Finite Group Databasebeta

← C2×#22577791 #31791222

Subgroups — C2×#22577791 #31791222

Order 17536.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#14555499C211yesmaximal, characteristic, complemented
2C22×#14555498C211yesmaximal, characteristic, complemented
3C22×#14555495C211yesmaximal, characteristic, complemented
4C2×#17984084C211yesmaximal, characteristic, complemented
5C2×#17984075C211yesmaximal, characteristic, complemented
6#22577791C218yesmaximal, direct product factor
7C2×#17983998C211yesmaximal, characteristic, complemented
8C22×D8×C137C211yesmaximal, characteristic, complemented
9C22×Dic274C2211yescharacteristic
10C22×D548C2211yescharacteristic
11C22×C22⋊D137—22no—
12C2×#14555499—24no—
13C2×#14555499C2214yescomplemented
14C2×#14555498—24no—
15C2×#14555498C2214yescomplemented
16C2×#14555495—24no—
17C2×#14555495C2214yescomplemented
18C22×#11764040C2211yescharacteristic
19#17984084—22no—
20#17984084C2214yescomplemented
21#17984075—22no—
22#17984075C2214yescomplemented
23C2×C22⋊Dic137—22no—
24C2×C4×Dic137C2211yescharacteristic
25#17983998C2214yescomplemented
26C22×D4×C137C2211yescharacteristic
27C22×D4×C137C2211yescharacteristic
28C2×D8×C137—24no—
29C2×D8×C137C2214yescomplemented
30C22×C8×C137C2211yescharacteristic
31C23×D137—21no—
32C2×Dic274—21no—
33C2×Dic274—22no—
34C2×Dic274—22no—
35C2×Dic274C2312yes—
36C2×D548—22no—
37C2×D548—22no—
38C2×D548—21no—
39C2×D548C2312yes—
40C2×C22⋊D137—28no—
41C2×C22⋊D137—48no—
42C22×Dic137—21no—
43C22×Dic137D412yes—
44#14555499—28no—
45#14555498—28no—
46#14555495—28no—
47C2×#11764040D414yescomplemented
48C2×#11764040C2312yes—
49C22⋊Dic137—28no—
50C4×Dic137D412yes—
51C4×Dic137C2312yes—
52C24×C137—21no—
53C24×C137—21no—
54C2×D4×C137—22no—
55C2×D4×C137—22no—
56C2×D4×C137—22no—
57C2×D4×C137—22no—
58C2×D4×C137—21no—
59C2×D4×C137—21no—
60C2×D4×C137C2312yes—
61C2×D4×C137C2312yes—
62C22×C4×C137C2311yescharacteristic
63D8×C137—28no—
64C2×C8×C137D414yescomplemented
65C2×C8×C137C2312yes—
66C22×D137—22no—
67C22×D137—42no—
68C22×D137—44no—
69Dic274—22no—
70Dic274—22no—
71Dic274—42no—
72D548—22no—
73D548—22no—
74D548—42no—
75C22⋊D137—416no—
76C2×Dic137—24no—
77C2×Dic137—22no—
78C2×Dic137—42no—
79C2×Dic137C2×D414yes—
80#11764040C2×D414yescomplemented
81C23×C137—22no—
82C23×C137—22no—
83C23×C137—42no—
84C23×C137—42no—
85C23×C137—44no—
86C23×C137—44no—
87C23×C137C2×D411yescharacteristic
88D4×C137—22no—
89D4×C137—22no—
90D4×C137—22no—
91D4×C137—22no—
92D4×C137—42no—
93D4×C137—42no—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D412yes—
96C2×C4×C137C2×D412yes—
97C2×C4×C137C2411yescharacteristic
98C8×C137C2×D414yescomplemented
99C22×C137—42no—
100C22×C137—42no—
101C22×C137—44no—
102C22×C137—44no—
103C22×C137—84no—
104C22×C137—84no—
105C22×C137#12712yes—
106C22×C137AGL(1,ℤ/8)14yes—
107C22×C137C22×D411yescharacteristic
108C2×D137—42no—
109C2×D137—44no—
110C2×D137—84no—
111C4×C137#12712yes—
112C4×C137C22×D411yescharacteristic
113C4×C137C22×D411yescharacteristic
114Dic137—24no—
115Dic137—42no—
116C2×C137—82no—
117C2×C137—82no—
118C2×C137#49614yescomplemented
119C2×C137C2×#12711yescharacteristic
120C2×C137C2×AGL(1,ℤ/8)12yes—
121D137—82no—
122C137C2×#49611yescharacteristic, complemented, Sylow
123C2×#496—1371nomaximal, Sylow
124C22×SD16—1371no—
125C22×SD16—1371no—
