Finite Group Databasebeta

← C2×#23757942 #34113614

Subgroups — C2×#23757942 #34113614

Order 20096.

1736 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 573 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×#14578955C211yesmaximal, characteristic, complemented
2#23757942C2116yesmaximal, direct product factor
3C2×#19097767C213yesmaximal, complemented
4C2×#19097745C213yesmaximal, complemented
5C2×#19097742C213yesmaximal, complemented
6C2×Q8×Dic157C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D157C213yesmaximal, complemented
8C22×#41×C157C211yesmaximal, characteristic, complemented
9C2×#14578955—28no—
10C2×#14578955C2218yescomplemented
11C22×D628C2213yes—
12C22×C22⋊D157C2213yes—
13C22×C22⋊D157C2216yes—
14C22×C4×D157C2213yes—
15#19097767C22124yescomplemented
16#19097745C22124yescomplemented
17#19097742C22124yescomplemented
18Q8×Dic157C2218yescomplemented
19C4×C22⋊D157C22124yescomplemented
20C2×C4⋊Dic157C2213yes—
21C2×#14578933C2211yescharacteristic
22C2×C22⋊Dic157C2213yes—
23C2×#14578931C2213yes—
24C2×C4×Dic157C2213yes—
25C2×#41×C157—28no—
26C2×#41×C157C2218yescomplemented
27C22×Q8×C157C2211yescharacteristic
28C22×D4×C157C2213yes—
29C23×C4×C157C2213yes—
30C23×D157C2313yes—
31#14578955—232no—
32C2×D628—212no—
33C2×D628C23112yes—
34C2×C22⋊D157—212no—
35C2×C22⋊D157—412no—
36C2×C22⋊D157C23112yes—
37C2×C22⋊D157C23124yes—
38C22×Dic157C2313yes—
39C22×Dic157C2311yescharacteristic
40C22×Dic157C2311yescharacteristic
41C2×C4×D157—212no—
42C2×C4×D157C23112yes—
43C4⋊Dic157C23112yes—
44#14578933C2314yes—
45C22⋊Dic157C23112yes—
46#14578931C23112yes—
47C4×Dic157C23112yes—
48C24×C157C2313yes—
49#41×C157—232no—
50C2×Q8×C157D418yescomplemented
51C2×Q8×C157C2314yes—
52C2×D4×C157—212no—
53C2×D4×C157C23112yes—
54C22×C4×C157—212no—
55C22×C4×C157C2311yescharacteristic
56C22×C4×C157C23112yes—
57C22×C4×C157C2313yes—
58C22×D157—26no—
59C22×D157—26no—
60C22×D157—26no—
61C22×D157C2416yes—
62D628—224no—
63C22⋊D157—224no—
64C22⋊D157—424no—
65C2×Dic157—22no—
66C2×Dic157—26no—
67C2×Dic157#4114yescomplemented
68C2×Dic157C2412yes—
69C2×Dic157C2416yes—
70C2×Dic157C2412yes—
71C4×D157—224no—
72C23×C157—26no—
73C23×C157—26no—
74C23×C157—26no—
75C23×C157C2416yes—
76C23×C157C2411yescharacteristic
77Q8×C157C2×D4116yescomplemented
78D4×C157—224no—
79C2×C4×C157—22no—
80C2×C4×C157—224no—
81C2×C4×C157C2×D4112yes—
82C2×C4×C157C2412yes—
83C2×C4×C157C2416yes—
84C22×C157—26no—
85C22×C157—26no—
86C22×C157—26no—
87C22×C157—412no—
88C22×C157C22×D412yes—
89C22×C157C2×#4112yes—
90C22×C157H(5,𝔽2)12yes—
91C22×C157C2511yescharacteristic
92C2×D157—26no—
93C2×D157—26no—
94C2×D157—26no—
95C2×D157—412no—
96C4×C157—22no—
97C4×C157C22×D4112yes—
98Dic157—22no—
99Dic157—26no—
100Dic157C2×#4114yescomplemented
101C2×C157—46no—
102C2×C157#55014yescomplemented
103C2×C157C23×D411yescharacteristic
104C2×C157C22×#4111yescharacteristic
105C2×C157C2×H(5,𝔽2)11yescharacteristic
106D157—46no—
107C157C2×#55011yescharacteristic, complemented, Sylow
108C2×#550—1571nomaximal, Sylow
109C22×#41—1571no—
110C22×#41D15711yescharacteristic, complemented
111#550—15716no—
