Finite Group Databasebeta

← #31780775

Subgroups — #31780775

Order 17536.

380 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 95 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×D4×Dic137C211yesmaximal, characteristic
2C2×#17984003C211yesmaximal, characteristic
3#22577261C212yesmaximal, complemented
4#22577169C212yesmaximal, complemented
5C2×#99×C137C211yesmaximal, characteristic
6C23×Dic137C411yescharacteristic, complemented
7C23×Dic137C2211yescharacteristic
8D4×Dic137—22no—
9D4×Dic137—22no—
10C2×C4⋊Dic137C411yescharacteristic, complemented
11C2×#14555489—22no—
12C2×C22⋊Dic137C411yescharacteristic, complemented
13C2×C22⋊Dic137C2212yes—
14C2×C22⋊Dic137C2211yescharacteristic
15C2×C4×Dic137C411yescharacteristic, complemented
16#17984003—22no—
17#17983995—22no—
18C22×D4×C137C2211yescharacteristic
19C2×C22⋊C4×C137C2212yes—
20#99×C137—22no—
21C22×Dic137—21no—
22C22×Dic137—21no—
23C22×Dic137—22no—
24C22×Dic137—42no—
25C22×Dic137—42no—
26C22×Dic137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2×C411yescharacteristic
29C22×Dic137C2×C411yescharacteristic
30C22×Dic137D412yes—
31C22×Dic137D412yes—
32C4⋊Dic137—22no—
33#14555489—44no—
34C22⋊Dic137—22no—
35C22⋊Dic137—22no—
36C22⋊Dic137—22no—
37C22⋊Dic137—22no—
38C4×Dic137—21no—
39C4×Dic137—21no—
40C24×C137C2×C411yescharacteristic
41C24×C137C2311yescharacteristic
42C2×D4×C137—22no—
43C2×D4×C137—22no—
44C2×D4×C137D412yes—
45C2×D4×C137Q812yes—
46C22×C4×C137—22no—
47C22×C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—22no—
49C22⋊C4×C137—22no—
50C2×Dic137—24no—
51C2×Dic137—22no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—41no—
55C2×Dic137—42no—
56C2×Dic137—42no—
57C2×Dic137—42no—
58C2×Dic137—41no—
59C2×Dic137—41no—
60C2×Dic137—42no—
61C2×Dic137—42no—
62C2×Dic137—42no—
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C22⋊C412yes—
65C2×Dic137C22⋊C411yescharacteristic
66C23×C137—21no—
67C23×C137—21no—
68C23×C137—41no—
69C23×C137—41no—
70C23×C137—41no—
71C23×C137—41no—
72C23×C137C4⋊C412yescomplemented
73C23×C137C22×C411yescharacteristic
74C23×C137C2×D412yes—
75C23×C137C2×D411yescharacteristic
76C23×C137C2×Q811yescharacteristic
77C23×C137#4112yes—
78D4×C137—24no—
79D4×C137—24no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—42no—
83C2×C4×C137—42no—
84C2×C4×C137—42no—
85C2×C4×C137C4⋊C412yescomplemented
86C22×C137—22no—
87C22×C137—21no—
88C22×C137—22no—
89C22×C137—21no—
90C22×C137—22no—
91C22×C137—41no—
92C22×C137—41no—
93C22×C137—42no—
94C22×C137—42no—
95C22×C137—42no—
96C22×C137—41no—
97C22×C137—41no—
98C22×C137C2×C22⋊C411yescharacteristic
99C22×C137C2×C4⋊C411yescharacteristic
100C22×C137#11711yescharacteristic
101C22×C137#12112yes—
102C22×C137C22⋊Q812yes—
103C4×C137—22no—
104C4×C137—82no—
105Dic137—21no—
106Dic137—22no—
107Dic137—41no—
108Dic137—41no—
109Dic137—82no—
110Dic137#9912yescomplemented
111C2×C137—22no—
112C2×C137—21no—
113C2×C137—21no—
114C2×C137—22no—
115C2×C137—42no—
116C2×C137#38011yescharacteristic
