Finite Group Databasebeta

← C2×#22577631 #31790348

Subgroups — C2×#22577631 #31790348

Order 17536.

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

normal subgroups only

This group has 564 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×#14555498C211yesmaximal, characteristic, complemented
2C22×C8×D137C211yesmaximal, characteristic, complemented
3C2×#17984068C211yesmaximal, characteristic, complemented
4C2×#17984059C211yesmaximal, characteristic
5#22577631C218yesmaximal, direct product factor
6C2×#17984019C211yesmaximal, characteristic, complemented
7C2×#17984013C211yesmaximal, characteristic, complemented
8C2×#102×C137C211yesmaximal, characteristic, complemented
9C22×Dic274C2211yescharacteristic
10C22×C22⋊D137—21no—
11C22×C4×D137C2211yescharacteristic
12C2×#14555498—24no—
13C2×#14555498C2214yescomplemented
14C22×#11764040C2211yescharacteristic
15C2×C8×D137—24no—
16C2×C8×D137C2214yescomplemented
17#17984068—22no—
18#17984068C2214yescomplemented
19#17984059—22no—
20#17984059C2214yes—
21C2×C4⋊Dic137—21no—
22C2×C4⋊Dic137C2211yescharacteristic
23C2×C22⋊Dic137—21no—
24C2×#14555487—21no—
25#17984019C2214yescomplemented
26#17984013C2214yescomplemented
27C22×D4×C137C2211yescharacteristic
28C22×C8×C137C2211yescharacteristic
29C2×C4⋊C4×C137C2211yescharacteristic
30#102×C137C2214yescomplemented
31C23×D137D411yescharacteristic
32C2×Dic274—21no—
33C2×Dic274—22no—
34C2×Dic274—22no—
35C2×Dic274C2312yes—
36C2×C22⋊D137—24no—
37C2×C22⋊D137—44no—
38C22×Dic137—21no—
39C22×Dic137—21no—
40C22×Dic137D411yescharacteristic
41C2×C4×D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137D412yes—
44C2×C4×D137C2312yes—
45#14555498—28no—
46C2×#11764040D414yescomplemented
47C2×#11764040C2312yes—
48C8×D137—28no—
49C4⋊Dic137—24no—
50C4⋊Dic137—21no—
51C4⋊Dic137C2312yes—
52C22⋊Dic137—24no—
53#14555487—24no—
54C24×C137—21no—
55C2×D4×C137—22no—
56C2×D4×C137—21no—
57C2×D4×C137—22no—
58C2×D4×C137C2312yes—
59C22×C4×C137—21no—
60C22×C4×C137C2311yescharacteristic
61C2×C8×C137—22no—
62C2×C8×C137C2312yes—
63C4⋊C4×C137—21no—
64C4⋊C4×C137C2312yes—
65C22×D137—22no—
66C22×D137—22no—
67C22×D137SD1614yescomplemented
68C22×D137C2×D412yes—
69Dic274—22no—
70Dic274—22no—
71Dic274—42no—
72C22⋊D137—48no—
73C2×Dic137—22no—
74C2×Dic137—22no—
75C2×Dic137—22no—
76C2×Dic137—42no—
77C2×Dic137—42no—
78C2×Dic137C2×D412yes—
79C4×D137—24no—
80C4×D137—22no—
81C4×D137—22no—
82#11764040C2×D414yescomplemented
83C23×C137—22no—
84C23×C137—42no—
85C23×C137—42no—
86C23×C137—42no—
87C23×C137C2×D411yescharacteristic
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×C137—22no—
98C22×C137—42no—
99C22×C137—42no—
100C22×C137—42no—
101C22×C137—84no—
102C22×C137#12112yes—
103C22×C137C2×SD1612yes—
104C22×C137#13512yes—
105C22×C137C22×D411yescharacteristic
106C2×D137—22no—
107C2×D137—22no—
108C2×D137—28no—
109C2×D137C2×SD1614yescomplemented
110C4×C137—42no—
111C4×C137#12112yes—
112C4×C137C22×D411yescharacteristic
113C4×C137C2×#4111yescharacteristic
114Dic137—22no—
115Dic137—42no—
116Dic137—42no—
117C2×C137—82no—