126C22×D8—1371no—
127C22×D8D13711yescharacteristic, complemented
128C2×#127—1371no—
129C2×#124—1371no—
130#496—1378no—
131C2×#97—1371no—
132C22×Q8—1371no—
133C22×D4—2742no—
134C22×D4—1371no—
135C22×D4C2×D13711yescharacteristic
136C22×D4C2×D13711yescharacteristic
137C2×SD16—2744no—
138C2×SD16—2744no—
139C2×SD16—1374no—
140C2×SD16—1374no—
141C2×D8—24no—
142C2×D8—2744no—
143C2×D8—1374no—
144C2×D8C2×D13714yescomplemented
145C22×C8—1371no—
146C22×C8C2×D13711yescharacteristic
147#127—2742no—
148#127—1374no—
149#124—2742no—
150#124—1374no—
151C2×C22⋊C4—2742no—
152C2×C42—1371no—
153#97—1374no—
154C24—21no—
155C24—21no—
156C24—2741no—
157C2×Q8—2741no—
158C2×Q8—2742no—
159C2×Q8—2742no—
160C2×Q8—1372no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—2748no—
168C2×D4—2742no—
169C2×D4—2741no—
170C2×D4—2742no—
171C2×D4—5488no—
172C2×D4—1372no—
173C2×D4C22×D13712yes—
174C2×D4C22×D13712yes—
175C22×C4—2741no—
176C22×C4—1372no—
177C22×C4C22×D13711yescharacteristic
178SD16—2748no—
179SD16—2748no—
180D8—28no—
181D8—2748no—
182C2×C8—1374no—
183C2×C8—1372no—
184C2×C8C22⋊D13714yescomplemented
185C2×C8C22×D13712yes—
186C22⋊C4—2748no—
187C42—1372no—
188C42—1372no—
189C23—22no—
190C23—22no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—44no—
195C23—2742no—
196C23—5482no—
197C23—5484no—
198C23D4×D13711yescharacteristic
199Q8—2742no—
200Q8—2742no—
201Q8—5482no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—2742no—
209D4—2742no—
210D4—54816no—
211D4—5482no—
212C2×C4—2744no—
213C2×C4—2742no—
214C2×C4—5482no—
215C2×C4—1374no—
216C2×C4D4×D13711yescharacteristic
217C2×C4D4×D13712yes—
218C2×C4C2×C22⋊D13712yes—
219C2×C4C23×D13711yescharacteristic
220C8—1374no—
221C8C2×C22⋊D13714yescomplemented
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—84no—
227C22—84no—
228C22—5484no—
229C22—5482no—
230C22—10964no—
231C22#1798408412yes—
232C22#1798411214yes—
233C22C2×D4×D13711yescharacteristic
234C4—2744no—
235C4—5482no—
236C4#1798408412yes—
237C4C2×D4×D13711yescharacteristic
238C4C22×C22⋊D13711yescharacteristic
239C2—82no—
240C2—82no—
241C2—10962no—
242C2#2257779114yesdirect product factor
243C2C2×#1798408411yescharacteristic
244C2C2×#1798411212yes—
subgroupG/H
centerC23D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×D8×C137C2
socleC23×C137C2×D4
hypercenterC22×D8D137
hypocenterC137C2×#496
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14555499C2semidirect
2. C22×#14555498C2semidirect
3. C22×#14555495C2semidirect
4. C2×#17984084C2semidirect
5. C2×#17984075C2semidirect
6. #22577791C2direct (N × H)
7. C2×#17983998C2semidirect
8. C22×D8×C137C2semidirect
13. C2×#14555499C22semidirect
15. C2×#14555498C22semidirect
17. C2×#14555495C22semidirect
20. #17984084C22semidirect
22. #17984075C22semidirect
25. #17983998C22semidirect
29. C2×D8×C137C22semidirect
47. C2×#11764040D4semidirect
64. C2×C8×C137D4semidirect
80. #11764040C2×D4semidirect
98. C8×C137C2×D4semidirect
118. C2×C137#496semidirect
122. C137C2×#496semidirect
127. C22×D8D137semidirect
144. C2×D8C2×D137semidirect
184. C2×C8C22⋊D137semidirect
221. C8C2×C22⋊D137semidirect
242. C2#22577791direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145554992
C22×#145554982
C22×#145554952
C2×#179840842
C2×#179840752
#225777912
C2×#179839982
C22×D8×C1372
C2×#496137