Finite Group Databasebeta

← C2×#22577647 #31790375

Subgroups — C2×#22577647 #31790375

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×#14555498C211yesmaximal, characteristic, complemented
2C22×#14555494C211yesmaximal, characteristic, complemented
3C2×#17984067C211yesmaximal, characteristic, complemented
4#22577647C218yesmaximal, direct product factor
5C2×#17984021C211yesmaximal, characteristic, complemented
6C2×#17984011C211yesmaximal, characteristic, complemented
7C2×#17984009C211yesmaximal, characteristic, complemented
8C2×#121×C137C211yesmaximal, characteristic
9C22×Dic274C2211yescharacteristic
10C2×#14555498—24no—
11C2×#14555498C2214yescomplemented
12C2×#14555494—24no—
13C2×#14555494C2214yescomplemented
14C22×#11764040C2211yescharacteristic
15C22×#11764040C2211yescharacteristic
16#17984067—22no—
17#17984067C2214yescomplemented
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489—21no—
20C2×C22⋊Dic137—21no—
21#17984021C2214yescomplemented
22#17984011C2214yescomplemented
23#17984009C2214yescomplemented
24C22×D4×C137—21no—
25C22×D4×C137C2211yescharacteristic
26C23×C4×C137C2211yescharacteristic
27#121×C137—22no—
28#121×C137C2214yes—
29C2×C4⋊C4×C137C2211yescharacteristic
30C2×C22⋊C4×C137—21no—
31C2×Dic274—21no—
32C2×Dic274—22no—
33C2×Dic274—22no—
34C2×Dic274C2312yes—
35C22×Dic137—21no—
36C22×Dic137—21no—
37#14555498—28no—
38#14555494—28no—
39C2×#11764040—22no—
40C2×#11764040D414yescomplemented
41C2×#11764040C2312yes—
42C2×#11764040C2312yes—
43C4⋊Dic137—21no—
44C4⋊Dic137C2312yes—
45#14555489—24no—
46C22⋊Dic137—24no—
47C24×C137—21no—
48C24×C137D411yescharacteristic
49C2×D4×C137—24no—
50C2×D4×C137—22no—
51C2×D4×C137—21no—
52C2×D4×C137—22no—
53C2×D4×C137—44no—
54C2×D4×C137C2312yes—
55C22×C4×C137—22no—
56C22×C4×C137—22no—
57C22×C4×C137—21no—
58C22×C4×C137D411yescharacteristic
59C22×C4×C137D412yes—
60C22×C4×C137C2312yes—
61C22×C4×C137C2311yescharacteristic
62C4⋊C4×C137—21no—
63C4⋊C4×C137C2312yes—
64C22⋊C4×C137—24no—
65Dic274—22no—
66Dic274—22no—
67Dic274—42no—
68C2×Dic137—22no—
69C2×Dic137—22no—
70C2×Dic137—42no—
71C2×Dic137—42no—
72#11764040—22no—
73#11764040C2×D414yescomplemented
74C23×C137—22no—
75C23×C137—22no—
76C23×C137—22no—
77C23×C137—22no—
78C23×C137—42no—
79C23×C137—42no—
80C23×C137—42no—
81C23×C137C2×D412yes—
82C23×C137C2×D411yescharacteristic
83D4×C137—22no—
84D4×C137—22no—
85D4×C137—48no—
86D4×C137—42no—
87C2×C4×C137—22no—
88C2×C4×C137—24no—
89C2×C4×C137—22no—
90C2×C4×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—42no—
93C2×C4×C137C2×D412yes—
94C2×C4×C137C2×D412yes—
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137#4112yes—
97C2×C4×C137C2411yescharacteristic
98C22×C137—22no—
99C22×C137—22no—
100C22×C137—22no—
101C22×C137—44no—
102C22×C137—42no—
103C22×C137—42no—
104C22×C137—42no—
105C22×C137—84no—
106C22×C137#12112yes—
107C22×C137AGL(1,ℤ/8)12yes—
108C22×C137#13712yes—
109C22×C137C22×D411yescharacteristic
110C4×C137—22no—
111C4×C137—42no—
112C4×C137#12112yes—
113C4×C137C22×D411yescharacteristic
114C4×C137C2×#4111yescharacteristic
115Dic137—42no—
116Dic137—42no—
117C2×C137—42no—
118C2×C137—82no—
119C2×C137#46814yescomplemented
120C2×C137C2×#12111yescharacteristic
