Finite Group Databasebeta

← C2×#14521654 #16859071

Subgroups — C2×#14521654 #16859071

Order 3968.

1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 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
1#14521654C2116yesmaximal, direct product factor
2C2×#11758451C211yesmaximal, characteristic, complemented
3C2×#11758447C211yesmaximal, characteristic, complemented
4C2×#11758471C214yesmaximal, complemented
5C2×#11758507C212yesmaximal, complemented
6C2×#11758461C214yesmaximal, complemented
7C2×C4×D124C212yesmaximal, complemented
8C2×C31×#117C211yesmaximal, characteristic, complemented
9C23×C4×C31C2211yescharacteristic
10C22×C22⋊D31C2214yes—
11C22×D124C2212yes—
12C22×D124C2214yes—
13C22×C4×D31C2214yes—
14C31×#117C2218yescomplemented
15C2×C4⋊C4×C31C2212yes—
16C2×C22⋊C4×C31C2212yes—
17C2×C42×C31C2211yescharacteristic
18C2×C42×C31C2211yescharacteristic
19#11758507C22116yescomplemented
20C2×(C2×D31)⋊C4C2214yes—
21C2×(C2×D31)⋊C4C2214yes—
22C2×C4⋊Dic31C2212yes—
23C2×Dic31⋊C4C2214yes—
24#11758471C22132yescomplemented
25#11758461C22132yescomplemented
26#11758451C2218yescomplemented
27C4×D124C22116yescomplemented
28#11758447C2218yescomplemented
29C24×C31C2311yescharacteristic
30C23×D31C2314yes—
31C22×C4×C31—24no—
32C22×C4×C31C2312yes—
33C22×C4×C31C2311yescharacteristic
34C22×C4×C31C2312yes—
35C22×C4×C31C2314yes—
36C22×C4×C31C2311yescharacteristic
37C2×C22⋊D31—48no—
38C2×C22⋊D31C23116yes—
39C22×Dic31C2314yes—
40C2×D124—28no—
41C2×D124—48no—
42C2×D124C2318yes—
43C2×D124C23116yes—
44C2×C4×D31—48no—
45C2×C4×D31C23116yes—
46C4⋊C4×C31C2318yes—
47C22⋊C4×C31C2318yes—
48C42×C31C2314yes—
49C42×C31C2314yes—
50(C2×D31)⋊C4C23116yes—
51(C2×D31)⋊C4C23116yes—
52C4⋊Dic31C2318yes—
53Dic31⋊C4C23116yes—
54C23×C31—22no—
55C23×C31—24no—
56C23×C31C2412yes—
57C23×C31C2411yescharacteristic
58C22×D31—28no—
59C22×D31—28no—
60C22×D31—28no—
61C22×D31C2418yes—
62C2×C4×C31—22no—
63C2×C4×C31—24no—
64C2×C4×C31—24no—
65C2×C4×C31—28no—
66C2×C4×C31#4114yes—
67C2×C4×C31C2414yes—
68C2×C4×C31C2412yes—
69C2×C4×C31C2414yes—
70C2×C4×C31C2412yes—
71C22⋊D31—416no—
72C2×Dic31—28no—
73C2×Dic31C2418yes—
74D124—216no—
75D124—416no—
76C4×D31—416no—
77C2×#540—311nomaximal, Sylow
78C22×C31—22no—
79C22×C31—24no—
80C22×C31—44no—
81C22×C31C2×#4112yes—
82C22×C31H(5,𝔽2)14yes—
83C22×C31C2511yescharacteristic
84C2×D31—28no—
85C2×D31—28no—
86C2×D31—28no—
87C2×D31—416no—
88C4×C31—22no—
89C4×C31—24no—
90C4×C31—24no—
91C4×C31C2×#4114yes—
92Dic31—28no—
93#540—3116no—
94C2×#127—311no—
95C2×#125—311no—
96C2×#123—314no—
97C2×#121—312no—
98C2×#121—314no—
99C2×C4×D4—312no—
100C2×#117D3111yescharacteristic, complemented
101C2×C31—42no—
102C2×C31#54014yescomplemented
103C2×C31C22×#4111yescharacteristic
