Finite Group Databasebeta

← #28499866

Subgroups — #28499866

Order 15104.

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

normal subgroups only

This group has 318 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#21434443C211yesmaximal, characteristic
2#347×D59C212yesmaximal, complemented
3#21422131C211yesmaximal, characteristic, complemented
4#21422128C212yesmaximal, complemented
5C59×#1356C211yesmaximal, characteristic, complemented
6#16837958C411yescharacteristic
7#16837953C411yescharacteristic
8M32×D59—22no—
9#16837770C411yescharacteristic
10#16837768C411yescharacteristic
11#16837767C2211yescharacteristic
12#16837763—22no—
13#16837397—22no—
14#97×D59C2211yescharacteristic
15#16837180C2212yes—
16#16837179—22no—
17C59×#491C2211yescharacteristic
18C59×#347C2212yes—
19C2×#11745689—21no—
20C2×C8×D59D412yes—
21#14273726—21no—
22#14273724C2×C411yescharacteristic
23#14273719C2×C411yescharacteristic
24#14273716—21no—
25#14273705—41no—
26#14273702—41no—
27#14273692—41no—
28#14273689—41no—
29#14273685—21no—
30C42×D59C2×C411yescharacteristic
31#14273667—22no—
32#14273666—24no—
33C16×D59—24no—
34#14273651—22no—
35#14273649—22no—
36#14273647—22no—
37#14273638C2×C411yescharacteristic
38#14273637D412yes—
39C4⋊Q8×C59C2×C411yescharacteristic
40#125×C59C2×C411yescharacteristic
41M32×C59—22no—
42#103×C59—22no—
43#97×C59C2311yescharacteristic
44C2×Q8×C59—21no—
45#11745689—42no—
46#11745689—42no—
47C2×D236—21no—
48C2×D236—41no—
49C2×D236—41no—
50C2×C4×D59—21no—
51C2×C4×D59—41no—
52C2×C4×D59C22⋊C411yescharacteristic
53C2×Dic118—41no—
54C2×C8×C59C2×D412yes—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—41no—
57C4⋊C4×C59—41no—
58C42×C59C22×C411yescharacteristic
59(C2×D59)⋊C4—42no—
60(C2×D59)⋊C4—41no—
61C4⋊Dic59—21no—
62Dic59⋊C4—41no—
63Dic59⋊C4—41no—
64C4×Dic59—21no—
65C4×Dic59—41no—
66C4×Dic59C22⋊C411yescharacteristic
67C2×C59⋊C8C22⋊C412yes—
68C8×D59—22no—
69C8×D59—22no—
70C16×C59—24no—
71C59⋊C16—24no—
72C22×D59—41no—
73C22×D59#10011yescharacteristic
74Q8×C59—42no—
75C2×C4×C59—21no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59C2×C22⋊C411yescharacteristic
79C2×Dic59—21no—
80C2×Dic59—41no—
81C2×Dic59#10011yescharacteristic
82D236—42no—
83D236—82no—
84D236—82no—
85C4×D59—81no—
86C4×D59—81no—
87C4×D59—81no—
88C4×D59#10412yes—
89C4×D59#10412yes—
90Dic118—82no—
91C8×C59—22no—
92C59⋊C8—22no—
93#11911—591nomaximal, Sylow
94C22×C59C2×#10011yescharacteristic
95C2×D59—41no—
96C2×D59—82no—
97C2×D59—81no—
98C2×D59#32912yes—
99C4×C59—41no—
100C4×C59—81no—
101C4×C59—81no—
102C4×C59C2×#10412yes—
103Dic59—41no—
104Dic59—81no—
105Dic59#32911yescharacteristic
106Dic59#32911yescharacteristic
107#3135—591no—
108C2×#347—592no—
109#1356—591no—
110#1356D5911yescharacteristic, complemented
111#1355—592no—
112C2×C59—21no—
113C2×C59C2×#32911yescharacteristic
114D59—22no—
115D59—161no—
116#533—591no—
117#532—591no—
118C2×M32—1182no—
119#491—591no—
120#491C2×D5911yescharacteristic
121#490—591no—
122#490—591no—
123#489—1182no—
124#416—1182no—
125C2×#97—591no—
126#347—592no—
127#347—1182no—
128#347C2×D5912yes—
129C59#1191111yescharacteristic, complemented, Sylow
130C2×#41—1181no—
131C22×C8—592no—
132C4⋊Q8—1181no—
133C4⋊Q8C4×D5911yescharacteristic
134#127—591no—
135#125—591no—
136#125C4×D5911yescharacteristic
137#124—1181no—
138C22⋊Q8—2361no—
139#121—2361no—
140C4×Q8—2361no—
141C4×D4—2361no—
142#117—1181no—
143C2×C42—591no—
144M32—22no—
145M32—1184no—
146M32—1182no—
147C2×C16—1184no—
148#103—22no—
149#103—1182no—
150#102—1182no—
151#102—1182no—
152#97—592no—
153#97—591no—
154#97C22×D5911yescharacteristic
155#41—2362no—
156#41—2362no—
157C2×Q8—21no—
158C2×Q8—2361no—
159C2×D4—1181no—
160C2×D4—2361no—
161C2×D4—2361no—
162C22×C4—591no—
163C22×C4—1181no—
164C22×C4—2361no—
165C2×C8—592no—
166C2×C8—1182no—
167C2×C8—1182no—
168C2×C8D4×D5912yes—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1181no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C22⋊C4—2362no—
176C22⋊C4—2361no—
177C42—591no—
178C42—1181no—
179C42—2361no—
180C42C2×C4×D5911yescharacteristic
181C16—24no—
182C16—1184no—
183C23—591no—
184C23—2361no—
185Q8—42no—
186Q8—4722no—
187D4—4722no—
188D4—4722no—
189D4—2362no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—591no—
194C2×C4—592no—
195C2×C4—592no—
196C2×C4—1181no—
197C2×C4—4721no—
198C2×C4—4721no—
199C2×C4—4721no—
200C2×C4—2361no—
201C2×C4C22⋊C4×D5911yescharacteristic
202C8—22no—
203C8—1182no—
204C22—592no—
205C22—4722no—
206C22—4721no—
207C22—2361no—
208C22#100×D5911yescharacteristic
209C4—41no—
210C4—81no—
211C4—81no—
212C4—591no—
213C4—591no—
214C4—4721no—
215C4—2361no—
216C4#104×D5912yes—
217C2—21no—
218C2—1182no—
219C2—9441no—
220C2#329×D5911yescharacteristic
subgroupG/H
centerC2#329×D59
commutatorC42×C59C22×C4
Frattini#97C22×D59
FittingC59×#1356C2
socleC2×C59C2×#329
prime radical#16837953C4
cosocle#97×C59C23
hypercenter#1356D59
hypocenterC59#11911
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
2. #347×D59C2semidirect
3. #21422131C2semidirect
4. #21422128C2semidirect
5. C59×#1356C2semidirect
110. #1356D59semidirect
129. C59#11911semidirect

Maximal subgroups

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

subgroupindex
#214344432
#347×D592
#214221312
#214221282
C59×#13562
#1191159