121C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C137C2×#13711yescharacteristic
123C137C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—1371nomaximal, Sylow
125C22×SD16—1371no—
126C22×M16—1371no—
127C2×C22⋊Q8—1371no—
128C2×#121D13711yescharacteristic
129#468—1378no—
130C2×#107—1371no—
131C2×#103—1371no—
132C2×#102—1371no—
133C22×Q8—1371no—
134C22×D4—21no—
135C22×D4C2×D13711yescharacteristic
136C23×C4C2×D13711yescharacteristic
137C2×SD16—2744no—
138C2×SD16—1374no—
139C2×M16—2744no—
140C2×M16—1374no—
141C22×C8—1371no—
142C22×C8—1371no—
143C22⋊Q8—2742no—
144C22⋊Q8—1374no—
145#121—22no—
146#121C2×D13714yes—
147C2×C4⋊C4—2741no—
148C2×C4⋊C4—1371no—
149C2×C4⋊C4C2×D13711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2741no—
152#107—1374no—
153#103—1374no—
154#102—1374no—
155C24—21no—
156C24C22⋊D13711yescharacteristic
157C2×Q8—2741no—
158C2×Q8—2742no—
159C2×Q8—2742no—
160C2×Q8—1372no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—44no—
166C2×D4C22×D13712yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4C22⋊D13712yes—
173C22×C4C22⋊D13711yescharacteristic
174C22×C4C22×D13712yes—
175C22×C4C22×D13711yescharacteristic
176SD16—2748no—
177M16—2748no—
178C2×C8—2742no—
179C2×C8—1372no—
180C2×C8—1374no—
181C2×C8—1372no—
182C4⋊C4—21no—
183C4⋊C4—2744no—
184C4⋊C4—2741no—
185C4⋊C4—1372no—
186C4⋊C4C22×D13712yes—
187C22⋊C4—24no—
188C22⋊C4—2744no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23C2×C22⋊D13712yes—
197C23C2×C22⋊D13711yescharacteristic
198Q8—2742no—
199Q8—2742no—
200Q8—5482no—
201D4—22no—
202D4—22no—
203D4—48no—
204D4—42no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—2742no—
212C2×C4—2742no—
213C2×C4—5482no—
214C2×C4—5482no—
215C2×C4D4×D13712yes—
216C2×C4C2×C22⋊D13712yes—
217C2×C4C2×C22⋊D13711yescharacteristic
218C2×C4#1455550912yes—
219C2×C4C23×D13711yescharacteristic
220C8—2742no—
221C8—1374no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—44no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22#1798406012yes—
231C22#1798411412yes—
232C22#1798411812yes—
233C22C22×C22⋊D13711yescharacteristic
234C4—22no—
235C4—42no—
236C4—5482no—
237C4—5482no—
238C4#1798406012yes—
239C4C2×D4×D13711yescharacteristic
240C4C2×#1455550911yescharacteristic
241C2—42no—
242C2—82no—
243C2#2257764714yesdirect product factor
244C2C2×#1798406011yescharacteristic
245C2C2×#1798411411yescharacteristic
246C2C2×#1798411811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radicalC2×#121×C137C2
socleC23×C137C2×D4
hypercenterC2×#121D137
hypocenterC137C2×#468
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14555498C2semidirect
2. C22×#14555494C2semidirect
3. C2×#17984067C2semidirect
4. #22577647C2direct (N × H)
5. C2×#17984021C2semidirect
6. C2×#17984011C2semidirect
7. C2×#17984009C2semidirect
11. C2×#14555498C22semidirect
13. C2×#14555494C22semidirect
17. #17984067C22semidirect
21. #17984021C22semidirect
22. #17984011C22semidirect
23. #17984009C22semidirect
40. C2×#11764040D4semidirect
73. #11764040C2×D4semidirect
119. C2×C137#468semidirect
123. C137C2×#468semidirect
243. C2#22577647direct (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×#145554942
C2×#179840672
#225776472
C2×#179840212
C2×#179840112
C2×#179840092
C2×#121×C1372
C2×#468137