Finite Group Databasebeta

← #31790298

Subgroups — #31790298

Order 17536.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984114C211yesmaximal, characteristic
2#22577977C211yesmaximal, characteristic, complemented
3#22577878C211yesmaximal, characteristic, complemented
4#22577620C214yesmaximal, complemented
5#22577606C214yesmaximal, complemented
6#22577429C211yesmaximal, characteristic, complemented
7#22577412C211yesmaximal, characteristic, complemented
8#22577400C211yesmaximal, characteristic, complemented
9C22×SD16×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11#17984114—24no—
12C2×#14555498C2212yes—
13C2×#14555495C2212yes—
14C2×#14555494C2211yescharacteristic
15#17984086C2212yescomplemented
16#17984077C2212yescomplemented
17#17984071—22no—
18#17984064—22no—
19D4×Dic137—22no—
20D4×Dic137C2214yes—
21#17984047C2211yescharacteristic
22#17984047C2211yescharacteristic
23C2×C22⋊Dic137—22no—
24#17984017C2214yescomplemented
25#17984013C2212yescomplemented
26#17984012C2214yescomplemented
27#17984010C2212yescomplemented
28C22×Q8×C137C2211yescharacteristic
29C22×D4×C137C2211yescharacteristic
30C2×SD16×C137—24no—
31C2×SD16×C137C2214yes—
32C22×C8×C137C2211yescharacteristic
33#14555510—22no—
34#14555510—22no—
35C2×Dic274C2311yescharacteristic
36C2×D548C2311yescharacteristic
37C2×C22⋊D137—21no—
38C22×Dic137—24no—
39C2×C4×D137—21no—
40#14555498—24no—
41#14555495—24no—
42#14555494—22no—
43C2×#11764040C2312yes—
44C4⋊Dic137D412yes—
45C4⋊Dic137C2312yes—
46#14555489—24no—
47C22⋊Dic137—24no—
48C22⋊Dic137—21no—
49C22⋊Dic137D412yes—
50#14555487—24no—
51C4×Dic137—21no—
52C4×Dic137C2312yes—
53C24×C137—21no—
54C2×Q8×C137—21no—
55C2×Q8×C137—22no—
56C2×Q8×C137—22no—
57C2×Q8×C137C2312yes—
58C2×D4×C137—21no—
59C2×D4×C137—22no—
60C2×D4×C137D414yes—
61C2×D4×C137C2312yes—
62C22×C4×C137—21no—
63C22×C4×C137C2311yescharacteristic
64SD16×C137—28no—
65C2×C8×C137—22no—
66C2×C8×C137C2312yes—
67C22×D137—21no—
68Dic274—21no—
69Dic274—21no—
70D548—21no—
71D548—21no—
72C22⋊D137—42no—
73C2×Dic137—22no—
74C2×Dic137—21no—
75C2×Dic137—84no—
76C2×Dic137C2×D414yes—
77C4×D137—42no—
78#11764040—22no—
79C23×C137—22no—
80C23×C137—24no—
81C23×C137—82no—
82C23×C137C2×D411yescharacteristic
83Q8×C137—22no—
84Q8×C137—22no—
85Q8×C137—42no—
86D4×C137—24no—
87D4×C137—22no—
88D4×C137C2×D414yes—
89C2×C4×C137—22no—
90C2×C4×C137—42no—
91C2×C4×C137C2×D411yescharacteristic
92C2×C4×C137C2×D412yes—
93C2×C4×C137#4112yes—
94C2×C4×C137C2411yescharacteristic
95C8×C137—22no—
96C22×C137—22no—
97C22×C137—24no—
98C22×C137—44no—
99C22×C137—84no—
100C22×C137#12112yes—
101C22×C137C22×D411yescharacteristic
102C2×D137—41no—
103C2×D137—42no—
104C4×C137—42no—
105C4×C137#12112yes—
106C4×C137C22×D411yescharacteristic
107C4×C137C2×#4111yescharacteristic
108Dic137—24no—
109Dic137—41no—
110Dic137—42no—
111C2×C137—22no—
112C2×C137—44no—
113C2×C137C2×#12111yescharacteristic
114C2×C137#57712yes—
115D137—81no—
116C137#304211yescharacteristic, complemented, Sylow
