Finite Group Databasebeta

← C2×#23757456 #34109478

Subgroups — C2×#23757456 #34109478

Order 20096.

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

normal subgroups only

This group has 708 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
1C2×#19097801C212yesmaximal, complemented
2C2×#19097769C211yesmaximal, characteristic, complemented
3#23757456C218yesmaximal, direct product factor
4C2×#19097698C212yesmaximal, complemented
5C2×#19097690C211yesmaximal, characteristic, complemented
6C2×2−1+4×C157C211yesmaximal, characteristic
7C22×Dic314C2211yescharacteristic
8#19097801—24no—
9#19097801C2218yescomplemented
10C2×#14578944—22no—
11C2×#14578942—22no—
12C2×#14578938C2212yes—
13#19097769—22no—
14#19097769C2214yescomplemented
15C2×#14578933—22no—
16C2×C4×Dic157C2211yescharacteristic
17#19097698C2218yescomplemented
18#19097690C2214yescomplemented
192−1+4×C157—24no—
202−1+4×C157—22no—
212−1+4×C157C2214yes—
22C2×#41×C157—22no—
23C2×#41×C157—22no—
24C2×#41×C157C2212yes—
25C22×Q8×C157—22no—
26C22×Q8×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314—22no—
29C2×Dic314—22no—
30C2×Dic314C2312yes—
31C22×Dic157—21no—
32C22×Dic157—21no—
33#14578944—28no—
34#14578942—28no—
35#14578938—22no—
36#14578938C2314yes—
37C2×#11929434—22no—
38#14578933—28no—
39C4×Dic157—21no—
40C4×Dic157C2312yes—
41#41×C157—28no—
42#41×C157—28no—
43#41×C157—28no—
44#41×C157—28no—
45#41×C157—22no—
46#41×C157—22no—
47#41×C157D414yes—
48#41×C157C2314yes—
49C2×Q8×C157—22no—
50C2×Q8×C157—28no—
51C2×Q8×C157—28no—
52C2×Q8×C157—22no—
53C2×Q8×C157—22no—
54C2×Q8×C157D412yes—
55C2×Q8×C157D412yes—
56C2×Q8×C157C2312yes—
57C2×D4×C157—22no—
58C2×D4×C157—22no—
59C2×D4×C157D412yes—
60C22×C4×C157—24no—
61C22×C4×C157—22no—
62C22×C4×C157D412yes—
63C22×C4×C157C2311yescharacteristic
64Dic314—24no—
65Dic314—42no—
66C2×Dic157—24no—
67C2×Dic157—22no—
68C2×Dic157—22no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71#11929434—24no—
72C23×C157—22no—
73C23×C157C2×D411yescharacteristic
74Q8×C157—28no—
75Q8×C157—22no—
76Q8×C157—24no—
77Q8×C157—24no—
78Q8×C157C2×D414yes—
79D4×C157—28no—
80D4×C157—24no—
81D4×C157—24no—
82D4×C157—22no—
83D4×C157C2×D414yes—
84C2×C4×C157—216no—
85C2×C4×C157—22no—
86C2×C4×C157—22no—
87C2×C4×C157—22no—
88C2×C4×C157—24no—
89C2×C4×C157—24no—
90C2×C4×C157C2×D414yes—
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D412yes—
93C2×C4×C157C2×D412yes—
94C2×C4×C157C2411yescharacteristic
95C22×C157—22no—
96C22×C157—24no—
97C22×C157—44no—
98C22×C157#12011yescharacteristic
99C22×C157#12011yescharacteristic
100C22×C157C22×D411yescharacteristic
101C4×C157—24no—
102C4×C157—24no—
103C4×C157#12012yes—
104C4×C157C22×D412yes—
105Dic157—24no—
106Dic157—42no—
107C2×C157—21no—
108C2×C157—21no—
109C2×C157—44no—
110C2×C157#45612yescomplemented
111C2×C157C2×#12011yescharacteristic
112C157C2×#45611yescharacteristic, complemented, Sylow
113C2×#456—1571nomaximal, Sylow
114C2×2−1+4D15711yescharacteristic
115C2×#137—1572no—
116C2×C4⋊Q8—1571no—
117#456—1578no—
118C2×#104—1572no—
119C2×#101—1571no—
1202−1+4—24no—
1212−1+4—22no—
1222−1+4C2×D15714yes—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41C2×D15712yes—
126C22×Q8—22no—
127C22×Q8—1571no—
128C22×Q8C2×D15711yescharacteristic
129#137—3144no—
130#137—1578no—
131C2×Q16—3142no—
132C2×SD16—3142no—
133C2×M16—1572no—
134C4⋊Q8—3142no—
135C4⋊Q8—1574no—
136C2×C4⋊C4—3142no—
137C2×C42—1571no—
138#104—1578no—
139#101—1574no—
140#41—28no—
141#41—28no—
142#41—28no—
143#41—28no—
144#41—22no—
145#41—22no—
146#41C22⋊D15714yes—
147#41C22×D15714yes—
148C2×Q8—22no—
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—3141no—
154C2×Q8—3142no—
155C2×Q8—3142no—
156C2×Q8—1572no—
157C2×Q8C22⋊D15712yes—
158C2×Q8C22⋊D15712yes—
159C2×Q8C22×D15712yes—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4C22⋊D15712yes—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—3141no—
166C22×C4—3141no—
167C22×C4C22⋊D15712yes—
168C22×C4C22×D15711yescharacteristic
169Q16—3148no—
170SD16—3148no—
171M16—3142no—
172M16—1574no—
173C2×C8—3142no—
174C4⋊C4—3148no—
175C42—3141no—
176C42—1572no—
177C23—22no—
178C23C2×C22⋊D15711yescharacteristic
179Q8—28no—
180Q8—22no—
181Q8—24no—
182Q8—24no—
183Q8—3144no—
184Q8—6282no—
185Q8C2×C22⋊D15714yes—
186D4—28no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4C2×C22⋊D15714yes—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—3144no—
198C2×C4—3142no—
199C2×C4—3142no—
200C2×C4—6281no—
201C2×C4—6281no—
202C2×C4C2×C22⋊D15714yes—
203C2×C4C2×C22⋊D15711yescharacteristic
204C2×C4C2×C22⋊D15712yes—
205C2×C4C2×C22⋊D15712yes—
206C2×C4C23×D15711yescharacteristic
207C8—3144no—
208C22—22no—
209C22—24no—
210C22—44no—
211C22#1909774011yescharacteristic
212C22#1909774011yescharacteristic
213C22C22×C22⋊D15711yescharacteristic
214C4—24no—
215C4—24no—
216C4—3144no—
217C4—6282no—
218C4#1909774012yes—
219C4C22×C22⋊D15712yes—
220C2—21no—
221C2—21no—
222C2—44no—
223C2#2375745612yesdirect product factor
224C2C2×#1909774011yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting, prime radicalC2×2−1+4×C157C2
socleC22×C157#120
hypercenterC2×2−1+4D157
hypocenterC157C2×#456
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C2×#19097769C2semidirect
3. #23757456C2direct (N × H)
4. C2×#19097698C2semidirect
5. C2×#19097690C2semidirect
9. #19097801C22semidirect
14. #19097769C22semidirect
17. #19097698C22semidirect
18. #19097690C22semidirect
110. C2×C157#456semidirect
112. C157C2×#456semidirect
223. C2#23757456direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×#190977692
#237574562
C2×#190976982
C2×#190976902
C2×2−1+4×C1572
C2×#456157