristic
216C2—21no—
217C2—2361no—
218C2#2142279011yescharacteristic
subgroupG/H
centerC2#21422790
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1366C2
socleC2×C59#1551
prime radicalC22×D59#100
cosocleC4×C8×C59C23
hypercenter#1366D59
hypocenterC59#11990
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
4. #21422157C2semidirect
5. #21422134C2semidirect
6. #21422134C2semidirect
7. C59×#1366C2semidirect
111. #1366D59semidirect
134. C59#11990semidirect

Maximal subgroups

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

subgroupindex
#214347562
#214259112
#214258602
#214221572
#214221342
C59×#13662
#1199059