Finite Group Databasebeta

← #28500205

Subgroups — #28500205

Order 15104.

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

normal subgroups only

This group has 292 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#21434747C211yesmaximal, characteristic
2#21425911C211yesmaximal, characteristic, complemented
3#21425860C211yesmaximal, characteristic, complemented
4#21422157C211yesmaximal, characteristic, complemented
5#21422132C211yesmaximal, characteristic, complemented
6#21422132C211yesmaximal, characteristic, complemented
7C59×#1366C211yesmaximal, characteristic, complemented
8#16838008C412yes—
9M32×D59—21no—
10#16837779C2211yescharacteristic
11#16837768C412yes—
12#16837656—21no—
13#16837639—21no—
14#16837513—21no—
15#16837484—21no—
16#16837356C2211yescharacteristic
17#16837223C2211yescharacteristic
18#16837223C2211yescharacteristic
19#16837222—21no—
20#16837183—21no—
21#16837179—21no—
22C59×#499C2211yescharacteristic
23C59×#364C2211yescharacteristic
24C59×#364C2211yescharacteristic
25C2×D472—21no—
26C2×D472—21no—
27C2×#11745659D411yescharacteristic
28C2×C8×D59D411yescharacteristic
29#14273724C2×C412yes—
30#14273712—42no—
31#14273702—22no—
32#14273702—42no—
33#14273689—22no—
34#14273684C2×C411yescharacteristic
35#14273667—21no—
36#14273667—21no—
37#14273666—41no—
38C16×D59—41no—
39#14273647—22no—
40#14273647—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×D236—22no—
51C2×D236—42no—
52C2×D236—42no—
53C2×C4×D59—42no—
54C2×C4×D59C22⋊C411yescharacteristic
55C2×C8×C59C2×D411yescharacteristic
56C2×C8×C59C2×D411yescharacteristic
57C4⋊C4×C59—22no—
58C4⋊C4×C59—42no—
59C42×C59C22×C411yescharacteristic
60(C2×D59)⋊C4—21no—
61(C2×D59)⋊C4—42no—
62(C2×D59)⋊C4—42no—
63Dic59⋊C4—21no—
64C4×Dic59C22⋊C411yescharacteristic
65C2×C59⋊C8C22⋊C411yescharacteristic
66C2×C59⋊C8C22⋊C411yescharacteristic
67D472—41no—
68D472—41no—
69D472—41no—
70D472—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—42no—
80C22×D59#10011yescharacteristic
81C2×C4×C59—21no—
82C2×C4×C59—42no—
83C2×C4×C59C2×C22⋊C411yescharacteristic
84C2×Dic59—21no—
85C2×Dic59#10011yescharacteristic
86D236—42no—
87D236—42no—
88D236—162no—
89D236—162no—
90C4×D59—21no—
91C4×D59—21no—
92C4×D59—162no—
93C8×C59#13511yescharacteristic
94C8×C59#13511yescharacteristic
95C8×C59AGL(1,ℤ/8)11yescharacteristic
96C8×C59#13711yescharacteristic
97C59⋊C8—21no—
98C59⋊C8—21no—
99#11989—591nomaximal, Sylow
100C22×C59C2×#10011yescharacteristic
101C2×D59—21no—
102C2×D59—41no—
103C2×D59—82no—
104C2×D59—82no—
105C2×D59—82no—
106C4×C59—41no—
107C4×C59—82no—
108C4×C59#41611yescharacteristic
109C4×C59#41711yescharacteristic
110Dic59—21no—
111Dic59—41no—
112#3176—591no—
113#2166—591no—
114#2153—591no—
115#1366—591no—
116#1366D5911yescharacteristic, complemented
117#1357—591no—
118#1357—591no—
119C2×C59—21no—
120C2×C59#155111yescharacteristic
121D59—41no—
122D59—162no—
123#540—592no—
124C2×M32—1181no—
125#499C2×D5911yescharacteristic
126#493—591no—
127#490—592no—
128#469—1181no—
129#466—1181no—
130#442—1181no—
131C4×D8—1181no—
132#405—591no—
133#364—591no—
134#364—591no—
135#364—1181no—
136#364C2×D5911yescharacteristic
137#364C2×D5911yescharacteristic
138#349—1181no—
139#347—1181no—
140C59#1198911yescharacteristic, complemented, Sylow
141C2×D8—1181no—
142C2×D8—1181no—
143C2×M16—591no—
144C22×C8—591no—
145#127—592no—
146#125C4×D5912yes—
147#123—2362no—
148#121—1182no—
149#121—2362no—
150C4×D4—1182no—
151#117—591no—
152M32—21no—
153M32—21no—
154M32—1181no—
155M32—1181no—
156M32—2361no—
157C2×C16—2361no—
158#107—21no—
159#106—21no—
160#102—1182no—
161#102—1182no—
162#97—591no—
163#97—591no—
164C4×C8—591no—
165C4×C8C22×D5911yescharacteristic
166C2×D4—1182no—
167C2×D4—2362no—
168C2×D4—2362no—
169C22×C4—591no—
170C22×C4—2362no—
171D8—2361no—
172D8—2361no—
173D8—2361no—
174D8—2361no—
175M16—1181no—
176M16—1181no—
177C2×C8—591no—
178C2×C8—591no—
179C2×C8—1181no—
180C2×C8—1181no—
181C2×C8D4×D5911yescharacteristic
182C2×C8D4×D5911yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—1181no—
186C22⋊C4—1181no—
187C22⋊C4—2362no—
188C22⋊C4—2362no—
189C42—591no—
190C42C2×C4×D5911yescharacteristic
191C16—41no—
192C16—41no—
193C16—2361no—
194C16—2361no—
195C23—591no—
196C23—2362no—
197D4—9442no—
198D4—9442no—
199D4—2362no—
200D4—2362no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—591no—
204C2×C4—1181no—
205C2×C4—1181no—
206C2×C4—1181no—
207C2×C4—9442no—
208C2×C4C22⋊C4×D5911yescharacteristic
209C8—1181no—
210C8—1181no—
211C8#1427374811yescharacteristic
212C8#1427375011yescharacteristic
213C8#1427375411yescharacteristic
214C8#1427375911yescharacteristic
215C22—1181no—
216C22—4722no—
217C22—4722no—
218C22—4722no—
219C22—2361no—
220C22#100×D5911yescharacteristic
221C4—41no—
222C4—82no—
223C4—1181no—
224C4—2361no—
225C4#1683739711yescharacteristic
226C4#1683740211yescharacteristic
227C2—21no—
228C2—9442no—
229C2—2361no—
230C2#2142279011yescharacteristic
subgroupG/H
centerC2#21422790
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1366C2
socleC2×C59#1551
prime radical#21434747C2
cosocleC4×C8×C59C23
hypercenter#1366D59
hypocenterC59#11989
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
2. #21425911C2semidirect
3. #21425860C2semidirect
4. #21422157C2semidirect
5. #21422132C2semidirect
6. #21422132C2semidirect
7. C59×#1366C2semidirect
116. #1366D59semidirect
140. C59#11989semidirect

Maximal subgroups

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

subgroupindex
#214347472
#214259112
#214258602
#214221572
#214221322
C59×#13662
#1198959