Finite Group Databasebeta

← #28501169

Subgroups — #28501169

Order 15104.

350 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426024C212yesmaximal
2#21425352C211yesmaximal, characteristic
3#21422114C212yesmaximal, complemented
4#21422038C211yesmaximal, characteristic, complemented
5C59×#3261C211yesmaximal, characteristic, complemented
6C22×D472C2211yescharacteristic
7C2×#14273670—22no—
8#16837406—21no—
9C2×#14273643—21no—
10#16837202—22no—
11C22⋊(C59⋊C16)C2212yes—
12#16837167C2211yescharacteristic
13C59×#541—21no—
14C59×#479—22no—
15C59×#470—21no—
16C59×#467C2212yes—
17C59×#462—22no—
18C59×#424C2211yescharacteristic
19C22×D236—21no—
20C2×D472—22no—
21C2×D472—42no—
22C2×#11745664—21no—
23#14273670—44no—
24C2×C59⋊C16—22no—
25#14273647—42no—
26#14273643—42no—
27C2×Q16×C59—21no—
28C2×Q16×C59D412yes—
29C2×M16×C59D411yescharacteristic
30C22×C8×C59C2311yescharacteristic
31C4⋊Q8×C59—21no—
32#126×C59—42no—
33#125×C59—21no—
34#125×C59—42no—
35C22⋊Q8×C59—22no—
36C4×Q8×C59—22no—
37#117×C59D411yescharacteristic
38#107×C59D412yes—
39#106×C59—21no—
40C4⋊C8×C59—22no—
41#103×C59—22no—
42#103×C59—22no—
43C23×D59—41no—
44C2×Q8×C59—22no—
45C22×C4×C59C2×D411yescharacteristic
46C2×D236—22no—
47C2×D236—41no—
48C2×D236—41no—
49C2×D236—41no—
50C2×D236—41no—
51C2×D236—41no—
52Q16×C59—42no—
53Q16×C59—81no—
54M16×C59—41no—
55C2×C8×C59—21no—
56C2×C8×C59—41no—
57C2×C8×C59C2×D412yes—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59D812yes—
64C22⋊C4×C59—21no—
65C22⋊C4×C59—42no—
66C42×C59—21no—
67C42×C59—42no—
68#11745664—41no—
69#11745664—41no—
70C2×C59⋊C8—41no—
71D472—24no—
72D472—42no—
73C59⋊C16—42no—
74C23×C59AGL(1,ℤ/8)11yescharacteristic
75C22×D59—42no—
76C22×D59—81no—
77C22×D59—81no—
78C22×D59—162no—
79Q8×C59—42no—
80Q8×C59—42no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59#12011yescharacteristic
87C2×C4×C59C2×D811yescharacteristic
88D236—42no—
89D236—42no—
90D236—81no—
91D236—81no—
92C8×C59—22no—
93C8×C59—41no—
94C59⋊C8—81no—
95#12165—591nomaximal, Sylow
96C22×C59—21no—
97C22×C59—22no—
98C22×C59#44711yescharacteristic
99C2×D59—82no—
100C2×D59—162no—
101C2×D59—162no—
102C2×D59—162no—
103C4×C59—21no—
104C4×C59—42no—
105C4×C59—82no—
106C4×C59—82no—
107C4×C59#45411yescharacteristic
108C4×C59B(4,𝔽2)11yescharacteristic
109#3261D5911yescharacteristic, complemented
110#2187—592no—
111#2045—591no—
112#1350—592no—
113#1323—591no—
114C2×C59—41no—
115C2×C59#159811yescharacteristic
116C2×C59#219312yes—
117D59—162no—
118C22×D8—591no—
119#541—21no—
120C2×SD32—1182no—
121#479—22no—
122#470—21no—
123#467C2×D5912yes—
124#462—22no—
125#424C2×D5911yescharacteristic
126#418—1181no—
127C2×#100—1181no—
128#357—1182no—
129C22⋊C16—592no—
130#341—591no—
131C59#1216511yescharacteristic, complemented, Sylow
132C22×D4—1181no—
133C2×Q16—21no—
134C2×Q16C22⋊D5912yes—
135C2×D8—1182no—
136C2×D8—2362no—
137C2×M16—1181no—
138C2×M16C22⋊D5911yescharacteristic
139C22×C8C22×D5911yescharacteristic
140C4⋊Q8—21no—
141#126—42no—
142#125—21no—
143#125—42no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146#117C22⋊D5911yescharacteristic
147SD32—2364no—
148C2×C16—1182no—
149#107C22⋊D5912yes—
150#106—21no—
151C4⋊C8—22no—
152#103—22no—
153#103—22no—
154#102—2362no—
155#100—2362no—
156C24—2361no—
157C2×Q8—22no—
158C2×D4—1182no—
159C2×D4—2361no—
160C2×D4—2361no—
161C2×D4—2361no—
162C2×D4—2361no—
163C2×D4—2361no—
164C22×C4C2×C22⋊D5911yescharacteristic
165Q16—42no—
166Q16—81no—
167D8—1184no—
168D8—2362no—
169M16—41no—
170M16—2361no—
171M16—2361no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—2361no—
175C2×C8C2×C22⋊D5912yes—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4#1174566912yes—
182C22⋊C4—21no—
183C22⋊C4—42no—
184C42—21no—
185C42—42no—
186C16—2362no—
187C23—9442no—
188C23—4721no—
189C23—4721no—
190C23—2362no—
191C23#1427375311yescharacteristic
192Q8—42no—
193Q8—42no—
194D4—4721no—
195D4—4721no—
196D4—2362no—
197D4—2362no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4#1427369611yescharacteristic
204C2×C4C2×#1174566911yescharacteristic
205C8—22no—
206C8—41no—
207C8—4721no—
208C22—22no—
209C22—21no—
210C22—9442no—
211C22—9442no—
212C22—9442no—
213C22—4722no—
214C22#1683753611yescharacteristic
215C4—21no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4#1683757511yescharacteristic
220C4#1683758911yescharacteristic
221C2—41no—
222C2—9442no—
223C2#2142303111yescharacteristic
224C2#2142606012yes—
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3261C2
socleC22×C59#447
prime radicalC22×D472C22
hypercenter#3261D59
hypocenterC59#12165
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #3261

Complements

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

NH ≅ G/Ntype
3. #21422114C2semidirect
4. #21422038C2semidirect
5. C59×#3261C2semidirect
109. #3261D59semidirect
131. C59#12165semidirect

Maximal subgroups

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

subgroupindex
#214260242
#214253522
#214221142
#214220382
C59×#32612
#1216559