Finite Group Databasebeta

← C2×#22577646 #31790374

Subgroups — C2×#22577646 #31790374

Order 17536.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14555493C211yesmaximal, characteristic, complemented
2C2×#17984064C211yesmaximal, characteristic, complemented
3C2×#17984059C211yesmaximal, characteristic
4#22577646C218yesmaximal, direct product factor
5C2×#17984022C211yesmaximal, characteristic, complemented
6C2×#17984012C211yesmaximal, characteristic, complemented
7C2×#17984009C211yesmaximal, characteristic, complemented
8C22×SD16×C137C211yesmaximal, characteristic, complemented
9C22×C22⋊D137—21no—
10C22×C4×D137C2211yescharacteristic
11C2×#14555493—24no—
12C2×#14555493C2214yescomplemented
13C22×#11764040C2211yescharacteristic
14#17984064—22no—
15#17984064C2214yescomplemented
16#17984059—22no—
17#17984059C2214yes—
18C2×C4⋊Dic137C2211yescharacteristic
19C2×C4⋊Dic137C2211yescharacteristic
20C2×#14555489—21no—
21C2×C22⋊Dic137—21no—
22C2×#14555487—21no—
23#17984022C2214yescomplemented
24#17984012C2214yescomplemented
25#17984009C2214yescomplemented
26C22×Q8×C137C2211yescharacteristic
27C22×D4×C137C2211yescharacteristic
28C2×SD16×C137—24no—
29C2×SD16×C137C2214yescomplemented
30C22×C8×C137C2211yescharacteristic
31C23×D137D411yescharacteristic
32C2×C22⋊D137—24no—
33C2×C22⋊D137—44no—
34C22×Dic137—21no—
35C22×Dic137—21no—
36C22×Dic137D411yescharacteristic
37C2×C4×D137—22no—
38C2×C4×D137—22no—
39C2×C4×D137D412yes—
40C2×C4×D137C2312yes—
41#14555493—28no—
42C2×#11764040—22no—
43C2×#11764040C2312yes—
44C4⋊Dic137—21no—
45C4⋊Dic137—21no—
46C4⋊Dic137C2312yes—
47C4⋊Dic137C2312yes—
48#14555489—24no—
49C22⋊Dic137—24no—
50#14555487—24no—
51C24×C137—21no—
52C2×Q8×C137—21no—
53C2×Q8×C137—22no—
54C2×Q8×C137—22no—
55C2×Q8×C137C2312yes—
56C2×D4×C137—22no—
57C2×D4×C137—21no—
58C2×D4×C137—22no—
59C2×D4×C137C2312yes—
60C22×C4×C137—21no—
61C22×C4×C137C2311yescharacteristic
62SD16×C137—28no—
63C2×C8×C137D414yescomplemented
64C2×C8×C137C2312yes—
65C22×D137—22no—
66C22×D137—22no—
67C22×D137—22no—
68C22×D137C2×D412yes—
69C22⋊D137—48no—
70C2×Dic137—22no—
71C2×Dic137—22no—
72C2×Dic137—22no—
73C2×Dic137—42no—
74C2×Dic137—42no—
75C2×Dic137C2×D412yes—
76C4×D137—24no—
77C4×D137—22no—
78C4×D137—22no—
79#11764040—22no—
80C23×C137—22no—
81C23×C137—42no—
82C23×C137—42no—
83C23×C137—42no—
84C23×C137C2×D411yescharacteristic
85Q8×C137—22no—
86Q8×C137—22no—
87Q8×C137—42no—
88D4×C137—22no—
89D4×C137—22no—
90D4×C137—42no—
91C2×C4×C137—22no—
92C2×C4×C137—42no—
93C2×C4×C137C2×D411yescharacteristic
94C2×C4×C137C2×D412yes—
95C2×C4×C137#4112yes—
96C2×C4×C137C2411yescharacteristic
97C8×C137C2×D414yescomplemented
98C22×C137—42no—
99C22×C137—42no—
100C22×C137—42no—
101C22×C137—84no—
102C22×C137#12112yes—
103C22×C137AGL(1,ℤ/8)12yes—
104C22×C137#13712yes—
105C22×C137C22×D411yescharacteristic
106C2×D137—22no—
107C2×D137—22no—
108C2×D137—22no—
109C2×D137—44no—
110C4×C137—42no—
111C4×C137#12112yes—
112C4×C137C22×D411yescharacteristic
113C4×C137C2×#4111yescharacteristic
114Dic137—22no—
115Dic137—42no—
116Dic137—42no—
117C2×C137—82no—
118C2×C137#46814yescomplemented
119C2×C137C2×#12111yescharacteristic
120C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C137C2×#13711yescharacteristic
122D137—42no—
123C137C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—1371nomaximal, Sylow
125C22×SD16D13711yescharacteristic, complemented
