Finite Group Databasebeta

← #28476422

Subgroups — #28476422

Order 15104.

262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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×#16837201C211yesmaximal, characteristic
2#21422822C211yesmaximal, characteristic
3C59×#1886C211yesmaximal, characteristic
4#16837459—21no—
5C2×#14273646—21no—
6#16837201—21no—
7#16837201C412yescomplemented
8#16837151—21no—
9#16837151C412yescomplemented
10C2×C4⋊Q8×C59C2211yescharacteristic
11C59×#519C411yescharacteristic
12C59×#419—21no—
13C2×#101×C59C411yescharacteristic
14C59×#339—21no—
15C2×#11745664—41no—
16C4⋊(C59⋊C8)—22no—
17#14273646—22no—
18#14273646—41no—
19C22⋊(C59⋊C8)—41no—
20#14273638—41no—
21C22×Q8×C59C2×C411yescharacteristic
22C22×Q8×C59D411yescharacteristic
23C2×M16×C59—21no—
24C4⋊Q8×C59—41no—
25C4⋊Q8×C59C2×C411yescharacteristic
26C4⋊Q8×C59C2×C411yescharacteristic
27C4⋊Q8×C59D411yescharacteristic
28C4⋊Q8×C59Q811yescharacteristic
29C4×Q8×C59—42no—
30#117×C59—21no—
31#117×C59—41no—
32C2×C4⋊C4×C59—41no—
33C2×C42×C59D411yescharacteristic
34#103×C59—42no—
35#101×C59—41no—
36C2×Q8×C59—21no—
37C2×Q8×C59—21no—
38C2×Q8×C59—41no—
39C2×Q8×C59—41no—
40C2×Q8×C59—41no—
41C2×Q8×C59—41no—
42C2×Q8×C59C4211yescharacteristic
43C2×Q8×C59C22⋊C411yescharacteristic
44C2×Q8×C59C4⋊C411yescharacteristic
45C2×Q8×C59C4⋊C411yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59C22⋊C411yescharacteristic
50M16×C59—41no—
51M16×C59—41no—
52C2×C8×C59—41no—
53C4⋊C4×C59—22no—
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C4⋊C4×C59—41no—
57C4⋊C4×C59—41no—
58C4⋊C4×C59—42no—
59C22⋊C4×C59—41no—
60C42×C59—21no—
61C42×C59—41no—
62C42×C59—42no—
63C42×C59C22⋊C411yescharacteristic
64C42×C59C4⋊C411yescharacteristic
65#11745664—42no—
66#11745664—81no—
67C2×C59⋊C8—42no—
68C2×C59⋊C8—81no—
69C23×C59#9911yescharacteristic
70Q8×C59—41no—
71Q8×C59—41no—
72Q8×C59—41no—
73Q8×C59—41no—
74Q8×C59—42no—
75Q8×C59—81no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—42no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59#9511yescharacteristic
91C2×C4×C59#9911yescharacteristic
92C8×C59—81no—
93C59⋊C8—82no—
94C59⋊C8—82no—
95#7483—591nomaximal, Sylow
96C22×C59—41no—
97C22×C59#34211yescharacteristic
98C22×C59AΣL(1,𝔽16)11yescharacteristic
99C22×C59#35211yescharacteristic
100C4×C59—22no—
101C4×C59—21no—
102C4×C59—21no—
103C4×C59—41no—
104C4×C59—41no—
105C4×C59—41no—
106C4×C59—41no—
107C4×C59—41no—
108C4×C59—82no—
109C2×#356—591no—
110#1886D5911yescharacteristic
111#1558—591no—
112C2×C59—21no—
113C2×C59—21no—
114C2×C59—21no—
115C2×C59#139011yescharacteristic
116C2×C4⋊Q8C2×D5911yescharacteristic
117#519Dic5911yescharacteristic
118#432—1181no—
119#419—21no—
120C2×#101—1181no—
121C2×#101Dic5911yescharacteristic
122#356—592no—
123#356—1181no—
124#339—21no—
125#333—592no—
126#333—1181no—
127C59#748311yescharacteristic, complemented, Sylow
128C22×Q8C2×Dic5911yescharacteristic
129C22×Q8C22⋊D5911yescharacteristic
130C2×M16—21no—
131C2×M16—2361no—
132C4⋊Q8—41no—
133C4⋊Q8Dic11811yescharacteristic
134C4⋊Q8C4×D5911yescharacteristic
135C4⋊Q8C4×D5911yescharacteristic
136C4⋊Q8D23611yescharacteristic
137C4×Q8—42no—
138#117—21no—
139#117—41no—
140C2×C4⋊C4—41no—
141C2×C42C22⋊D5911yescharacteristic
142C4⋊C8—1182no—
143#103—42no—
144#101—41no—
145#101—1182no—
146#101—2361no—
147C22⋊C8—2361no—
148#97—2361no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8C4×Dic5911yescharacteristic
156C2×Q8Dic59⋊C411yescharacteristic
157C2×Q8C4⋊Dic5911yescharacteristic
158C2×Q8(C2×D59)⋊C411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C22⋊Dic5911yescharacteristic
163M16—41no—
164M16—41no—
165M16—4721no—
166M16—2362no—
167C2×C8—41no—
168C2×C8—4721no—
169C2×C8—2362no—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C22⋊C4—41no—
177C42—21no—
178C42—41no—
179C42—42no—
180C42Dic59⋊C411yescharacteristic
181C42(C2×D59)⋊C411yescharacteristic
182C23#1427364211yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4C59⋊#9511yescharacteristic
204C2×C4#1427364211yescharacteristic
205C8—81no—
206C8—4722no—
207C8—4722no—
208C22—41no—
209C22#1683716911yescharacteristic
210C22#1683718911yescharacteristic
211C22#1683719211yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2142221411yescharacteristic
subgroupG/H
centerC2#21422214
commutatorC2×Q8×C59C42
FrattiniC2×C4⋊Q8C2×D59
FittingC59×#1886C2
socleC2×C59#1390
prime radicalC23×C59#99
cosocleC2×C4⋊Q8×C59C22
hypercenter#1886D59
hypocenterC59#7483
LCSG ▷ C2×Q8×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
7. #16837201C4semidirect
9. #16837151C4semidirect
127. C59#7483semidirect

Maximal subgroups

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

subgroupindex
C2×#168372012
#214228222
C59×#18862
#748359