Finite Group Databasebeta

← #31790267

Subgroups — #31790267

Order 17536.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#17984113C211yesmaximal, characteristic, complemented
2C2×#17984084C211yesmaximal, characteristic, complemented
3#41×Dic137C211yesmaximal, characteristic, complemented
4#22577601C214yesmaximal, complemented
5#22577596C214yesmaximal, complemented
6#22577426C211yesmaximal, characteristic, complemented
7#22577409C211yesmaximal, characteristic, complemented
8#22577408C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
10C22×D548C2211yescharacteristic
11C22×C22⋊D137—22no—
12#17984113—24no—
13C2×#14555497C2212yescomplemented
14C2×#14555495C2212yescomplemented
15C2×#14555494C2211yescharacteristic
16#17984084—21no—
17#17984084—21no—
18#17984084C2214yes—
19Q8×Dic137C2212yescomplemented
20D4×Dic137—22no—
21D4×Dic137C2212yescomplemented
22#17984047—21no—
23#17984047C2211yescharacteristic
24C2×C4×Dic137—21no—
25C2×C4×Dic137C2211yescharacteristic
26#17984017C2214yescomplemented
27#17984010C2214yescomplemented
28#17984009C2214yescomplemented
29C2×#41×C137C2211yescharacteristic
30C22×D4×C137C2211yescharacteristic
31AGL(1,ℤ/8)×C137—24no—
32C2×SD16×C137C2212yescomplemented
33C2×D8×C137C2212yescomplemented
34C2×M16×C137C2211yescharacteristic
35C23×D137—21no—
36C2×D548—22no—
37C2×D548—21no—
38C2×D548—22no—
39C2×D548C2312yes—
40C2×C22⋊D137—24no—
41C2×C22⋊D137—24no—
42C2×C22⋊D137—48no—
43C22×Dic137—22no—
44C22×Dic137D412yes—
45#14555497—24no—
46#14555495—24no—
47#14555494—22no—
48C2×#11764040C2312yes—
49C4⋊Dic137—22no—
50C4⋊Dic137C2312yes—
51C22⋊Dic137—22no—
52C22⋊Dic137—21no—
53C4×Dic137—22no—
54C4×Dic137—21no—
55C4×Dic137D411yescharacteristic
56C4×Dic137D411yescharacteristic
57C4×Dic137C2312yes—
58C24×C137—21no—
59#41×C137—22no—
60#41×C137D414yescomplemented
61C2×Q8×C137C2311yescharacteristic
62C2×D4×C137—21no—
63C2×D4×C137—22no—
64C2×D4×C137—21no—
65C2×D4×C137—22no—
66C2×D4×C137C2312yes—
67C2×D4×C137C2311yescharacteristic
68C22×C4×C137—21no—
69C22×C4×C137C2311yescharacteristic
70SD16×C137—24no—
71D8×C137—24no—
72M16×C137—22no—
73C2×C8×C137C2312yes—
74C22×D137—22no—
75C22×D137—44no—
76C22×D137—42no—
77D548—22no—
78D548—22no—
79D548—42no—
80C22⋊D137—48no—
81C22⋊D137—48no—
82C2×Dic137—24no—
83C2×Dic137—22no—
84C2×Dic137—44no—
85C2×Dic137C2×D412yes—
86C2×Dic137C2×D412yes—
87#11764040—22no—
88C23×C137—21no—
89C23×C137—22no—
90C23×C137—44no—
91C23×C137—42no—
92C23×C137C2×D411yescharacteristic
93Q8×C137—21no—
94Q8×C137C2×D412yescomplemented
95D4×C137—24no—
96D4×C137—22no—
97D4×C137—21no—
98D4×C137—22no—
99D4×C137—42no—
100D4×C137C2×D412yescomplemented
101C2×C4×C137—21no—
102C2×C4×C137—24no—
103C2×C4×C137C2×D411yescharacteristic
104C2×C4×C137C2×D412yes—
105C2×C4×C137#4112yes—
106C2×C4×C137C2411yescharacteristic
107C8×C137—22no—
108C22×C137—22no—
109C22×C137—22no—
110C22×C137—42no—
111C22×C137—42no—
112C22×C137—44no—
113C22×C137—84no—
114C22×C137#12112yes—
115C22×C137C22×D411yescharacteristic
116C2×D137—42no—
117C2×D137—44no—
118C2×D137—84no—
119C4×C137—22no—
120C4×C137#12112yes—
121C4×C137C22×D411yescharacteristic
122C4×C137C2×#4111yescharacteristic
123Dic137—22no—
124Dic137—42no—
125Dic137AGL(1,ℤ/8)14yescomplemented
126C2×C137—22no—
127C2×C137—42no—
128C2×C137—82no—
129C2×C137C2×#12111yescharacteristic
130C2×C137C2×AGL(1,ℤ/8)12yes—
131D137—82no—
132C137#303811yescharacteristic, complemented, Sylow
133#3038—1371nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1371no—