126C22×M16—1371no—
127C2×C22⋊Q8—1371no—
128C2×#121—1371no—
129#468—1378no—
130C2×#107—1371no—
131C2×#103—1371no—
132C2×#102—1371no—
133C22×Q8C2×D13711yescharacteristic
134C22×D4—2741no—
135C22×D4C2×D13711yescharacteristic
136C23×C4—1371no—
137C2×SD16—24no—
138C2×SD16C2×D13714yescomplemented
139C2×M16—2744no—
140C2×M16—1374no—
141C22×C8—1371no—
142C22×C8C2×D13711yescharacteristic
143C22⋊Q8—2742no—
144C22⋊Q8—1374no—
145#121—2742no—
146#121—1374no—
147C2×C4⋊C4—2741no—
148C2×C4⋊C4—1371no—
149C2×C4⋊C4—1371no—
150C2×C22⋊C4—2741no—
151C2×C22⋊C4—2741no—
152#107—1374no—
153#103—1374no—
154#102—1374no—
155C24—21no—
156C24—1371no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D13712yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—2744no—
165C2×D4—5484no—
166C2×D4C22×D13712yes—
167C22×C4—21no—
168C22×C4—2742no—
169C22×C4—2742no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4—1371no—
173C22×C4—1372no—
174C22×C4—1372no—
175C22×C4C22×D13711yescharacteristic
176SD16—28no—
177M16—2748no—
178C2×C8—2742no—
179C2×C8—1372no—
180C2×C8C22⋊D13714yescomplemented
181C2×C8C22×D13712yes—
182C4⋊C4—2744no—
183C4⋊C4—2741no—
184C4⋊C4—2741no—
185C4⋊C4—1372no—
186C4⋊C4—1372no—
187C22⋊C4—2744no—
188C22⋊C4—2744no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—2742no—
194C23—2742no—
195C23—2742no—
196C23—1372no—
197C23D4×D13711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—5488no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—2742no—
208C2×C4—2744no—
209C2×C4—2742no—
210C2×C4—2742no—
211C2×C4—2742no—
212C2×C4—2742no—
213C2×C4—5482no—
214C2×C4—5482no—
215C2×C4—1372no—
216C2×C4D4×D13711yescharacteristic
217C2×C4C2×C22⋊D13712yes—
218C2×C4#1455550912yes—
219C2×C4C23×D13711yescharacteristic
220C8—2742no—
221C8C2×C22⋊D13714yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—2742no—
227C22—2742no—
228C22—2742no—
229C22—5484no—
230C22#1798405912yes—
231C22#1798411512yes—
232C22#1798411712yes—
233C22C2×D4×D13711yescharacteristic
234C4—42no—
235C4—2742no—
236C4—5482no—
237C4—5482no—
238C4#1798405912yes—
239C4C22×C22⋊D13711yescharacteristic
240C4C2×#1455550911yescharacteristic
241C2—82no—
242C2—5482no—
243C2#2257764614yesdirect product factor
244C2C2×#1798405911yescharacteristic
245C2C2×#1798411511yescharacteristic
246C2C2×#1798411711yescharacteristic
subgroupG/H
centerC23D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×SD16×C137C2
socleC23×C137C2×D4
prime radicalC2×#17984059C2
hypercenterC22×SD16D137
hypocenterC137C2×#468
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14555493C2semidirect
2. C2×#17984064C2semidirect
4. #22577646C2direct (N × H)
5. C2×#17984022C2semidirect
6. C2×#17984012C2semidirect
7. C2×#17984009C2semidirect
8. C22×SD16×C137C2semidirect
12. C2×#14555493C22semidirect
15. #17984064C22semidirect
23. #17984022C22semidirect
24. #17984012C22semidirect
25. #17984009C22semidirect
29. C2×SD16×C137C22semidirect
63. C2×C8×C137D4semidirect
97. C8×C137C2×D4semidirect
118. C2×C137#468semidirect
123. C137C2×#468semidirect
125. C22×SD16D137semidirect
138. C2×SD16C2×D137semidirect
180. C2×C8C22⋊D137semidirect
221. C8C2×C22⋊D137semidirect
243. C2#22577646direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145554932
C2×#179840642
C2×#179840592
#225776462
C2×#179840222
C2×#179840122
C2×#179840092
C22×SD16×C1372
C2×#468137