112C2×#127—1573no—
113C2×#121—1573no—
114C2×#121—1573no—
115C2×C4×Q8—1571no—
116C2×C4×D4—1573no—
117C2×#41—28no—
118C2×#41—3148no—
119C2×#41—1578no—
120C2×#41C2×D15718yescomplemented
121C22×Q8C2×D15711yescharacteristic
122C22×D4—1573no—
123C22×D4—1576no—
124C22×D4—1573no—
125C22×D4C2×D15713yes—
126C23×C4—1573no—
127C23×C4C2×D15713yes—
128#127—15724no—
129#121—15724no—
130#121—15724no—
131C4×Q8—1578no—
132C4×D4—15724no—
133C2×C4⋊C4—1571no—
134C2×C4⋊C4—1573no—
135C2×C22⋊C4—1573no—
136C2×C22⋊C4—1573no—
137C2×C42—1573no—
138C24—1573no—
139C24C22×D15713yes—
140#41—232no—
141#41—31432no—
142C2×Q8C22⋊D15718yescomplemented
143C2×Q8C22×D15714yes—
144C2×D4—212no—
145C2×D4—31412no—
146C2×D4—31412no—
147C2×D4—62812no—
148C2×D4—15712no—
149C2×D4—15724no—
150C2×D4—15712no—
151C2×D4C22×D157112yes—
152C22×C4—212no—
153C22×C4—31412no—
154C22×C4—1573no—
155C22×C4—1571no—
156C22×C4—15712no—
157C22×C4—1571no—
158C22×C4C22×D157112yes—
159C22×C4C22×D15711yescharacteristic
160C22×C4C22×D15713yes—
161C4⋊C4—1574no—
162C4⋊C4—15712no—
163C22⋊C4—15712no—
164C22⋊C4—15712no—
165C42—15712no—
166C23—26no—
167C23—26no—
168C23—26no—
169C23—3146no—
170C23—3146no—
171C23—3146no—
172C23—1576no—
173C23C23×D15716yes—
174C23C23×D15711yescharacteristic
175Q8C2×C22⋊D157116yescomplemented
176D4—224no—
177D4—31424no—
178D4—31424no—
179D4—62824no—
180C2×C4—22no—
181C2×C4—224no—
182C2×C4—3142no—
183C2×C4—3146no—
184C2×C4—31424no—
185C2×C4—1576no—
186C2×C4—1572no—
187C2×C4—1574no—
188C2×C4—1572no—
189C2×C4C2×C22⋊D157112yes—
190C2×C4C23×D15712yes—
191C2×C4C23×D15716yes—
192C22—26no—
193C22—26no—
194C22—26no—
195C22—412no—
196C22—3146no—
197C22—3146no—
198C22—3146no—
199C22—62812no—
200C22C22×C22⋊D15712yes—
201C22#41×D15712yes—
202C22#1909781712yes—
203C22C24×D15711yescharacteristic
204C4—22no—
205C4—3142no—
206C4—3146no—
207C4—1574no—
208C4C22×C22⋊D157112yes—
209C2—46no—
210C2—6286no—
211C2#2375794214yesdirect product factor
212C2C23×C22⋊D15711yescharacteristic
213C2C2×#41×D15711yescharacteristic
214C2C2×#1909781711yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC22×#41×C157C2
socleC23×C157C24
hypercenterC22×#41D157
hypocenterC157C2×#550
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. #23757942C2direct (N × H)
3. C2×#19097767C2semidirect
4. C2×#19097745C2semidirect
5. C2×#19097742C2semidirect
6. C2×Q8×Dic157C2semidirect
7. C2×C4×C22⋊D157C2semidirect
8. C22×#41×C157C2semidirect
10. C2×#14578955C22semidirect
15. #19097767C22semidirect
16. #19097745C22semidirect
17. #19097742C22semidirect
18. Q8×Dic157C22semidirect
19. C4×C22⋊D157C22semidirect
26. C2×#41×C157C22semidirect
50. C2×Q8×C157D4semidirect
67. C2×Dic157#41semidirect
77. Q8×C157C2×D4semidirect
100. Dic157C2×#41semidirect
102. C2×C157#550semidirect
107. C157C2×#550semidirect
110. C22×#41D157semidirect
120. C2×#41C2×D157semidirect
142. C2×Q8C22⋊D157semidirect
175. Q8C2×C22⋊D157semidirect
211. C2#23757942direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145789552
#237579422
C2×#190977672
C2×#190977452
C2×#190977422
C2×Q8×Dic1572
C2×C4×C22⋊D1572
C22×#41×C1572
C2×#550157