Finite Group Databasebeta

← #28500204

Subgroups — #28500204

Order 15104.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434746C211yesmaximal, characteristic
2#21425912C211yesmaximal, characteristic, complemented
3#21422158C212yesmaximal, complemented
4#21422132C212yesmaximal, complemented
5C59×#2153C211yesmaximal, characteristic, complemented
6#16838007C412yes—
7#16837804C2211yescharacteristic
8#16837779C2211yescharacteristic
9#16837767C412yes—
10#16837655—21no—
11#16837638—21no—
12#16837515—21no—
13C4×D472—21no—
14#16837223C2212yes—
15#16837184—21no—
16#16837180—21no—
17C2×M32×C59—21no—
18C59×#405C2211yescharacteristic
19C59×#364—21no—
20C59×#364C2212yes—
21C2×D472—21no—
22C2×D472—21no—
23#14273724C2×C412yes—
24#14273719C2×C412yes—
25#14273712—42no—
26#14273701—22no—
27#14273700—42no—
28C4×D236—22no—
29#14273667—22no—
30#14273661—21no—
31#14273659—21no—
32#14273649—22no—
33#14273649—22no—
34C2×M16×C59D411yescharacteristic
35C22×C8×C59D411yescharacteristic
36#117×C59C2×C411yescharacteristic
37M32×C59—22no—
38M32×C59—41no—
39C2×C16×C59—41no—
40#97×C59C2×C411yescharacteristic
41#97×C59D411yescharacteristic
42C4×C8×C59D411yescharacteristic
43C4×C8×C59C2311yescharacteristic
44C22×C4×C59C22⋊C411yescharacteristic
45C2×C22⋊D59—42no—
46C2×D236—22no—
47C2×D236—42no—
48C2×C4×D59—42no—
49M16×C59—22no—
50C2×C8×C59—21no—
51C2×C8×C59—21no—
52C2×C8×C59C22⋊C411yescharacteristic
53C2×C8×C59C22⋊C411yescharacteristic
54C2×C8×C59C2×D411yescharacteristic
55C2×C8×C59C2×D411yescharacteristic
56C4⋊C4×C59—21no—
57C22⋊C4×C59—21no—
58C42×C59C22⋊C411yescharacteristic
59C42×C59C22×C411yescharacteristic
60(C2×D59)⋊C4—42no—
61(C2×D59)⋊C4—42no—
62C4⋊Dic59—22no—
63Dic59⋊C4—42no—
64D472—42no—
65D472—41no—
66D472—41no—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59#10011yescharacteristic
70C22×D59—42no—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59#10011yescharacteristic
76C2×C4×C59C2×C22⋊C411yescharacteristic
77C22⋊D59—162no—
78C2×Dic59—42no—
79D236—42no—
80D236—42no—
81D236—162no—
82C4×D59—162no—
83C8×C59—21no—
84C8×C59—21no—
85C8×C59#13512yes—
86C8×C59AGL(1,ℤ/8)11yescharacteristic
87C8×C59#13711yescharacteristic
88#11989—591nomaximal, Sylow
89C22×C59—21no—
90C22×C59—41no—
91C22×C59C2×#10011yescharacteristic
92C2×D59—82no—
93C2×D59—82no—
94C2×D59—82no—
95C4×C59—21no—
96C4×C59—41no—
97C4×C59—41no—
98C4×C59#41611yescharacteristic
99C4×C59#41711yescharacteristic
100Dic59—82no—
101#3176—591no—
102#2166—591no—
103#2153D5911yescharacteristic, complemented
104#1366—592no—
105#1357—592no—
106C2×C59—21no—
107C2×C59—41no—
108C2×C59#155111yescharacteristic
109D59—162no—
110#540—592no—
111C2×M32—21no—
112#499—591no—
113#493—591no—
114#490—592no—
115#469—1181no—
116#466—1181no—
117#442—1181no—
118C4×D8—1181no—
119#405C2×D5911yescharacteristic
120#364—21no—
121#364—592no—
122#364C2×D5912yes—
123#349—1181no—
124#347—1181no—
125C59#1198911yescharacteristic, complemented, Sylow
126C2×D8—1181no—
127C2×D8—1181no—
128C2×M16D23611yescharacteristic
129C22×C8C22⋊D5911yescharacteristic
130#127—592no—
131#125—592no—
132#123—2362no—
133#121—1182no—
134#121—2362no—
135C4×D4—1182no—
136#117C4×D5911yescharacteristic
137M32—22no—
138M32—41no—
139M32—1182no—
140C2×C16—41no—
141#107—1181no—
142#106—1181no—
143#102—1182no—
144#102—1182no—
145#97C4×D5911yescharacteristic
146#97D23611yescharacteristic
147C4×C8C22⋊D5911yescharacteristic
148C4×C8C22×D5911yescharacteristic
149C2×D4—1182no—
150C2×D4—2362no—
151C2×D4—2362no—
152C22×C4—2362no—
153C22×C4(C2×D59)⋊C411yescharacteristic
154D8—2362no—
155D8—2361no—
156D8—2361no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8(C2×D59)⋊C411yescharacteristic
161C2×C8(C2×D59)⋊C411yescharacteristic
162C2×C8C2×D23611yescharacteristic
163C2×C8C2×C22⋊D5911yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—1182no—
166C4⋊C4—2362no—
167C22⋊C4—21no—
168C22⋊C4—2362no—
169C22⋊C4—2362no—
170C42(C2×D59)⋊C411yescharacteristic
171C42C2×C4×D5911yescharacteristic
172C16—42no—
173C16—2362no—
174C23—2362no—
175C23#1427364311yescharacteristic
176D4—9442no—
177D4—9442no—
178D4—2362no—
179D4—2362no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—9442no—
185C2×C4—2362no—
186C2×C4#1427364311yescharacteristic
187C2×C4C2×(C2×D59)⋊C411yescharacteristic
188C8—21no—
189C8—21no—
190C8#1427374912yes—
191C8#1427375311yescharacteristic
192C8#1427375811yescharacteristic
193C22—21no—
194C22—41no—
195C22—4722no—
196C22—4722no—
197C22—4722no—
198C22C2×#1427364311yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—4722no—
203C4#1683739911yescharacteristic
204C4#1683740111yescharacteristic
205C2—21no—
206C2—41no—
207C2—9442no—
208C2#2142278911yescharacteristic
subgroupG/H
centerC2#21422789
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#2153C2
socleC2×C59#1551
prime radical#21434746C2
cosocleC4×C8×C59C23
hypercenter#2153D59
hypocenterC59#11989
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #21425912C2semidirect
3. #21422158C2semidirect
4. #21422132C2semidirect
5. C59×#2153C2semidirect
103. #2153D59semidirect
125. C59#11989semidirect

Maximal subgroups

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

subgroupindex
#214347462
#214259122
#214221582
#214221322
C59×#21532
#1198959