Finite Group Databasebeta

← C22×#41×D97 #27156532

Subgroups — C22×#41×D97 #27156532

Order 12416.

6520 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095512C211yesmaximal, characteristic, complemented
2C22×#13095511C213yesmaximal, complemented
3C22×#13095510C213yesmaximal, complemented
4C2×#41×D97C2148yesmaximal, direct product factor
5C22×Q8×D97C211yesmaximal, characteristic, complemented
6C22×D4×D97C213yesmaximal, complemented
7C23×C4×D97C213yesmaximal, complemented
8C22×#41×C97C211yesmaximal, characteristic, complemented
9C24×D97C2213yes—
10C2×#13095512C22124yescomplemented
11C2×#13095511C22172yescomplemented
12C2×#13095510C22172yescomplemented
13#41×D97C221256yesdirect product factor
14C22×Dic194C2213yescomplemented
15C2×Q8×D97C22124yescomplemented
16C22×D388C2213yescomplemented
17C22×C22⋊D97C2216yescomplemented
18C2×D4×D97C22172yescomplemented
19C23×Dic97C2213yescomplemented
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2213yes—
22C22×C4×D97C2213yescomplemented
23C22×C4×D97C2213yescomplemented
24C22×C4×D97C22172yescomplemented
25C2×#41×C97C22124yescomplemented
26C22×Q8×C97C2211yescharacteristic, complemented
27C22×D4×C97C2213yescomplemented
28C23×C4×C97C2213yescomplemented
29C23×D97—224no—
30C23×D97C23136yes—
31C23×D97C2313yes—
32C23×D97C2311yescharacteristic
33#13095512C23164yescomplemented
34#13095511C231192yescomplemented
35#13095510C231192yescomplemented
36C2×Dic194C23136yescomplemented
37Q8×D97C23164yescomplemented
38C2×D388C23136yescomplemented
39C2×C22⋊D97C23172yescomplemented
40D4×D97C231192yescomplemented
41C22×Dic97C2311yescharacteristic
42C22×Dic97C2313yes—
43C22×Dic97C23136yescomplemented
44C2×C4×D97C23112yes—
45C2×C4×D97C23136yes—
46C2×C4×D97C231192yescomplemented
47C2×C4×D97C23136yescomplemented
48C2×C4×D97C23136yescomplemented
49C24×C97C2313yes—
50#41×C97C23164yescomplemented
51C2×Q8×C97C23112yescomplemented
52C2×D4×C97C23136yescomplemented
53C22×C4×C97C2313yes—
54C22×C4×C97C2311yescharacteristic
55C22×C4×C97C23136yescomplemented
56C23×C97—212no—
57C23×C97C24118yes—
58C23×C97C2411yescharacteristic
59C22×D97—212no—
60C22×D97—2144no—
61C22×D97#4118yesdirect product factor
62C22×D97C24148yes—
63C22×D97C24118yes—
64C22×D97C2416yes—
65Q8×C97C24116yescomplemented
66D4×C97C24148yescomplemented
67C2×C4×C97C2416yes—
68C2×C4×C97C24118yes—
69C2×C4×C97C24148yescomplemented
70C22⋊D97C24196yescomplemented
71C2×Dic97C2416yes—
72C2×Dic97C24118yes—
73C2×Dic97C24148yescomplemented
74D388C24148yescomplemented
75C4×D97C24116yes—
76C4×D97C24148yes—
77C4×D97C24148yescomplemented
78C4×D97C24148yescomplemented
79Dic194C24148yescomplemented
80C22×C97—236no—
81C22×C97C2×#4114yescomplemented
82C22×C97C25112yes—
83C22×C97C2513yes—
84C2×D97—236no—
85C2×D97—296no—
86C2×D97C2×#41124yesdirect product factor
87C2×D97C25112yes—
88C2×D97C2514yes—
89C4×C97C25112yes—
90C4×C97C2514yes—
91Dic97C25112yes—
92Dic97C2514yes—
93C2×C97—212no—
94C2×C97C22×#4116yescomplemented
95C2×C97C2611yescharacteristic
96D97—212no—
97D97C22×#4118yesdirect product factor
98C23×#41—971nomaximal, Sylow
99C97C23×#4111yescharacteristic, complemented, Sylow
100C22×#41—9748no—
101C22×#41—971no—
102C22×#41—973no—
103C22×#41—973no—
104C22×#41D9711yescharacteristic, direct product factor
105C23×Q8—971no—
106C23×D4—973no—
107C24×C4—973no—
108C25—973no—
109C2×#41—97256no—
110C2×#41—9724no—
111C2×#41—9772no—
112C2×#41—9772no—
113C2×#41C2×D97124yesdirect product factor
114C22×Q8—9724no—
115C22×Q8—973no—
116C22×Q8C2×D9711yescharacteristic, complemented
117C22×D4—976no—
118C22×D4—9772no—
119C22×D4—973no—
120C22×D4C2×D9713yescomplemented
121C23×C4—973no—
122C23×C4—973no—
123C23×C4—973no—
124C23×C4—971no—