117#3042—1371nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1371no—
119C22×SD16D13711yescharacteristic, complemented
120#536—1371no—
121#518—1371no—
122#463—1374no—
123#461—1374no—
124#424—1371no—
125#419—1371no—
126#416—1371no—
127C2×#41—1371no—
128C22×Q8C2×D13711yescharacteristic
129C22×D4C2×D13711yescharacteristic
130AGL(1,ℤ/8)—2744no—
131C2×SD16—24no—
132C2×SD16—1372no—
133C2×SD16C2×D13714yes—
134C2×D8—1372no—
135C2×M16—1371no—
136C22×C8C2×D13711yescharacteristic
137C4⋊Q8—1372no—
138#124—1372no—
139#123—2742no—
140C22⋊Q8—2742no—
141C4×D4—2742no—
142C4×D4—1374no—
143#117—1371no—
144#117—1371no—
145C2×C22⋊C4—2742no—
146C4⋊C8—1374no—
147#103—1374no—
148#103—1372no—
149#102—1372no—
150C24—21no—
151#41—2742no—
152#41—2742no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1371no—
157C2×Q8C22×D13712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—2741no—
161C2×D4—1371no—
162C2×D4C22⋊D13714yes—
163C2×D4C22×D13712yes—
164C22×C4—21no—
165C22×C4—2741no—
166C22×C4—2744no—
167C22×C4C22×D13711yescharacteristic
168SD16—28no—
169SD16—2744no—
170D8—2744no—
171M16—2742no—
172C2×C8—22no—
173C2×C8—1372no—
174C2×C8C22×D13712yes—
175C4⋊C4—2744no—
176C4⋊C4—1372no—
177C4⋊C4—1372no—
178C22⋊C4—2744no—
179C22⋊C4—2741no—
180C22⋊C4—2744no—
181C22⋊C4—1372no—
182C42—2741no—
183C42—1372no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2741no—
188C23D4×D13711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2741no—
193Q8—2741no—
194D4—24no—
195D4—22no—
196D4—2741no—
197D4—2741no—
198D4—5482no—
199D4C2×C22⋊D13714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—2742no—
203C2×C4—2741no—
204C2×C4—5482no—
205C2×C4—10964no—
206C2×C4—1374no—
207C2×C4D4×D13711yescharacteristic
208C2×C4C2×C22⋊D13712yes—
209C2×C4#1455550912yes—
210C2×C4C23×D13711yescharacteristic
211C8—22no—
212C8—2742no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—5482no—
218C22—5481no—
219C22#1798406012yes—
220C22C2×D4×D13711yescharacteristic
221C4—42no—
222C4—2744no—
223C4—5482no—
224C4—5481no—
225C4#1798406012yes—
226C4C22×C22⋊D13711yescharacteristic
227C4C2×#1455550911yescharacteristic
228C2—22no—
229C2—44no—
230C2—10961no—
231C2C2×#1798406011yescharacteristic
232C2#2257820912yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×SD16×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984114C2
hypercenterC22×SD16D137
hypocenterC137#3042
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #22577977C2semidirect
3. #22577878C2semidirect
4. #22577620C2semidirect
5. #22577606C2semidirect
6. #22577429C2semidirect
7. #22577412C2semidirect
8. #22577400C2semidirect
9. C22×SD16×C137C2semidirect
15. #17984086C22semidirect
16. #17984077C22semidirect
24. #17984017C22semidirect
25. #17984013C22semidirect
26. #17984012C22semidirect
27. #17984010C22semidirect
116. C137#3042semidirect
119. C22×SD16D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
#225779772
#225778782
#225776202
#225776062
#225774292
#225774122
#225774002
C22×SD16×C1372
#3042137