Finite Group Databasebeta

← #31781331

Subgroups — #31781331

Order 17536.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#22577962C211yesmaximal, characteristic
2C2×#17984003C211yesmaximal, characteristic, complemented
3C2×#17984002C212yesmaximal, complemented
4#22577169C212yesmaximal, complemented
5#518×C137C211yesmaximal, characteristic, complemented
6C22×C22⋊D137C2211yescharacteristic
7#17984071—21no—
8#17984069C412yescomplemented
9#17984060—22no—
10#17984057—21no—
11#17984056C412yescomplemented
12C2×C22⋊Dic137C2212yes—
13C2×C22⋊Dic137C2211yescharacteristic
14#17984003—22no—
15#17984002—24no—
16C22×D4×C137C2211yescharacteristic
17C4×D4×C137—24no—
18#117×C137—21no—
19C2×C22⋊C4×C137—21no—
20C2×C22⋊C4×C137C2212yes—
21C23×D137C2×C411yescharacteristic, complemented
22C2×C22⋊D137—22no—
23C2×C22⋊D137—22no—
24C2×C22⋊D137—21no—
25C2×C22⋊D137—21no—
26C2×C22⋊D137—22no—
27C22×Dic137—21no—
28C22×Dic137—22no—
29C22×Dic137C2×C411yescharacteristic, complemented
30#14555489—22no—
31C22⋊Dic137—22no—
32C22⋊Dic137—21no—
33C22⋊Dic137—22no—
34C22⋊Dic137—22no—
35C22⋊Dic137C2×C412yescomplemented
36#14555487—22no—
37C24×C137D411yescharacteristic
38C24×C137C2311yescharacteristic
39C2×D4×C137—22no—
40C2×D4×C137—22no—
41C2×D4×C137C2×C412yes—
42C2×D4×C137D412yes—
43C22×C4×C137—22no—
44C22×C4×C137—22no—
45C22×C4×C137D411yescharacteristic
46C4⋊C4×C137—22no—
47C22⋊C4×C137—22no—
48C22⋊C4×C137—22no—
49C22⋊C4×C137D414yescomplemented
50C42×C137—22no—
51C22×D137—21no—
52C22×D137—22no—
53C22×D137—81no—
54C22⋊D137—41no—
55C22⋊D137—42no—
56C22⋊D137—41no—
57C22⋊D137—82no—
58C2×Dic137—22no—
59C2×Dic137—21no—
60C2×Dic137—22no—
61C2×Dic137—42no—
62C2×Dic137—81no—
63C2×Dic137—82no—
64C23×C137—21no—
65C23×C137—21no—
66C23×C137—81no—
67C23×C137—81no—
68C23×C137C22⋊C412yes—
69C23×C137C22×C411yescharacteristic
70C23×C137C2×D411yescharacteristic
71C23×C137C2×D412yes—
72C23×C137C2×D411yescharacteristic
73C23×C137#4112yes—
74D4×C137—24no—
75D4×C137—24no—
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137—24no—
79C2×C4×C137—82no—
80C2×C4×C137—82no—
81C2×C4×C137C22⋊C412yes—
82C22×C137—22no—
83C22×C137—42no—
84C22×C137—41no—
85C22×C137—41no—
86C22×C137—41no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137—42no—
90C22×C137—81no—
91C22×C137C2×C22⋊C411yescharacteristic
92C22×C137C4×D412yes—
93C22×C137#12011yescharacteristic
94C22×C137#12112yes—
95C22×C137#12311yescharacteristic
96C2×D137—42no—
97C2×D137—41no—
98C2×D137—82no—
99C4×C137—22no—
100C4×C137—44no—
101Dic137—42no—
102Dic137—41no—
103Dic137—82no—
104C2×C137—21no—
105C2×C137—22no—
106C2×C137—21no—
107C2×C137—22no—
108C2×C137—21no—
109C2×C137—42no—
110C2×C137#38611yescharacteristic
111D137—81no—
112C137#191311yescharacteristic, complemented, Sylow
113#1913—1371nomaximal, Sylow
114#534—1371no—
115#518D13711yescharacteristic, complemented
116C2×#99—1371no—
117C2×#99—1372no—
118#342—1372no—
119C22×D4—1371no—
120C22×D4C2×D13711yescharacteristic
121#123—2741no—
122#123—1372no—
123#121—2742no—
124#120—2741no—
125#120—1372no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—1372no—
130C2×C22⋊C4—1371no—
131C2×C22⋊C4C2×D13712yes—
132#99—2742no—
133#99—2744no—
134C24—1371no—
135C24C22⋊D13711yescharacteristic
136C24C22×D13711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—2742no—
140C2×D4—2741no—
141C2×D4—2741no—
142C2×D4—2742no—
143C2×D4—2742no—
144C2×D4C4×D13712yes—
145C2×D4D54812yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—2741no—
149C22×C4—2742no—
150C22×C4—1371no—
151C22×C4C22⋊D13711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—2742no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—2742no—
157C22⋊C4—2741no—
158C22⋊C4—2742no—
159C22⋊C4—2742no—
160C22⋊C4—2742no—
161C22⋊C4—1372no—
162C22⋊C4C22⋊D13714yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—2741no—
169C23—2742no—
170C23—10961no—
171C23#1455548712yes—
172C23C2×C4×D13711yescharacteristic
173C23C2×C22⋊D13711yescharacteristic
174C23C2×C22⋊D13712yes—
175C23C2×D54811yescharacteristic
176C23#1455551012yes—
177D4—24no—
178D4—24no—
179D4—5481no—
180D4—5482no—
181D4—5481no—
182D4—10962no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—2742no—
189C2×C4—2741no—
190C2×C4—2742no—
191C2×C4—5482no—
192C2×C4—10961no—
193C2×C4—10962no—
194C2×C4#1455548712yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—5482no—
205C22—5481no—
206C22—10962no—
207C22C2×#1455548711yescharacteristic
208C22C4×C22⋊D13712yes—
209C22#1798405711yescharacteristic
210C22#1798406212yes—
211C22#1798407111yescharacteristic
212C4—22no—
213C4—44no—
214C4—5482no—
215C4—5481no—
216C4—10962no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—10961no—
224C2#2257728211yescharacteristic
subgroupG/H
centerC2#22577282
commutatorC23×C137C22×C4
FrattiniC24C22×D137
Fitting#518×C137C2
socleC2×C137#386
prime radical#22577962C2
cosocleC24×C137C23
hypercenter#518D137
hypocenterC137#1913
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#17984003C2semidirect
3. C2×#17984002C2semidirect
4. #22577169C2semidirect
5. #518×C137C2semidirect
8. #17984069C4semidirect
11. #17984056C4semidirect
21. C23×D137C2×C4semidirect
29. C22×Dic137C2×C4semidirect
35. C22⋊Dic137C2×C4semidirect
49. C22⋊C4×C137D4semidirect
112. C137#1913semidirect
115. #518D137semidirect
162. C22⋊C4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
#225779622
C2×#179840032
C2×#179840022
#225771692
#518×C1372
#1913137