125C23×C4—973no—
126C23×C4—9772no—
127C23×C4C2×D9713yescomplemented
128C24—971no—
129C24—9736no—
130C24—973no—
131C24—19424no—
132C24C22×D9713yes—
133#41—9764no—
134#41—97192no—
135#41—97192no—
136#41C22×D97164yesdirect product factor
137C2×Q8—9764no—
138C2×Q8—9736no—
139C2×Q8C22×D97112yescomplemented
140C2×D4—9772no—
141C2×D4—97192no—
142C2×D4—9736no—
143C2×D4C22×D97136yescomplemented
144C22×C4—971no—
145C22×C4—973no—
146C22×C4—9736no—
147C22×C4—9712no—
148C22×C4—9736no—
149C22×C4—9736no—
150C22×C4—9736no—
151C22×C4—97192no—
152C22×C4C22×D9713yes—
153C22×C4C22×D9711yescharacteristic
154C22×C4C22×D97136yescomplemented
155C23—212no—
156C23—978no—
157C23—976no—
158C23—9748no—
159C23—9718no—
160C23—194144no—
161C23—19412no—
162C23C23×D97118yes—
163C23C23×D9711yescharacteristic
164Q8—9748no—
165Q8C23×D97116yescomplemented
166D4—9796no—
167D4—9748no—
168D4C23×D97148yescomplemented
169C2×C4—976no—
170C2×C4—9718no—
171C2×C4—9748no—
172C2×C4—9716no—
173C2×C4—9748no—
174C2×C4—9748no—
175C2×C4—9748no—
176C2×C4C23×D9716yes—
177C2×C4C23×D97118yes—
178C2×C4C23×D97148yescomplemented
179C22—236no—
180C22—9724no—
181C22—974no—
182C22—9712no—
183C22—19496no—
184C22—19436no—
185C22#41×D9714yesdirect product factor
186C22C24×D97112yes—
187C22C24×D9713yes—
188C4—974no—
189C4—9712no—
190C4C24×D9714yes—
191C4C24×D97112yes—
192C2—212no—
193C2—978no—
194C2—19412no—
195C2C2×#41×D9716yesdirect product factor
196C2C25×D9711yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC22×#41×C97C2
socleC23×C97C24
hypercenterC22×#41D97
hypocenterC97C23×#41
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. C22×#13095511C2semidirect
3. C22×#13095510C2semidirect
4. C2×#41×D97C2direct (N × H)
5. C22×Q8×D97C2semidirect
6. C22×D4×D97C2semidirect
7. C23×C4×D97C2semidirect
8. C22×#41×C97C2semidirect
10. C2×#13095512C22semidirect
11. C2×#13095511C22semidirect
12. C2×#13095510C22semidirect
13. #41×D97C22direct (N × H)
14. C22×Dic194C22semidirect
15. C2×Q8×D97C22semidirect
16. C22×D388C22semidirect
17. C22×C22⋊D97C22semidirect
18. C2×D4×D97C22semidirect
19. C23×Dic97C22semidirect
22. C22×C4×D97C22semidirect
23. C22×C4×D97C22semidirect
24. C22×C4×D97C22semidirect
25. C2×#41×C97C22semidirect
26. C22×Q8×C97C22semidirect
27. C22×D4×C97C22semidirect
28. C23×C4×C97C22semidirect
33. #13095512C23semidirect
34. #13095511C23semidirect
35. #13095510C23semidirect
36. C2×Dic194C23semidirect
37. Q8×D97C23semidirect
38. C2×D388C23semidirect
39. C2×C22⋊D97C23semidirect
40. D4×D97C23semidirect
43. C22×Dic97C23semidirect
46. C2×C4×D97C23semidirect
47. C2×C4×D97C23semidirect
48. C2×C4×D97C23semidirect
50. #41×C97C23semidirect
51. C2×Q8×C97C23semidirect
52. C2×D4×C97C23semidirect
55. C22×C4×C97C23semidirect
61. C22×D97#41direct (N × H)
65. Q8×C97C24semidirect
66. D4×C97C24semidirect
69. C2×C4×C97C24semidirect
70. C22⋊D97C24semidirect
73. C2×Dic97C24semidirect
74. D388C24semidirect
77. C4×D97C24semidirect
78. C4×D97C24semidirect
79. Dic194C24semidirect
81. C22×C97C2×#41semidirect
86. C2×D97C2×#41direct (N × H)
94. C2×C97C22×#41semidirect
97. D97C22×#41direct (N × H)
99. C97C23×#41semidirect
104. C22×#41D97direct (N × H)
113. C2×#41C2×D97direct (N × H)
116. C22×Q8C2×D97semidirect
120. C22×D4C2×D97semidirect
127. C23×C4C2×D97semidirect
136. #41C22×D97direct (N × H)
139. C2×Q8C22×D97semidirect
143. C2×D4C22×D97semidirect
154. C22×C4C22×D97semidirect
165. Q8C23×D97semidirect
168. D4C23×D97semidirect
178. C2×C4C23×D97semidirect
185. C22#41×D97direct (N × H)
195. C2C2×#41×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
C22×#130955112
C22×#130955102
C2×#41×D972
C22×Q8×D972
C22×D4×D972
C23×C4×D972
C22×#41×C972
C23×#4197