118C2×C137#46514yescomplemented
119C2×C137C2×#12111yescharacteristic
120C2×C137C22×SD1611yescharacteristic
121C2×C137C2×#13511yescharacteristic
122D137—24no—
123C137C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1371nomaximal, Sylow
125C22×SD16—1371no—
126C23×C8—1371no—
127C2×C22⋊Q8—1371no—
128C2×#121—1371no—
129#465—1378no—
130C2×#106—1371no—
131C2×#103—1371no—
132C2×#102D13711yescharacteristic, complemented
133C22×Q8—1371no—
134C22×D4—2741no—
135C22×D4C2×D13711yescharacteristic
136C23×C4—1371no—
137C2×SD16—2744no—
138C2×SD16—1374no—
139C22×C8—2744no—
140C22×C8—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⋊C4C2×D13711yescharacteristic
150C2×C22⋊C4—2741no—
151C2×C22⋊C4—2741no—
152#106—1374no—
153#103—1374no—
154#102C2×D13714yescomplemented
155C24—21no—
156C24—1371no—
157C2×Q8—2741no—
158C2×Q8—2742no—
159C2×Q8—2742no—
160C2×Q8—1372no—
161C2×D4—22no—
162C2×D4—21no—
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—2748no—
177C2×C8—22no—
178C2×C8—2748no—
179C2×C8—1374no—
180C2×C8—1372no—
181C2×C8C22×D13712yes—
182C4⋊C4—21no—
183C4⋊C4—2744no—
184C4⋊C4—2741no—
185C4⋊C4—1372no—
186C4⋊C4C22×D13712yes—
187C22⋊C4—2744no—
188C22⋊C4—2744no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—2742no—
194C23—2742no—
195C23—1374no—
196C23—1372no—
197C23D4×D13711yescharacteristic
198Q8—2742no—
199Q8—2742no—
200Q8—5482no—
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×C4D4×D13712yes—
218C2×C4#1455551012yes—
219C2×C4C23×D13711yescharacteristic
220C8—22no—
221C8—1374no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—2742no—
227C22—2748no—
228C22—2742no—
229C22—1374no—
230C22#1798406112yes—
231C22SD16×D13712yes—
232C22#1798410712yes—
233C22C2×D4×D13711yescharacteristic
234C4—42no—
235C4—2742no—
236C4—5482no—
237C4—5482no—
238C4#1798406112yes—
239C4C2×D4×D13711yescharacteristic
240C4C2×#1455551011yescharacteristic
241C2—82no—
242C2—2744no—
243C2#2257763114yesdirect product factor
244C2C2×#1798406111yescharacteristic
245C2C2×SD16×D13711yescharacteristic
246C2C2×#1798410711yescharacteristic
subgroupG/H
centerC23D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC23×C137C2×D4
prime radicalC2×#17984059C2
hypercenterC2×#102D137
hypocenterC137C2×#465
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14555498C2semidirect
2. C22×C8×D137C2semidirect
3. C2×#17984068C2semidirect
5. #22577631C2direct (N × H)
6. C2×#17984019C2semidirect
7. C2×#17984013C2semidirect
8. C2×#102×C137C2semidirect
13. C2×#14555498C22semidirect
16. C2×C8×D137C22semidirect
18. #17984068C22semidirect
25. #17984019C22semidirect
26. #17984013C22semidirect
30. #102×C137C22semidirect
46. C2×#11764040D4semidirect
67. C22×D137SD16semidirect
82. #11764040C2×D4semidirect
109. C2×D137C2×SD16semidirect
118. C2×C137#465semidirect
123. C137C2×#465semidirect
132. C2×#102D137semidirect
154. #102C2×D137semidirect
243. C2#22577631direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145554982
C22×C8×D1372
C2×#179840682
C2×#179840592
#225776312
C2×#179840192
C2×#179840132
C2×#102×C1372
C2×#465137