117C2×C137C2×#9911yescharacteristic
118C2×C137#41011yescharacteristic
119C137#179911yescharacteristic, complemented, Sylow
120#1799—1371nomaximal, Sylow
121C2×C4×D4—1371no—
122C2×#99—1371no—
123C2×#99D13711yescharacteristic
124#380—1372no—
125#342—1372no—
126C22×D4C2×D13711yescharacteristic
127C23×C4—1371no—
128C23×C4—1371no—
129C4×D4—2742no—
130C4×D4—2742no—
131C2×C4⋊C4—2742no—
132C2×C4⋊C4—1371no—
133C2×C22⋊C4—1372no—
134C2×C22⋊C4—1371no—
135C2×C22⋊C4—1371no—
136C2×C22⋊C4C2×D13712yes—
137C2×C42—1371no—
138#99—22no—
139#99—2742no—
140#95—2742no—
141C24C4×D13711yescharacteristic
142C24C22×D13711yescharacteristic
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22⋊D13712yes—
146C2×D4Dic27412yes—
147C22×C4—22no—
148C22×C4—2741no—
149C22×C4—2741no—
150C22×C4—2742no—
151C22×C4—5482no—
152C22×C4—5482no—
153C22×C4—1371no—
154C22×C4—1372no—
155C22×C4—1371no—
156C22×C4—1371no—
157C22×C4—1371no—
158C22×C4—1372no—
159C22×C4C4×D13711yescharacteristic
160C4⋊C4—2742no—
161C4⋊C4—5484no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—2742no—
165C22⋊C4—2742no—
166C22⋊C4—2742no—
167C22⋊C4—2742no—
168C42—2741no—
169C42—2741no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#1455548912yescomplemented
177C23C2×C4×D13711yescharacteristic
178C23D4×D13712yes—
179C23C2×C22⋊D13711yescharacteristic
180C23C2×Dic27411yescharacteristic
181C23#1455550912yes—
182D4—24no—
183D4—24no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—2744no—
190C2×C4—2742no—
191C2×C4—2741no—
192C2×C4—2741no—
193C2×C4—5482no—
194C2×C4—5481no—
195C2×C4—5482no—
196C2×C4—5482no—
197C2×C4—5481no—
198C2×C4—5481no—
199C2×C4—5482no—
200C2×C4—5482no—
201C2×C4—5482no—
202C2×C4—1371no—
203C2×C4—1372no—
204C2×C4—1371no—
205C2×C4#1455548912yescomplemented
206C22—22no—
207C22—21no—
208C22—22no—
209C22—21no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22C22⋊C4×D13711yescharacteristic
219C22C2×#1455548911yescharacteristic
220C22#1798404411yescharacteristic
221C22#1798406012yes—
222C22#1798406512yes—
223C4—22no—
224C4—82no—
225C4—2741no—
226C4—2742no—
227C4—5481no—
228C4—5481no—
229C4—10962no—
230C4—1372no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2#2257726111yescharacteristic
237C2#99×D13711yescharacteristic
238C2#2257737511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC23×C137C22×C4
FrattiniC24C22×D137
FittingC2×#99×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D4×C137C22
cosocleC24×C137C23
hypercenterC2×#99D137
hypocenterC137#1799
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #22577261C2semidirect
4. #22577169C2semidirect
6. C23×Dic137C4semidirect
10. C2×C4⋊Dic137C4semidirect
12. C2×C22⋊Dic137C4semidirect
15. C2×C4×Dic137C4semidirect
72. C23×C137C4⋊C4semidirect
85. C2×C4×C137C4⋊C4semidirect
110. Dic137#99semidirect
119. C137#1799semidirect
176. C23#14555489semidirect
205. C2×C4#14555489semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1372
C2×#179840032
#225772612
#225771692
C2×#99×C1372
#1799137