Finite Group Databasebeta

← #31790080

Subgroups — #31790080

Order 17536.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984136C211yesmaximal, characteristic
2C2×#17984119C211yesmaximal, characteristic, complemented
3#22577977C211yesmaximal, characteristic, complemented
4#22577564C214yesmaximal, complemented
5#22577557C214yesmaximal, complemented
6#22577412C211yesmaximal, characteristic, complemented
7#22577400C211yesmaximal, characteristic, complemented
8#22577369C211yesmaximal, characteristic, complemented
9C22×Q16×C137C211yesmaximal, characteristic, complemented
10#17984136—24no—
11#17984136—24no—
12C2×#14555511—21no—
13C2×#14555511C2212yes—
14C2×#14555510—22no—
15C2×#14555510C2211yescharacteristic
16C2×#14555510C2211yescharacteristic
17C2×Q8×D137—21no—
18C2×Q8×D137C2212yes—
19#17984119—24no—
20C2×#14555500C2212yes—
21C2×#14555497C2212yes—
22C2×#14555494C2211yescharacteristic
23#17984086—21no—
24#17984077—21no—
25#17984071—22no—
26#17984064C2214yes—
27#17984047C2211yescharacteristic
28#17984013C2212yes—
29#17984012C2214yes—
30#17984010C2212yes—
31#17984000C2214yes—
32C22×Q8×C137C2211yescharacteristic
33C22×Q8×C137C2211yescharacteristic
34C2×Q16×C137—24no—
35C2×Q16×C137C2214yes—
36C22×C8×C137C2211yescharacteristic
37#14555511—28no—
38#14555511—24no—
39#14555511—24no—
40#14555510—216no—
41#14555510—22no—
42#14555510—22no—
43#14555510—22no—
44#14555510D414yes—
45C2×Dic274—22no—
46C2×Dic274D411yescharacteristic
47C2×Dic274C2311yescharacteristic
48Q8×D137—28no—
49Q8×D137—24no—
50Q8×D137—24no—
51C2×D548—22no—
52C2×D548D411yescharacteristic
53C2×D548C2311yescharacteristic
54C2×C22⋊D137—21no—
55C2×C22⋊D137D412yes—
56C2×C4×D137—22no—
57C2×C4×D137—22no—
58C2×C4×D137—21no—
59C2×C4×D137C2312yes—
60#14555500—24no—
61#14555497—24no—
62#14555494—22no—
63C2×#11764040C2312yes—
64C4⋊Dic137C2312yes—
65#14555489—24no—
66C22⋊Dic137—21no—
67#14555487—24no—
68C4×Dic137—21no—
69C2×Q8×C137—21no—
70C2×Q8×C137—22no—
71C2×Q8×C137—21no—
72C2×Q8×C137—22no—
73C2×Q8×C137—22no—
74C2×Q8×C137D414yes—
75C2×Q8×C137C2312yes—
76C2×Q8×C137C2312yes—
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137C2311yescharacteristic
80Q16×C137—28no—
81C2×C8×C137—22no—
82C2×C8×C137C2312yes—
83C22×D137—21no—
84C22×D137C2×D412yes—
85Dic274—28no—
86Dic274—22no—
87Dic274—21no—
88Dic274C2×D412yes—
89D548—28no—
90D548—22no—
91D548—21no—
92D548C2×D412yes—
93C22⋊D137—24no—
94C22⋊D137—24no—
95C2×Dic137—21no—
96C2×Dic137—22no—
97C2×Dic137C2×D412yes—
98C4×D137—28no—
99C4×D137—28no—
100C4×D137—24no—
101C4×D137—22no—
102C4×D137—22no—
103#11764040—22no—
104C23×C137C2×D411yescharacteristic
105Q8×C137—24no—
106Q8×C137—22no—
107Q8×C137—22no—
108Q8×C137—22no—
109Q8×C137—42no—
110Q8×C137C2×D414yes—
111C2×C4×C137—24no—
112C2×C4×C137—22no—
113C2×C4×C137—22no—
114C2×C4×C137—42no—
115C2×C4×C137C2×D411yescharacteristic
116C2×C4×C137C2×D412yes—
117C2×C4×C137C2×D412yes—
118C2×C4×C137C2411yescharacteristic
119C8×C137—22no—
120C22×C137—22no—
121C22×C137#12012yes—
122C22×C137C22×D411yescharacteristic
123C2×D137—22no—
124C2×D137—22no—
125C2×D137—24no—
126C2×D137—42no—
127C4×C137—24no—
128C4×C137—42no—
129C4×C137#12012yes—
130C4×C137C22×D411yescharacteristic
131C4×C137C22×D411yescharacteristic
132Dic137—22no—
133Dic137—22no—
134Dic137—42no—