104C2×C31C2×H(5,𝔽2)12yes—
105D31—48no—
106C22×D4—314no—
107C22×D4—312no—
108C22×D4—314no—
109C23×C4—314no—
110C23×C4C2×D3111yescharacteristic
111#127—318no—
112#125—318no—
113#123—3132no—
114#121—3116no—
115#121—3132no—
116C4×D4—3116no—
117#117C2×D3118yescomplemented
118C2×C4⋊C4—312no—
119C2×C4⋊C4—314no—
120C2×C4⋊C4C2×D3112yes—
121C2×C22⋊C4—314no—
122C2×C22⋊C4—314no—
123C2×C22⋊C4C2×D3112yes—
124C2×C42C2×D3111yescharacteristic
125C2×C42C2×D3111yescharacteristic
126C31C2×#54011yescharacteristic, complemented, Sylow
127C24—314no—
128C24C22×D3111yescharacteristic
129C2×D4—3116no—
130C2×D4—318no—
131C2×D4—3116no—
132C2×D4—628no—
133C2×D4—1248no—
134C2×D4—1248no—
135C22×C4—24no—
136C22×C4—314no—
137C22×C4—3116no—
138C22×C4—1248no—
139C22×C4C22×D3114yes—
140C22×C4C22×D3112yes—
141C22×C4C22×D3112yes—
142C22×C4C22×D3111yescharacteristic
143C22×C4C22×D3111yescharacteristic
144C4⋊C4—3116no—
145C4⋊C4—318no—
146C4⋊C4C22×D3118yes—
147C22⋊C4—3116no—
148C22⋊C4—3116no—
149C22⋊C4C22×D3118yes—
150C42C22×D3114yes—
151C42C22×D3114yes—
152C23—22no—
153C23—24no—
154C23—318no—
155C23—628no—
156C23—628no—
157C23—628no—
158C23C23×D3112yes—
159C23C23×D3111yescharacteristic
160D4—6216no—
161D4—12416no—
162D4—12416no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—318no—
168C2×C4—628no—
169C2×C4—12416no—
170C2×C4#9249514yes—
171C2×C4C23×D3114yes—
172C2×C4C23×D3114yes—
173C2×C4C23×D3112yes—
174C2×C4C23×D3112yes—
175C22—22no—
176C22—24no—
177C22—44no—
178C22—628no—
179C22—628no—
180C22—628no—
181C22—12416no—
182C22C2×#9249512yes—
183C22#1175856614yes—
184C22C24×D3111yescharacteristic
185C4—22no—
186C4—24no—
187C4—24no—
188C4—628no—
189C4C2×#9249514yes—
190C2—42no—
191C2—1248no—
192C2#1452165414yesdirect product factor
193C2C22×#9249511yescharacteristic
194C2C2×#1175856612yes—
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC2×C31×#117C2
socleC23×C31C24
hypercenterC2×#117D31
hypocenterC31C2×#540
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #14521654C2direct (N × H)
2. C2×#11758451C2semidirect
3. C2×#11758447C2semidirect
4. C2×#11758471C2semidirect
5. C2×#11758507C2semidirect
6. C2×#11758461C2semidirect
7. C2×C4×D124C2semidirect
8. C2×C31×#117C2semidirect
14. C31×#117C22semidirect
19. #11758507C22semidirect
24. #11758471C22semidirect
25. #11758461C22semidirect
26. #11758451C22semidirect
27. C4×D124C22semidirect
28. #11758447C22semidirect
100. C2×#117D31semidirect
102. C2×C31#540semidirect
117. #117C2×D31semidirect
126. C31C2×#540semidirect
192. C2#14521654direct (N × H)

Maximal subgroups

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

subgroupindex
#145216542
C2×#117584512
C2×#117584472
C2×#117584712
C2×#117585072
C2×#117584612
C2×C4×D1242
C2×C31×#1172
C2×#54031