Finite Group Databasebeta

← #28499233

Subgroups — #28499233

Order 15104.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#21423851C211yesmaximal, characteristic, complemented
3#21422088C212yesmaximal, complemented
4#21422082C212yesmaximal, complemented
5C59×#2166C211yesmaximal, characteristic, complemented
6#16838007—21no—
7#16837804C411yescharacteristic, complemented
8#16837779C411yescharacteristic, complemented
9#16837767—21no—
10#16837655C2212yes—
11#16837638—21no—
12#16837515C412yescomplemented
13C4×D472—21no—
14#16837357—21no—
15C59⋊#338C2211yescharacteristic
16#16837159C2211yescharacteristic
17#16837158—21no—
18C59×#405C2211yescharacteristic
19C59×#364—21no—
20C59×#349C2212yes—
21C59×#347—21no—
22C2×D472—21no—
23C2×D472C2×C412yes—
24C2×#11745664—21no—
25#14273724—21no—
26#14273719—21no—
27#14273712—42no—
28#14273701—22no—
29#14273700—42no—
30C4×D236—22no—
31#14273686—21no—
32#14273661—21no—
33#14273659C2×C412yes—
34#14273649—22no—
35#14273649—22no—
36#14273637—22no—
37C8×Dic59—22no—
38C2×M16×C59C2311yescharacteristic
39C22×C8×C59D411yescharacteristic
40#117×C59D411yescharacteristic
41M32×C59—22no—
42#97×C59C2×C411yescharacteristic
43#97×C59D411yescharacteristic
44C4×C8×C59C2×C411yescharacteristic
45C4×C8×C59D411yescharacteristic
46C22×C4×C59C2×D411yescharacteristic
47C2×C22⋊D59—42no—
48C2×D236—22no—
49C2×D236—42no—
50C2×C4×D59—42no—
51M16×C59—41no—
52C2×C8×C59—21no—
53C2×C8×C59—21no—
54C2×C8×C59C22⋊C411yescharacteristic
55C2×C8×C59C22⋊C411yescharacteristic
56C2×C8×C59C22×C411yescharacteristic
57C2×C8×C59C2×D411yescharacteristic
58C4⋊C4×C59—21no—
59C22⋊C4×C59—21no—
60C42×C59C22⋊C411yescharacteristic
61C42×C59C22⋊C411yescharacteristic
62C22⋊Dic59—41no—
63(C2×D59)⋊C4—42no—
64(C2×D59)⋊C4—42no—
65C4⋊Dic59—22no—
66C4⋊Dic59—41no—
67Dic59⋊C4—42no—
68C4×Dic59—22no—
69#11745664—41no—
70#11745664—41no—
71C2×C59⋊C8—22no—
72D472—42no—
73D472—41no—
74D472—41no—
75C16×C59—42no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77C22×D59—42no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59C2×C22⋊C411yescharacteristic
83C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
84C22⋊D59—162no—
85C2×Dic59—42no—
86C2×Dic59—42no—
87D236—42no—
88D236—42no—
89D236—162no—
90C4×D59—162no—
91C8×C59—21no—
92C8×C59—21no—
93C8×C59—21no—
94C8×C59#10411yescharacteristic
95C8×C59#10411yescharacteristic
96C59⋊C8—42no—
97#11770—591nomaximal, Sylow
98C22×C59—21no—
99C22×C59—41no—
100C22×C59#41811yescharacteristic
101C2×D59—82no—
102C2×D59—82no—
103C2×D59—82no—
104C4×C59—21no—
105C4×C59—41no—
106C4×C59—41no—
107C4×C59#41011yescharacteristic
108C4×C59C2×#10411yescharacteristic
109Dic59—82no—
110Dic59—82no—
111#3176—591no—
112#2166D5911yescharacteristic, complemented
113#1747—591no—
114#1341—592no—
115#1339—592no—
116C2×C59—21no—
117C2×C59—41no—
118C2×C59#150511yescharacteristic
119D59—162no—
120#540—1181no—
121#499—591no—
122#493—591no—
123#490—1181no—
124#469—592no—
125#466—1181no—
126#442—592no—
127C4×D8—1181no—
128#405—1181no—
129#405C2×D5911yescharacteristic
130#364—21no—
131#349C2×D5912yes—
132#347—21no—
133#338—591no—
134H(3,ℤ/4)—591no—
135H(3,ℤ/4)—1181no—
136C59#1177011yescharacteristic, complemented, Sylow
137C2×D8—592no—
138C2×D8—1181no—
139C2×M16—1181no—
140C2×M16C22×D5911yescharacteristic
141C22×C8D23611yescharacteristic
142#127—1181no—
143#125—1181no—
144#123—2362no—
145#121—1182no—
146#121—2362no—
147C4×D4—1182no—
148#117—1181no—
149#117D23611yescharacteristic
150M32—22no—
151#107—1181no—
152#106—592no—
153#102—1182no—
154#102—1182no—
155#97—1182no—
156#97C4×D5911yescharacteristic
157#97C22⋊D5911yescharacteristic
158C4×C8—1182no—
159C4×C8C4×D5911yescharacteristic
160C4×C8C22⋊D5911yescharacteristic
161C2×D4—1182no—
162C2×D4—2362no—
163C2×D4—2362no—
164C22×C4—2362no—
165C22×C4C2×D23611yescharacteristic
166D8—2362no—
167D8—2361no—
168D8—2361no—
169M16—41no—
170M16—2361no—
171M16—2361no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1182no—
175C2×C8(C2×D59)⋊C411yescharacteristic
176C2×C8(C2×D59)⋊C411yescharacteristic
177C2×C8C2×C4×D5911yescharacteristic
178C2×C8C2×C22⋊D5911yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—1182no—
181C4⋊C4—2361no—
182C4⋊C4—2362no—
183C22⋊C4—21no—
184C22⋊C4—2362no—
185C22⋊C4—2361no—
186C22⋊C4—2362no—
187C42—1182no—
188C42(C2×D59)⋊C411yescharacteristic
189C42(C2×D59)⋊C411yescharacteristic
190C16—42no—
191C23—2362no—
192C23#1427375511yescharacteristic
193D4—9442no—
194D4—9442no—
195D4—2362no—
196D4—2362no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—9442no—
202C2×C4—2362no—
203C2×C4—2362no—
204C2×C4C2×(C2×D59)⋊C411yescharacteristic
205C2×C4#1427375511yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—2362no—
210C8#1427365311yescharacteristic
211C8#1427365311yescharacteristic
212C22—21no—
213C22—41no—
214C22—4722no—
215C22—4722no—
216C22—4722no—
217C22#1683740811yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—4722no—
222C4—4722no—
223C4#1683737511yescharacteristic
224C4C2×#1427365311yescharacteristic
225C2—21no—
226C2—41no—
227C2—9442no—
228C2#2142260511yescharacteristic
subgroupG/H
centerC2#21422605
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2166C2
socleC2×C59#1505
prime radical#21434746C2
cosocleC2×M16×C59C23
hypercenter#2166D59
hypocenterC59#11770
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #21423851C2semidirect
3. #21422088C2semidirect
4. #21422082C2semidirect
5. C59×#2166C2semidirect
7. #16837804C4semidirect
8. #16837779C4semidirect
12. #16837515C4semidirect
112. #2166D59semidirect
136. C59#11770semidirect

Maximal subgroups

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

subgroupindex
#214347462
#214238512
#214220882
#214220822
C59×#21662
#1177059