135C2×C137—22no—
136C2×C137C2×#12011yescharacteristic
137C2×C137#57812yes—
138D137—42no—
139D137—42no—
140C137#301111yescharacteristic, complemented, Sylow
141#3011—1371nomaximal, Sylow
142C2×2−1+4—1371no—
143C2×#137—1371no—
144C22×Q16D13711yescharacteristic, complemented
145#536—1371no—
146#452—1374no—
147#451—1374no—
148#419—1371no—
149#416—1371no—
150#408—1371no—
1512−1+4—2744no—
1522−1+4—2744no—
153C2×#41—2742no—
154C2×#41—2741no—
155C2×#41—1372no—
156C2×#41—1371no—
157C2×#41—1371no—
158C22×Q8—2741no—
159C22×Q8—1372no—
160C22×Q8C2×D13711yescharacteristic
161C22×Q8C2×D13711yescharacteristic
162#137—2744no—
163C2×Q16—24no—
164C2×Q16—1372no—
165C2×Q16C2×D13714yes—
166C2×SD16—1372no—
167C2×M16—1371no—
168C22×C8C2×D13711yescharacteristic
169C4⋊Q8—2741no—
170#124—2741no—
171#123—2742no—
172C22⋊Q8—1374no—
173#117—1371no—
174#103—1374no—
175#103—1372no—
176#102—1372no—
177C22⋊C8—1374no—
178#41—27416no—
179#41—2748no—
180#41—2744no—
181#41—2744no—
182#41—2742no—
183#41—2742no—
184#41—2742no—
185#41—1374no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—2742no—
192C2×Q8—2748no—
193C2×Q8—2744no—
194C2×Q8—2744no—
195C2×Q8—1371no—
196C2×Q8—1371no—
197C2×Q8C22⋊D13714yes—
198C2×Q8C22×D13712yes—
199C2×Q8C22×D13712yes—
200C2×D4—2742no—
201C2×D4—2741no—
202C2×D4—1372no—
203C2×D4—1371no—
204C2×D4—1371no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2742no—
208C22×C4—2742no—
209C22×C4—2741no—
210C22×C4—1372no—
211C22×C4C22×D13711yescharacteristic
212Q16—28no—
213Q16—2744no—
214SD16—2744no—
215M16—2742no—
216C2×C8—22no—
217C2×C8—1372no—
218C2×C8C22×D13712yes—
219C4⋊C4—2744no—
220C4⋊C4—1372no—
221C22⋊C4—2744no—
222C22⋊C4—2741no—
223C42—2741no—
224C23—2741no—
225C23—1372no—
226C23D4×D13711yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—2748no—
233Q8—2742no—
234Q8—2741no—
235Q8—1372no—
236Q8C2×C22⋊D13714yes—
237D4—2748no—
238D4—2744no—
239D4—2744no—
240D4—2742no—
241D4—2741no—
242D4—1372no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2748no—
248C2×C4—2748no—
249C2×C4—2741no—
250C2×C4—2744no—
251C2×C4—2742no—
252C2×C4—2742no—
253C2×C4—2742no—
254C2×C4—1372no—
255C2×C4D4×D13711yescharacteristic
256C2×C4D4×D13712yes—
257C2×C4C2×C22⋊D13712yes—
258C2×C4C23×D13711yescharacteristic
259C8—22no—
260C8—2742no—
261C22—22no—
262C22—2744no—
263C22—2742no—
264C22—2742no—
265C22—5482no—
266C22#1798405612yes—
267C22C2×D4×D13711yescharacteristic
268C4—24no—
269C4—42no—
270C4—2742no—
271C4—2742no—
272C4—5482no—
273C4#1798405612yes—
274C4C2×D4×D13711yescharacteristic
275C4C22×C22⋊D13711yescharacteristic
276C2—22no—
277C2—5482no—
278C2—5482no—
279C2C2×#1798405611yescharacteristic
280C2#2257821612yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×Q16×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984136C2
hypercenterC22×Q16D137
hypocenterC137#3011
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#17984119C2semidirect
3. #22577977C2semidirect
4. #22577564C2semidirect
5. #22577557C2semidirect
6. #22577412C2semidirect
7. #22577400C2semidirect
8. #22577369C2semidirect
9. C22×Q16×C137C2semidirect
140. C137#3011semidirect
144. C22×Q16D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841362
C2×#179841192
#225779772
#225775642
#225775572
#225774122
#225774002
#225773692
C22×Q16×C1372
#3011137