135C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
136C2×#127—1371no—
137C4×#41—1371no—
138#460—1374no—
139#459—1374no—
140#423—1371no—
141#418—1371no—
142#418—1371no—
143C2×#41C2×D13711yescharacteristic
144C22×D4—2742no—
145C22×D4—1371no—
146C22×D4C2×D13711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2744no—
149C2×SD16—1372no—
150C2×SD16C2×D13712yescomplemented
151C2×D8—1372no—
152C2×D8C2×D13712yescomplemented
153C2×M16—1371no—
154C2×M16C2×D13711yescharacteristic
155#127—2741no—
156#127—2741no—
157#127—1374no—
158C4×Q8—1372no—
159C4×D4—2742no—
160C4×D4—1372no—
161#117—2741no—
162#117—1371no—
163C2×C42—2741no—
164C2×C42—1371no—
165C4⋊C8—1374no—
166#102—1374no—
167#102—1374no—
168C24—21no—
169C24—2741no—
170#41—22no—
171#41C22⋊D13714yescomplemented
172C2×Q8C22×D13711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2744no—
178C2×D4—2744no—
179C2×D4—2741no—
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—5488no—
183C2×D4—1372no—
184C2×D4C22×D13712yes—
185C2×D4C22×D13711yescharacteristic
186C22×C4—21no—
187C22×C4—2742no—
188C22×C4—1372no—
189C22×C4C22×D13711yescharacteristic
190SD16—24no—
191SD16—2744no—
192D8—24no—
193D8—2744no—
194M16—22no—
195M16—2742no—
196C2×C8—1372no—
197C2×C8C22×D13712yes—
198C4⋊C4—2742no—
199C4⋊C4—1372no—
200C22⋊C4—2741no—
201C22⋊C4—2742no—
202C42—2742no—
203C42—2741no—
204C42—1371no—
205C42—1371no—
206C42—1372no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—2742no—
212C23—5484no—
213C23—5482no—
214C23D4×D13711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D13712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—2742no—
223D4—2742no—
224D4—5488no—
225D4—5488no—
226D4—5482no—
227D4C2×C22⋊D13712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2744no—
231C2×C4—2742no—
232C2×C4—5484no—
233C2×C4—1372no—
234C2×C4—1372no—
235C2×C4D4×D13711yescharacteristic
236C2×C4C2×C22⋊D13712yes—
237C2×C4#1455550912yes—
238C2×C4C23×D13711yescharacteristic
239C8—22no—
240C8—2742no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—5484no—
248C22—5482no—
249C22—10964no—
250C22#1798406012yes—
251C22C2×D4×D13711yescharacteristic
252C4—22no—
253C4—2742no—
254C4—5482no—
255C4—1374no—
256C4#1798406012yes—
257C4C22×C22⋊D13711yescharacteristic
258C4C2×#1455550911yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—10962no—
263C2C2×#1798406011yescharacteristic
264C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×AGL(1,ℤ/8)×C137C2
socleC22×C137C22×D4
hypercenterC2×AGL(1,ℤ/8)D137
hypocenterC137#3038
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984113C2semidirect
2. C2×#17984084C2semidirect
3. #41×Dic137C2semidirect
4. #22577601C2semidirect
5. #22577596C2semidirect
6. #22577426C2semidirect
7. #22577409C2semidirect
8. #22577408C2semidirect
9. C2×AGL(1,ℤ/8)×C137C2semidirect
13. C2×#14555497C22semidirect
14. C2×#14555495C22semidirect
19. Q8×Dic137C22semidirect
21. D4×Dic137C22semidirect
26. #17984017C22semidirect
27. #17984010C22semidirect
28. #17984009C22semidirect
32. C2×SD16×C137C22semidirect
33. C2×D8×C137C22semidirect
60. #41×C137D4semidirect
94. Q8×C137C2×D4semidirect
100. D4×C137C2×D4semidirect
125. Dic137AGL(1,ℤ/8)semidirect
132. C137#3038semidirect
135. C2×AGL(1,ℤ/8)D137semidirect
150. C2×SD16C2×D137semidirect
152. C2×D8C2×D137semidirect
171. #41C22⋊D137semidirect
216. Q8C2×C22⋊D137semidirect
227. D4C2×C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841132
C2×#179840842
#41×Dic1372
#225776012
#225775962
#225774262
#225774092
#225774082
C2×AGL(1,ℤ/8)×C1372
#3038137